mstDIF provides a collection of methods for the detection of differential item functioning (DIF) in multistage tests using an item response theory framework. It contains three types of methods. The first type is based on logistic regression, the second type is based on the mstSIB method, and the third type consists of a family of score-based DIF tests. In this brief tutorial, we illustrate the application of each method.
After the mstDIF package has been installed, we load it by the following command.
To illustrate the functions of this package, we use an artificial dataset that is part of mstDIF. This dataset consists of 1000 respondents that responded to a multistage test. This test used a (1,2,2) design: All test takers first worked on a module of 7 items. Based on their estimated ability parameter after completing this module, they worked on a easier or a more difficult module next. After this second module, their ability parameter was estimated again and they were either an easy or a difficult module. We load this toy example using the following code:
This dataset is a list with seven elements. We will use six of them:
resp <- toydata$resp
group_categ <- toydata$group_categ
group_cont <- toydata$group_cont
it <- toydata$it
theta_est <- toydata$theta_est
see_est <- toydata$see_estThe matrix resp contains the response matrix, with 0 corresponding to incorrect and 1 corresponding to correct responses. Missing responses are denoted by NA. group_categ is a vector that indicates an artificial person covariate. 0 indicates that a respondent is a member of the reference group, and 1 that a respondent is a member of the focal group. group_cont is a continuous person covariate, which takes on integer values between 20 and 60; this variable aims at simulating an age variable. it contains a matrix with the item parameters, where the first column corresponds to the discrimination parameters and the second column to the difficulty parameters of the 35 items used in this test. theta_est and see_est are the estimated ability parameters and their standard errors for the individual test takers, respectively.
We want to check whether the item parameters are stable between the focal and reference groups. We use the various methods of mstDIF for this purpose. We are now ready to apply our first method in the next section.
Using the results from the previous section, we are now able to apply the logistic regression DIF test. We do this by the following command, where we also transform group_categ into a categorical variable. The command uses three arguments: resp is a data frame which contains the response matrix (where rows correspond to respondents and columns to items), DIF_covariate is a factor which determines the membership to the focal and reference groups, and theta is a vector of ability parameter estimates for the respondents.
log_reg_DIF <- mstDIF(resp, DIF_covariate = factor(group_categ), method = "logreg",
theta = theta_est)This results in an mstDIF-object. Printing the object
gives us information about the test and the data.
log_reg_DIF
#> Differential Item Functioning (DIF) Detection Test
#> Method: DIF-test using Logistic Regression
#> Test: Likelihood Ratio Test
#> DIF covariate: factor(group_categ)
#> Data: resp
#> Items: 35
#> Persons: 1000Using the summary-method returns a data frame with
item-wise test information. In the logistic regression method, three
tests are computed per item. A test to detect uniform DIF, a test to
detect non-uniform DIF and a global test that is sensitive to both
uniform and non-uniform DIF. By default only the results of the global
tests are returned. Using the DIF_type-argument one of more
tests can be selected per item. Check ?"mstDIF-Methods" for
more information.
