| Title: | Basic and Advanced Statistical Power Analysis |
| Version: | 0.9.4 |
| Date: | 2023-10-13 |
| Author: | Zhiyong Zhang [aut, cre], Yujiao Mai [aut], Miao Yang [ctb], Ziqian Xu [ctb], Conor McNamara [ctb] |
| Maintainer: | Zhiyong Zhang <johnnyzhz@gmail.com> |
| Depends: | R (≥ 3.2.5), MASS, grDevices, graphics, lme4, lavaan, parallel, PearsonDS |
| Imports: | dplyr, stats |
| License: | GPL (≥ 3) |
| Description: | This is a collection of tools for conducting both basic and advanced statistical power analysis including correlation, proportion, t-test, one-way ANOVA, two-way ANOVA, linear regression, logistic regression, Poisson regression, mediation analysis, longitudinal data analysis, structural equation modeling and multilevel modeling. It also serves as the engine for conducting power analysis online at https://webpower.psychstat.org. |
| URL: | https://webpower.psychstat.org |
| Encoding: | UTF-8 |
| LazyLoad: | yes |
| LazyData: | yes |
| NeedsCompilation: | no |
| Packaged: | 2023-10-13 16:34:44 UTC; ziqianxu |
| Repository: | CRAN |
| Date/Publication: | 2023-10-13 20:20:02 UTC |
Basic and Advanced Statistical Power Analysis
Description
This is a collection of tools for conducting both basic and advanced statistical power analysis including correlation, proportion, t-test, one-way ANOVA, two-way ANOVA, linear regression, logistic regression, Poisson regression, mediation analysis, longitudinal data analysis, structural equation modeling and multilevel modeling. It also serves as the engineer for conducting power analysis online at https://webpower.psychstat.org.
Details
This is a collection of tools for conducting both basic and advanced statistical power analysis including correlation, proportion, t-test, one-way ANOVA, two-way ANOVA, linear regression, logistic regression, Poisson regression, mediation analysis, longitudinal data analysis, structural equation modeling and multilevel modeling. It also serves as the engineer for conducting power analysis online at https://webpower.psychstat.org.
References
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Example Data For CRT With 2 Arms
Description
ID. The identification number of the subjects.
cluster. The cluster number.
score. The score of the subject.
group. The group number.
Usage
CRT2
Format
An object of class data.frame with 8 rows and 4 columns.
Examples
# ID cluster score group
# 1 1 6 0
# 2 1 2 0
# 3 2 6 1
# 4 2 5 1
# 5 3 1 0
# 6 3 4 0
# 7 4 6 1
# 8 4 4 1
Example Data For CRT With 3 Arms
Description
ID. The identification number of the subjects.
cluster. The cluster number.
score. The score of the subject.
group. The group number.
Usage
CRT3
Format
An object of class data.frame with 30 rows and 4 columns.
Examples
# id cluster score group
# 1 1 1.93 0
# 2 1 1.51 0
# 3 1 2.13 0
# 4 1 2.96 0
# 5 1 3.84 0
# 6 2 3.36 1
# 7 2 3.13 1
# 8 2 1.71 1
# 9 2 3.1 1
# 10 2 2.53 1
# 11 3 2.01 2
# 12 3 4.73 2
# 13 3 3.34 2
# 14 3 0.11 2
# 15 3 3.6 2
# 16 4 2 0
# 17 4 1.99 0
# 18 4 1.89 0
# 19 4 2.25 0
# 20 4 1.83 0
# 21 5 3.03 1
# 22 5 2.08 1
# 23 5 1.5 1
# 24 5 3.18 1
# 25 5 1.92 1
# 26 6 3.49 2
# 27 6 3.08 2
# 28 6 4.54 2
# 29 6 2.34 2
# 30 6 4.33 2
Example Data For MRT With 2 Arms
Description
ID. The identification number of the subjects.
cluster. The cluster number.
score. The score of the subject.
group. The group number.
Usage
MRT2
Format
An object of class data.frame with 16 rows and 4 columns.
Examples
#Example data for MRT with 2 arms
# id cluster score group
# 1 1 6 0
# 2 1 2 0
# 3 1 3 1
# 4 1 3 1
# 5 2 6 0
# 6 2 10 0
# 7 2 7 1
# 8 2 6 1
# 9 3 6 0
# 10 3 5 0
# 11 3 4 1
# 12 3 4 1
# 13 4 1 0
# 14 4 8 0
# 15 4 10 1
# 16 4 -2 1
Example Data For MRT With 3 Arms
Description
ID. The identification number of the subjects.
cluster. The cluster number.
score. The score of the subject.
group. The group number.
Usage
MRT3
Format
An object of class data.frame with 24 rows and 4 columns.
Examples
# id cluster score group
# 1 1 2 0
# 2 1 3 0
# 3 1 2 1
# 4 1 0 1
# 5 1 3 2
# 6 1 2 2
# 7 2 1 0
# 8 2 4 0
# 9 2 2 1
# 10 2 3 1
# 11 2 3 2
# 12 2 1 2
# 13 3 1 0
# 14 3 4 0
# 15 3 1 1
# 16 3 1 1
# 17 3 2 2
# 18 3 0 2
# 19 4 4 0
# 20 4 3 0
# 21 4 1 1
# 22 4 3 1
# 23 4 3 2
# 24 4 3 2
Estimate multilevel effect size from data
Description
Estimate multilevel effect size from data
Usage
estCRT2arm(file)
estCRT3arm(file)
estMRT2arm(file)
estMRT3arm(file)
Arguments
file |
a data file |
Solve A Single Equation
Description
The function searches in an interval for a root (i.e., zero) of the function f with respect to its first argument. The argument interval is for the input of x, the corresponding outcome interval will be used as the interval to be searched in.
Usage
nuniroot(f, interval, maxlength = 100)
Arguments
f |
Function for which the root is sought. |
interval |
A vector containing the end-points of the interval to be searched for the root. |
maxlength |
The number of vaulue points in the interval to be searched. It is 100 by default. |
Value
A list with at least four components: root and f.root give the location of the root and the value of the function evaluated at that point. iter and estim.prec give the number of iterations used and an approximate estimated precision for root. (If the root occurs at one of the endpoints, the estimated precision is NA.)
Examples
f <- function(x) 1+x-0.5*x^2
interval <- c(-3,6)
nuniroot(f,interval)
Plot the power curve for Latent Change Score Models
Description
This function is used to plot the power analysis results for Latent Change Score Models.
Usage
## S3 method for class 'lcs.power'
plot(x, parameter, ...)
Arguments
x |
Data to plot. |
parameter |
Parameters for features of the plot. |
... |
Extra arguments. It is not required. |
References
Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
To plot Statistical Power Curve
Description
This function is used to plot the power curves generated by webpower.
Usage
## S3 method for class 'webpower'
plot(x, xvar = NULL, yvar = NULL, xlab = NULL,
ylab = NULL, ...)
Arguments
x |
Objects of power analysis. |
xvar |
The variable name used as the x (horizontal) axis. It is not required. |
yvar |
The variable name used as the y (vertical) axis. It is not required. |
xlab |
The label for the x axis. It is not required. |
ylab |
The label for the y axis. It is not required. |
... |
Extra arguments. It is not required. |
Value
The plot.
Examples
res <- wp.correlation(n=seq(50,100,10),r=0.3, alternative="two.sided")
plot(res)
To Print Statistical Power Analysis Results
Description
This function is used to summary the power analysis results.
Usage
## S3 method for class 'webpower'
print(x, ...)
Arguments
x |
Object of power analysis. It is an object returned by a webpower function such as wp.anova(). |
... |
Extra arguments. It is not required. |
Value
The printing of the input object of power analysis.
Examples
res <- wp.correlation(n=50,r=0.3, alternative="two.sided")
print(res)
Calculate the Effect Size for SEM
Description
This function is for calculating SEM effect size.
Usage
sem.effect.size(full.model.pop, reduced.model)
Arguments
full.model.pop |
Full model (under the alternative hypothesis) with population parameters. |
reduced.model |
Reduced model (under the null hypothesis) lavaan specification. |
Value
An object of the power analysis.
delta |
Effect size. |
df |
Degrees of freedom |
RMSEA |
RMSEA |
References
Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
full.model.pop <-'
y1 ~ 0.4*x
y2 ~ 0.5*x + 0.2*y1
y3 ~ 0.4*x
y4 ~ 0.4*y1 + 0.4*y2 + 0.4*y3
y1 ~~ 0.84*y1
y2 ~~ 0.61*y2
y3 ~~ 0.84*y3
y4 ~~ 0.27*y4
'
reduced.model <-'
y1 ~ x
y2 ~ x
y3 ~ x
y4 ~ y1 + y3
'
sem.effect.size(full.model.pop, reduced.model)
Summary Statistical Power Analysis Results
Description
This function is used to summary the power analysis results.
Usage
## S3 method for class 'power'
summary(object, ...)
Arguments
object |
Object of power analysis. It is an object returned by a webpower function for SEM based on Monte Carlo methods with class = 'power'. |
... |
Extra arguments. It is not required. |
Value
The summary of the input object of power analysis.
Statistical Power Analysis for One-way ANOVA
Description
One-way analysis of variance (one-way ANOVA) is a technique used to compare means of two or more groups (e.g., Maxwell & Delaney, 2003). The ANOVA tests the null hypothesis that samples in two or more groups are drawn from populations with the same mean values. The ANOVA analysis typically produces an F-statistic, the ratio of the bewteen-group variance to the within-group variance.
Usage
wp.anova(k = NULL, n = NULL, f = NULL, alpha = 0.05, power = NULL,
type = c("overall", "two.sided", "greater", "less"))
Arguments
k |
Number of groups. |
n |
Sample size. |
f |
Effect size. We use the statistic f as the measure of effect size for one-way ANOVA as in Cohen (1988). Cohen defined the size of effect as: small 0.1, medium 0.25, and large 0.4. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
type |
Type of test ( |
Value
An object of the power analysis.
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective (Vol. 1). Psychology Press.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power for the overall test of one-way ANOVA:
wp.anova(f=0.25,k=4, n=100, alpha=0.05)
# Power for One-way ANOVA
#
# k n f alpha power
# 4 100 0.25 0.05 0.5181755
#
# NOTE: n is the total sample size (overall)
# URL: http://psychstat.org/anova
#To calculate the power curve with a sequence of sample sizes:
res <- wp.anova(f=0.25, k=4, n=seq(100,200,10), alpha=0.05)
res
# Power for One-way ANOVA
#
# k n f alpha power
# 4 100 0.25 0.05 0.5181755
# 4 110 0.25 0.05 0.5636701
# 4 120 0.25 0.05 0.6065228
# 4 130 0.25 0.05 0.6465721
# 4 140 0.25 0.05 0.6837365
# 4 150 0.25 0.05 0.7180010
# 4 160 0.25 0.05 0.7494045
# 4 170 0.25 0.05 0.7780286
# 4 180 0.25 0.05 0.8039869
# 4 190 0.25 0.05 0.8274169
# 4 200 0.25 0.05 0.8484718
#
# NOTE: n is the total sample size (overall)
# URL: http://psychstat.org/anova
#To plot the power curve:
plot(res, type='b')
#To estimate the sample size with a given power:
wp.anova(f=0.25,k=4, n=NULL, alpha=0.05, power=0.8)
# Power for One-way ANOVA
#
# k n f alpha power
# 4 178.3971 0.25 0.05 0.8
#
# NOTE: n is the total sample size (overall)
# URL: http://psychstat.org/anova
#To estimate the minimum detectable effect size with a given power:
wp.anova(f=NULL,k=4, n=100, alpha=0.05, power=0.8)
# Power for One-way ANOVA
#
# k n f alpha power
# 4 100 0.3369881 0.05 0.8
#
# NOTE: n is the total sample size (overall)
# URL: http://psychstat.org/anova
#To conduct power analysis for a contrast one-way ANOVA:
wp.anova(f=0.25,k=4, n=100, alpha=0.05, type='two.sided')
# Power for One-way ANOVA
#
# k n f alpha power
# 4 100 0.25 0.05 0.6967142
#
# NOTE: n is the total sample size (contrast, two.sided)
# URL: http://psychstat.org/anova
#To calculate the power curve with a sequence of sample sizes:
res <- wp.anova(f=seq(0.1, 0.8, 0.1), k=4, n=100, alpha=0.05)
res
# Power for One-way ANOVA
#
# k n f alpha power
# 4 100 0.1 0.05 0.1128198
# 4 100 0.2 0.05 0.3452612
# 4 100 0.3 0.05 0.6915962
# 4 100 0.4 0.05 0.9235525
# 4 100 0.5 0.05 0.9911867
# 4 100 0.6 0.05 0.9995595
# 4 100 0.7 0.05 0.9999908
# 4 100 0.8 0.05 0.9999999
#
# NOTE: n is the total sample size (overall)
# URL: http://psychstat.org/anova
Statistical Power Analysis for One-way ANOVA with Binary Data
Description
The power analysis procedure for one-way ANOVA with binary data is introduced by Mai and Zhang (2017). One-way ANOVA with binary data is used for comparing means of three or more groups of binary data. Its outcome variable is supposed to follow Bernoulli distribution. And its overall test uses a likelihood ratio test statistics.
Usage
wp.anova.binary(k = NULL, n = NULL, V = NULL, alpha = 0.05,
power = NULL)
Arguments
k |
Number of groups. |
n |
Sample size. |
V |
Effect size. See the research by Mai and Zhang (2017) for details. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
Value
An object of the power analysis.
References
Mai, Y., & Zhang, Z. (2017). Statistical Power Analysis for Comparing Means with Binary or Count Data Based on Analogous ANOVA. In L. A. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W.-C. Wang (Eds.), Quantitative Psychology - The 81st Annual Meeting of the Psychometric Society, Asheville, North Carolina, 2016: Springer.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=100,V=0.15,alpha=0.05)
# One-way Analogous ANOVA with Binary Data
#
# k n V alpha power
# 4 100 0.15 0.05 0.5723443
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovabinary
#To generate a power curve given a sequence of sample sizes:
res <- wp.anova.binary(k=4,n=seq(100,200,10),V=0.15,alpha=0.05,power=NULL)
res
# One-way Analogous ANOVA with Binary Data
#
# k n V alpha power
# 4 100 0.15 0.05 0.5723443
# 4 110 0.15 0.05 0.6179014
# 4 120 0.15 0.05 0.6601594
# 4 130 0.15 0.05 0.6990429
# 4 140 0.15 0.05 0.7345606
# 4 150 0.15 0.05 0.7667880
# 4 160 0.15 0.05 0.7958511
# 4 170 0.15 0.05 0.8219126
# 4 180 0.15 0.05 0.8451603
# 4 190 0.15 0.05 0.8657970
# 4 200 0.15 0.05 0.8840327
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovabinary
#To plot the power curve:
plot(res)
#To calculate the required sample size for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=NULL,V=0.15,power=0.8, alpha=0.05)
# One-way Analogous ANOVA with Binary Data
#
# k n V alpha power
# 4 161.5195 0.15 0.05 0.8
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovabinary
#To calculate the minimum detectable effect size for one-way ANOVA (overall test) with binary data:
wp.anova.binary(k=4,n=100,V=NULL,power=0.8, alpha=0.05)
# One-way Analogous ANOVA with Binary Data
#
# k n V alpha power
# 4 100 0.1906373 0.05 0.8
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovabinary
#To generate a power curve given a sequence of effect sizes:
wp.anova.binary(k=4,n=100,V=seq(0.1,0.5,0.05),alpha=0.05,power=NULL)
# One-way Analogous ANOVA with Binary Data
#
# k n V alpha power
# 4 100 0.10 0.05 0.2746396
# 4 100 0.15 0.05 0.5723443
# 4 100 0.20 0.05 0.8402271
# 4 100 0.25 0.05 0.9659434
# 4 100 0.30 0.05 0.9961203
# 4 100 0.35 0.05 0.9997729
# 4 100 0.40 0.05 0.9999933
# 4 100 0.45 0.05 0.9999999
# 4 100 0.50 0.05 1.0000000
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovabinary
Statistical Power Analysis for One-way ANOVA with Count Data
Description
The power analysis procedure for one-way ANOVA with count data is introduced by Mai and Zhang (2017). One-way ANOVA with count data is used for comparing means of three or more groups of binary data. Its outcome variable is supposed to follow Poisson distribution. And its overall test uses a likelihood ratio test statistics.
