1. Assume two tests are administered to two groups. Each test consists of 30 items. The first 30 items were assigned to Group 1, and the last 30 items to Group 2.
a) Calibrate the data for each group separately, and check the item fit separately. Can the model adequately fit each item? Use a significance level of 0.01, a Bonferroni correction, and df = 10.
b) Plot the problematic items found in a) and describe the issue with each item.
c) Remove the problematic items you find in a), and calibrate the remaining items separately for each group.
d) Redo c), but this time assume the two groups are equivalent, and calibrate the remaining items jointly.
e) Compare the item parameter estimates obtained in c) and d). Are the results expected or not?
Use the (updated) plot.errorbars1.R to plot the problematic items.
Use hwa3_Pr2.csv and the mirt package for Problem 2.
a) Fit the 3PL model to the data WITHOUT any priors.
b) Redo a), but this time specify priors for the guessing parameters ONLY with a prior mean appropriate for multiple-choice type items with four options.
c) Compare the item parameter estimates with and without priors
For the remaining problems, use the mirt package unless stated otherwise.
For Problem 3, use hwa3_3groups.csv. This file contains test data of three groups, and each group has 5000 examinees. Group 1 answered Items 1-30; Group 2 Items 21-50; and Group 3 Items 41-70. (Note the overlaps.)
3. Calibrate the test concurrently using the 3PL model without assuming that the three groups are identical.
a) Compare the means and variances of the three groups. Which group is the ablest? Most variable?
b) Inspect the estimated item parameters. Which item(s), if any, may be problematic?
For Problem 4, use hwa3_Pr4_1.csv and hwa3_Pr4_2.csv. These two files contain response data from two non-equivalent groups, and the first 10 items in each file are the anchor items.
4. Calibrate the data separately, and place Group 2's estimates on the scale of Group 1.
a) Use the irt.link() function to compute the scaling constants for the mean-sigma and Stocking- Lord methods.
b) Compute by hand the scaling constants for the robust mean-sigma method.
c) Put the item parameters of Group 2 on the scale of Group 1, but this time by fixing the common item parameter estimates to those of Group 1.
d) Based on a), what are the mean and variance of the ability distribution of Group 2? What are these values based on b) and c)?
e) Based on d), how similar or different are the ability distributions of Groups 1 and 2?
Use hwa3_Pr5_1.csv, hwa3_Pr5_2.csv, hwa3_constraints.csv, hwa3_itmattrib1.csv and hwa3_itmattrib2.csv for Problem 5.
Files hwa3_Pr5_1.csv and hwa3_Pr5_2.csv contain response data from two non-equivalent groups. The first 10 items in each dataset are the anchor items.
In the item attribute files, ITEMTYPE = 1 when the item is a unique item, and ITEMTYPE = 2 when the item is an anchor item.
5.
a) Calibrate the two tests concurrently. Specify the following priors for item parameters:
a1 ~ logN(0, 0.5);
d ~ N(0,1); and g ~ N(-1.39,1).
What are the group mean and variance of Group 2 (Group 1 as the reference group)?
b) Construct a 30-item test using only the 100 items in Group 1 that would be optimal for the theta levels of -1, 0, and 1. Use the maximum Fisher information as the item selection criterion and the constraints in hwa3_constraints.csv.
c) Repeat the process using only the 100 items in Group 2.
d) Finally, pool all the items from the two groups, and construct a 30-item test with the same constraints and specifications.
e) Compare the three tests in terms of the information they provide. Use the lower and upper limits of -4 and 4, respectively, with the theta increment of 0.01.
• Note that the combined item pool has less than 200 items.
• Use the info.with.itempar.R to compute the information.
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