gt4ireval: Generalizability Theory for Information Retrieval Evaluation
Julián Urbano
2017-03-06
gt4ireval is a package to measure the reliability of an Information Retrieval test collection. It allows users to estimate reliability using Generalizability Theory (Brennan 2001) and map those estimates onto well-known indicators such as Kendall \(\tau\) correlation or sensitivity. For background information and details, the reader is referred to (Urbano, Marrero, and Martín 2013).
Loading Data
Once loaded, gt4ireval needs initial evaluation data to run a G-study and the corresponding D-study. These data need to be in a standard data frame or matrix, where columns correspond to systems and rows correspond to queries. For this vignette, let us use data from the TREC-3 Ad hoc track.
## [1] 50 40
## sys1 sys2 sys3 sys4 sys5
## 1 0.2830 0.5163 0.4810 0.5737 0.5184
## 2 0.0168 0.5442 0.3987 0.2964 0.6115
## 3 0.0746 0.2769 0.3002 0.2459 0.3803
## 4 0.1828 0.6622 0.6164 0.4291 0.6556
## 5 0.0181 0.3670 0.3762 0.1095 0.2465
If your data is transposed (i.e. columns correspond to queries and rows correspond to systems), you can get the correct format with the t function: data <- t(data).
G-Study
To run a G-study with the initial data we have, we simply call function g.study.
##
## Summary of G-Study
##
## Systems Queries Interaction
## ----------- ----------- -----------
## Variance 0.0071668 0.022642 0.01092
## Variance(%) 17.596 55.593 26.811
## ---
## Mean Sq. 0.36926 0.91661 0.01092
## Sample size 40 50 2000
Additionally, we can tell the function to ignore the systems with lowest average effectiveness scores by setting parameter drop. For instance, we can ignore the bottom 25% of systems.
adhoc3.g <- gstudy(adhoc3, drop = .25)
adhoc3.g
##
## Summary of G-Study
##
## Systems Queries Interaction
## ----------- ----------- -----------
## Variance 0.0028117 0.028093 0.010152
## Variance(%) 6.8482 68.425 24.727
## ---
## Mean Sq. 0.15074 0.85296 0.010152
## Sample size 30 50 1500
The summary shows the estimated variance components: variance due to the system effect \(\hat\sigma_s^2=0.0028\), due to the query effect \(\hat\sigma_q^2=0.0281\), and due to the system-query interaction effect \(\hat\sigma_e^2=0.0102\). The second row shows the same values but as a fraction of the total variance. The third row shows the estimated Mean Squares for each component, and finally the sample size in each case. In our example, we have 30 systems and 50 queries as initial data.
D-Study
The results from the G-study above can now be used to run a D-study. First, let us estimate the stability of the current collection (50 queries).
##
## Summary of D-Study
##
## Call:
## queries = 50
## stability = 0.95
## alpha = 0.025
##
## Stability:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Queries Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 50 0.93265 0.89311 0.96287 0.78613 0.66141 0.88039
##
## Required number of queries:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Stability Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 0.95 69 37 114 259 130 487
The summary first shows how dstudy was called. In particular, it tells us that the target number of queries is \(n_q'=50\) (set by default from the G-study initial data), the target stability is \(\pi=0.95\) (set by default), and the confidence level is \(\alpha=0.025\) (set by default). Next are the estimated stability scores; the relative stability with 50 queries is \(\text{E}\hat\rho^2=0.93265\) with a 95% confidence interval of \([0.89311, 0.96287]\), and the absolute stability is \(\hat\Phi=0.78613\) with a 95% confidence interval of \([0.66141, 0.88039]\). Regarding the required number of queries to reach the target stability, the estimate is \(\hat{n}_q'=69\) with a 95% confidence interval of \([37, 114]\) to reach \(\text{E}\rho^2=\pi\), and \(\hat{n}_q'=259\) with a 95% confidence interval of \([130, 487]\) to reach \(\Phi=\pi\).