Usage
wp.anova.count(k = NULL, n = NULL, V = NULL, alpha = 0.05,
power = NULL)
Arguments
k |
Number of groups. |
n |
Sample size. |
V |
Effect size. See the research by Mai and Zhang (2017) for details. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
Value
An object of the power analysis.
References
Mai, Y., & Zhang, Z. (2017). Statistical Power Analysis for Comparing Means with Binary or Count Data Based on Analogous ANOVA. In L. A. van der Ark, M. Wiberg, S. A. Culpepper, J. A. Douglas, & W.-C. Wang (Eds.), Quantitative Psychology - The 81st Annual Meeting of the Psychometric Society, Asheville, North Carolina, 2016: Springer.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=100,V=0.148,alpha=0.05)
# One-way Analogous ANOVA with Count Data
#
# k n V alpha power
# 4 100 0.148 0.05 0.5597441
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovacount
#To generate a power curve given sequence of sample sizes:
res <- wp.anova.count(k=4,n=seq(100,200,10),V=0.148,alpha=0.05,power=NULL)
res
# One-way Analogous ANOVA with Count Data
#
# k n V alpha power
# 4 100 0.148 0.05 0.5597441
# 4 110 0.148 0.05 0.6049618
# 4 120 0.148 0.05 0.6470911
# 4 130 0.148 0.05 0.6860351
# 4 140 0.148 0.05 0.7217782
# 4 150 0.148 0.05 0.7543699
# 4 160 0.148 0.05 0.7839101
# 4 170 0.148 0.05 0.8105368
# 4 180 0.148 0.05 0.8344142
# 4 190 0.148 0.05 0.8557241
# 4 200 0.148 0.05 0.8746580
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovacount
#To plot the power curve:
plot(res)
#To calculate the required sample size for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=NULL,V=0.148,power=0.8, alpha=0.05)
# One-way Analogous ANOVA with Count Data
#
# k n V alpha power
# 4 165.9143 0.148 0.05 0.8
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovacount
#To calculate the minimum detectable effect size for one-way ANOVA (overall test) with count data:
wp.anova.count(k=4,n=100,V=NULL,power=0.8, alpha=0.05)
# One-way Analogous ANOVA with Count Data
#
# k n V alpha power
# 4 100 0.1906373 0.05 0.8
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovacount
#To generate a power curve given a sequence of effect sizes:
res <- wp.anova.count(k=5,n=100,V=seq(0.1,0.5,0.05),alpha=0.05,power=NULL)
res
# One-way Analogous ANOVA with Count Data
#
# k n V alpha power
# 5 100 0.10 0.05 0.3200744
# 5 100 0.15 0.05 0.6634861
# 5 100 0.20 0.05 0.9118531
# 5 100 0.25 0.05 0.9893643
# 5 100 0.30 0.05 0.9994549
# 5 100 0.35 0.05 0.9999887
# 5 100 0.40 0.05 0.9999999
# 5 100 0.45 0.05 1.0000000
# 5 100 0.50 0.05 1.0000000
#
# NOTE: n is the total sample size
# URL: http://psychstat.org/anovacount
Statistical Power Curve for Bivariate Latent Change Score Models based on Monte Carlo Simulation
Description
A longitudinal design often involves data collection on multiple variables from multiple participants at multiple times. Growth curve models (GCM) are structural equation models for longitudinal data analysis (McArdle & Epstein, 1987; McArdle & Nesselroade, 2014). Latent change score models (LCSM) combine difference equations with growth curves to investigate change in longitudinal studies . LCSM provied an efficient way to model nonlinear trajectory (e.g., McArdle, 2000; McArdle & Hamagami, 2001; Hamagami et al., 2010). This function is used to conduct power analysis for bivariate LCSMs based on a Monte Carlo method ( a method also used by Muthén & Muthén, 2002; Thoemmes et al., 2010; Zhang & Wang, 2009; Zhang, 2014). For each Monte Carlo replication, the Maximum likelihood ratio test is used for the model, while the Wald test is used for the parameter test. The method can obtain the power for testing each individual parameter of the models such as the change rate and coupling parameters.
Usage
wp.blcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
varey = 1, vary0 = 1, varys = 1, vary0ys = 0, alpha = 0.05,
betax = 0, mx0 = 0, mxs = 0, varex = 1, varx0 = 1, varxs = 1,
varx0xs = 0, varx0y0 = 0, varx0ys = 0, vary0xs = 0, varxsys = 0,
gammax = 0, gammay = 0, ...)
Arguments
N |
Sample size. It is 100 by default. |
T |
Number of measurement occasions. It is 5 by default. |
R |
Number of replications for the Monte Carlo simulation. It is 1000 by default. |
betay |
Parameter in the model: The compound rate of change for variable y. Its default value is 0. |
my0 |
Parameter in the model: Mean of the initial latent score for variable y. Its default value is 0. |
mys |
Parameter in the model: Mean of the linear constant effect for variable y. Its default value is 0. |
varey |
Parameter in the model: Variance of the measurement error/uniqueness score for variable y. Its default value is 1. |
vary0 |
Parameter in the model: Variance of the initial latent score for variable y. Its default value is 1. |
varys |
Parameter in the model: Variance of the linear constant effect for variable y. Its default value is 0. |
vary0ys |
Parameter in the model: Covariance of the initial latent score and the linear constant effect for variable y. Its default value is 0. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
betax |
Parameter in the model: The compound rate of change for variable x. Its default value is 0. |
mx0 |
Parameter in the model: Mean of the initial latent score for variable x. Its default value is 0. |
mxs |
Parameter in the model: Mean of the linear constant effect for variable x. Its default value is 0. |
varex |
Parameter in the model: Variance of the measurement error/uniqueness score for variable x. Its default value is 1. |
varx0 |
Parameter in the model: Variance of the initial latent score for variable x. Its default value is 1. |
varxs |
Parameter in the model: Variance of the linear constant effect for variable x. Its default value is 0. |
varx0xs |
Parameter in the model: Covariance of the initial latent score and the linear constant effect for variable x. Its default value is 0. |
varx0y0 |
Parameter in the model: Covariance of the initial latent scores for y and x. Its default value is 0. |
varx0ys |
Parameter in the model: Covariance of the initial latent score for x and the linear constant effect for y. Its default value is 0. |
vary0xs |
Parameter in the model: Covariance of the initial latent score for y and the linear constant effect for x. Its default value is 0. |
varxsys |
Parameter in the model: Covariance of the linear constant effects for y and x. Its default value is 0. |
gammax |
Coupling parameter in the model: The effect of variable x on the change score of variable y. Its default value is 0. |
gammay |
Coupling parameter in the model: The effect of variable y on the change score of variable x. Its default value is 0. |
... |
Extra arguments. It is not required. |
Value
An object of the power analysis. The output of the R function includes 4 main pieces of information for each parameter in the model. The first is the Monte Carlo estimate (mc.est). It is calculated as the mean of the R sets of parameter estimates from the simulated data. Note that the Monte Carlo estimates should be close to the population parameter values used in the model. The second is the Monte Carlo standard deviation (mc.sd), which is calculated as the standard deviation of the R sets of parameter estimates. The third is the Monte Carlo standard error (mc.se), which is obtained as the average of the R sets of standard error estimates of the parameter estimates. Lastly, mc.power is the statistical power for each parameter.
References
Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
## Not run:
#To conduct power analysis for a bivariate LCSM with sample size equal to 100:
wp.blcsm(N=100, T=5, R=1000, betay=0.08, my0=20, mys=1.5, varey=9,
vary0=3, varys=1, vary0ys=0, alpha=0.05, betax=0.2, mx0=20, mxs=5,
varex=9, varx0=3, varxs=1, varx0xs=0, varx0y0=1, varx0ys=0,
vary0xs=0, varxsys=0, gammax=0, gammay=-.1)
# pop.par mc.est mc.sd mc.se mc.power N T
# betax 0.20 0.230 0.260 0.187 0.241 100 5
# betay 0.08 0.164 0.572 0.435 0.081 100 5
# gammax 0.00 -0.033 0.234 0.178 0.112 100 5
# gammay -0.10 -0.175 0.641 0.458 0.075 100 5
# mx0 20.00 20.004 0.336 0.326 1.000 100 5
# mxs 5.00 5.933 7.848 5.615 0.167 100 5
# my0 20.00 20.019 0.346 0.326 1.000 100 5
# mys 1.50 0.451 6.933 5.321 0.156 100 5
# varex 9.00 8.941 0.744 0.732 1.000 100 5
# varey 9.00 8.939 0.749 0.720 1.000 100 5
# varx0 3.00 3.029 1.243 1.222 0.739 100 5
# varx0xs 0.00 -0.210 0.768 0.767 0.030 100 5
# varx0y0 1.00 1.052 0.840 0.835 0.226 100 5
# varx0ys 0.00 -0.012 0.668 0.601 0.017 100 5
# varxs 0.60 2.343 6.805 2.687 0.090 100 5
# varxsys 0.00 0.072 3.559 1.740 0.019 100 5
# vary0 3.00 2.951 1.423 1.245 0.684 100 5
# vary0xs 0.00 0.198 2.263 1.629 0.031 100 5
# vary0ys 0.00 -0.371 1.970 1.511 0.106 100 5
# varys 0.05 1.415 3.730 2.096 0.024 100 5
#To conduct power analysis for a bivariate LCSM with sample size equal to 500:
wp.blcsm(N=500, T=5, R=1000, betay=0.08, my0=20, mys=1.5, varey=9,
vary0=3, varys=1, vary0ys=0, alpha=0.05, betax=0.2, mx0=20
, mxs=5, varex=9, varx0=3, varxs=1, varx0xs=0, varx0y0=1,
varx0ys=0, vary0xs=0, varxsys=0, gammax=0, gammay=-.1)
# pop.par mc.est mc.sd mc.se mc.power N T
# betax 0.20 0.2009 0.031 0.031 1.000 500 5
# betay 0.08 0.0830 0.070 0.068 0.199 500 5
# gammax 0.00 -0.0014 0.030 0.029 0.057 500 5
# gammay -0.10 -0.1022 0.072 0.073 0.271 500 5
# mx0 20.00 19.9911 0.145 0.145 1.000 500 5
# mxs 5.00 5.0308 0.939 0.942 1.000 500 5
# my0 20.00 19.9999 0.143 0.146 1.000 500 5
# mys 1.50 1.4684 0.889 0.885 0.420 500 5
# varex 9.00 8.9836 0.340 0.328 1.000 500 5
# varey 9.00 8.9961 0.341 0.328 1.000 500 5
# varx0 3.00 3.0052 0.524 0.523 1.000 500 5
# varx0xs 0.00 -0.0144 0.222 0.230 0.047 500 5
# varx0y0 1.00 1.0064 0.360 0.360 0.808 500 5
# varx0ys 0.00 -0.0012 0.199 0.201 0.051 500 5
# varxs 1.00 1.0312 0.180 0.189 1.000 500 5
# varxsys 0.00 0.0028 0.161 0.163 0.045 500 5
# vary0 3.00 2.9777 0.519 0.547 1.000 500 5
# vary0xs 0.00 0.0072 0.286 0.294 0.035 500 5
# vary0ys 0.00 -0.0135 0.252 0.257 0.043 500 5
# varys 1.00 1.0246 0.260 0.253 0.999 500 5
## End(Not run)
Statistical Power Analysis for Correlation
Description
This function is for power analysis for correlation. Correlation measures whether and how a pair of variables are related. The Pearson Product Moment correlation coefficient (r) is adopted here. The power calculation for correlation is conducted based on Fisher's z transformation of Pearson correlation coefficent (Fisher, 1915, 1921).
Usage
wp.correlation(n = NULL, r = NULL, power = NULL, p = 0, rho0 = 0,
alpha = 0.05, alternative = c("two.sided", "less", "greater"))
Arguments
n |
Sample size. |
r |
Effect size or correlation. According to Cohen (1988), a correlation coefficient of 0.10, 0.30, and 0.50 are considered as an effect size of "small", "medium", and "large", respectively. |
power |
Statistical power. |
p |
Number of variables to partial out. |
rho0 |
Null correlation coefficient. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
alternative |
Direction of the alternative hypothesis ( |
Value
An object of the power analysis.
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika, 10(4), 507-521.
Fisher, R. A. (1921). On the probable error of a coefficient of correlation deduced from a small sample. Metron, 1, 3-32.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
wp.correlation(n=50,r=0.3, alternative="two.sided")
# Power for correlation
#
# n r alpha power
# 50 0.3 0.05 0.5728731
#
# URL: http://psychstat.org/correlation
#To calculate the power curve with a sequence of sample sizes:
res <- wp.correlation(n=seq(50,100,10),r=0.3, alternative="two.sided")
res
# Power for correlation
#
# n r alpha power
# 50 0.3 0.05 0.5728731
# 60 0.3 0.05 0.6541956
# 70 0.3 0.05 0.7230482
# 80 0.3 0.05 0.7803111
# 90 0.3 0.05 0.8272250
# 100 0.3 0.05 0.8651692
#
# URL: http://psychstat.org/correlation
#To plot the power curve:
plot(res, type='b')
#To estimate the sample size with a given power:
wp.correlation(n=NULL, r=0.3, power=0.8, alternative="two.sided")
# Power for correlation
#
# n r alpha power
# 83.94932 0.3 0.05 0.8
#
# URL: http://psychstat.org/correlation
#To estimate the minimum detectable effect size with a given power:
wp.correlation(n=NULL,r=0.3, power=0.8, alternative="two.sided")
# Power for correlation
#
# n r alpha power
# 83.94932 0.3 0.05 0.8
#
# URL: http://psychstat.org/correlation
#
#To calculate the power curve with a sequence of effect sizes:
res <- wp.correlation(n=100,r=seq(0.05,0.8,0.05), alternative="two.sided")
res
# Power for correlation
#
# n r alpha power
# 100 0.05 0.05 0.07854715
# 100 0.10 0.05 0.16839833
# 100 0.15 0.05 0.32163978
# 100 0.20 0.05 0.51870091
# 100 0.25 0.05 0.71507374
# 100 0.30 0.05 0.86516918
# 100 0.35 0.05 0.95128316
# 100 0.40 0.05 0.98724538
# 100 0.45 0.05 0.99772995
# 100 0.50 0.05 0.99974699
# 100 0.55 0.05 0.99998418
# 100 0.60 0.05 0.99999952
# 100 0.65 0.05 0.99999999
# 100 0.70 0.05 1.00000000
# 100 0.75 0.05 1.00000000
# 100 0.80 0.05 1.00000000
#
# URL: http://psychstat.org/correlation
Statistical Power Analysis for Cluster Randomized Trials with 2 Arms
Description
Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 2 arms (i.e., a treatment and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).