For instance, when we want the information form all the tests, we can use:
summary(log_reg_DIF, DIF_type = "all")
#> overall_stat overall_p_value overall_eff_size uniform_stat
#> Item_8 12.16875795 0.002278179 0.0241827373 1.161337e+01
#> Item_33 9.72315310 0.007738274 0.0237115902 8.855648e+00
#> Item_30 9.53189957 0.008514797 0.0229798361 3.440068e-01
#> Item_32 6.80924287 0.033219393 0.0223673719 6.267057e+00
#> Item_5 6.67415852 0.035540611 0.0067044801 6.452145e+00
#> Item_25 6.37952893 0.041181570 0.0144723882 4.259595e+00
#> Item_7 5.71821721 0.057319832 0.0073594481 1.978559e+00
#> Item_21 5.66930097 0.058739052 0.0160079717 5.200306e+00
#> Item_24 4.99009162 0.082492672 0.0114843336 4.380913e+00
#> Item_3 3.18153017 0.203769651 0.0035582868 3.165964e+00
#> Item_28 2.99557723 0.223624133 0.0069983109 2.809478e+00
#> Item_1 2.83699541 0.242077416 0.0028920528 7.697821e-01
#> Item_16 2.81621943 0.244605220 0.0072479034 1.498498e-01
#> Item_27 1.88748532 0.389168581 0.0041230220 1.642642e+00
#> Item_14 1.77810651 0.411044723 0.0038642187 5.441687e-01
#> Item_23 1.48314314 0.476364686 0.0030051788 4.000434e-01
#> Item_15 1.39785298 0.497118680 0.0034687560 9.509201e-01
#> Item_19 1.37728431 0.502257593 0.0036391944 3.203909e-01
#> Item_22 1.19139830 0.551177077 0.0025834418 9.250174e-01
#> Item_10 1.09360706 0.578796963 0.0024268684 1.073037e+00
#> Item_6 1.04667169 0.592540624 0.0011398236 1.031821e+00
#> Item_13 0.73290590 0.693188750 0.0016357837 9.430860e-02
#> Item_12 0.65517407 0.720660564 0.0011895057 3.424121e-03
#> Item_4 0.62353169 0.732152944 0.0008186510 4.791145e-01
#> Item_17 0.56848892 0.752582647 0.0015896194 4.729859e-01
#> Item_2 0.56122811 0.755319792 0.0005672282 5.556867e-01
#> Item_29 0.55470127 0.757788745 0.0013967037 5.448385e-01
#> Item_9 0.44689166 0.799758214 0.0008250358 2.355315e-01
#> Item_26 0.29721920 0.861905536 0.0005740969 1.966462e-01
#> Item_35 0.29198078 0.864166007 0.0007965819 2.910217e-01
#> Item_11 0.26293458 0.876807955 0.0005536073 2.136855e-01
#> Item_31 0.15422487 0.925785757 0.0004047319 1.487255e-01
#> Item_18 0.12917702 0.937453137 0.0005307723 7.395971e-02
#> Item_20 0.09413638 0.954022340 0.0002471444 9.981562e-04
#> Item_34 0.04655953 0.976989117 0.0001236534 2.952516e-02
#> uniform_p_value uniform_eff_size non-uniform_stat non-uniform_p_value
#> Item_8 0.0006547933 2.309012e-02 0.555386043 0.456125273
#> Item_33 0.0029218344 2.161678e-02 0.867505551 0.351646804
#> Item_30 0.5575254755 8.378380e-04 9.187892728 0.002436212
#> Item_32 0.0123003302 2.059874e-02 0.542186170 0.461528116
#> Item_5 0.0110818446 6.482177e-03 0.222013133 0.637510275
#> Item_25 0.0390292238 9.681787e-03 2.119933861 0.145392823
#> Item_7 0.1595423285 2.551203e-03 3.739658523 0.053135303
#> Item_21 0.0225829096 1.469181e-02 0.468994760 0.493449990
#> Item_24 0.0363435304 1.008793e-02 0.609178867 0.435097149
#> Item_3 0.0751880488 3.540905e-03 0.015566464 0.900709092
#> Item_28 0.0937088854 6.564652e-03 0.186099614 0.666182897
#> Item_1 0.3802844848 7.855324e-04 2.067213356 0.150496286
#> Item_16 0.6986789051 3.868717e-04 2.666369601 0.102489567
#> Item_27 0.1999634020 3.588984e-03 0.244843623 0.620729475
#> Item_14 0.4607102056 1.183866e-03 1.233937830 0.266642588
#> Item_23 0.5270668604 8.113754e-04 1.083099774 0.298005149
#> Item_15 0.3294852126 2.360941e-03 0.446932911 0.503795076