Function dstudy can be called with multiple values for \(n_q'\), \(\pi\) and \(\alpha\) to study trends. For instance, we can indicate several query set sizes by setting parameter queries.
dstudy(adhoc3.g, queries = seq(20, 200, 20))
##
## Summary of D-Study
##
## Call:
## queries = 20 40 60 80 100 120 140 160 180 200
## stability = 0.95
## alpha = 0.025
##
## Stability:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Queries Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 20 0.84707 0.76971 0.91208 0.5952 0.43864 0.74647
## 40 0.91721 0.86987 0.95402 0.74624 0.6098 0.85483
## 60 0.94324 0.90931 0.96887 0.81519 0.70097 0.8983
## 80 0.95682 0.93041 0.97647 0.85468 0.75761 0.92174
## 100 0.96515 0.94354 0.98109 0.88026 0.79621 0.93639
## 120 0.97079 0.9525 0.98419 0.89819 0.8242 0.94643
## 140 0.97486 0.95901 0.98642 0.91144 0.84543 0.95373
## 160 0.97793 0.96395 0.98809 0.92165 0.86209 0.95927
## 180 0.98033 0.96783 0.9894 0.92974 0.8755 0.96363
## 200 0.98227 0.97095 0.99045 0.93632 0.88654 0.96715
##
## Required number of queries:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Stability Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 0.95 69 37 114 259 130 487
The output above shows the estimated stability scores, with confidence intervals, for various query set sizes. For example, we have \(\text{E}\hat\rho^2=0.96515\) with 100 queries, and \(\hat\Phi\in[0.88654, 0.96715]\) with 95% confidence when having 200 queries. Similarly, we may indicate several target stability scores by setting parameter stability.
dstudy(adhoc3.g, stability = c(0.8, 0.85, 0.9, 0.95, 0.97, 0.99))
##
## Summary of D-Study
##
## Call:
## queries = 50
## stability = 0.8 0.85 0.9 0.95 0.97 0.99
## alpha = 0.025
##
## Stability:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Queries Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 50 0.93265 0.89311 0.96287 0.78613 0.66141 0.88039
##
## Required number of queries:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Stability Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 0.8 15 8 24 55 28 103
## 0.85 21 11 34 78 39 146
## 0.9 33 18 54 123 62 231
## 0.95 69 37 114 259 130 487
## 0.97 117 63 194 440 220 828
## 0.99 358 191 593 1347 673 2534
The output above shows that the estimated number of queries to reach \(\text{E}\rho^2=0.97\) is 117, while 123 are required to reach \(\Phi=0.9\). Finally, we can also indicate several confidence levels for the computation of confidence intervals by setting parameter alpha.
dstudy(adhoc3.g, alpha = c(0.005, 0.025, 0.05))
##
## Summary of D-Study
##
## Call:
## queries = 50
## stability = 0.95
## alpha = 0.005 0.025 0.05
##
## Stability:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Alpha Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 0.005 0.93265 0.87737 0.96967 0.78613 0.61466 0.9023
## 0.025 0.93265 0.89311 0.96287 0.78613 0.66141 0.88039
## 0.05 0.93265 0.90062 0.95901 0.78613 0.68417 0.86796
##
## Required number of queries:
## Erho2 Phi
## ----------------------------------- -----------------------------------
## Alpha Expected Lower Upper Expected Lower Upper
## ----------- ----------- ----------- ----------- ----------- ----------- -----------
## 0.005 69 30 133 259 103 596
## 0.025 69 37 114 259 130 487
## 0.05 69 41 105 259 145 439
The summary above shows that with 50 queries a 99% confidence interval for \(\text{E}\rho^2\) is \([0.87737, 0.96967]\), and a 90% confidence interval on the number of queries to reach \(\Phi=0.95\) is \([145, 439]\).