Usage
wp.crt2arm(n = NULL, f = NULL, J = NULL, icc = NULL, power = NULL,
alpha = 0.05, alternative = c("two.sided", "one.sided"), interval = NULL)
Arguments
n |
Sample size. It is the number of individuals within each cluster. |
f |
Effect size. It specifies either the main effect of treatment, or the mean difference between the treatment clusters and the control clusters. |
J |
Number of clusters / sides. It tells how many clusters are considered in the study design. At least two clusters are required. |
icc |
Intra-class correlation. ICC is calculated as the ratio of between-cluster variance to the total variance. It quantifies the degree to which two randomly drawn observations within a cluster are correlated. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
alternative |
Type of the alternative hypothesis ( |
interval |
A vector containing the end-points of the interval to be searched for the root. |
Value
An object of the power analysis.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.crt2arm(f = 0.6, n = 20, J = 10, icc = 0.1, alpha = 0.05, power = NULL)
# Cluster randomized trials with 2 arms
#
# J n f icc power alpha
# 10 20 0.6 0.1 0.5901684 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt2arm
#To generate a power curve given a sequence of sample sizes:
res <- wp.crt2arm(f = 0.6, n = seq(20,100,10), J = 10,
icc = 0.1, alpha = 0.05, power = NULL)
res
# Cluster randomized trials with 2 arms
#
# J n f icc power alpha
# 10 20 0.6 0.1 0.5901684 0.05
# 10 30 0.6 0.1 0.6365313 0.05
# 10 40 0.6 0.1 0.6620030 0.05
# 10 50 0.6 0.1 0.6780525 0.05
# 10 60 0.6 0.1 0.6890755 0.05
# 10 70 0.6 0.1 0.6971076 0.05
# 10 80 0.6 0.1 0.7032181 0.05
# 10 90 0.6 0.1 0.7080217 0.05
# 10 100 0.6 0.1 0.7118967 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt2arm
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.crt2arm(f = 0.8, n = NULL, J = 10,
icc = 0.1, alpha = 0.05, power = 0.8)
# Cluster randomized trials with 2 arms
#
# J n f icc power alpha
# 10 16.02558 0.8 0.1 0.8 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt2arm
Statistical Power Analysis for Cluster Randomized Trials with 3 Arms
Description
Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).
Usage
wp.crt3arm(n = NULL, f = NULL, J = NULL, icc = NULL, power = NULL,
alpha = 0.05, alternative = c("two.sided", "one.sided"),
type = c("main", "treatment", "omnibus"), interval = NULL)
Arguments
n |
Sample size. It is the number of individuals within each cluster. |
f |
Effect size. It specifies one of the three types of effects: the main effect of treatment, the mean difference between the treatment clusters, and the control clusters. |
J |
Number of clusters / sides. It tells how many clusters are considered in the study design. At least two clusters are required. |
icc |
Intra-class correlation. ICC is calculated as the ratio of between-cluster variance to the total variance. It quantifies the degree to which two randomly drawn observations within a cluster are correlated. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
alternative |
Type of the alternative hypothesis ( |
type |
Type of effect ( |
interval |
A vector containing the end-points of the interval to be searched for the root. |
Value
An object of the power analysis.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.crt3arm(f = 0.5, n = 20, J = 10, icc = 0.1, alpha = 0.05, power = NULL)
# Cluster randomized trials with 3 arms
#
# J n f icc power alpha
# 10 20 0.5 0.1 0.3940027 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt3arm
#To generate a power curve given a sequence of sample sizes:
res <- wp.crt3arm(f = 0.5, n = seq(20, 100, 10), J = 10,
icc = 0.1, alpha = 0.05, power = NULL)
res
# Cluster randomized trials with 3 arms
#
# J n f icc power alpha
# 10 20 0.5 0.1 0.3940027 0.05
# 10 30 0.5 0.1 0.4304055 0.05
# 10 40 0.5 0.1 0.4513376 0.05
# 10 50 0.5 0.1 0.4649131 0.05
# 10 60 0.5 0.1 0.4744248 0.05
# 10 70 0.5 0.1 0.4814577 0.05
# 10 80 0.5 0.1 0.4868682 0.05
# 10 90 0.5 0.1 0.4911592 0.05
# 10 100 0.5 0.1 0.4946454 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt3arm
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.crt3arm(f = 0.8, n = NULL, J = 10, icc = 0.1, alpha = 0.05, power = 0.8)
# Cluster randomized trials with 3 arms
#
# J n f icc power alpha
# 10 27.25145 0.8 0.1 0.8 0.05
#
# NOTE: n is the number of subjects per cluster.
# URL: http://psychstat.org/crt3arm
Effect size calculatator based on raw data for Cluster Randomized Trials with 2 Arms
Description
This function is for effect size and ICC calculation for CRT with 2 arms based on empirical data. Cluster randomized trials (CRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). The Effect size f specifies the main effect of treatment, the mean difference between the treatment clusters and the control clusters. This function is used to calculate the effect size with a input data set.
Usage
wp.effect.CRT2arm(file)
Arguments
file |
The input data set. The first column of the data is the ID variable, the second column represents cluster, the third column is the outcome variable, and the fourth column is the condition variable (0 for control, 1 for condition). |
Value
A list including effect size f and ICC.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Examples
#Empirical data set CRT2:
CRT2
#ID cluster score group
#1 1 6 0
#2 1 2 0
#3 2 6 1
#4 2 5 1
#5 3 1 0
#6 3 4 0
#7 4 6 1
#8 4 4 1
#To calculate the effect size and ICC based on empirical data
wp.effect.CRT2arm (CRT2)
# Effect size for CRT2arm
#
# f ICC
# 1.264911 -0.5
#
# NOTE: f is the effect size.
# URL: http://psychstat.org/crt2arm
Effect size calculatator based on raw data for Cluster Randomized Trials with 3 Arms
Description
This function is for effect size and ICC calculation for Cluster randomized trials (CRT) with 3 arms based on empirical data. CRT are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from CRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in second level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). The Effect size f specifies the main effect of treatment, the mean difference between the treatment clusters and the control clusters. This function is used to calculate the effect size with a input data set.
Usage
wp.effect.CRT3arm(file)
Arguments
file |
The input data set. The first column of the data is the ID variable, the second column represents cluster, the third column is the outcome variable, and the fourth column is the condition variable (0 for control, 1 for treatment1, 2 for treatment2). |
Value
A list including effect size f1, f2, f3, and ICC.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Examples
#To calculate the effect sizes based on empirical data
wp.effect.CRT3arm (CRT3)
# Effect size for CRT3arm
#
# f1 f2 f3 ICC
# 0.6389258 -0.6189113 0.3931397 -0.019794
#
# NOTE: f1 for treatment main effect;
# f2 for difference between two treatments;
# f3 for effect size of omnibus test.
# URL: http://psychstat.org/crt3arm
Effect size calculatator based on raw data for Multisite Randomized Trials with 2 Arms
Description
This function is for effect size calculation for Multisite randomized trials (MRT) with 2 arms based on empirical data. MRTs are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierachical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then mutiple indicators will be used. Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "site" type tests the variance of cluster/site means; and (3) The "variance" type tests variance of treatment effects. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). This function is used to calculate the effect size with a input data set.
Usage
wp.effect.MRT2arm(file)
Arguments
file |
The input data set. The first column of the data is the ID variable, the second column represents cluster, the third column is the outcome variable, and the fourth column is the condition variable (0 for control, 1 for condition). |
Value
A list including effect size f.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Examples
#To calculate the effect size based on empirical data
wp.effect.MRT2arm (MRT2)
# Effect size for MRT2arm
#
# f
# -0.2986755
#
# NOTE: f is the effect size.
# URL: http://psychstat.org/mrt2arm
Effect size calculatator based on raw data for Multisite Randomized Trials with 3 Arms
Description
This function is for effect size calculation for Multisite randomized trials (MRT) with 3 arms based on empirical data. MRTs are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a contral arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierachical linear model, where the indicator variable for reatment assignment is included in first level. If a study contains multiple treatments, then mutiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "treatment" type tests the difference between the two treaments; and (3) The "omnibus" type tests whether the three arms are all equivalent. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013). This function is used to calculate the effect size with a input data set.
Usage
wp.effect.MRT3arm(file)
Arguments
file |
The input data set. The first column of the data is the ID variable, the second column represents cluster, the third column is the outcome variable, and the fourth column is the condition variable (0 for control, 1 for treatment1, 2 for treatment2). |
Value
A list including effect size f.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Examples
#To calculate the effect size and ICC based on empirical data
# wp.effect.MRT3arm(MRT3)
# Effect size for MRT3arm
#
# f1 f2
# -0.6214215 -0.355098
#
# NOTE: f1 for treatment main effect;
# f2 for comparing the two treatments.
# URL: http://psychstat.org/mrt3arm
Power analysis for two-way, three-way and k-way ANOVA
Description
Power analysis for two-way, three-way and k-way ANOVA
Usage
wp.kanova(n = NULL, ndf = NULL, f = NULL, ng = NULL, alpha = 0.05, power = NULL)
Arguments
n |
Sample size |
ndf |
Numerator degrees of freedom |
f |
Effect size |
ng |
Number of groups |
alpha |
Significance level |
power |
Statistical power |
References
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
########## Chapter 6. two-way and three-way ANOVA #############
## Main effect of two-way ANOVA
wp.kanova(n=120, ndf=2, f=0.2, alph=0.05, ng=6)
## Interaction effect of two-way ANOVA
wp.kanova(n=120, ndf=2, f=0.4, alph=0.05, ng=6)
## Interaction effect of three-way ANOVA
wp.kanova(n=360, ndf=4, f=0.3, alph=0.05, ng=18)
Statistical Power Curve for Univariate Latent Change Score Models based on Monte Carlo Simulation
Description
A longitudinal design often involves data collection on multiple variables from multiple participants at multiple times. Growth curve models (GCM) are structural equation models for longitudinal data analysis (McArdle & Epstein, 1987; McArdle & Nesselroade, 2014). Latent change score models (LCSM) combine difference equations with growth curves to investigate change in longitudinal studies . LCSM provied an efficient way to model nonlinear trajectory (e.g., McArdle, 2000; McArdle & Hamagami, 2001; Hamagami et al., 2010). This function is used to conduct power analysis for univariate LCSMs based on a Monte Carlo method ( a method also used by Muthén & Muthén, 2002; Thoemmes et al., 2010; Zhang & Wang, 2009; Zhang, 2014). For each Monte Carlo replication, the Maximum likelihood ratio test is used for the model, while the Wald test is used for the parameter test. The method can obtain the power for testing each individual parameter of the models such as the change rate and coupling parameters.
Usage
wp.lcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
varey = 1, vary0 = 1, varys = 1, vary0ys = 0, alpha = 0.05, ...)
Arguments
N |
Sample size. It is 100 by default. |
T |
Number of measurement occasions. It is 5 by default. |
R |
Number of replications for the Monte Carlo simulation. It is 1000 by default. |
betay |
Parameter in the model: The compound rate of change. Its default value is 0. |
my0 |
Parameter in the model: Mean of the initial latent score. Its default value is 0. |
mys |
Parameter in the model: Mean of the linear constant effect. Its default value is 0. |
varey |
Parameter in the model: Variance of the measurement error/uniqueness score. Its default value is 1. |
vary0 |
Parameter in the model: Variance of the initial latent score. Its default value is 1. |
varys |
Parameter in the model: Variance of the linear constant effect. Its default value is 0. |
vary0ys |
Parameter in the model: Covariance of the initial latent score and the linear constant effect. Its default value is 0. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
... |
Extra arguments. It is not required. |
Value
An object of the power analysis. The output of the R function includes 4 main pieces of information for each parameter in the model. The first is the Monte Carlo estimate (mc.est). It is calculated as the mean of the R sets of parameter estimates from the simulated data. Note that the Monte Carlo estimates should be close to the population parameter values used in the model. The second is the Monte Carlo standard deviation (mc.sd), which is calculated as the standard deviation of the R sets of parameter estimates. The third is the Monte Carlo standard error (mc.se), which is obtained as the average of the R sets of standard error estimates of the parameter estimates. Lastly, mc.power is the statistical power for each parameter.
References
Zhang, Z., & Liu, H. (2018). Sample Size and Measurement Occasion Planning for Latent Change Score Models through Monte Carlo Simulation. In E. Ferrer, S. M. Boker, and K. J. Grimm (Eds.) Advances in Longitudinal Models for Multivariate Psychology: A Festschrift for Jack McArdle.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
## Not run:
#Power analysis for a univariate LCSM
#Power for each parameter given sample size, number of measurement occasions,
# true effect (true values of parameters), and significance level:
wp.lcsm(N = 100, T = 5, R = 1000, betay = 0.1, my0 = 20, mys = 1.5,
varey = 9, vary0 = 2.5, varys = .05, vary0ys = 0, alpha = 0.05)
# pop.par mc.est mc.sd mc.se mc.power N T
# betay 0.10 0.103 0.043 0.044 0.664 100 5
# my0 20.00 19.999 0.324 0.319 1.000 100 5
# mys 1.50 1.418 1.106 1.120 0.274 100 5
# varey 9.00 8.961 0.724 0.732 1.000 100 5
# vary0 2.50 2.463 1.151 1.139 0.583 100 5
# vary0ys 0.00 -0.004 0.408 0.403 0.048 100 5
# varys 0.05 0.053 0.173 0.175 0.050 100 5
#
# #To calculate the Type I error rate and power for parameters
wp.lcsm(N = 100, T = 5, R = 1000, betay = 0, my0 = 0, mys = 0,
varey = 1, vary0 = 1, varys = 1, vary0ys = 0,alpha = 0.05)
# pop.par mc.est mc.sd mc.se mc.power N T
# betay 0 0.001 0.056 0.056 0.046 100 5
# my0 0 0.001 0.129 0.126 0.056 100 5
# mys 0 0.002 0.105 0.105 0.044 100 5
# varey 1 0.994 0.083 0.081 1.000 100 5
# vary0 1 0.990 0.236 0.230 1.000 100 5
# vary0ys 0 -0.005 0.136 0.136 0.044 100 5
# varys 1 1.006 0.227 0.227 1.000 100 5
# To generate a power curve for different sample sizes for a univariate LCSM
res <- wp.lcsm(N = seq(100, 200, 10), T = 5, R = 1000, betay = 0.1,
my0 = 20, mys = 1.5, varey = 9, vary0 = 2.5,
varys = .05, vary0ys = 0, alpha = 0.05)
#plot(res, parameter='betay')
#plot(res, parameter='mys')
# To generate a power curve for different numbers of occasions for a univariate LCSM
res <- wp.lcsm(N = 100, T = 4:10, R = 1000, betay = 0.1, my0 = 20, mys = 1.5,
varey = 9, vary0 = 2.5, varys = .05, vary0ys = 0, alpha = 0.05)
#plot(res, parameter='betay')
#plot(res, parameter='mys')
## End(Not run)
Statistical Power Analysis for Logistic Regression
Description
This function is for Logistic regression models. Logistic regression is a type of generalized linear models where the outcome variable follows Bernoulli distribution. Here, Maximum likelihood methods is used to estimate the model parameters. The estimated regression coefficent is assumed to follow a normal distribution. A Wald test is use to test the mean difference between the estimated parameter and the null parameter (tipically the null hypothesis assumes it equals 0). The procedure introduced by Demidenko (2007) is adopted here for computing the statistical power.
