Item Trace Lines

The three approaches to accounting for extreme response style attempt to model the interaction an individual has with the response scale for Likert type items. This interaction can be visualized using category response curves. Category response curves, also known as trace lines, are used to plot the propensity to endorse each response option across the substantive trait continuum. Category response curves plot the continuum of one trait at a time. The category response curves of propensity versus the substantive trait given a value of ERS can be analyzed for multidimensional models with one substantive trait and one ERS trait. The resulting trace lines indicate the probability to select each response option along the substantive trait continuum for individuals with the same ERS tendency.

Trace lines can be generated for every item and for every value of ERS tendencies for each model. For example, the interaction of individuals with a high extreme response tendency with item 10 and the interaction of individuals with a low extreme response tendency with item 10 were considered. The extreme response trait estimates for individual 5 were used to represent high extreme response tendencies and the extreme response trait estimates for individual 5210 were used to represent low extreme response tendencies.

Item parameters and person parameters were estimated simultaneously using PROC MCMC in SAS. The item parameter estimates for items 1 – 10 are displayed in Appendix A. The item parameter estimates were derived by taking the mean of the post burn-in iterations of the MCMC procedure estimating the posterior distribution. Table 18 displays the item parameter estimates for item 10. For comparison purposes, the item parameters estimates for the MNRM found by Bolt and Newton (2011) when two scales were considered (value of science and enjoyment of science) were .291, 2.146, 1.005, and -3.444 for intercepts 𝑐 , … , 𝑐 , respectively.

The item parameters are not equivalent due to different estimation techniques; however, the estimates are approximately equal for the same item and model.

Table 18. Item 10 parameter estimates under the IRTree Model, the MNRM, and the MPCM

Method Parameter Estimate

IRTree Model 𝑎₁ 2.4318 𝑏₁ 1.5603 𝑎₂ 2.0147 𝑏₂ -1.7175 𝜈 .0910 MNRM 𝑐₁ .3465 𝑐2 2.3176 𝑐3 .9360 𝑐4 -3.6001 MPCM 𝑎 1.8370 𝑏 .6589 𝜏1 -1.5487 𝜏2 .0494 𝜏₃ 1.4993

Figure 23 displays the trace plots of item 10 for an individual with an extreme response tendency trait of 𝜃𝐸𝑅𝑆 = 1.99 (IRTree Model), 𝜃𝐸𝑅𝑆 = 1.24 (MNRM), and 𝜔 = .48 (MPCM). The vertical axis is the propensity to select each category for item 10. The horizontal axis represents the substantive trait continuum. Response option trace lines are labeled with their quantitative value (1- Strongly Disagree, 2-Disagree, 3-Agree, and, 4-Strongly Agree). The vertical line within each plot represents the placement of individual 5’s substantive trait estimate. Across the substantive trait continuum for the IRTree Model, the response option with the highest probability of selection is either 1 or 4. An individual with an extreme response trait of

regardless of the value of their substantive trait. Individuals with substantive traits above 0.5 are likely to select 4-Strongly Agree and individuals with substantive traits below 0.5 are likely to select 1-Strongly Disagree.

Under the MNRM, individuals with an extreme response trait of 1.24 and substantive traits above 1 are most likely to select 4-Strongly Agree on item 10. Individuals with an extreme response trait equal to 1.24 and a substantive trait less than 0 are most likely to select 1-Strongly

Disagree on item 10. Unlike the results from the IRTree Model, there exists a group of individuals

who are most likely to endorse a non-extreme category. Individuals with substantive trait values between 0 and 1 are most likely to endorse one of the middle categories (2 and 3) more than any other option on item 10.

The MPCM was analyzed for individuals with extreme response tendencies of .48. For the MPCM, individuals with substantive traits below 0 or above 1.5 are most likely to endorse an extreme category. A non-extreme category has the highest probability of selection for individuals with substantive trait estimates between 0 and 1.5.

Figure 23. Item 10 trace plots for an individual with an extreme response tendency trait of 𝜽𝑬𝑹𝑺= 𝟏. 𝟗𝟗 (IRTree Model), 𝜽𝑬𝑹𝑺= 𝟏. 𝟐𝟒 (MNRM), and 𝝎 =. 𝟒𝟖 (MPCM)

The trace plots for item 10 related to individuals with extreme response tendencies of 𝜃𝐸𝑅𝑆 = −1.28 (IRTree Model), 𝜃𝐸𝑅𝑆= −0.65 (MNRM), and 𝜔 = 2.27 (MPCM) are displayed in Figure 24. The vertical line on each plot represents the trait estimate of individual 5210. The plots are used to illustrate individuals who have a very low tendency for extreme responses. The trace lines for item 10 under the IRTree Model show, regardless of the substantive trait value, either 2-Disagree or 3-Agree has the highest propensity of endorsement.

Individuals with low extreme response tendencies (𝜔 = 2.27) under the MPCM, tend to select either 3-Agree or 2-Disagree with the highest probability along the substantive trait continuum. The exception is for substantive trait values near -3. The substantive trait overpowers the tendency to be extreme response averse for individuals with very low presence of the substantive trait. In other words, even though the individual has a very low extreme response tendency (2.27), an extreme category (1-Strongly Disagree) has highest probability of selection.

The trace plot developed under the MNRM for individuals with 𝜃𝐸𝑅𝑆 = −0.65 exhibits a unique pattern compared to the trace plot using 𝜔 = 2.27 for the MPCM and the trace plot using 𝜃_𝐸𝑅𝑆 = −1.28 for the IRTree Model. Individuals with substantive trait values greater than 3 endorse category 4 with the highest probability even though these individuals do not have a high extreme response tendency. Individuals with substantive trait estimates greater than 0.8 but less than 3 have the highest propensity to select 3-Agree. Individuals with substantive trait estimates between -1.5 and 0.8 have the highest propensity towards selecting 2-Disagree on item 10. Even with a low tendency to select extreme responses, individuals with substantive trait values less than -1.5 tend to select 1-Strongly Disagree.

Figure 24. Item 10 trace plots for an individual with an extreme response tendency trait of 𝜽𝑬𝑹𝑺= −𝟏. 𝟐𝟖 (IRTree Model), 𝜽𝑬𝑹𝑺= −𝟎. 𝟔𝟓 (MNRM), and 𝝎 = 𝟐. 𝟐𝟕 (MPCM)