In wbelzak/regDIF: Regularized Differential Item Functioning

knitr::opts_chunk$set(echo = TRUE, fig.width=7, fig.height=4)

This vignette introduces regDIF() to the general user by providing common use cases.

Using LASSO regularization to evaluate measurement bias in a 2-parameter logistic IRT model.

In this example, data in ida were generated to mimic an integrative data analysis, where data are pooled across multiple studies and the measurement model is evaluated for between-study and within-study (e.g., gender, age) measurement bias. These data include 6 item responses (binary) and 3 background characteristics -- namely, age (continuous, centered), gender (categorical, effect-coded), and study (categorical, effect-coded).

DIF was generated to be on items 2 (age, gender, study), 3 (age, gender, study), 4 (age), and 5 (gender, study), for both intercepts and slopes.

library(regDIF)
head(ida)

The item response data must first be separated from the predictor data (background variables) before running regDIF(). A single value of the tuning parameter, tau = 2, is then fit to the data.

item.data <- regDIF::ida[,1:6]
pred.data <- regDIF::ida[,7:9]
fit <- regDIF(item.data, pred.data, tau = 2)

summary(fit)

The summary() function shows that no DIF effects remain in the model. Only the latent variable parameters and base item parameters, which were not penalized at all, are estimated to be non-zero. This is shown by using the coef method.

coef(fit)

Now that the data have been properly specified in regDIF, a more thorough investigation of DIF is warranted. The regDIF() function defaults to estimating 100 values of the tuning parameter, starting with a value large enough to penalize all DIF effects to zero. However, for brevity, only 10 values of tau are specified with the num.tau argument and we reduce the tolerance for convergence using the control argument.

fit2 <- regDIF(item.data, pred.data, num.tau = 10, control = list(tol = 1e-3))

fit2

By printing the model object, 10 rows of results appear, one for each value of the tuning parameter. The first thing to notice is that 4 rows are missing. This occurs because regDIF() automatically stops model-fitting when a small value of tau would produce a non-identified model. In focusing attention to the BIC column, it is evident that the smallest value occurs well before the model would be non-identified. This is an encouraging result. The summary() function may be used again, which, with multiple values of the tuning parameter fit to the data, produces non-zero DIF effects corresponding to the model with the minimum value of BIC.

summary(fit2)

A plot of the regularization path also shows the remaining DIF effects.

plot(fit2)

To produce other model results, the fit2 object contains lists of the impact (latent variable) parameters, base (intercept and slope) item parameters, and DIF parameters for all values of the tuning parameter. For instance, the impact parameters are printed below.

fit2$impact

EAP scores and standard deviations may also be produced.

lapply(fit2$eap, head)

Finally, when data include a large number of items, observations, and predictors, regDIF() can run relatively slowly. An alternative approach, which yields much faster results, is to provide an observed proxy for the latent scores. In the case of binary data, this might be sum scores. Note that using observed proxy scores is identical to performing a multivariate regression, where the item responses are regressed on the proxy scores and background variables. (The proxy scores are simultaneously regressed on the background variables as well.)

fit3 <- regDIF(item.data, pred.data, prox.data = rowSums(item.data), num.tau = 20)

summary(fit3)

The results show more DIF on both the intercepts (Items 3 and 6) and slopes (Items 2, 3, and 5).

More modeling possibilities with regDIF.

In addition to LASSO, other penalty functions are possible with regDIF(). For instance, the elastic net penalty combines LASSO and ridge functions, which is useful when many correlated predictors are evaluated for DIF. The elastic net is controlled by a second tuning parameter, alpha, and defaults to alpha = 1, corresponding to the LASSO penalty. In contrast, alpha = 0 corresponds to the ridge penalty. When alpha is between 0 and 1, however, the elastic net is used to perform DIF selection. For brevity, observed proxy scores are used in all model fitting below.

fit_net <- regDIF(item.data, pred.data, prox.data = rowSums(item.data), num.tau = 20, alpha = .5)

summary(fit_net)

The final elastic net results yield the same DIF effects as the LASSO results, although the amount of penalization is greater for elastic net (i.e., larger tau).

Other penalty functions include the minimax concave penalty (MCP) and the group extensions of LASSO and MCP, which penalize the intercept and slope DIF effects in tandem. The group LASSO function is shown below.

fit_grp_mcp <- regDIF(item.data, pred.data, prox.data = rowSums(item.data), num.tau = 20, pen.type = "grp.mcp")

summary(fit_grp_mcp)

Although the MCP results appear largely the same as the LASSO results, the group MCP function included both the intercept and slope for each background variable remaining in the final model.

In summary, the regDIF R package provides a flexible implementation of using regularization to identify DIF across multiple background characteristics.

Please reach out to wbelzak@gmail.com for any questions, and remember to cite regDIF in your work. Thank you kindly!