Usage
wp.logistic(n = NULL, p0 = NULL, p1 = NULL, alpha = 0.05,
power = NULL, alternative = c("two.sided", "less", "greater"),
family = c("Bernoulli", "exponential", "lognormal", "normal", "Poisson",
"uniform"), parameter = NULL)
Arguments
n |
Sample size. |
p0 |
Prob(Y=1|X=0): the probobility of observieng 1 for the outcome variable Y when the predictor X equals 0. |
p1 |
Prob(Y=1|X=1): the probobility of observieng 1 for the outcome variable Y when the predictor X equals 1. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
alternative |
Direction of the alternative hypothesis ( |
family |
Distribution of the predictor ( |
parameter |
Corresponding parameter for the predictor's distribution. The default is 0.5 for "Bernoulli", 1 for "exponential", (0,1) for "lognormal" or "normal", 1 for "Poisson", and (0,1) for "uniform". |
Value
An object of the power analysis.
References
Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.logistic(n = 200, p0 = 0.15, p1 = 0.1, alpha = 0.05,
power = NULL, family = "normal", parameter = c(0,1))
# Power for logistic regression
#
# p0 p1 beta0 beta1 n alpha power
# 0.15 0.1 -1.734601 -0.4626235 200 0.05 0.6299315
#
# URL: http://psychstat.org/logistic
#To generate a power curve given a sequence of sample sizes:
res <- wp.logistic(n = seq(100,500,50), p0 = 0.15, p1 = 0.1, alpha = 0.05,
power = NULL, family = "normal", parameter = c(0,1))
res
# Power for logistic regression
#
# p0 p1 beta0 beta1 n alpha power
# 0.15 0.1 -1.734601 -0.4626235 100 0.05 0.3672683
# 0.15 0.1 -1.734601 -0.4626235 150 0.05 0.5098635
# 0.15 0.1 -1.734601 -0.4626235 200 0.05 0.6299315
# 0.15 0.1 -1.734601 -0.4626235 250 0.05 0.7264597
# 0.15 0.1 -1.734601 -0.4626235 300 0.05 0.8014116
# 0.15 0.1 -1.734601 -0.4626235 350 0.05 0.8580388
# 0.15 0.1 -1.734601 -0.4626235 400 0.05 0.8998785
# 0.15 0.1 -1.734601 -0.4626235 450 0.05 0.9302222
# 0.15 0.1 -1.734601 -0.4626235 500 0.05 0.9518824
#
# URL: http://psychstat.org/logistic
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.logistic(n = NULL, p0 = 0.15, p1 = 0.1, alpha = 0.05,
power = 0.8, family = "normal", parameter = c(0,1))
# Power for logistic regression
#
# p0 p1 beta0 beta1 n alpha power
# 0.15 0.1 -1.734601 -0.4626235 298.9207 0.05 0.8
#
# URL: http://psychstat.org/logistic
Statistical Power Analysis for SEM Based on Chi-square Difference Test
Description
This function is for SEM power analysis based on the chi-square difference test.
Usage
wp.mc.chisq.diff(full.model.pop, full.model,
reduced.model, N=100, R=1000, alpha=0.05)
Arguments
full.model.pop |
Full model (under the alternative hypothesis) with population parameters. |
full.model |
Full model (under the alternative hypothesis) lavaan specification. |
reduced.model |
Reduced model (under the null hypothesis) lavaan specification. |
N |
Sample size. |
R |
Number of Monte Carlo replications. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
Value
An object of the power analysis.
power |
Statistical power. |
df |
Degrees of freedom |
chi.diff |
Chi-square differences between the reduced model and the full model |
References
Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
set.seed(20220722)
full.model.pop <-'
y1 ~ 0.4*x
y2 ~ 0.5*x + 0.2*y1
y3 ~ 0.4*x
y4 ~ 0.4*y1 + 0.4*y2 + 0.4*y3
y1 ~~ 0.84*y1
y2 ~~ 0.61*y2
y3 ~~ 0.84*y3
y4 ~~ 0.27*y4
'
full.model <-'
y1 ~ x
y2 ~ x + y1
y3 ~ x
y4 ~ y1 + y2 + y3
'
reduced.model <-'
y1 ~ x
y2 ~ x
y3 ~ x
y4 ~ y1 + y3
'
wp.mc.chisq.diff(full.model.pop, full.model, reduced.model)
Statistical Power Analysis for Structural Equation Modeling / Mediation based on Monte Carlo Simulation
Description
Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. A mediation model can be viewed as a SEM model. Funtions wp.sem.chisq and wp.sem.rmsea provide anlytical solutions of power analysis for SEM. Function wp.mediation provides anlytical solutions of power analysis for a simple mediatoin model. This function provides a solution based on Monte Carlo simulation (see Zhang, 2014). If the model is a mediation, Sobel test is used for the mediation / indirect effects. The solution is extended from the general framework for power analysis for complex mediation models using Monte Carlo simulation in Mplus (Muthén & Muthén, 2011) proposed by Thoemmes et al. (2010). We extended the framework in two ways. First, the method allows the specification of nonnormal data in the Monte Carlo simulation and can thereby reflect more closely practical data collection. Second, the function wp.mc.sem.basic of a free, open-source R package, WebPower, is developed to ease power anlysis for mediation models using the proposed method.
Usage
wp.mc.sem.basic(model, indirect = NULL, nobs = 100, nrep = 1000,
alpha = 0.95, skewness = NULL, kurtosis = NULL, ovnames = NULL,
se = "default", estimator = "default", parallel = "no",
ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL, ...)
Arguments
model |
Model specified using lavaan syntax. More about model specification can be found in Rosseel (2012). |
indirect |
Indirect effect difined using lavaan syntax. |
nobs |
Sample size. |
nrep |
Number of replications for the Monte Carlo simulation. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
skewness |
A sequence of skewnesses of the observed variables. |
kurtosis |
A sequence of kurtosises of the observed variables. |
ovnames |
Names of the observed variables in the model. |
se |
The method for calculatating the standard errors. Its default method "default" is regular standard errors. More about methods specification standard errors calculatationcan be found in Rosseel (2012). |
estimator |
Estimator. It is Maxmum likelihood estimator by default. More about estimator specification can be found in Rosseel (2012). |
parallel |
Parallel computing ( |
ncore |
Number of processors used for parallel computing. By default, ncore = Sys.getenv('NUMBER_OF_PROCESSORS'). |
cl |
Number of clusters. It is NULL by default. When it is NULL, the program will detect the number of clusters automatically. |
... |
Extra arguments. It is not required. |
Value
An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.
References
MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological methods, 1(2), 130.
Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.
Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50(1), 83-90.
Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.
Zhang, Z. (2014). Monte Carlo based statistical power analysis for mediation models: Methods and software. Behavior research methods, 46(4), 1184-1198.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
set.seed(20220722)
#To calculate power for mediation based on Monte Carlo simulation when Sobel test is used:
#To specify the model
demo ="
y ~ cp*x + start(0)*x + b*m + start(0.39)* m
m ~ a*x + start(0.39)*x
x ~~ start(1)*x
m ~~ start(1)*m
y ~~ start(1)*y
"
#To specify the indirect effects
mediation = "
ab := a*b
abc:= a*b + cp
"
#To calculate power for mediation using regular standard errors
## change nrep to at least 1,000 in real data analysis
sobel.regular = wp.mc.sem.basic(model=demo, indirect=mediation, nobs=100, nrep=10,
skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"))
#To calculate power for mediation using robust standard errors
sobel.robust = wp.mc.sem.basic(model=demo, indirect=mediation, nobs=100, nrep=10,
skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"), se="robust")
#To print the power for mediation based on Sobel test using regular standard errors:
summary(sobel.regular)
# Basic information:
#
# Esimation method ML
# Standard error standard
# Number of requested replications 1000
# Number of successful replications 1000
#
# True Estimate MSE SD Power Coverage
# Regressions:
# y ~
# x (cp) 0.000 0.003 0.106 0.107 0.045 0.955
# m (b) 0.390 0.387 0.099 0.113 0.965 0.919
# m ~
# x (a) 0.390 0.389 0.100 0.101 0.976 0.953
# Variances:
# x 1.000 0.995 0.141 0.139 1.000 0.936
# m 1.000 0.981 0.139 0.137 1.000 0.923
# y 1.000 0.968 0.137 0.330 1.000 0.560
#
# Indirect/Mediation effects:
# ab 0.152 0.150 0.056 0.060 0.886 0.928
# abc 0.152 0.153 0.106 0.109 0.305 0.948
#To print the power analysis results for mediation based on Sobel test using robust standard errors:
summary(sobel.robust)
# Basic information:
#
# Esimation method ML
# Standard error robust.sem
# Number of requested replications 1000
# Number of successful replications 1000
#
# True Estimate MSE SD Power Coverage
# Regressions:
# y ~
# x (cp) 0.000 -0.003 0.106 0.113 0.055 0.945
# m (b) 0.390 0.398 0.111 0.119 0.972 0.927
# m ~
# x (a) 0.390 0.389 0.099 0.101 0.974 0.939
#
# Intercepts:
# y 0.000 0.000 0.100 0.104 0.058 0.942
# m 0.000 0.000 0.100 0.105 0.054 0.946
# x 0.000 -0.004 0.100 0.104 0.066 0.934
#
# Variances:
# x 1.000 0.991 0.138 0.140 1.000 0.930
# m 1.000 0.976 0.135 0.135 1.000 0.915
# y 1.000 1.002 0.281 0.365 0.981 0.805
#
# Indirect/Mediation effects:
# ab 0.152 0.156 0.060 0.064 0.870 0.900
# abc 0.152 0.153 0.108 0.117 0.303 0.936
Statistical Power Analysis for Structural Equation Modeling / Mediation based on Monte Carlo Simulation: bootstrap method
Description
Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. A mediation model can be viewed as a SEM model. Funtions wp.sem.chisq and wp.sem.rmsea provide anlytical solutions of power analysis for SEM. Function wp.mediation provides anlytical solutions of power analysis for a simple mediatoin model. This function provides a solution based on Monte Carlo simulation (see Zhang, 2014) and a bootstrap method for testing the indirect /mediation effects. The solution is extended from the general framework for power analysis for complex mediation models using Monte Carlo simulation in Mplus (Muthén & Muthén, 2011) proposed by Thoemmes et al. (2010). We extended the framework in three ways. First, we proposes a general method to conduct power analysis for mediation models based on the bootstrap method. The method is still based on Monte Carlo simulation but uses the bootstrap method to test mediation effects. Second, the method allows the specification of nonnormal data in the Monte Carlo simulation and can thereby reflect more closely practical data collection. Third, the function wp.mc.sem.boot of a free, open-source R package, WebPower, is developed to ease power anlysis for mediation models using the proposed method.
Usage
wp.mc.sem.boot(model, indirect = NULL, nobs = 100, nrep = 1000,
nboot = 1000, alpha = 0.95, skewness = NULL, kurtosis = NULL,
ovnames = NULL, se = "default", estimator = "default",
parallel = "no", ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL,
...)
Arguments
model |
Model specified using lavaan syntax. More about model specification can be found in Rosseel (2012). |
indirect |
Indirect effect difined using lavaan syntax. |
nobs |
Sample size. It is 100 by default. |
nrep |
Number of replications for the Monte Carlo simulation. It is 1000 by default. |
nboot |
Number of replications for the bootstrap to test the specified parameter (e.g., mediation). It is 1000 by default. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
skewness |
A sequence of skewnesses of the observed variables. It is not required. |
kurtosis |
A sequence of kurtosises of the observed variables. It is not required. |
ovnames |
Names of the observed variables in the model. It is not required. |
se |
The method for calculatating the standard errors. Its default method "default" is regular standard errors. More about methods specification standard errors calculatationcan be found in Rosseel (2012). |
estimator |
Estimator. It is Maxmum likelihood estimator by default. More about estimator specification can be found in Rosseel (2012). |
parallel |
Parallel computing ( |
ncore |
Number of processors used for parallel computing. By default, ncore = Sys.getenv('NUMBER_OF_PROCESSORS'). |
cl |
Number of clusters. It is NULL by default. When it is NULL, the program will detect the number of clusters automatically. |
... |
Extra arguments. It is not required. |
Value
An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.
References
Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.
Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.
