# Computers & Education

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4 432 A.C. Carle / Computers & Education 53 (2009) component for the class-level random effect to zero. This approach fits a taxonomy of simpler two-level longitudinal growth models that does not model variation among classes within teachers (Singer & Willett, 2003). Alternatively, a negative covariance component can result from unbalanced data (Singer & Willett, 2003). For example, not all professors taught the same number of classes at each time. Thus, one should first empirically explore whether the negative covariance resulted from unbalanced data. If analyses in balanced data reveal significant variation among classes within teachers, this suggests that the negative covariance results from unbalanced data. In this case, one should fit the more complex model using the balanced data. However, if analyses in balanced data reveal little variation among classes within teachers, this suggests fitting the reduced two-level models (Singer & Willett, 2003). I took this approach. I first created a balanced dataset and selected from the larger sample only those instructors with multiple measures at every data collection wave (n = 446). I then fit the unconditional three-level model in this subsample. The variance component describing residual variance among classes within teachers did not differ significantly from zero (^r 2 eclass = 0.02, SE = 0.05, p < 0.33). This suggested that the negative variance component for the class-level random effect in the full sample primarily resulted because most teachers received similar ratings across their classes. This supported setting the variance component for the class-level random effect to zero and fitting a taxonomy of reduced two-level longitudinal growth models. I did this. All subsequent models used the reduced two-level taxonomy. Model 1: Unconditional Means Model: Model 1 included no predictors and supplied a baseline for future model comparisons (Singer & Willett, 2003). The variance components showed statistically significant variance associated with teachers (^r 2 0 = 0.219, SE = 0.01, p < 0.001) and statistically significant residual variance (^r 2 s = 0.188, SE = 0.003, p < 0.001). The intraclass correlation coefficient formed from the variance components showed that approximately 54% SETE variance arose from differences among teachers. Table 1 summarizes these results. Model 2: Unconditional Growth Model: Model 2, the standard unconditional growth model (Singer & Willett, 2003), incorporated fixed and random effects for linear time. This resulted in statistically significant fit improvement (D 2 vð3þ = 94.70, p <.01), as well as reduced AIC and BIC. Model 2 s variance components revealed statistically significant variance associated with teachers (^r 2 0 ¼ 0:23, SE = 0.01, p < 0.01) and statistically significant residual variance (^r 2 e = 0.180, SE = 0.003, p < 0.01). The linear fixed effect of time differed significantly and positively from zero, but slightly (c 10 = 0.008, SE = 0.002, p < 0.01). The variance component associated with time indicated small, statistically significant, systematic differences among linear trends across instructors (^r 2 1 = 0.002, SE < 0.001, p < 0.001), and the residual covariance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi component differed significantly from zero (^r 2 01 ¼ 0:006, SE = 0.002, p < 0.01). Expressed as a correlation coefficient, ^q p 0 p 1 ¼ ^r 2 0 ^r2 1 ¼ 0:27, it suggested a small to moderate (Cohen, 1988) negative correlation between initial status and rates of change. Teachers with initially low intercepts improved relatively quickly compared to individuals with higher intercepts. Finally, adding the fixed and random effects for time accounted for 4% of the explainable variance. Model 3: Quadratic Effect of Time: Before exploring additional predictors, I examined the possibility that change across time did not follow a rectilinear (straight line) form. Results did not support this contention. The estimate associated with the fixed effect of quadratic time did not differ significantly from zero, the addition of the quadratic effects did not significantly reduce the deviance statistic (Dv 2 ð4þ ¼4:6, p = 0.20), and the AIC and BIC increased. Thus, I dropped this term. No remaining analyses included quadratic effects. Model 4: Effect of Instruction Mode: Model 4 examined overall differences in initial status across instruction mode by adding a fixed effect for instruction to Model 2. Model 4 explored the possibility that online and face-to-face classes differed in their overall rates of change. It Table 1 Results of selected growth models examining professors SETEs across seven semesters, instruction mode (face-to-face and online), and minority status. a Parameter Unconditional means Unconditional growth fixed and random Reduced model Initial status Intercept c * (SE) (0.015) (0.018) (0.019) Mode c (SE) (0.138) Ethnicity c * (SE) (0.048) Mode ethnicity c (SE) (0.294) Rate of change Intercept c * (SE) (0.003) Mode c (SE) (0.030) Ethnicity c (SE) (0.007) Mode ethnicity time c (SE) (0.064) Level 1 (residual) Within person r 2 e * * * (SE) Residual (0.003) (0.003) Level 2 Initial status r * * * (SE) (0.014) (0.013) Rate of change r * * (SE) (0.000) (0.000) Rate of change covariance r * * (SE) (0.002) (0.002) * p < a Mode reference group = face-to-face. Ethnicity reference group = majority.

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