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Applying a multiple group causal indicator modeling framework to the reading comprehension skills of third, seventh, and tenth grade students

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This study demonstrates the utility of applying a causal indicator modeling framework to investigate important predictors of reading comprehension in third, seventh, and tenth grade students. The results indicated that a 4-factor multiple indicator multiple indicator cause (MIMIC) model of reading comprehension provided adequate fit at each grade level. This model included latent predictor constructs of decoding, verbal reasoning, nonverbal reasoning, and working memory and accounted for a large portion of the reading comprehension variance (73–87 %) across grade levels. Verbal reasoning contributed the most unique variance to reading comprehension at all grade levels. In addition, we fit a multiple group 4-factor MIMIC model to investigate the relative stability (or variability) of the predictor contributions to reading comprehension across development (i.e., grade levels). The results revealed that the contributions of verbal reasoning, nonverbal reasoning, and working memory to reading comprehension were stable across the three grade levels. Decoding was the only predictor that could not be constrained to be equal across grade levels. The contribution of decoding skills to reading comprehension was higher in third grade and then remained relatively stable between seventh and tenth grade. These findings illustrate the feasibility of using MIMIC models to explain individual differences in reading comprehension across the development of reading skills.

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Acknowledgments

Support for this manuscript was provided by Grant Number P50 HD052120 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development awarded to the second and third authors. Support was also provided by Predoctoral Interdisciplinary Research Training Fellowships R305B04074 and R305B090021 from the Institute of Education Sciences awarded to the first author.

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  1. Department of Psychology, Florida State University, 1107 W. Call Street, Tallahassee, FL, 32306-4301, USA

    Elizabeth L. Tighe, Richard K. Wagner & Christopher Schatschneider

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  1. Elizabeth L. Tighe
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Correspondence to Elizabeth L. Tighe.

Appendix: Establishing measurement invariance in multiple group causal indicator models

Appendix: Establishing measurement invariance in multiple group causal indicator models

Measurement invariance (MI) refers to scores on a latent construct having the same operational definition across different groups, time points, or methods of measurement administration (Meade & Lautenschlager, 2004). To establish MI across the structural components of our three independent groups (third, seventh, and tenth grade), we followed a six-step process and inspected and compared model fit indices at each step (Cheung & Rensvold, 2002; Kline, 2011). It is important to note that the steps proceed in hierarchical fashion such that subsequent levels impose additional equivalence constraints and therefore, indicate stronger MI.

Step one is to ensure that the same general model fits reasonably well at all grade levels. As demonstrated in our results section, a 4-factor MIMIC model provided adequate fit at each grade level. Inspecting the model fit indices and factor loadings of the observed measures onto the latent constructs also showed similar loadings across the three grades. Thus, we can explore building a multiple group MIMIC model.

Step two is to obtain a configural or baseline model (labeled M0 in Table 6) by fitting the 4-factor MIMIC model across the three grades without any invariances (or cross-group equality constraints). By default for a multiple group model, Mplus imposes cross-group equality constraints on factor loadings and intercepts. We are unable to relax the cross-group equality constraints on intercepts because this would lead to an unidentified model. However, we were able to run a baseline model that freed up the default cross-group constraints on factor loadings. This model is referred to as a baseline model (M0) because this model was compared with subsequent models in our later steps of testing for MI. Comparing Chi square values from the three models (one per grade) and M0 demonstrated that the baseline model is merely the sum of Chi square values and degrees of freedom from each of the individual models.

Table 6 Testing for multiple group measurement invariance of the structural components
Full size table

Step three is to test for construct-level metric invariance or equal factor loadings. To achieve this, we constrained all unstandardized factor loadings to be equal across the three groups (Table 6). A Chi square difference test between this model (M1) and the baseline model (M0) revealed a non-significant difference [M1–M0 χ2(12) = 15.39, p = .221]. Inspection of the model fit indices revealed an increase in RMSEA and TLI values in the constrained M1 model, indicating better model fit. Therefore, we established invariance of factor loadings, which indicates that the latent constructs (decoding, verbal reasoning, nonverbal reasoning, and working memory) are equivalent across grades.

Step four is to test for the equivalence of construct variances and covariances. To do this, we imposed constraints on factor loadings and factor variances and covariances (Table 6). A Chi square difference test between this model (M2) and the Step three model (M1) revealed no significant differences [M2–M1 χ2(8) = 1.37, p = .994]. The model fit indices show increased model fit for the RMSEA, TLI, and CFI values. Thus, cross-group equivalence of factor variances and covariances is supported.

Step five is to test for invariance of the correlations among our latent constructs (decoding, verbal reasoning, nonverbal reasoning, and working memory). To do this, cross-group equality constraints were imposed on factor loadings, factor covariances, and correlations (Table 6). A Chi square difference test between this model (M3) and the model in Step four (M2) revealed no significant differences [M3–M2 χ2(12) = 18.05, p = .114]. Inspection of the model fit indices revealed increased model fit for the RMSEA and TLI values. Thus, cross-group equivalence of predictor correlations is supported.

Step six is to test for invariance of the residual variances. To do this, cross-group equality constraints were enforced on factor loadings, factor variances and covariances, construct correlations, and residual variances (Table 6). A Chi square difference test between this model (M4) and the model in Step five (M3) revealed a significant difference [M4–M3(22) = 48.76, p < .001]. Inspection of the model fit indices demonstrated decreased model fit for the RMSEA, CFI, and TLI values. Although model fit was still in an overall acceptable range (Hu & Bentler, 1998), we could not declare cross-group equivalence of residual variances. Thus, we utilized model M3 as our baseline model to test for the stability of the estimates of the four causal indicator pathways.

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Tighe, E.L., Wagner, R.K. & Schatschneider, C. Applying a multiple group causal indicator modeling framework to the reading comprehension skills of third, seventh, and tenth grade students. Read Writ 28, 439–466 (2015). https://doi.org/10.1007/s11145-014-9532-1

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