#> Item_19 0.5713727937 8.476235e-04 1.056893364 0.303924808
#> Item_22 0.3361610042 2.006304e-03 0.266380887 0.605769901
#> Item_10 0.3002604570 2.381263e-03 0.020570025 0.885956482
#> Item_6 0.3097314014 1.123659e-03 0.014850906 0.903006496
#> Item_13 0.7587695051 2.106055e-04 0.638597303 0.424219191
#> Item_12 0.9533376271 6.220204e-06 0.651749953 0.419487720
#> Item_4 0.4888236954 6.290873e-04 0.144417214 0.703928602
#> Item_17 0.4916167151 1.322721e-03 0.095503024 0.757294420
#> Item_2 0.4560033932 5.616290e-04 0.005541451 0.940659546
#> Item_29 0.4604343634 1.371885e-03 0.009862717 0.920891175
#> Item_9 0.6274522746 4.349099e-04 0.211360132 0.645703429
#> Item_26 0.6574416815 3.798687e-04 0.100573023 0.751143065
#> Item_35 0.5895665701 7.939661e-04 0.000959122 0.975293708
#> Item_11 0.6438939599 4.499329e-04 0.049249086 0.824375110
#> Item_31 0.6997563210 3.903023e-04 0.005499388 0.940884776
#> Item_18 0.7856563491 3.039111e-04 0.055217304 0.814221442
#> Item_20 0.9747961394 2.620834e-06 0.093138224 0.760224832
#> Item_34 0.8635720757 7.841480e-05 0.017034374 0.896158354
#> non-uniform_eff_size N
#> Item_8 1.092617e-03 576
#> Item_33 2.094808e-03 450
#> Item_30 2.214200e-02 450
#> Item_32 1.768627e-03 450
#> Item_5 2.223030e-04 1000
#> Item_25 4.790601e-03 550
#> Item_7 4.808245e-03 1000
#> Item_21 1.316159e-03 424
#> Item_24 1.396404e-03 550
#> Item_3 1.738230e-05 1000
#> Item_28 4.336591e-04 550
#> Item_1 2.106520e-03 1000
#> Item_16 6.861032e-03 424
#> Item_27 5.340379e-04 550
#> Item_14 2.680353e-03 576
#> Item_23 2.193803e-03 550
#> Item_15 1.107815e-03 424
#> Item_19 2.791571e-03 424
#> Item_22 5.771377e-04 550
#> Item_10 4.560528e-05 576
#> Item_6 1.616427e-05 1000
#> Item_13 1.425178e-03 576
#> Item_12 1.183285e-03 576
#> Item_4 1.895637e-04 1000
#> Item_17 2.668984e-04 424
#> Item_2 5.599138e-06 1000
#> Item_29 2.481869e-05 450
#> Item_9 3.901259e-04 576
#> Item_26 1.942282e-04 550
#> Item_35 2.615830e-06 450
#> Item_11 1.036744e-04 576
#> Item_31 1.442965e-05 450
#> Item_18 2.268612e-04 424
#> Item_20 2.445235e-04 424
#> Item_34 4.523864e-05 450This output can be read as follows: Each rows corresponds to an item, and each column to information on this item. Items with a lower p-value are presented first. Focusing on the global DIF tests, the following information is given:
overall_stat the test statisticoverall_p_value the \(p\)-valueoverall_eff_size the effect size (Nagelkerke’s R
squared)N The number of respondents answering this item.Note that most DIF tests only contain a global test per item, and effect sizes are only available for the logistic regression method in the current version of mstDIF.
By inspecting the p-values in the second column, we see that there is an indication for an overall DIF effect in three items, which are labeled as Item_8, Item_33 and Item_30. In these three items, the p-values are below 0.05. However, the effect sizes are very small. An inspection of the columns uniform_p_value and non-uniform_p_value would indicate that the DIF effect of items 8 and 33 is overall uniform, while it is rather non-uniform for item 30. However, given the large size of the item set, these effects could also be random fluctuations in the sample and therefore false positive. We could either a) correct for multiple testing or b) form hypotheses which items we would like to test for DIF.