Zhang, Z. (2014). Monte Carlo based statistical power analysis for mediation models: Methods and software. Behavior research methods, 46(4), 1184-1198.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
set.seed(20220722)
#To specify the model
demo ="
y ~ cp*x + start(0)*x + b*m + start(0.39)* m
m ~ a*x + start(0.39)*x
x ~~ start(1)*x
m ~~ start(1)*m
y ~~ start(1)*y
"
#To specify the indirect effects
mediation = "
ab := a*b
abc:= a*b + cp
"
#Power for mediation based on MC method when bootstrap method is used to test the effects:
# change nrep and nboot to at least 1,000 in real analysis
mediation.boot = wp.mc.sem.boot(model=demo, indirect=mediation, nobs=100,
nrep=10, nboot=10,
skewness=c(0, 0, 1.3), kurtosis=c(0,0,10), ovnames=c("x","m","y"))
#To print the power analysis results
summary(mediation.boot)
#Example: Power for Simple Mediation Analysis
ex1model <- "
math ~ c*ME + start(0)*ME + b*HE + start(0.39)*HE
HE ~ a*ME + start(0.39)*ME
"
indirect <- "ab:=a*b"
# change nrep and nboot to at least 1,000 in real analysis
boot.normal <- wp.mc.sem.boot(ex1model,indirect, 50, nrep=10, nboot=10)
summary(boot.normal)
boot.non.normal <- wp.mc.sem.boot(ex1model,indirect, 100, nrep=10, nboot=10,
skewness=c(-0.3, -0.7, 1.3),
kurtosis=c(1.5, 0, 5), ovnames=c('ME','HE','math'))
summary(boot.non.normal)
#Example: Multiple Group Mediation Analysis (Moderated Mediation)
ex3model <- "
y ~ start(c(0.283, 0.283))*x + c(c1,c2)*x + start(c(0.36, 0.14))*m + c(b1,b2)*m
m ~ start(c(0.721, 0.721))*x + c(a1,a2)*x
m =~ c(1,1)*m1 + start(c(0.8, 0.8))*m2 + start(c(0.8, 0.8))*m3
x ~~ start(c(0.25, 0.25))*x
y ~~ start(c(0.81, 0.95))*y
m ~~ start(c(0.87, 0.87))*m
m1 ~~ start(c(0.36, 0.36))*m1
m2 ~~ start(c(0.36, 0.36))*m2
m3 ~~ start(c(0.36, 0.36))*m3
"
# med1 and med2 are the mediation effect for group1 and group2, respectively.
indirect <- "
med1 := a1*b1
med2 := a2*b2
diffmed := a1*b1 - a2*b2
"
# change nrep and nboot to at least 1,000 in real analysis
bootstrap <- wp.mc.sem.boot(ex3model, indirect, nobs=c(400,200),
nrep=10, nboot=10)
summary(bootstrap)
#Example: A Longitudinal Mediation Model
ex4model <- "
x2 ~ start(.9)*x1 + x*x1
x3 ~ start(.9)*x2 + x*x2
m2 ~ start(.3)*x1 + a*x1 + start(.3)*m1 + m*m1
m3 ~ start(.3)*x2 + a*x2 + start(.3)*m2 + m*m2
y2 ~ start(.3)*m1 + b*m1 + start(.7)*y1 + y*y1
y3 ~ start(.3)*m2 + b*m2 + start(.7)*y2 + y*y2 + start(0)*x1 + c*x1
x1 ~~ start(.37)*m1
x1 ~~ start(.27)*y1
y1 ~~ start(.2278)*m1
x2 ~~ start(.19)*x2
x3 ~~ start(.19)*x3
m2 ~~ start(.7534)*m2
m3 ~~ start(.7534)*m3
y2 ~~ start(.3243)*y2
y3 ~~ start(.3243)*y3
"
indirect <- "ab := a*b"
# change nrep and nboot to at least 1,000 in real analysis
set.seed(10)
bootstrap <- wp.mc.sem.boot(ex4model, indirect, nobs=500, nrep=10, nboot=10)
summary(bootstrap)
Statistical Power Curve for Structural Equation Modeling / Mediation based on Monte Carlo Simulation
Description
A power curve is useful to graphically display how power changes with sample size (e.g., Zhang & Wang). This function is to generate a power curve for SEM based on Monte Carlo simulation, either using Sobel test or bootstrap method to test the indirect / mediation effects if applicable.
Usage
wp.mc.sem.power.curve(model, indirect = NULL, nobs = 100, type = "basic",
nrep = 1000, nboot = 1000, alpha = 0.95, skewness = NULL,
kurtosis = NULL, ovnames = NULL, se = "default",
estimator = "default", parallel = "no",
ncore = Sys.getenv("NUMBER_OF_PROCESSORS"), cl = NULL, ...)
Arguments
model |
Model specified using lavaan syntax. More about model specification can be found in Rosseel (2012). |
indirect |
Indirect effect difined using lavaan syntax. |
nobs |
Sample size. It is 100 by default. |
type |
The method used to test the indirect effects ( |
nrep |
Number of replications for the Monte Carlo simulation. It is 1000 by default. |
nboot |
Number of replications for the bootstrap to test the specified parameter (e.g., mediation). It is 1000 by default. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
skewness |
A sequence of skewnesses of the observed variables. It is not required. |
kurtosis |
A sequence of kurtosises of the observed variables. It is not required. |
ovnames |
Names of the observed variables in the model. It is not required. |
se |
The method for calculatating the standard errors. Its default method "default" is regular standard errors. More about methods specification standard errors calculatationcan be found in Rosseel (2012). |
estimator |
Estimator. It is Maxmum likelihood estimator by default. More about estimator specification can be found in Rosseel (2012). |
parallel |
Parallel computing ( |
ncore |
Number of processors used for parallel computing. By default, ncore = Sys.getenv('NUMBER_OF_PROCESSORS'). |
cl |
Number of clusters. It is NULL by default. When it is NULL, the program will detect the number of clusters automatically. |
... |
Extra arguments. It is not required. |
Value
An object of the power analysis. The power for all parameters in the model as well as the indirect effects if specified.
References
Rosseel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Ghent, Belgium: Ghent University.
Thoemmes, F., MacKinnon, D. P., & Reiser, M. R. (2010). Power analysis for complex mediational designs using Monte Carlo methods. Structural Equation Modeling, 17(3), 510-534.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
set.seed(20220722)
#To specify the model
ex2model ="
ept ~ start(0.4)*hvltt + b*hvltt + start(0)*age + start(0)*edu + start(2)*R
hvltt ~ start(-0.35)*age + a*age +c*edu + start(0.5)*edu
R ~ start(-0.06)*age + start(0.2)*edu
R =~ 1*ws + start(0.8)*ls + start(0.5)*lt
age ~~ start(30)*age
edu ~~ start(8)*edu
age ~~ start(-2.8)*edu
hvltt ~~ start(23)*hvltt
R ~~ start(14)*R
ws ~~ start(3)*ws
ls ~~ start(3)*ls
lt ~~ start(3)*lt
ept ~~ start(3)*ept
"
#To specify the indirect effects
indirect = "ind1 := a*b + c*b"
nobs <- seq(100, 2000, by =200)
#To calculate power curve:
# change nrep and nboot to at least 1,000 in real analysis
power.curve = wp.mc.sem.power.curve(model=ex2model, indirect=indirect,
nobs=nobs, type='boot',
nrep=10, nboot=10)
Power analysis for t-test based on Monte Carlo simulation
Description
Power analysis for t-test based on Monte Carlo simulation
Usage
wp.mc.t(n = NULL, R0 = 1e+05, R1 = 1000, mu0 = 0, mu1 = 0,
sd = 1, skewness = 0, kurtosis = 3, alpha = 0.05,
type = c("two.sample", "one.sample", "paired"),
alternative = c("two.sided", "less", "greater"))
Arguments
n |
Sample size |
R0 |
Number of replications under the null |
R1 |
Number of replications |
mu0 |
Population mean under the null |
mu1 |
Population mean under the alternative |
sd |
Standard deviation |
skewness |
Skewness |
kurtosis |
kurtosis |
alpha |
Significance level |
type |
Type of anlaysis |
alternative |
alternative hypothesis |
References
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
set.seed(20220722)
########## Chapter 16. Monte Carlo t-test #############
wp.mc.t(n=20 , mu0=0, mu1=0.5, sd=1, skewness=0,
kurtosis=3, type = c("one.sample"), alternative = c("two.sided"))
wp.mc.t(n=40 , mu0=0, mu1=0.3, sd=1, skewness=1,
kurtosis=6, type = c("paired"), alternative = c("greater"))
wp.mc.t(n=c(15, 15), mu1=c(0.2, 0.5), sd=c(0.2, 0.5),
skewness=c(1, 2), kurtosis=c(4, 6), type = c("two.sample"), alternative = c("less"))
Statistical Power Analysis for Simple Mediation
Description
This function is for mediation models. Mediation models can be used to investigate the underlying mechanisms related to why an input variable x influences an output variable y (e.g., Hayes, 2013; MacKinnon, 2008). The mediation effect is calculated as a*b, where a is the path coefficent from the predictor x to the mediator m, and b is the path coefficent from the mediator m to the outcome variable y. Sobel test statistic (Sobel, 1982) is used to test whether the mediation effect is significantly different from zero.
Usage
wp.mediation(n = NULL, power = NULL, a = 0.5, b = 0.5, varx = 1,
vary = 1, varm = 1, alpha = 0.05, interval = NULL)
Arguments
n |
Sample size. |
power |
Statistical power. |
a |
Coefficient from x to m. The default value is 0.5. |
b |
Coefficient from m to y. The default value is 0.5. |
varx |
Variance of x. The default value is 1. |
vary |
Variance of y. The default value is 1. |
varm |
Variance of m. The default value is 1. |
alpha |
significance level chosen for the test. It equals 0.05 by default. |
interval |
A vector containing the end-points of the interval to be searched for the root. |
Value
An object of the power analysis.
References
Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford Press.
MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Routledge.
Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological methodology, 13, 290-312.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.mediation(n = 100, power = NULL, a = 0.5, b = 0.5,
varx = 1, vary = 1, varm = 1, alpha = 0.05)
# Power for simple mediation
#
# n power a b varx varm vary alpha
# 100 0.9337271 0.5 0.5 1 1 1 0.05
#
# URL: http://psychstat.org/mediation
#To generate a power curve given a sequence of sample sizes:
res <- wp.mediation(n = seq(50,100,5), power = NULL, a = 0.5, b = 0.5,
varx = 1, vary = 1, varm = 1, alpha = 0.05)
res
# Power for simple mediation
#
# n power a b varx varm vary alpha
# 50 0.6877704 0.5 0.5 1 1 1 0.05
# 55 0.7287681 0.5 0.5 1 1 1 0.05
# 60 0.7652593 0.5 0.5 1 1 1 0.05
# 65 0.7975459 0.5 0.5 1 1 1 0.05
# 70 0.8259584 0.5 0.5 1 1 1 0.05
# 75 0.8508388 0.5 0.5 1 1 1 0.05
# 80 0.8725282 0.5 0.5 1 1 1 0.05
# 85 0.8913577 0.5 0.5 1 1 1 0.05
# 90 0.9076417 0.5 0.5 1 1 1 0.05
# 95 0.9216744 0.5 0.5 1 1 1 0.05
# 100 0.9337271 0.5 0.5 1 1 1 0.05
#
# URL: http://psychstat.org/mediation
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.mediation(n = NULL, power = 0.9, a = 0.5, b = 0.5,
varx = 1, vary = 1, varm = 1, alpha = 0.05)
# Power for simple mediation
#
# n power a b varx varm vary alpha
# 87.56182 0.9 0.5 0.5 1 1 1 0.05
#
# URL: http://psychstat.org/mediation
#To calculate the minimum detectable effect size of one coefficent given power and sample size:
wp.mediation(n = 100, power = 0.9, a = NULL, b = 0.5,
varx = 1, vary = 1, varm = 1, alpha = 0.05)
# Power for simple mediation
#
# n power a b varx varm vary alpha
# 100 0.9 0.7335197 0.5 1 1 1 0.05
#
# URL: http://psychstat.org/mediation
Power analysis for longitudinal data analysis
Description
The two functions are adapted from the R package longpower by Michael C. Donohue. More will be added later.