Using the Returned Objects
Both gstudy and dstudy return objects with all results from the analysis so they can be used in subsequent computations. In fact, object adhoc3.g above contains all the G-study results, and it is provided to function d.study. The full list of available data in both objects can be obtained with function names.
adhoc3.g <- gstudy(adhoc3, drop = 0.25)
names(adhoc3.g)
## [1] "n.s" "n.q" "var.s" "var.q" "var.e" "em.s" "em.q" "em.e" "call"
## [1] 0.002811699
adhoc3.d <- dstudy(adhoc3.g, queries = seq(10, 100, 10), stability = seq(0.5, 0.99, .05))
names(adhoc3.d)
## [1] "Erho2" "Phi" "n.q_Erho2" "n.q_Phi"
## [5] "Erho2.lwr" "Erho2.upr" "Phi.lwr" "Phi.upr"
## [9] "n.q_Erho2.lwr" "n.q_Erho2.upr" "n.q_Phi.lwr" "n.q_Phi.upr"
## [13] "call"
## [1] 0.7347152 0.8470730 0.8925725 0.9172057 0.9326493 0.9432373 0.9509485
## [8] 0.9568151 0.9614284 0.9651511
cbind(lwr = adhoc3.d$n.q_Phi.lwr, upr = adhoc3.d$n.q_Phi.upr)
## lwr upr
## [1,] 26 7
## [2,] 32 9
## [3,] 39 11
## [4,] 48 13
## [5,] 60 16
## [6,] 77 21
## [7,] 103 28
## [8,] 146 39
## [9,] 231 62
## [10,] 487 130
With all these data we can for instance plot the estimated \(\text{E}\hat\rho^2\) score, with a 95% confidence interval, as a function of the number of queries in the collection.
xx <- seq(10, 200, 5)
adhoc3.d <- dstudy(adhoc3.g, queries = xx)
plot(xx, adhoc3.d$Erho2,
yaxs = "i", ylim = c(0.75, 1), lwd = 2, type = "l",
xlab = "Number of queries", ylab = "Relative stability")
lines(xx, adhoc3.d$Erho2.lwr) # lower confidence limit
lines(xx, adhoc3.d$Erho2.upr) # upper confidence limit
grid()
Mapping G-Theory onto Data-based Indicators
Finally, the following functions can be used to map stability indicators from Generalizability Theory onto well-known data-based indicators (see (Urbano, Marrero, and Martín 2013) for details):
gt2tau and gt2tauAP map \(\text{E}\rho^2\) onto Kendall \(\tau\) correlation and \(AP\) correlation coefficients.gt2power, gt2minor and gt2major map \(\text{E}\rho^2\) onto expected power, minor conflict rate and major conflict rate of 2-tailed t-tests.gt2asens and gt2rsens map \(\text{E}\rho^2\) and \(\Phi\) onto absolute and relative sensitivity, respectively.gt2rmse maps \(\Phi\) onto rooted mean squared error.
## [1] 0.8641168
## [1] 0.1238861
The results show that the estimated rank correlation at \(\text{E}\rho^2=0.95\) is \(\hat\tau=0.86412\), and that the relative sensitivity at \(\Phi=0.8\) is estimated as \(\hat\delta_r=12.389\%\). In order to map the stability of a certain D-study, we can simply use the returned dstudy object. These functions can be used for instance to plot the estimated \(\hat\tau\) correlation as a function of the query set size.
xx <- seq(10, 200, 5)
adhoc3.d <- dstudy(adhoc3.g, queries = xx)
plot(xx, gt2tau(adhoc3.d$Erho2),
yaxs = "i", ylim = c(0.5, 1), lwd = 2, type = "l",
xlab = "Number of queries", ylab = "Kendall rank correlation")
lines(xx, gt2tau(adhoc3.d$Erho2.lwr)) # lower confidence limit
lines(xx, gt2tau(adhoc3.d$Erho2.upr)) # upper confidence limit
grid()
In any case, the user is strongly advised to take these mappings with a grain of salt (see Fig. 3 in (Urbano, Marrero, and Martín 2013)).
Acknowledgements
This work was supported by an A4U postdoctoral grant and a Juan de la Cierva postdoctoral fellowship.
References
Brennan, Robert L. 2001. Generalizability Theory. Springer.
Urbano, Julián, Mónica Marrero, and Diego Martín. 2013. “On the Measurement of Test Collection Reliability.” In International Acm Sigir Conference on Research and Development in Information Retrieval, 393–402.