We carry out the second DIF test, which is the mstSIB procedure. The respective command requires four arguments. The first argument is the response matrix resp, the second argument DIF_covariate is a factor that indicates the membership to the focal and reference group, and the final two arguments are theta and see. Whereas theta contains estimates of the ability parameters, see contains the standard errors of the ability parameters. We run the second DIF test by running:
mstSIB_DIF <- mstDIF(resp, DIF_covariate = factor(group_categ), method = "mstsib",
theta = theta_est, see = see_est)
mstSIB_DIF
#> Differential Item Functioning (DIF) Detection Test
#> Method: SIB test for DIF in MST
#> Test: SIB-test
#> DIF covariate: factor(group_categ)
#> Data: resp
#> Items: 35
#> Persons: 1000As in the first test, printing the test given detailed information on the test and the underlying data set. By applying summary, we get the individual p-values:
summary(mstSIB_DIF)
#> stat p_value N
#> Item_32 0.053326967 2.024588e-36 450
#> Item_25 -0.104163154 1.151095e-14 550
#> Item_5 -0.068129925 3.448207e-07 1000
#> Item_33 -0.122530904 3.505046e-05 450
#> Item_21 -0.097249381 1.557694e-04 424
#> Item_7 0.042461528 1.786444e-04 1000
#> Item_8 0.129955506 1.461819e-02 576
#> Item_14 -0.046968712 2.163451e-02 576
#> Item_11 -0.016487268 3.537402e-02 576
#> Item_4 -0.004426816 9.884723e-02 1000
#> Item_3 0.047124831 1.076104e-01 1000
#> Item_2 -0.021864399 1.478652e-01 1000
#> Item_28 0.055640837 1.706851e-01 550
#> Item_19 0.028802851 1.750301e-01 424
#> Item_18 -0.002701141 1.930745e-01 424
#> Item_6 0.033373013 2.163196e-01 1000
#> Item_15 0.061048096 3.200998e-01 424
#> Item_30 0.041791752 3.604481e-01 450
#> Item_1 0.021816674 3.815204e-01 1000
#> Item_26 -0.041927073 4.175809e-01 550
#> Item_24 -0.072836077 4.431133e-01 550
#> Item_35 -0.028356221 4.709391e-01 450
#> Item_16 -0.018072989 4.999245e-01 424
#> Item_9 -0.017760091 6.030239e-01 576
#> Item_12 -0.015659338 6.079199e-01 576
#> Item_10 -0.025964475 6.327634e-01 576
#> Item_27 0.042820257 6.487289e-01 550
#> Item_31 0.034512576 6.993907e-01 450
#> Item_22 0.034949483 7.053106e-01 550
#> Item_29 -0.019122147 8.394662e-01 450
#> Item_13 0.010666372 8.423829e-01 576
#> Item_20 0.006413806 8.977554e-01 424
#> Item_17 -0.006816036 9.120287e-01 424
#> Item_23 0.006519237 9.143417e-01 550
#> Item_34 0.003871662 9.626978e-01 450We see that the p-values of 9 items (5, 7, 8, 11, 14, 21, 25, 32 and 33) are below 0.05, indicating a DIF effect for these items. N again indicates the number of respondents responding to the respective item. As can be seen, the DIF tests of mstSIB and logistic regression do not always agree in their results. We move on to the third DIF test, which is a score-based DIF test.