Usage
wp.mmrm(N = NULL, Ra = NULL, ra = NULL, sigmaa = NULL,
Rb = NULL, rb = NULL, sigmab = NULL, lambda = 1,
delta = NULL, alpha = 0.05, power = NULL,
alternative = c("two.sided", "one.sided"))
wp.mmrm.ar1(N = NULL, rho = NULL, ra = NULL, sigmaa = NULL,
rb = NULL, sigmab = NULL, lambda = 1, times = 1:length(ra),
delta = NULL, alpha = 0.05, power = NULL,
alternative = c("two.sided", "one.sided"))
Arguments
N |
N |
Ra |
Ra |
ra |
ra |
sigmaa |
sigmaa |
Rb |
Rb |
rb |
rb |
sigmab |
sigmab |
lambda |
lambda |
delta |
delta |
alpha |
Significance level |
power |
Statistical power |
alternative |
alternative hypothesis |
rho |
rho |
times |
times |
References
Lu, K., Luo, X., Chen, P.-Y. (2008) Sample size estimation for repeated measures analysis in randomized clinical trials with missing data. International Journal of Biostatistics, 4, (1)
model14
Description
power analysis of model 14 in Introduction to Mediation, Moderation, and Conditional Process Analysis
Usage
wp.modmed.m14(
a1,
cp,
b1,
d1,
b2,
sige12,
sige22,
n,
sigx_w,
sigx2 = 1,
sigw2 = 1,
nrep = 1000,
alpha = 0.05,
b = 1000,
nb = n,
MCrep = 1000,
w_value = 0,
power_method = "product",
simulation_method = "percentile",
ncore = 1,
pop.cov = NULL,
mu = NULL,
varnames = c("y", "x", "w", "m", "mw")
)
Arguments
a1 |
regression coefficient of mediator (m) on predictor (x) |
cp |
regression coefficient of outcome (y) on predictor (x) |
b1 |
regression coefficient of outcome (y) on mediator (m) |
d1 |
regression coefficient of outcome (y) on moderator (w) |
b2 |
regression coefficient of outcome (y) on the product (mw) |
sige12 |
variance of error in the first regression equation |
sige22 |
variance of error in the second regression equation |
n |
sample size |
sigx_w |
covariance between predictor (x) and moderator (w) |
sigx2 |
variance of predictor (x) |
sigw2 |
variance of moderator (w) |
nrep |
number of replications for finding power |
alpha |
type 1 error rate |
b |
number of bootstrap iterations used when simulation method is "percentile" |
nb |
bootstrap sample size, default to n, used when simulation method is "percentile" |
MCrep |
number of repetitions used for finding distribution when simulation method is "MC" |
w_value |
moderator level |
power_method |
"product" for using the indirect effect value in power calculation, or "joint" for using joint significance in power calculation |
simulation_method |
"percentile" for using percentile bootstrap CI in finding significance of mediation, or "MC" for using Monte Carlo CI in finding significance of mediation |
ncore |
number of cores to use, default is 1, when ncore > 1, parallel is used |
pop.cov |
covariance matrix, default to NULL if using the regression coefficient approach |
mu |
mean vector, default to NULL if using the regression coefficient approach |
varnames |
name of variables for the covariance matrix |
Value
power of indirect effect, direct effect, and moderation
References
Xu, Z., Gao, F., Fa, A., Qu, W., & Zhang, Z. (2023). Statistical Power Analysis and Sample Size Planning for Moderated Mediation Models. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
test = wp.modmed.m14(a1 = 0.2, cp = 0.2, b1 = 0.5, d1 = 0.5, b2 = 0.2, sigx2 = 1,
sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5, n = 50,
w_value = 0.5, simulation_method = "MC",
nrep = 1000, alpha = 0.05, b = 1000, ncore = 1)
print(test)
model15
Description
power analysis of model 15 in Introduction to Mediation, Moderation, and Conditional Process Analysis
Usage
wp.modmed.m15(
a1,
cp,
b1,
b2,
d1,
d2,
sige12,
sige22,
sigx_w,
n,
sigx2 = 1,
sigw2 = 1,
nrep = 1000,
alpha = 0.05,
b = 1000,
nb = n,
w_value = 0,
power_method = "product",
simulation_method = "percentile",
ncore = 1,
pop.cov = NULL,
mu = NULL,
MCrep = 1000,
varnames = c("y", "x", "w", "m", "xw", "mw")
)
Arguments
a1 |
regression coefficient of mediator (m) on predictor (x) |
cp |
regression coefficient of outcome (y) on predictor (x) |
b1 |
regression coefficient of outcome (y) on mediator (m) |
b2 |
regression coefficient of outcome (y) on the product (mw) |
d1 |
regression coefficient of outcome (y) on moderator (w) |
d2 |
regression coefficient of outcome (y) on the product (xw) |
sige12 |
variance of error in the first regression equation |
sige22 |
variance of error in the second regression equation |
sigx_w |
covariance between predictor (x) and moderator (w) |
n |
sample size |
sigx2 |
variance of predictor (x) |
sigw2 |
variance of moderator (w) |
nrep |
number of replications for finding power |
alpha |
type 1 error rate |
b |
number of bootstrap iterations used when simulation method is "percentile" |
nb |
bootstrap sample size, default to n, used when simulation method is "percentile" |
w_value |
moderator level |
power_method |
"product" for using the indirect effect value in power calculation, or "joint" for using joint significance in power calculation |
simulation_method |
"percentile" for using percentile bootstrap CI in finding significance of mediation, or "MC" for using Monte Carlo CI in finding significance of mediation |
ncore |
number of cores to use, default is 1, when ncore > 1, parallel is used |
pop.cov |
covariance matrix, default to NULL if using the regression coefficient approach |
mu |
mean vector, default to NULL if using the regression coefficient approach |
MCrep |
number of repetitions used for finding distribution when simulation method is "MC" |
varnames |
name of variables for the covariance matrix |
Value
power of indirect effect, direct effect, and moderation
References
Xu, Z., Gao, F., Fa, A., Qu, W., & Zhang, Z. (2023). Statistical Power Analysis and Sample Size Planning for Moderated Mediation Models. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
test = wp.modmed.m15(a1 = 0.6, cp = 0.2, b1 = 0.3, b2 = 0.2, d1 = 0.2, d2 = 0.1,
sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.4,
w_value = 0.3, simulation_method = "MC",
n = 50, nrep = 1000, alpha = 0.05, ncore = 1)
print(test)
model58
Description
power analysis of model 58 in Introduction to Mediation, Moderation, and Conditional Process Analysis
Usage
wp.modmed.m58(
c1,
a1,
c2,
d1,
b1,
b2,
cp,
sige12,
sige22,
sigx_w,
n,
sigx2 = 1,
sigw2 = 1,
nrep = 1000,
alpha = 0.05,
b = 1000,
nb = n,
w_value = 0,
power_method = "product",
MCrep = 1000,
ncore = 1,
simulation_method = "percentile",
pop.cov = NULL,
mu = NULL,
varnames = c("x", "w", "m", "xw", "mw", "y")
)
Arguments
c1 |
regression coefficient of outcome (m) on moderator (w) |
a1 |
regression coefficient of mediator (m) on predictor (x) |
c2 |
regression coefficient of outcome (m) on the product (xw) |
d1 |
regression coefficient of outcome (y) on moderator (w) |
b1 |
regression coefficient of outcome (y) on mediator (m) |
b2 |
regression coefficient of outcome (y) on the product (mw) |
cp |
regression coefficient of outcome (y) on predictor (x) |
sige12 |
variance of error in the first regression equation |
sige22 |
variance of error in the second regression equation |
sigx_w |
covariance between predictor (x) and moderator (w) |
n |
sample size |
sigx2 |
variance of predictor (x) |
sigw2 |
variance of moderator (w) |
nrep |
number of replications for finding power |
alpha |
type 1 error rate |
b |
number of bootstrap iterations used when simulation method is "percentile" |
nb |
bootstrap sample size, default to n, used when simulation method is "percentile" |
w_value |
moderator level |
power_method |
"product" for using the indirect effect value in power calculation, or "joint" for using joint significance in power calculation |
MCrep |
number of repetitions used for finding distribution when simulation method is "MC" |
ncore |
number of cores to use, default is 1, when ncore > 1, parallel is used |
simulation_method |
"percentile" for using percentile bootstrap CI in finding significance of mediation, or "MC" for using Monte Carlo CI in finding significance of mediation |
pop.cov |
covariance matrix, default to NULL if using the regression coefficient approach |
mu |
mean vector, default to NULL if using the regression coefficient approach |
varnames |
name of variables for the covariance matrix |
Value
power of indirect effect, direct effect, and moderation
References
Xu, Z., Gao, F., Fa, A., Qu, W., & Zhang, Z. (2023). Statistical Power Analysis and Sample Size Planning for Moderated Mediation Models. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
test = wp.modmed.m58(c1 = 0.2, a1 = 0.2, c2 = 0.1, b2 = 0.1,
b1 = 0.2, cp = 0.2, d1 = 0.2, w_value = 0.3, simulation_method = "MC",
sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5,
n = 50, nrep = 1000, alpha = 0.05, ncore = 1)
print(test)
model7
Description
power analysis of model 7 in Introduction to Mediation, Moderation, and Conditional Process Analysis
Usage
wp.modmed.m7(
a1,
cp,
b1,
c1,
c2,
sige12,
sige22,
sigx_w,
n,
sigx2 = 1,
sigw2 = 1,
nrep = 1000,
alpha = 0.05,
b = 1000,
nb = n,
w_value = 0,
power_method = "product",
simulation_method = "percentile",
MCrep = 1000,
ncore = 1,
pop.cov = NULL,
mu = NULL,
varnames = c("y", "x", "w", "m", "xw")
)
Arguments
a1 |
regression coefficient of mediator (m) on predictor (x) |
cp |
regression coefficient of outcome (y) on predictor (x) |
b1 |
regression coefficient of outcome (y) on mediator (m) |
c1 |
regression coefficient of mediator (m) on moderator (w) |
c2 |
regression coefficient of mediator (m) on the product (xw) |
sige12 |
variance of error in the first regression equation |
sige22 |
variance of error in the second regression equation |
sigx_w |
covariance between predictor (x) and moderator (w) |
n |
sample size |
sigx2 |
variance of predictor (x) |
sigw2 |
variance of moderator (w) |
nrep |
number of replications for finding power |
alpha |
type 1 error rate |
b |
number of bootstrap iterations used when simulation method is "percentile" |
nb |
bootstrap sample size, default to n, used when simulation method is "percentile" |
w_value |
moderator level |
power_method |
"product" for using the indirect effect value in power calculation, or "joint" for using joint significance in power calculation |
simulation_method |
"percentile" for using percentile bootstrap CI in finding significance of mediation, or "MC" for using Monte Carlo CI in finding significance of mediation |
MCrep |
number of repetitions used for finding distribution when simulation method is "MC" |
ncore |
number of cores to use, default is 1, when ncore > 1, parallel is used |
pop.cov |
covariance matrix, default to NULL if using the regression coefficient approach |
mu |
mean vector, default to NULL if using the regression coefficient approach |
varnames |
name of variables for the covariance matrix |
Value
power of indirect effect, direct effect, and moderation
References
Xu, Z., Gao, F., Fa, A., Qu, W., & Zhang, Z. (2023). Statistical Power Analysis and Sample Size Planning for Moderated Mediation Models. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
# usage of wp.modmed.m7
test = wp.modmed.m7(a1 = 0.39, cp = 0.2, b1 = 0.3, c1 = 0.39,
c2 = 0.2, sigx2 = 1, sigw2 = 1, sige12 = 1,
sige22 = 1, sigx_w = 0.5, n = 50, nrep = 1000, simulation_method = "MC",
alpha = 0.05, MCrep = 1000, ncore = 1)
print(test)
model8
Description
power analysis of model 8 in Introduction to Mediation, Moderation, and Conditional Process Analysis
Usage
wp.modmed.m8(
a1,
cp,
b1,
c1,
d1,
c2,
d2,
sige12,
sige22,
sigx_w,
n,
sigx2 = 1,
sigw2 = 1,
nrep = 1000,
alpha = 0.05,
b = 1000,
nb = n,
w_value = 0,
power_method = "product",
simulation_method = "percentile",
ncore = 1,
MCrep = 1000,
pop.cov = NULL,
mu = NULL,
varnames = c("y", "x", "w", "m", "xw")
)
Arguments
a1 |
regression coefficient of mediator (m) on predictor (x) |
cp |
regression coefficient of outcome (y) on predictor (x) |
b1 |
regression coefficient of outcome (y) on mediator (m) |
c1 |
regression coefficient of mediator (m) on moderator (w) |
d1 |
regression coefficient of mediator (y) on moderator (w) |
c2 |
regression coefficient of mediator (m) on the product (xw) |
d2 |
regression coefficient of mediator (y) on the product (xw) |
sige12 |
variance of error in the first regression equation |
sige22 |
variance of error in the second regression equation |
sigx_w |
covariance between predictor (x) and moderator (w) |
n |
sample size |
sigx2 |
variance of predictor (x) |
sigw2 |
variance of moderator (w) |
nrep |
number of replications for finding power |
alpha |
type 1 error rate |
b |
number of bootstrap iterations used when simulation method is "percentile" |
nb |
bootstrap sample size, default to n, used when simulation method is "percentile" |
w_value |
moderator level |
power_method |
"product" for using the indirect effect value in power calculation, or "joint" for using joint significance in power calculation |
simulation_method |
"percentile" for using percentile bootstrap CI in finding significance of mediation, or "MC" for using Monte Carlo CI in finding significance of mediation |
ncore |
number of cores to use, default is 1, when ncore > 1, parallel is used |
MCrep |
number of repetitions used for finding distribution when simulation method is "MC" |
pop.cov |
covariance matrix, default to NULL if using the regression coefficient approach |
mu |
mean vector, default to NULL if using the regression coefficient approach |
varnames |
name of variables for the covariance matrix |
Value
power of indirect effect, direct effect, and moderation
References
Xu, Z., Gao, F., Fa, A., Qu, W., & Zhang, Z. (2023). Statistical Power Analysis and Sample Size Planning for Moderated Mediation Models. Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
# usage of wp.modmed.m8
test = wp.modmed.m8(a1 = 0.2, cp = 0.2, b1 = 0.2,
c1 = 0.2, d1=0.2, c2 = 0.2, d2 = 0.2,
sigx2 = 1, sigw2 = 1, sige12 = 1, sige22 = 1, sigx_w = 0.5,
simulation_method = "MC",
n = 50, nrep = 1000, alpha = 0.05, MCrep = 1000, ncore = 1)
print(test)
Statistical Power Analysis for Multisite Randomized Trials with 2 Arms
Description
Multisite randomized trials (MRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 2 arms (i.e., a treatment and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "site" type tests the variance of cluster/site means; and (3) The "variance" type tests variance of treatment effects. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).
Usage
wp.mrt2arm(n = NULL, f = NULL, J = NULL, tau00 = NULL, tau11 = NULL,
sg2 = NULL, power = NULL, alpha = 0.05, alternative = c("two.sided",
"one.sided"), type = c("main", "site", "variance"), interval = NULL)
Arguments
n |
Sample size. It is the number of individuals within each cluster. |
f |
Effect size. It specifies the main effect of treatment, the mean difference between the treatment clusters/sites and the control clusters/sites. Effect size must be positive. |
J |
Number of clusters / sites. It tells how many clusters are considered in the study design. At least two clusters are required. |
tau00 |
Variance of cluster/site means. It is one of the residual variances in the second level. Its value must be positive. |
tau11 |
Variance of treatment effects across sites. It is one of the residual variances in the second level. Its value must be positive. |
sg2 |
Level-one error Variance. The residual variance in the first level. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
alternative |
Type of the alternative hypothesis ( |
type |
Type of effect ( |
interval |
A vector containing the end-points of the interval to be searched for the root. |
Value
An object of the power analysis.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
#For main effect
wp.mrt2arm(n = 45, f = 0.5, J = 20, tau11 = 0.5,
sg2 = 1.25, alpha = 0.05, power = NULL)
# Power analysis for Multileve model Multisite randomized trials with 2 arms
#
# J n f tau11 sg2 power alpha
# 20 45 0.5 0.5 1.25 0.8583253 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt2arm
#For variance of treament effect
wp.mrt2arm(n = 45, f = 0.5, J = 20, tau11 = 0.5,
sg2 = 1.25, alpha = 0.05, power = NULL, type = "variance")
# Power analysis for Multileve model Multisite randomized trials with 2 arms
#
# J n f tau11 sg2 power alpha
# 20 45 0.5 0.5 1.25 0.9987823 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt2arm
#For testing site variablity
res<- wp.mrt2arm(n = 45, J = 20, tau00 = 0.5,
sg2 = 1.25, alpha = 0.05, power = NULL, type = "site")
# Power analysis for Multileve model Multisite randomized trials with 2 arms
#
# J n tau00 sg2 power alpha
# 20 45 0.5 1.25 0.9999999 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt2arm
#To generate a power curve given a sequence of sample sizes:
res <- wp.mrt2arm(n = seq(10,50,5), f = 0.5, J = 20, tau11 = 0.5,
sg2 = 1.25, alpha = 0.05, power = NULL)
# Power analysis for Multileve model Multisite randomized trials with 2 arms
#
# J n f tau11 sg2 power alpha
# 20 10 0.5 0.5 1.25 0.6599499 0.05
# 20 15 0.5 0.5 1.25 0.7383281 0.05
# 20 20 0.5 0.5 1.25 0.7818294 0.05
# 20 25 0.5 0.5 1.25 0.8090084 0.05
# 20 30 0.5 0.5 1.25 0.8274288 0.05
# 20 35 0.5 0.5 1.25 0.8406659 0.05
# 20 40 0.5 0.5 1.25 0.8506049 0.05
# 20 45 0.5 0.5 1.25 0.8583253 0.05
# 20 50 0.5 0.5 1.25 0.8644864 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt2arm
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.mrt2arm(n = NULL, f = 0.5, J = 20, tau11 = 0.5,
sg2 = 1.25, alpha = 0.05, power = 0.8)
# Power analysis for Multileve model Multisite randomized trials with 2 arms
#
# J n f tau11 sg2 power alpha
# 20 23.10086 0.5 0.5 1.25 0.8 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt2arm
Statistical Power Analysis for Multisite Randomized Trials with 3 Arms
Description
Multisite randomized trials (MRT) are a type of multilevel design for the situation when the entire cluster is randomly assigned to either a treatment arm or a control arm (Liu, 2013). The data from MRT can be analyzed in a two-level hierarchical linear model, where the indicator variable for treatment assignment is included in first level. If a study contains multiple treatments, then multiple indicators will be used. This function is for designs with 3 arms (i.e., two treatments and a control). Three types of tests are considered in the function: (1) The "main" type tests treatment main effect; (2) The "treatment" type tests the difference between the two treatments; and (3) The "omnibus" type tests whether the three arms are all equivalent. Details leading to power calculation can be found in Raudenbush (1997) and Liu (2013).
