Hierarchical regressions using structural equation

To further strengthen the findings of the hierarchical regression models using IBM SPSS Statistics 22, the same hierarchical regression models were conducted using MPlus Version 7 and the technique of Cholesky factorisation with phantom factors in a latent variable model (de Jong, 1999) was applied. In the SEM path models, the dependent variable was the latent factor of children’s arithmetic scores. The independent variables included the latent factors for magnitude comparison and inhibition task. In Cholesky factorisation, the individual steps of the hierarchical regression were coded as phantom latent factors which were then regressed onto arithmetic outcome scores.

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In the first hierarchical regression model (Figure 6.8), the phantom latent factor magnitude comparison was entered first and the phantom latent factor

including magnitude comparison and inhibition was entered in the second step. Both phantom latent factors were then regressed onto arithmetic performance at Time 3. The path model provided an acceptable fit to the data, χ2 (71) = 97.323, p = .021,

RMSEA = .057 (90% CI = .023 - .083), CFI = .976, SRMR = .043, explaining 46.4%

of variance. Both phantom latent factors significantly predicted arithmetic meaning that step one, magnitude comparison, as well as step two, magnitude comparison and inhibition, were significant.

Figure 6.8. Hierarchical SEM regression model of arithmetic at Time 3. Step 1: Magnitude

Comparison. Step 2: Magnitude Comparison and Inhibition. * p < .05. ** p < .01.

The second hierarchical regression model (Figure 6.9) was conducted with the reverse order: The inhibition score was coded as the first phantom latent factor (step one) and inhibition and magnitude comparison coded as the second phantom latent factor (step two). The path model provided an acceptable fit to the data, χ2 (71) = 97.323, p = .021, RMSEA = .057 (90% CI = .023 - .083), CFI = .976, SRMR = .043, explaining 46.4% of variance. The phantom latent factor for step two

(inhibition and magnitude comparison) was the only unique predictor of arithmetic performance. The phantom latent factor including only inhibition did not

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Figure 6.9. Hierarchical SEM regression model of arithmetic at Time 3. Step 1: Inhibition.

Step 2: Inhibition and Magnitude Comparison. * p < .05. ** p < .01.

In other words, the results of the SEM path models confirmed the findings of the SPSS hierarchical regressions that inhibition did not significantly explain

variance in arithmetic performance at Time 3 once performance on magnitude comparison tasks had been taken into account.

6.2.2.2.2 Relationship between inhibition, ANS and arithmetic at Time 4. Similar to Time 3, the first hierarchical regression model (Figure 6.10) included the phantom latent factor magnitude comparison at Time 4 which was entered first and the phantom latent factor including magnitude comparison and inhibition which was entered in the second step. Both phantom latent factors were then regressed onto arithmetic performance at Time 4. The path model provided a moderate fit to the data, χ2 (98) = 142.120, p = .002, RMSEA = .063 (90% CI = .038 - .085), CFI = .961, SRMR = .058, explaining 39.3% of variance. Again, both phantom latent factors were significantly predicting arithmetic suggesting that step one, magnitude comparison, as well as step two, magnitude comparison and inhibition, were significantly predicting arithmetic score.

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Figure 6.10. Hierarchical SEM regression model of arithmetic at Time 4. Step 1: Magnitude

Comparison. Step 2: Magnitude Comparison and Inhibition. * p < .05. ** p < .01.

Figure 6.11. Hierarchical SEM regression model of arithmetic at Time 4. Step 1: Inhibition.

Step 2: Inhibition and Magnitude Comparison. * p < .05. ** p < .01.

The second hierarchical regression model (Figure 6.11) was conducted in reverse order: The inhibition score was coded as the first phantom latent factor (step one) and inhibition and magnitude comparison coded as the second phantom latent factor (step two). The path model provided an acceptable fit to the data, χ2 (98) = 142.120, p = .002, RMSEA = .063 (90% CI = .038 - .085), CFI = .961, SRMR = .058, explaining 39.3% of variance. The phantom latent factor for step two (inhibition and

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magnitude comparison) was the only unique predictor of arithmetic performance. The phantom latent factor including only inhibition did not contribute significantly.

These findings confirmed the Time 3 results and the Time 4 results of the SPSS hierarchical regression models that inhibition did not significantly explain variance in arithmetic performance at Time 4 once performance on magnitude comparison tasks had been taken into account.

6.2.3 Conclusion.

Contrary to the findings of Gilmore et al. (2013) showing that children’s inhibitory control predicted arithmetic after controlling for performance on dot comparison, this study showed the reverse pattern. It is worth mentioning that the sample in Gilmore et al. (2013) were older and, despite the broad age range (seven to ten years), the analyses did not control for age. Also, this study assessed a different tsk of a GoNoGo inhibition task than in Gilmore et al. (2013) using the NEPS-II inhibition subtask (Korkman, Kirk, and Kemp, 1998), a GoNoGo test. Both inhibition tasks are a GoNoGo inhibition test but it may be possible that they measure different aspects of inhibition, thus causing the contrasting results.

The finding of the current study was that children’s performance on inhibition tasks did not explain variance of arithmetic scores once performance on nonsymbolic magnitude comparison has been accounted for. Both conventional hierarchical regression models and SEM phantom latent factor regressions confirmed that the ANS is more important in the development of early arithmetic. It must be mentioned, that the model fit indices of the path models were moderate at best. Unique variance per predictor was low indicating that shared variance is substantial. Neither

inhibitory control nor nonsymbolic magnitude comparison may play the most important role in predicting arithmetic. These results support the findings from the longitudinal prediction in Chapter 5 that transcoding skills, children’s understanding of the Arabic numeral system, was the only stable longitudinal precursor of early arithmetic skills.

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