The third test is an analytical score-based DIF. This test uses the mstDIF command and can be applied to dRm objects which are generated by the RM command of eRm as well as SingleGroupObjects and MultiGroupObjects that can be generated with the mirt package. In its simplest version, it requires three arguments. The first argument is object, which is the object obtained from eRm or mirt. The second is DIF_covariate, which is again used as a person covariate that is used to test for DIF. In contrast to the logistic regression test and mstSIB, this argument can also be a metric variable. Finally, setting the third argument, method, to “analytical”, determines that an analytical test is used. To apply this test, we first estimate a 2PL model with the mirt package:
We now apply the analytical score-based DIF test:
sc_DIF <- mstDIF(mirt_model, DIF_covariate = factor(group_categ), method = "analytical")
sc_DIF
#> Differential Item Functioning (DIF) Detection Test
#> Method: Asymptotic score-based DIF test
#> Test: Lagrange Multiplier Test for Unordered Groups
#> DIF covariate: factor(group_categ)
#> Data: NULL
#> Items: 35
#> Persons: 1000As with the other tests, printing the object returns information on the test and the underlying dataset. Since we applied the test to a mirt object, the Data are given as NULL. The test statistic depends on the type of covariate that is used in the DIF test. In the case of a discrete, unordered person covariate, the used test statistic leads to a Lagrange Multiplier test for unordered groups. As with the other tests, we get p-values via the summary command:
summary(sc_DIF)
#> stat p_value N
#> Item_33 9.906090082 0.007061872 450
#> Item_8 9.896468123 0.007095929 576
#> Item_30 9.032859053 0.010927972 450
#> Item_32 6.992653625 0.030308508 450
#> Item_5 6.275120280 0.043388531 1000
#> Item_7 5.965298885 0.050658439 1000
#> Item_21 5.847563724 0.053730103 424
#> Item_24 4.909636758 0.085878791 550
#> Item_16 4.514169336 0.104655145 424
#> Item_25 4.003240644 0.135116174 550
#> Item_28 3.054264732 0.217157504 550
#> Item_1 2.858982161 0.239430742 1000
#> Item_3 2.506000875 0.285646446 1000
#> Item_27 2.487716089 0.288269912 550
#> Item_14 2.075146170 0.354313528 576
#> Item_23 1.964683191 0.374433300 550
#> Item_15 1.091405215 0.579434525 424
#> Item_17 1.056119576 0.589748097 424
#> Item_19 0.897735949 0.638350372 424
#> Item_6 0.897261524 0.638501814 1000
#> Item_4 0.854106018 0.652428967 1000
#> Item_10 0.822730948 0.662744671 576
#> Item_13 0.806095883 0.668280060 576
#> Item_9 0.743360294 0.689574770 576
#> Item_35 0.741449239 0.690233992 450
#> Item_2 0.674499242 0.713730655 1000
#> Item_18 0.600214048 0.740738940 424
#> Item_29 0.582602393 0.747290563 450
#> Item_22 0.571962850 0.751276571 550
#> Item_12 0.513515239 0.773555686 576
#> Item_31 0.414089485 0.812983274 450
#> Item_34 0.337143285 0.844870733 450
#> Item_20 0.283572933 0.867806542 424
#> Item_26 0.255243514 0.880186240 550
#> Item_11 0.009199239 0.995410943 576Similar to the logistic regression test, we obtain p-values below 0.05 for the five items 5, 8, 30, 32 and 33. To prevent an increased rate of false positive results, we could again a) correct for multiple testing or b) define hypotheses which items we want to test for DIF before we carry out the tests. From a technical perspective, these analytical DIF tests assume that all other items are DIF free. It is possible to explicitly define a set of anchor item to weaken this assumption, but this goes beyond the scope of this vignette.
