The impact of measurement noninvariance across time and group in longitudinal item response modeling (Nov, 10.1007/s12564-023-09907-4, 2023)

초록

In this article, under the subheading titled, ‘Growth and item parameters’, the symbols ‘β^0,β^1,β^2’were missed. It should have been The parameter estimates and corresponding standard errors of the growth trajectory expressed using the fixed intercept and slopes are presented in Table 5. By contrast with Model 1, in Model 2, group differences were allowed in the intercept and slopes. In particular, in Model 2, β^0 = 0.462 indicates the initial status of males, and the initial status of females was higher than that of males by the estimated difference of, β^0f= 0.093. However, this difference did not differ significantly from zero at the 5% level. The growth rate (or slope) of males while attending middle school was not significant (β^1 = 0.014), but the difference in growth between females and males was significant (β^1f= 0.090). In other words, for female students, awareness and planning with respect to future career changed or grew significantly (β^1+β^1f= 0.104), while the corresponding change was not significant for male students in middle school. On the other hand, the slope for this value among males attending high school (β^2 = 0.159) was positive and significant at the 5% level. For females, the growth rate was estimated to be 0.259 (β^2+β^2f= 0.159 + 0.1), and the difference in the slope between females and males (β^2f = 0.1) differed from zero at the 5% level. Model 2 revealed the different growth patterns for males and females while attending middle school and high school. The estimated item difficulties in Model 2 are presented in the left panel of Fig. 1. In Model 2, as measurement invariance was assumed, the item difficulties remained the same across multiple time points and between males and females. Figure 1 shows that in both the occupation selection and the career path items, the first four items (Items 1–4 and 8–11) showed positive estimates of difficulties, and the last three items (Items 5–7 and 12–14) had negative estimates. In Model 3, which incorporated differential item difficulties across multiple time points (measurement noninvariance across time), similar patterns in estimated growth parameters were found, as was also the case for Model 2. For instance, the difference in initial status between females and males was not significant, but the difference in the slope during middle school was significant. The difference in growth rate over the course of high school (β^2f = 0.079) was estimated to be slightly lower than that in Model 2 (β^2f = 0.1), and it was significant at the 10% level (p = 0.062). Item difficulties were allowed to vary by time point but were the same between females and males. There was substantial measurement noninvariance across time and the patterns were similar to those of Model 5, as shown below. In Model 4, gender differences in item difficulties (measurement noninvariance across groups) were included. One interesting finding was that the difference in initial status between females and males (β^0f= 0.140) was estimated greater than that between the initial statuses in Model 2 (β^0f = 0.097) and Model 3 (β^0f = 0.098). Furthermore, unlike in Models 2 and 3, the difference in Model 4 significantly differed from zero at the 5% level; that is, the initial status of females was significantly greater than that of males. The right panel in Fig. 1 presents estimated item difficulties for males (γ^0i) and females (γ^0i+γ^(female)i). For the two aspects, the first four items (Items 1–4 and 8–11) were more difficult for females, and the last three items (Items 5–7 and 12–14) were more difficult for males. In fact, with the exception of Item 9 (γ^(female)9 = 0.046), the differences in item difficulty estimates between females and males were significant at the 5% level, which suggests a nontrivial measurement noninvariance across groups. In addition, when the estimated item difficulties in Model 2 were compared with those in Model 4, the item difficulties in Model 2 appeared to be close to the average values for item difficulties for males and females in Model 4 (e.g., δ^1 = 0.769 in Model 2 and δ^1 = 0.589 for males and δ^1 = 0.970 for females in Model 4). In Model 5, which was the best-fitting model, item difficulties were modeled in occasion- and group-variant variables. Figure 2 presents a graphical representation of estimated latent mean changes for males and females. As in Model 4, a significant difference was found in initial status between females and males was found (β^0f = 0.142). The results from empirical data analysis suggest that, when gender differences in item difficulties are not modeled properly (Models 2 and 3), the differences in the initial status between females and males was underestimated. The estimated growth trajectories indicate the presence of distinct nonlinear growth patterns, characterized by varying slopes across the middle school and high school periods. Notably, gender-based differences were observed in the estimated means of vocational maturity. In particular, the estimated means for female students demonstrated a significant increase during middle school, while those of the male students remained relatively constant (see Fig. 2). In addition, using unweighted means from responses to the 14 items as dependent variables, a supplementary analysis was conducted to compare the LIRM approach to the composite score method. According to this analysis, five continuous outcomes, ranging from 0 to 1, were specified for five time points, respectively, and the latent growth curve model was employed. As presented in the final column of Table 5, the difference in initial status between females and males (β^0f = 0.016) was not statistically significant, echoing the results of the LIRM analysis without measurement noninvariance (Model 2 and Model 3). Finally, Fig. 3 presents the estimated item difficulties across the five time points for males and females in Model 5. For example, the estimated difficulty differences (γ^(t=m)i) between the first time point and the following time points for Items 1, 2, and 4 were negative and were significantly different from zero, suggesting that these items became easier over time. By contrast, Items 10 and 12 became more difficult relative to baseline difficulty. Note that, for each item, the gap between females and males was modeled as constant across time. Here as well, in these two respects, the first four items were more difficult for females, and the last three items were more difficult for males; with the exception of Item 9, the differences between females and males were significant. The original article has been corrected.