Usage
wp.mrt3arm(n = NULL, f1 = NULL, f2 = NULL, J = NULL, tau = NULL,
sg2 = NULL, power = NULL, alpha = 0.05, alternative = c("two.sided",
"one.sided"), type = c("main", "treatment", "omnibus"), interval = NULL)
Arguments
n |
Sample size. It is the number of individuals within each cluster. |
f1 |
Effect size for treatment main effect. Effect size must be positive. |
f2 |
Effect size for the difference between two treatments. Effect size must be positive. |
J |
Number of clusters / sites. It tells how many clusters are considered in the study design. At least two clusters are required. |
tau |
Variance of treatment effects across sites/clusters. |
sg2 |
Level-one error Variance. The residual variance in the first level. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
alternative |
Type of the alternative hypothesis ( |
type |
Type of effect ( |
interval |
A vector containing the end-points of the interval to be searched for the root. |
Value
An object of the power analysis.
References
Liu, X. S. (2013). Statistical power analysis for the social and behavioral sciences: basic and advanced techniques. Routledge.
Raudenbush, S. W. (1997). Statistical analysis and optimal design for cluster randomized trials. Psychological Methods, 2(2), 173.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
#For main effect
wp.mrt3arm(n = 30, f1 = 0.43, J = 20, tau = 0.4,
sg2 = 2.25, alpha = 0.05, power = NULL)
# Multisite randomized trials with 3 arms
#
# J n f1 tau sg2 power alpha
# 20 30 0.43 0.4 2.25 0.8066964 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt3arm
#For tesing difference between effects
wp.mrt3arm(n = 30, f2 = 0.2, J = 20, tau = 0.4, sg2 = 2.25,
alpha = 0.05, power = NULL, type="treatment")
# Multisite randomized trials with 3 arms
#
# J n f2 tau sg2 power alpha
# 20 30 0.2 0.4 2.25 0.2070712 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt3arm
#For testing site variablity
wp.mrt3arm(n = 30, f1=0.43, f2 = 0.2, J = 20, tau = 0.4, sg2 = 2.25,
alpha = 0.05, power = NULL, type="omnibus")
# Multisite randomized trials with 3 arms
#
# J n f1 f2 tau sg2 power alpha
# 20 30 0.43 0.2 0.4 2.25 0.7950757 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt3arm
#To generate a power curve given a sequence of numbers of sites/clusters:
res <- wp.mrt3arm(n = 30, f2 = 0.2, J = seq(20,120,10), tau = 0.4,
sg2 = 2.25, alpha = 0.05, power = NULL, type="treatment")
res
# Multisite randomized trials with 3 arms
#
# J n f2 tau sg2 power alpha
# 20 30 0.2 0.4 2.25 0.2070712 0.05
# 30 30 0.2 0.4 2.25 0.2953799 0.05
# 40 30 0.2 0.4 2.25 0.3804554 0.05
# 50 30 0.2 0.4 2.25 0.4603091 0.05
# 60 30 0.2 0.4 2.25 0.5337417 0.05
# 70 30 0.2 0.4 2.25 0.6001544 0.05
# 80 30 0.2 0.4 2.25 0.6593902 0.05
# 90 30 0.2 0.4 2.25 0.7116052 0.05
# 100 30 0.2 0.4 2.25 0.7571648 0.05
# 110 30 0.2 0.4 2.25 0.7965644 0.05
# 120 30 0.2 0.4 2.25 0.8303690 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt3arm
#To plot the power curve:
plot(res, "J", "power")
#To calculate the required sample size given power and effect size:
wp.mrt3arm(n = NULL, f1 = 0.43, J = 20, tau = 0.4,
sg2 = 2.25, alpha = 0.05, power = 0.8)
# Multisite randomized trials with 3 arms
#
# J n f1 tau sg2 power alpha
# 20 28.61907 0.43 0.4 2.25 0.8 0.05
#
# NOTE: n is the number of subjects per cluster
# URL: http://psychstat.org/mrt3arm
Statistical Power Analysis for Poisson Regression
Description
This function is for Poisson regression models. Poisson regression is a type of generalized linear models where the outcomes are usually count data. Here, Maximum likelihood methods is used to estimate the model parameters. The estimated regression coefficent is assumed to follow a normal distribution. A Wald test is used to test the mean difference between the estimated parameter and the null parameter (tipically the null hypothesis assumes it equals 0). The procedure introduced by Demidenko (2007) is adopted here for computing the statistical power.
Usage
wp.poisson(n = NULL, exp0 = NULL, exp1 = NULL, alpha = 0.05,
power = NULL, alternative = c("two.sided", "less", "greater"),
family = c("Bernoulli", "exponential", "lognormal", "normal", "Poisson",
"uniform"), parameter = NULL, subdivisions=200L,
i.method=c("numerical", "MC"), mc.iter=20000)
Arguments
n |
Sample size. |
exp0 |
The base rate under the null hypothesis. It always takes positive value. See the article by Demidenko (2007) for details. |
exp1 |
The relative increase of the event rate. It is used for calculatation of the effect size. See the article by Demidenko (2007) for details. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
alternative |
Direction of the alternative hypothesis ( |
family |
Distribution of the predictor ( |
parameter |
Corresponding parameter for the predictor's distribution. The default is 0.5 for "Bernoulli", 1 for "exponential", (0,1) for "lognormal" or "normal", 1 for "Poisson", and (0,1) for "uniform". |
subdivisions |
Number of divisions for integration |
i.method |
Integration method |
mc.iter |
Number of iterations for Monte Carlo integration |
Value
An object of the power analysis.
References
Demidenko, E. (2007). Sample size determination for logistic regression revisited. Statistics in medicine, 26(18), 3385-3397.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.poisson(n = 4406, exp0 = 2.798, exp1 = 0.8938, alpha = 0.05,
power = NULL, family = "Bernoulli", parameter = 0.53)
# Power for Poisson regression
#
# n power alpha exp0 exp1 beta0 beta1 paremeter
# 4406 0.9999789 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
#
# URL: http://psychstat.org/poisson
#To generate a power curve given a sequence of sample sizes:
res <- wp.poisson(n = seq(800, 1500, 100), exp0 = 2.798, exp1 = 0.8938,
alpha = 0.05, power = NULL, family = "Bernoulli", parameter = 0.53)
res
# Power for Poisson regression
#
# n power alpha exp0 exp1 beta0 beta1 paremeter
# 800 0.7324097 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 900 0.7813088 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1000 0.8224254 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1100 0.8566618 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1200 0.8849241 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1300 0.9080755 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1400 0.9269092 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
# 1500 0.9421344 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
#
# URL: http://psychstat.org/poisson
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.poisson(n = NULL, exp0 = 2.798, exp1 = 0.8938, alpha = 0.05,
power = 0.8, family = "Bernoulli", parameter = 0.53)
# Power for Poisson regression
#
# n power alpha exp0 exp1 beta0 beta1 paremeter
# 943.2628 0.8 0.05 2.798 0.8938 1.028905 -0.1122732 0.53
#
# URL: http://psychstat.org/poisson
Extract Population Value Table
Description
This function is used to extract population value table for parameters form an power analysis object for SEM based on Monte Carlo methods (class = 'power').
Usage
wp.popPar(object)
Arguments
object |
Object of power analysis. It is an object returned by a webpower function for SEM based on Monte Carlo methods with class = 'power'. |
Value
Population value table of parameters from the input object of power analysis.
Statistical Power Analysis for Tests of Proportions
Description
Tests of proportions are a technique used to compare proportions of success or agreement in one or two samples. The one-sample test of proportion tests the null proportion of success, usually 0.5. The two-sample test of proportions tests the null hypothesis that the two samples are drawn from populations with the same proportion of success. A z-test is used to evaluate whether the given difference in proportions is statistical significantly different from the null hypothesis. The power calculation is based on the arcsine transformation of the proportion (see Cohen, 1988, p.548).
Usage
wp.prop(h = NULL, n1 = NULL, n2 = NULL, alpha = 0.05, power = NULL,
type = c("1p", "2p", "2p2n"), alternative = c("two.sided", "less",
"greater"))
Arguments
h |
Effect size of the proportion comparison. Cohen (1992) suggested that effect size values of 0.2, 0.5, and 0.8 represent "small", "medium", and "large" effect sizes, repectively. |
n1 |
Sample size of the first group. |
n2 |
Sample size of the second group if applicable. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
type |
Type of comparison ( |
alternative |
Direction of the alternative hypothesis ( |
Value
An object of the power analysis.
References
Cohen, J. (1992). A power primer. Psychological bulletin, 112(1), 155.
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the power for two groups of proportion with unequal sample size:
wp.prop(h=0.52,n1=35,n2=50,alternative="greater",type="2p2n")
# Power for two-sample proportion (unequal n)
#
# h n1 n2 alpha power
# 0.52 35 50 0.05 0.7625743
#
# NOTE: Sample size for each group
# URL: http://psychstat.org/prop2p2n
#To calculate the power curve with a sequence of sample sizes:
res <- wp.prop(h=0.52,n1=seq(10,100,10),alternative="greater",type="1p")
res
# Power for one-sample proportion test
#
# h n alpha power
# 0.52 10 0.05 0.4998128
# 0.52 20 0.05 0.7519557
# 0.52 30 0.05 0.8855706
# 0.52 40 0.05 0.9499031
# 0.52 50 0.05 0.9789283
# 0.52 60 0.05 0.9914150
# 0.52 70 0.05 0.9965928
# 0.52 80 0.05 0.9986772
# 0.52 90 0.05 0.9994960
# 0.52 100 0.05 0.9998111
#
# URL: http://psychstat.org/prop
#To plot the power curve:
plot(res, type='b')
#To estimate the sample size with a given power:
wp.prop(h=0.52,n1=NULL,power=0.8,alternative="greater",type="1p")
# Power for one-sample proportion test
#
# h n alpha power
# 0.52 22.86449 0.05 0.8
#
# URL: http://psychstat.org/prop
#To estimate the minimum detectable effect size with a given power:
wp.prop(h=NULL,n1=35,power=0.8,alternative="greater",type="1p")
# Power for one-sample proportion test
#
# h n alpha power
# 0.4202907 35 0.05 0.8
#
# URL: http://psychstat.org/prop
#To calculate the power curve with a sequence of effect sizes:
wp.prop(h=seq(0.1, 0.8, 0.1),n1=100,alternative="greater",type="1p")
# Power for one-sample proportion test
#
# h n alpha power
# 0.1 100 0.05 0.2595110
# 0.2 100 0.05 0.6387600
# 0.3 100 0.05 0.9123145
# 0.4 100 0.05 0.9907423
# 0.5 100 0.05 0.9996034
# 0.6 100 0.05 0.9999934
# 0.7 100 0.05 1.0000000
# 0.8 100 0.05 1.0000000
#
# URL: http://psychstat.org/prop
Statistical Power Analysis for Linear Regression
Description
This function is for power analysis for regression models. Regression is a statistical technique for examining the relationship between one or more independent variables (or predictors) and one dependent variable (or the outcome). Regression provides an F-statistic that can be formulated using the ratio between variation in the outcome variable that is explained by the predictors and the unexplained variation (Cohen, 1988)). The test statistc can also be experessed in terms of caomparison between Full and Reduced models (Maxwell & Delaney, 2003).
Usage
wp.regression(n = NULL, p1 = NULL, p2 = 0, f2 = NULL, alpha = 0.05,
power = NULL, type=c("regular", "Cohen"))
Arguments
n |
Sample size. |
p1 |
Number of predictors in the full model. |
p2 |
Number of predictors in the reduced moedel, it is 0 by default. See the book by Maxwell and Delaney (2003) for the definition of the reduced model. |
f2 |
Effect size. We use the statistic f2 as the measure of effect size for linear regression proposed by Cohen(1988, p.410). Cohen discussed the effect size in three different cases. The calculation of f2 can be generalized using the idea of a full model and a reduced model by Maxwell and Delaney (2003). |
alpha |
significance level chosen for the test. It equals 0.05 by default. |
power |
Statistical power. |
type |
If set to "Cohen", the formula used in the Cohen's book will be used (not recommended). |
Value
An object of the power analysis
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Maxwell, S. E., & Delaney, H. D. (2004). Designing experiments and analyzing data: A model comparison perspective (Vol. 1). Psychology Press.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.regression(n = 100, p1 = 3, f2 = 0.1, alpha = 0.05, power = NULL)
# Power for multiple regression
#
# n p1 p2 f2 alpha power
# 100 3 0 0.1 0.05 0.7420463
#
# URL: http://psychstat.org/regression
#To generate a power curve given a sequence of sample sizes:
res <- wp.regression(n = seq(50,300,50), p1 = 3, f2 = 0.1,
alpha = 0.05, power = NULL)
res
# Power for multiple regression
#
# n p1 p2 f2 alpha power
# 50 3 0 0.1 0.05 0.4077879
# 100 3 0 0.1 0.05 0.7420463
# 150 3 0 0.1 0.05 0.9092082
# 200 3 0 0.1 0.05 0.9724593
# 250 3 0 0.1 0.05 0.9925216
# 300 3 0 0.1 0.05 0.9981375
#
# URL: http://psychstat.org/regression
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.regression(n = NULL, p1 = 3, f2 = 0.1, alpha = 0.05, power = 0.8)
# Power for multiple regression
#
# n p1 p2 f2 alpha power
# 113.0103 3 0 0.1 0.05 0.8
#
# URL: http://psychstat.org/regression
#The statistical power given sample size and effect size when controling two predictors:
wp.regression(n = 100, p1 = 3, p2 = 2, f2 = 0.1429, alpha = 0.05, power = NULL)
# Power for multiple regression
#
# n p1 p2 f2 alpha power
# 100 3 2 0.1429 0.05 0.9594695
#
# URL: http://psychstat.org/regression
# To generate a power curve given a sequence of effect sizes:
res <- wp.regression(n = 50, p1 = 3, f2 = seq(0.05,0.5,0.05),
alpha = 0.05, power = NULL)
res
# Power for multiple regression
#
# n p1 p2 f2 alpha power
# 50 3 0 0.05 0.05 0.2164842
# 50 3 0 0.10 0.05 0.4077879
# 50 3 0 0.15 0.05 0.5821296
# 50 3 0 0.20 0.05 0.7210141
# 50 3 0 0.25 0.05 0.8220164
# 50 3 0 0.30 0.05 0.8906954
# 50 3 0 0.35 0.05 0.9350154
# 50 3 0 0.40 0.05 0.9624324
# 50 3 0 0.45 0.05 0.9788077
# 50 3 0 0.50 0.05 0.9883012
#
# URL: http://psychstat.org/regression
Statistical Power Analysis for Repeated Measures ANOVA
Description
Repeated-measures ANOVA can be used to compare the means of a sequence of measurements (e.g., O'brien & Kaiser, 1985). In a repeated-measures design, evey subject is exposed to all different treatments, or more commonly measured across different time points. Power analysis for (1) the within-effect test about the mean difference among measurements by default. If the subjects are from more than one group,the power analysis is also available for (2) the between-effect test about mean difference among groups and (3) the interaction effect test of the measurements and groups.