In contrast to the logistic regression and mstSIB DIF test, score-based tests also allow to test continuous and ordinal person covariates for DIF effects. We will demonstrate this feature with the group_cont covariate:
sc_DIF_2 <- mstDIF(mirt_model, DIF_covariate = group_cont, method = "analytical")
sc_DIF_2
#> Differential Item Functioning (DIF) Detection Test
#> Method: Asymptotic score-based DIF test
#> Test: Double Maximum Test
#> DIF covariate: group_cont
#> Data: NULL
#> Items: 35
#> Persons: 1000As usual, we can investigate the results for the individual items with:
summary(sc_DIF_2)
#> stat p_value N
#> Item_5 1.3659237 0.09353094 1000
#> Item_2 1.3287680 0.11364405 1000
#> Item_12 1.2791132 0.14592279 576
#> Item_18 1.2504890 0.16762084 424
#> Item_7 1.1718791 0.24006446 1000
#> Item_32 1.1674586 0.24472636 450
#> Item_21 1.0864634 0.34151641 424
#> Item_35 1.0344161 0.41465048 450
#> Item_8 1.0124496 0.44779182 576
#> Item_23 1.0101888 0.45127109 550
#> Item_31 1.0066371 0.45676157 450
#> Item_3 0.9674889 0.51904522 1000
#> Item_9 0.9667843 0.52019204 576
#> Item_11 0.9454645 0.55522588 576
#> Item_10 0.9383952 0.56696119 576
#> Item_26 0.9248651 0.58953776 550
#> Item_14 0.9213109 0.59548694 576
#> Item_27 0.9179728 0.60107926 550
#> Item_24 0.9025400 0.62696387 550
#> Item_17 0.8997145 0.63170271 424
#> Item_20 0.8818305 0.66162553 424
#> Item_1 0.8493214 0.71521842 1000
#> Item_30 0.8349971 0.73825373 450
#> Item_29 0.8286536 0.74830002 450
#> Item_33 0.8172372 0.76610118 450
#> Item_28 0.7851220 0.81383178 550
#> Item_34 0.7831228 0.81667025 450
#> Item_16 0.7775574 0.82448038 424
#> Item_19 0.7612028 0.84660497 424
#> Item_15 0.7529646 0.85725090 424
#> Item_6 0.7415213 0.87144470 1000
#> Item_13 0.7336175 0.88082673 576
#> Item_25 0.7150150 0.90147702 550
#> Item_22 0.7047585 0.91196892 550
#> Item_4 0.6031870 0.98040977 1000As can be seen, there are no significant DIF effects.
Finally, we apply permutation and bootstrap DIF tests. In contrast to the other DIF tests presented in this vignette, these tests make use of the item parameters used during the presentation of the adaptive tests. Technically, these tests aim at testing the hypothesis that the true item parameters are invariant and correspond to the values used in the presentation of the adaptive test. These item parameters are stored in the it matrix. We start our application of these tests by explicitly storing the discrimination and difficulty parameters in separate vectors:
We can now apply the bootstrap DIF test by the following command:
bootstrap_DIF <- mstDIF(resp = resp, DIF_covariate = group_categ, method = "bootstrap",
a = discr, b = diff, decorrelate = F)
#> Estimating: 4pl model ...
#> type = wle
#> Estimation finished!After starting this command, the person parameters are calculated again using the PP package. We get notified that the estimation was finished. Printing the resulting object again gives details on the underlying data and test:
bootstrap_DIF
#> Differential Item Functioning (DIF) Detection Test
#> Method: Bootstrap score-based DIF test with 1000 samples
#> Test: Double Maximum Test
#> DIF covariate: group_categ
#> Data: resp
#> Items: 35
#> Persons: 1000Using the summary command, we get the p-values:
summary(bootstrap_DIF)
#> stat p_value N
#> Item_8 41.43535 0.006 576
