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When fitting longitudinal CFA-based models (say one latent factor measured twice using the same indicators), it is customary to correlate the residual variances at time 1 with the residual variances from the same model at time 2. I often need a citation for this practice so here we go:Â
“Measurement residual variance contains variance due to measurement error as well as variance unique to an indicator but not common to the other indicators of the factor. Because measurement error is random, by definition, the measurement error component of the variance of the measurement residual cannot be correlated with other factors. The Specific variance component on the other hand... may have auto-correlation over time”
“if not taken into account, the stability of the model may be overestimated...... Although the inclusion of correlated measurement residuals.. could be decided on the basis of significance tests, they are generally included in longitudinal models a priori (Mheaton, Muthen, Alwin, & Summers, 1977). Except for the cost of the degrees of freedom, there is typically no harm in including these estimates.Â
-Â Newsom, J. T. (2015). Longitudinal structural equation modeling: A comprehensive introduction. Routledge. LINK
BIC is a model fit statistic where a smaller number (closer to 0) is considered to be a better fitting model. When comparing two models, and the models don’t have to be nested, you can calculate the difference in the BIC values to get a quick idea of how much better one fits than the other. Raftery (1995) put forth some guidelines for model evaluation. The rule of thumb table is below:Â
Raftery, A. E. (1995). Bayesian model selection in social research. Sociological methodology, 25, 111-164.
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When comparing the fit of two nested models, you can use the Chi Square difference test, but it’s very susceptible to inflation when sample sizes are large. Hancock & Mueller (2013) suggest instead of relying on ONE fit index, use several and select the best model after looking at all of them. Bollen and Long (1983) pointed out, “The test statistics and fit indices are very beneficial, but they are no replacement for sound judgment and substantive expertise”
Alternate criteria include:
RMSEA: Improve (closer to 0) by less than .015 considered not important. (Chen, 2007)
CFI: Improve (closer to 1.0)Â by less than 0.01Â considered not important. (Cheung and Rensvold, 2002)
Bollen, K. A., & Long, J. S. (1993). Testing structural equation models (Vol. 154). Sage.Â
Chen, F. F. (2007). Sensitivity of goodness of fit indexes to lack of measurement invariance. Structural equation modeling, 14(3), 464-504.
Cheung, G. W., & Rensvold, R. B. (2002). Evaluating goodness-of-fit indexes for testing measurement invariance. Structural equation modeling, 9(2), 233-255.Â
Hancock, G. R., & Mueller, R. O. (Eds.). (2013).Structural equation modeling: A second course. Iap.
I found a fantastic website made by MRC Cognition and Brain Sciences Unit that includes general rules of thumb for various effect sizes at to what constitutes a small, medium, or large effect. They also provide the citations where those came from. I have some of their text below. See here for the full page.Â
The scales of magnitude are taken from Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates (see also here). The scales of magnitude for partial ω2 are taken from Table 2.2 of Murphy and Myors (2004).There is also a table of effect size magnitudes at the back of Kotrlik JW and Williams HA (2003) here. An overview of commonly used effect sizes in psychology is given by Vacha-Haase and Thompson (2004).
Kraemer and Thiemann (1987, p.54 and 55) use the same effect size values (which they call delta) for both intra-class correlations and Pearson correlations. This implies the below rules of thumb from Cohen (1988) for magnitudes of effect sizes for Pearson correlations could also be used for intra-class correlations. It should be noted, however, that the intra-class correlation is computed from a repeated measures ANOVA whose usual effect size (given below) is partial eta-squared. In addition, Shrout and Fleiss (1979) discuss different types of intra-class correlation coefficient and how their magnitudes can differ.
The general rules of thumb given by Cohen are for eta-squared, which uses the total sum of squares in the denominator, but these would arguably apply more to partial eta-squared than to eta-squared. This is because partial eta-squared in factorial ANOVA arguably more closely approximates what eta-squared would have been for the factor had it been a one-way ANOVA and it is presumably a one-way ANOVA which gave rise to Cohen's rules of thumb.
CLICK HERE FOR Effect Sizes Table Rules of Thumb Table
References
Cohen, J. (1988) Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.
Field, A. (2013) Discovering statistics using IBM SPSS Statistics. Fourth Edition. Sage:London.
Green, SB, Salkind, NJ & Akey, TM (1997). Using SPSS for Windows:Analyzing and understanding data. Upper Saddle River, NJ:
Haddock, CK, Rinkdskopf, D. & Shadish, C. (1998) Using odds ratios as effect sizes for meta-analysis of dichotomous data: A primer on methods and issues. Psychological Methods 3 339-353.
Howell, DC (2013). Statistical methods for psychology. 8th Edition. International Edition. Wadsworth:Belmont,CA.
Kotrlik, JW and Williams, HA (2003) The incorporation of effect size in information technology, learning, and performance research. Information Techology, Learning, and Performance Journal 21(1) 1-7.
Kraemer HC and Thiemann S (1987) How many subjects? Statistical power analysis in research. Sage:London. In CBSU library.
Murphy KR and Myors B (2004) Statistical power analysis: A Simple and General Model for Traditional and Modern Hypothesis Tests (2nd ed.). Lawrence Erlbaum, Mahwah NJ. (Alternative rule s of thumb for effect sizes to those from Cohen are given here in Table 2.2).
Preacher, KJ and Kelley, K (2011) Effect size measures for mediation models: quantitative strategies for communicating indirect effects. Psychological Methods 16(2) 93-115.
Shrout, PE and Fleiss, JL (1979) Intraclass Correlations: Uses in Assessing Rater Reliability, Psychological Bulletin, 86 (2) 420-428. (A good primer showing how anova output can be used to compute ICCs).
Tabachnick, BG and Fidell, LS (2007) Using multivariate statistics. Fifth Edition. Pearson Education:London.
Vacha-Haase, T and Thompson, B (2004) How to estimate and interpret various effect ssizes. Journal of Counseling Psychology 51(4) 473-481.