# Interactions and Specific Hypotheses

• Gideon J. Mellenbergh
Chapter

## Abstract

A factor is an independent variable. A factorial design completely crosses two or more factors. It is an efficient design that simultaneously studies main effects of factors and their interactions. A distinction is made between factors that can be manipulated by researchers and factors that cannot be manipulated. Usually, the effects of manipulable factors are causally interpreted. Nonmanipulable factors are included for two different reasons, but they should not be dichotomized. First, to increase the precision of parameter estimates and the power of statistical tests. The nonmanipulable factor is included as a blocking variable in a randomized block design (if its values are known before participants are assigned to conditions) or as a covariate in the analysis of the data. Second, to study their relations with manipulable factors, but these relations should not be causally interpreted. The statistical analysis of factorial design data has to be tuned to the type of dependent variable (DV). Usually, ANOVA or ANCOVA are applied to (approximately) continuous DVs, but these methods make strong assumptions. Akritas et al.’s (J Am Stat Assoc 92:3375–3384, 1997) nonparametric method is more appropriate to analyze (approximately) continuous and ranked DVs. The preferred methods for dichotomous, nominal-categorical, and ordinal-categorical DVs are the logit, baseline-category, and cumulative logit models, respectively. Often, researchers apply omnibus statistical tests to factorial design data. These tests mainly fit into exploratory research. Confirmatory studies prespecify specific hypotheses. The proper methods to test these hypotheses are planned comparisons of conditions.

## Keywords

Baseline splitting Cumulative splitting Dichotomization Factorial design Interaction effect Linear contrast Logit model Main effect Manipulable and nonmanipulable factors Planned comparisons

## References

1. Agresti, A. (2002). Categorical data analysis (2nd ed.). Hoboken, NJ: Wiley.Google Scholar
2. Akritas, M. G., Arnold, S. F., & Brunner, E. (1997). Nonparametric hypotheses and rank statistics for unbalanced factorial designs. Journal of the American Statistical Association, 92, 3375–3384.Google Scholar
3. Allison, D. B., Gorman, B. S., & Primavera, L. H. (1993). Some of the most common questions asked of statistical consultants: Our favorite responses and recommended readings. Genetic, Social, and General Psychology Monographs, 119, 153–185.Google Scholar
4. Bracht, G. H., & Glass, G. V. (1968). The external validity of experiments. American Educational Research Journal, 5, 437–474.Google Scholar
5. Clinch, J. J., & Keselman, H. J. (1982). Parametric alternatives to the analysis of variance. Journal of Educational Statistics, 7, 207–214.Google Scholar
6. Cohen, J. (1983). The cost of dichotomization. Applied Psychological Measurement, 7, 249–253.Google Scholar
7. DeCoster, J., Iselin, A. M. R., & Gallucci, M. (2009). A conceptual and empirical examination of justifications for dichotomization. Psychological Methods, 14, 349–366.
8. Glass, G. V., Peckham, P. D., & Sanders, J. R. (1972). Consequences of failure to meet assumptions underlying the analysis of variance and analysis of covariance. Review of Educational Research, 42, 237–288.Google Scholar
9. Hochberg, Y. (1988). A sharper Bonferroni procedure for multiple tests of significance. Biometrika, 75, 800–802.Google Scholar
10. Humphreys, L. G., & Fleishman, A. (1974). A pseudo-orthogonal and other analysis of variance designs involving individual difference variables. Journal of Educational Psychology, 66, 464–472.Google Scholar
11. Jaccard, J., Becker, M. A., & Wood, G. (1984). Pairwise multiple comparison procedures. Psychological Bulletin, 96, 589–596.Google Scholar
12. Keppel, G., & Wickens, T. D. (2004). Design and analysis: A researcher’s handbook (4th ed.). Upper Saddle River, NJ: Pearson.Google Scholar
13. MacCallum, R. C., Zhang, S., Preacher, K. J., & Rucker, D. D. (2002). On the practice of dichotomization of quantitative variables. Psychological Methods, 7, 19–40.
14. Ramsey, P. H. (2002). Comparison of closed testing procedures for pairwise testing of means. Psychological Methods, 7, 504–523.
15. Toothaker, L. E., & Newman, D. (1994). Nonparametric competitors to the two-way ANOVA. Journal of Educational and Behavioral Statistics, 19, 237–273.Google Scholar
16. Wilcox, R. R. (2012). Introduction to robust estimation and testing (3rd ed.). Amsterdam, The Netherlands: Elsevier.Google Scholar