#> Item_33 35.32980 0.013 450
#> Item_32 30.50668 0.036 450
#> Item_30 31.35700 0.052 450
#> Item_21 28.94987 0.058 424
#> Item_5 43.40601 0.082 1000
#> Item_16 27.69800 0.088 424
#> Item_25 28.84286 0.116 550
#> Item_24 28.25936 0.155 550
#> Item_22 27.19333 0.176 550
#> Item_7 37.26782 0.187 1000
#> Item_9 27.92269 0.218 576
#> Item_29 23.71674 0.223 450
#> Item_20 22.45761 0.254 424
#> Item_26 25.50687 0.320 550
#> Item_1 31.06844 0.446 1000
#> Item_28 22.58680 0.459 550
#> Item_23 22.57854 0.473 550
#> Item_11 21.29707 0.483 576
#> Item_14 23.01867 0.497 576
#> Item_15 18.65121 0.544 424
#> Item_3 28.14860 0.573 1000
#> Item_27 19.50961 0.609 550
#> Item_19 16.99457 0.683 424
#> Item_6 25.41009 0.714 1000
#> Item_31 16.70285 0.715 450
#> Item_35 16.22799 0.768 450
#> Item_10 16.43764 0.835 576
#> Item_18 13.13709 0.878 424
#> Item_13 15.50530 0.887 576
#> Item_12 16.06358 0.924 576
#> Item_17 13.01244 0.936 424
#> Item_2 20.48589 0.946 1000
#> Item_34 12.05401 0.970 450
#> Item_4 15.74184 0.992 1000We see that items 8, 30, 32 and 33 show p-values below 0.0, similar to the analytical score-based test. As with the other tests, we could either correct for multiple testing or define hypotheses beforehand to prevent an increased rate of false positive results. As was the case with the analytical score-based tests, we can also test continuous and ordinal person covariates for DIF. We demonstrate this type of analysis with the group_cont covariate:
bootstrap_DIF_2 <- mstDIF(resp = resp, DIF_covariate = group_cont, method = "bootstrap",
a = discr, b = diff, decorrelate = F)
#> Estimating: 4pl model ...
#> type = wle
#> Estimation finished!bootstrap_DIF_2
#> Differential Item Functioning (DIF) Detection Test
#> Method: Bootstrap score-based DIF test with 1000 samples
#> Test: Double Maximum Test
#> DIF covariate: group_cont
#> Data: resp
#> Items: 35
#> Persons: 1000The results of this analysis are:
summary(bootstrap_DIF_2)
#> stat p_value N
#> Item_18 34.00477 0.007 424
#> Item_5 47.12704 0.046 1000
#> Item_2 44.09327 0.053 1000
#> Item_35 28.95484 0.074 450
#> Item_32 25.54950 0.117 450
#> Item_7 39.03102 0.155 1000
#> Item_23 25.68833 0.314 550
#> Item_12 26.29353 0.319 576
#> Item_20 21.53959 0.322 424
#> Item_16 21.58565 0.340 424
#> Item_26 24.04213 0.405 550
#> Item_10 23.00194 0.409 576
#> Item_1 31.87110 0.417 1000
#> Item_27 22.10265 0.433 550
#> Item_3 29.74656 0.493 1000
#> Item_9 22.30023 0.523 576
#> Item_30 19.56794 0.548 450
#> Item_31 18.43406 0.586 450
#> Item_14 21.12367 0.601 576
#> Item_29 18.01397 0.603 450
#> Item_21 17.53189 0.646 424
#> Item_24 18.52007 0.672 550
#> Item_33 17.51145 0.674 450
#> Item_11 18.35649 0.735 576
#> Item_17 15.68125 0.742 424
#> Item_34 16.06359 0.762 450
#> Item_19 15.93857 0.765 424
#> Item_8 18.00935 0.776 576
#> Item_13 17.69414 0.783 576
#> Item_25 16.05803 0.797 550
#> Item_6 24.09979 0.809 1000
#> Item_15 15.36422 0.817 424
#> Item_28 17.04354 0.850 550
#> Item_22 16.08331 0.856 550
#> Item_4 16.80331 0.983 1000We find significant DIF effects for items 5 and 18.
The permutation based DIF test works analogously. We therefore just demonstrate the commands and their output:
permutation_DIF <- mstDIF(resp = resp, DIF_covariate = group_categ, method = "permutation",
a = discr, b = diff, decorrelate = F)
#> Estimating: 4pl model ...