Usage
wp.rmanova(n = NULL, ng = NULL, nm = NULL, f = NULL, nscor = 1,
alpha = 0.05, power = NULL, type = 0)
Arguments
n |
Sample size. |
ng |
Number of groups. |
nm |
Number of measurements. |
f |
Effect size. We use the statistic f as the measure of effect size for repeated-measures ANOVA as in Cohen(1988, p.275). |
nscor |
Nonsphericity correction coefficient. The nonsphericity correction coefficient is a measure of the degree of sphericity in the population. A coefficient of 1 means sphericity is met, while a coefficient less than 1 means not met. The samller value of the coefficient means the further departure from sphericity. The lowest value of the coefficient is 1/(nm-1) where nm is the total number of measurements. Two viable approaches for computing the empirical nonsphericity correction coefficient are sggested. One is by Greenhouse and Geisser (1959), the other is by Huynh and Feldt (1976). |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
type |
Type of analysis ( |
Value
An object of the power analysis
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Greenhouse, S. W., & Geisser, S. (1959). On methods in the analysis of profile data. Psychometrika, 24(2), 95-112.
Huynh, H., & Feldt, L. S. (1976). Estimation of the Box correction for degrees of freedom from sample data in randomized block and split-plot designs. Journal of educational statistics, 1(1), 69-82.
O'brien, R. G., & Kaiser, M. K. (1985). MANOVA method for analyzing repeated measures designs: an extensive primer. Psychological bulletin, 97(2), 316.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power for repeated-measures ANOVA:
wp.rmanova(n=30, ng=3, nm=4, f=0.36, nscor=0.7)
# Repeated-measures ANOVA analysis
#
# n f ng nm nscor alpha power
# 30 0.36 3 4 0.7 0.05 0.2674167
#
# NOTE: Power analysis for between-effect test
# URL: http://psychstat.org/rmanova
#To generate a power curve given a sequence of sample sizes:
res <- wp.rmanova(n=seq(30,150,20), ng=3, nm=4, f=0.36, nscor=0.7)
# Repeated-measures ANOVA analysis
#
# n f ng nm nscor alpha power
# 30 0.36 3 4 0.7 0.05 0.2674167
# 50 0.36 3 4 0.7 0.05 0.4386000
# 70 0.36 3 4 0.7 0.05 0.5894599
# 90 0.36 3 4 0.7 0.05 0.7110142
# 110 0.36 3 4 0.7 0.05 0.8029337
# 130 0.36 3 4 0.7 0.05 0.8691834
# 150 0.36 3 4 0.7 0.05 0.9151497
#
# NOTE: Power analysis for between-effect test
# URL: http://psychstat.org/rmanova
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.rmanova(n=NULL, ng=3, nm=4, f=0.36, power=0.8, nscor=0.7)
# Repeated-measures ANOVA analysis
#
# n f ng nm nscor alpha power
# 109.2546 0.36 3 4 0.7 0.05 0.8
#
# NOTE: Power analysis for between-effect test
# URL: http://psychstat.org/rmanova
#To calculate the minimum detectable effect size given power and sample size:
wp.rmanova(n=30, ng=3, nm=4, f=NULL, power=0.8, nscor=0.7)
# Repeated-measures ANOVA analysis
#
# n f ng nm nscor alpha power
# 30 0.716768 3 4 0.7 0.05 0.8
#
# NOTE: Power analysis for between-effect test
# URL: http://psychstat.org/rmanova
# To generate a power curve given a sequence of effec sizes:
wp.rmanova(n=30, ng=3, nm=4, f=seq(0.1,0.5,0.05), nscor=0.7)
# Repeated-measures ANOVA analysis
#
# n f ng nm nscor alpha power
# 30 0.10 3 4 0.7 0.05 0.06442235
# 30 0.15 3 4 0.7 0.05 0.08327886
# 30 0.20 3 4 0.7 0.05 0.11101678
# 30 0.25 3 4 0.7 0.05 0.14853115
# 30 0.30 3 4 0.7 0.05 0.19640404
# 30 0.35 3 4 0.7 0.05 0.25460008
# 30 0.40 3 4 0.7 0.05 0.32223192
# 30 0.45 3 4 0.7 0.05 0.39746082
# 30 0.50 3 4 0.7 0.05 0.47757523
#
# NOTE: Power analysis for between-effect test
# URL: http://psychstat.org/rmanova
Statistical Power Analysis for Structural Equation Modeling based on Chi-Squared Test
Description
Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. Two methods are widely used in power analysis for SEM. One is based on the likelihood ratio test proposed by Satorra and Saris (1985). The other is based on RMSEA proposed by MacCallum et al. (1996). This function is for SEM power analysis based on the likelihood ratio test.
Usage
wp.sem.chisq(n = NULL, df = NULL, effect = NULL, power = NULL,
alpha = 0.05)
Arguments
n |
Sample size. |
df |
Degrees of freedom. The degrees of freedom of the chi-squared test. |
effect |
Effect size. It specifies the population misfit of a SEM model, which is the difference between two SEM models: a full model (Mf) and a reduced model (Mr). A convienient way to get the effect size is to fit the reduced model using SEM software such R package 'lavaan' (Rossel, 2012). Then the effect size is calculated as the chi-squared statistics dividing by the sample size. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
Value
An object of the power analysis.
References
Satorra, A., & Saris, W. E. (1985). Power of the likelihood ratio test in covariance structure analysis. Psychometrika, 50(1), 83-90.
Rossel, Y. (2012). Lavaan: An R package for structural equation modeling and more. Version 0.5–12 (BETA). Retrieved from http://users. ugent. be/~ yrosseel/lavaan/lavaanIntroduction.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.sem.chisq(n = 100, df = 4, effect = 0.054, power = NULL, alpha = 0.05)
# Power for SEM (Satorra & Saris, 1985)
#
# n df effect power alpha
# 100 4 0.054 0.4221152 0.05
#
# URL: http://psychstat.org/semchisq
#To generate a power curve given a sequence of sample sizes:
res <- wp.sem.chisq(n = seq(100,600,100), df = 4,
effect = 0.054, power = NULL, alpha = 0.05)
res
# Power for SEM (Satorra & Saris, 1985)
#
# n df effect power alpha
# 100 4 0.054 0.4221152 0.05
# 200 4 0.054 0.7510630 0.05
# 300 4 0.054 0.9145660 0.05
# 400 4 0.054 0.9750481 0.05
# 500 4 0.054 0.9935453 0.05
# 600 4 0.054 0.9984820 0.05
#
# URL: http://psychstat.org/semchisq
#To plot the power curve:
plot(res)
#To generate a power curve given a sequence of alphas:
res <- wp.sem.chisq(n = 100, df = 4, effect = 0.054, power = NULL,
alpha = c(0.001, 0.005, 0.01, 0.025, 0.05))
res
# Power for SEM (Satorra & Saris, 1985)
#
# n df effect power alpha
# 100 4 0.054 0.06539478 0.001
# 100 4 0.054 0.14952768 0.005
# 100 4 0.054 0.20867087 0.010
# 100 4 0.054 0.31584011 0.025
# 100 4 0.054 0.42211515 0.050
#
# URL: http://psychstat.org/semchisq
#To calculate the required sample size given power and effect size:
wp.sem.chisq(n = NULL, df = 4, effect = 0.054, power = 0.8, alpha = 0.05)
# Power for SEM (Satorra & Saris, 1985)
#
# n df effect power alpha
# 222.0238 4 0.054 0.8 0.05
#
# URL: http://psychstat.org/semchisq
#To calculate the minimum detectable effect size of one coefficent given power and sample size:
wp.sem.chisq(n = 100, df = 4, effect = NULL, power = 0.8, alpha = 0.05)
# Power for SEM (Satorra & Saris, 1985)
#
# n df effect power alpha
# 100 4 0.1205597 0.8 0.05
#
# URL: http://psychstat.org/semchisq
Statistical Power Analysis for Structural Equation Modeling based on RMSEA
Description
Structural equation modeling (SEM) is a multivariate technique used to analyze relationships among observed and latent variables. It can be viewed as a combination of factor analysis and multivariate regression analysis. Two methods are widely used in power analysis for SEM. One is based on the likelihood ratio test proposed by Satorra and Saris (1985). The other is based on RMSEA proposed by MacCallum et al. (1996). This function is for SEM power analysis based on RMSEA.
Usage
wp.sem.rmsea(n = NULL, df = NULL, rmsea0 = NULL, rmsea1 = NULL,
power = NULL, alpha = 0.05, type = c("close", "notclose"))
Arguments
n |
Sample size. |
df |
Degrees of freedom. The degrees of freedom of the chi-squared test. |
rmsea0 |
RMSEA for H0. It usually euquals zero. |
rmsea1 |
RMSEA for H1. |
power |
Statistical power. |
alpha |
significance level chosed for the test. It equals 0.05 by default. |
type |
Close fit or non-clase fit ( |
Value
An object of the power analysis.
References
MacCallum, R. C., Browne, M. W., & Sugawara, H. M. (1996). Power analysis and determination of sample size for covariance structure modeling. Psychological methods, 1(2), 130.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the statistical power given sample size and effect size:
wp.sem.rmsea (n = 100, df = 4, rmsea0 = 0,
rmsea1 = 0.116, power = NULL, alpha = 0.05)
# Power for SEM based on RMSEA
#
# n df rmsea0 rmsea1 power alpha
# 100 4 0 0.116 0.4208173 0.05
#
# URL: http://psychstat.org/rmsea
#To generate a power curve given a sequence of sample sizes:
res <- wp.sem.rmsea (n = seq(100,600,100), df = 4, rmsea0 = 0,
rmsea1 = 0.116, power = NULL, alpha = 0.05)
res
# Power for SEM based on RMSEA
#
# n df rmsea0 rmsea1 power alpha
# 100 4 0 0.116 0.4208173 0.05
# 200 4 0 0.116 0.7494932 0.05
# 300 4 0 0.116 0.9135968 0.05
# 400 4 0 0.116 0.9746240 0.05
# 500 4 0 0.116 0.9933963 0.05
# 600 4 0 0.116 0.9984373 0.05
#
# URL: http://psychstat.org/rmsea
#To plot the power curve:
plot(res)
#To calculate the required sample size given power and effect size:
wp.sem.rmsea (n = NULL, df = 4, rmsea0 = 0,
rmsea1 = 0.116, power = 0.8, alpha = 0.05)
# Power for SEM based on RMSEA
#
# n df rmsea0 rmsea1 power alpha
# 222.7465 4 0 0.116 0.8 0.05
#
# URL: http://psychstat.org/rmsea
# #To calculate the minimum detectable effect size of rmsea1 given power and sample size:
wp.sem.rmsea (n = 100, df = 4, rmsea0 = 0,
rmsea1 = NULL, power = 0.8, alpha = 0.05)
# Power for SEM based on RMSEA
#
# n df rmsea0 rmsea1 power alpha
# 100 4 0 0.1736082 0.8 0.05
#
# URL: http://psychstat.org/rmsea
Statistical Power Analysis for t-Tests
Description
A t-test is a statistical hypothesis test in which the test statistic follows a Student's t distribution if the null hypothesis is true and follows a non-central t distribution if the alternative hypothesis is true. The t test can assess the statistical significance of (1) the difference between population mean and a specific value, (2) the difference between two independent populaion means, and (3) difference between means of matched paires.
Usage
wp.t(n1 = NULL, n2 = NULL, d = NULL, alpha = 0.05, power = NULL,
type = c("two.sample", "one.sample", "paired", "two.sample.2n"),
alternative = c("two.sided", "less", "greater"),
tol = .Machine$double.eps^0.25)
Arguments
n1 |
Sample size of the first group. |
n2 |
Sample size of the second group if applicable. |
d |
Effect size. See the book by Cohen (1988) for details. |
alpha |
Significance level chosed for the test. It equals 0.05 by default. |
power |
Statistical power. |
type |
Type of comparison ( |
alternative |
Direction of the alternative hypothesis ( |
tol |
tolerance in root solver. |
Value
An object of the power analysis.
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed). Hillsdale, NJ: Lawrence Erlbaum Associates.
Zhang, Z., & Yuan, K.-H. (2018). Practical Statistical Power Analysis Using Webpower and R (Eds). Granger, IN: ISDSA Press.
Examples
#To calculate the power for one sample t-test given sample size and effect size:
wp.t(n1=150, d=0.2, type="one.sample")
# One-sample t-test
#
# n d alpha power
# 150 0.2 0.05 0.682153
#
# URL: http://psychstat.org/ttest
#To calculate the power for paired t-test given sample size and effect size:
wp.t(n1=40, d=-0.4, type="paired", alternative="less")
# Paired t-test
#
# n d alpha power
# 40 -0.4 0.05 0.7997378
#
# NOTE: n is number of *pairs*
# URL: http://psychstat.org/ttest
#To estimate the required sample size given power and effect size for paired t-test :
wp.t(d=0.4, power=0.8, type="paired", alternative="greater")
# Paired t-test
#
# n d alpha power
# 40.02908 0.4 0.05 0.8
#
# NOTE: n is number of *pairs*
# URL: http://psychstat.org/ttest
#To estimate the power for balanced two-sample t-test given sample size and effect size:
wp.t(n1=70, d=0.3, type="two.sample", alternative="greater")
# Two-sample t-test
#
# n d alpha power
# 70 0.3 0.05 0.5482577
#
# NOTE: n is number in *each* group
# URL: http://psychstat.org/ttest
#To estimate the power for unbalanced two-sample t-test given sample size and effect size:
wp.t(n1=30, n2=40, d=0.356, type="two.sample.2n", alternative="two.sided")
# Unbalanced two-sample t-test
#
# n1 n2 d alpha power
# 30 40 0.356 0.05 0.3064767
#
# NOTE: n1 and n2 are number in *each* group
# URL: http://psychstat.org/ttest2n
#To estimate the power curve for unbalanced two-sample t-test given a sequence of effect sizes:
res <- wp.t(n1=30, n2=40, d=seq(0.2,0.8,0.05), type="two.sample.2n",
alternative="two.sided")
res
# Unbalanced two-sample t-test
#
# n1 n2 d alpha power
# 30 40 0.20 0.05 0.1291567
# 30 40 0.25 0.05 0.1751916
# 30 40 0.30 0.05 0.2317880
# 30 40 0.35 0.05 0.2979681
# 30 40 0.40 0.05 0.3719259
# 30 40 0.45 0.05 0.4510800
# 30 40 0.50 0.05 0.5322896
# 30 40 0.55 0.05 0.6121937
# 30 40 0.60 0.05 0.6876059
# 30 40 0.65 0.05 0.7558815
# 30 40 0.70 0.05 0.8151817
# 30 40 0.75 0.05 0.8645929
# 30 40 0.80 0.05 0.9040910
#
# NOTE: n1 and n2 are number in *each* group
# URL: http://psychstat.org/ttest2n
#To plot a power curve:
plot(res, xvar='d', yvar='power')