#> type = wle
#> Estimation finished!permutation_DIF_2 <- mstDIF(resp = resp, DIF_covariate = group_cont, method = "permutation",
a = discr, b = diff, decorrelate = F)
#> Estimating: 4pl model ...
#> type = wle
#> Estimation finished!The results for the categorical covariate are:
summary(permutation_DIF)
#> stat p_value N
#> Item_8 41.43535 0.005 576
#> Item_33 35.32980 0.019 450
#> Item_30 31.35700 0.040 450
#> Item_32 30.50668 0.041 450
#> Item_21 28.94987 0.059 424
#> Item_5 43.40601 0.077 1000
#> Item_16 27.69800 0.089 424
#> Item_25 28.84286 0.107 550
#> Item_24 28.25936 0.147 550
#> Item_22 27.19333 0.195 550
#> Item_9 27.92269 0.215 576
#> Item_7 37.26782 0.216 1000
#> Item_29 23.71674 0.234 450
#> Item_20 22.45761 0.253 424
#> Item_26 25.50687 0.307 550
#> Item_1 31.06844 0.436 1000
#> Item_23 22.57854 0.468 550
#> Item_14 23.01867 0.500 576
#> Item_28 22.58680 0.502 550
#> Item_11 21.29707 0.503 576
#> Item_15 18.65121 0.522 424
#> Item_27 19.50961 0.603 550
#> Item_3 28.14860 0.610 1000
#> Item_31 16.70285 0.710 450
#> Item_19 16.99457 0.714 424
#> Item_6 25.41009 0.746 1000
#> Item_35 16.22799 0.778 450
#> Item_10 16.43764 0.854 576
#> Item_18 13.13709 0.874 424
#> Item_13 15.50530 0.913 576
#> Item_12 16.06358 0.920 576
#> Item_17 13.01244 0.927 424
#> Item_2 20.48589 0.938 1000
#> Item_34 12.05401 0.966 450
#> Item_4 15.74184 0.994 1000The results for the continuous covariate are:
summary(permutation_DIF_2)
#> stat p_value N
#> Item_18 34.00477 0.016 424
#> Item_5 47.12704 0.038 1000
#> Item_2 44.09327 0.056 1000
#> Item_35 28.95484 0.071 450
#> Item_32 25.54950 0.141 450
#> Item_7 39.03102 0.151 1000
#> Item_12 26.29353 0.294 576
#> Item_20 21.53959 0.295 424
#> Item_23 25.68833 0.309 550
#> Item_16 21.58565 0.350 424
#> Item_26 24.04213 0.371 550
#> Item_10 23.00194 0.403 576
#> Item_1 31.87110 0.404 1000
#> Item_27 22.10265 0.470 550
#> Item_3 29.74656 0.500 1000
#> Item_9 22.30023 0.520 576
#> Item_30 19.56794 0.527 450
#> Item_31 18.43406 0.596 450
#> Item_14 21.12367 0.621 576
#> Item_29 18.01397 0.621 450
#> Item_21 17.53189 0.643 424
#> Item_11 18.35649 0.682 576
#> Item_24 18.52007 0.717 550
#> Item_33 17.51145 0.722 450
#> Item_17 15.68125 0.736 424
#> Item_19 15.93857 0.762 424
#> Item_34 16.06359 0.767 450
#> Item_6 24.09979 0.787 1000
#> Item_8 18.00935 0.788 576
#> Item_13 17.69414 0.789 576
#> Item_15 15.36422 0.802 424
#> Item_25 16.05803 0.816 550
#> Item_22 16.08331 0.847 550
#> Item_28 17.04354 0.859 550
#> Item_4 16.80331 0.986 1000The results are very similar to those of the bootstrap DIF test.
In this vignette, we illustrated the use of the various tests included in the mstDIF package. The available tests include logistic regression, the mstSIB test, analytical score-based tests, bootstrap score-based tests and permutation score-based tests. For the three types of score-based tests, we further demonstrated their application to test a continuous covariate for DIF.