**Leech, N. L., Barrett, K. C., & Morgan, G. A. (2015). Factorial ANOVA and ANCOVA (Chapter 9). In**

*IBM SPSS for Intermediate Statistics: Use and Interpretation*(5th edition, pp. 188–212). New York: Routledge.[“Factorial ANOVA and ANCOVA” (p 188) …]

“… these inferential statistics have two or more independent variables and one scale (normally distributed) dependent variable.

Factorial ANOVAis used when there is a small number of categorical independent variables (usually two or three), and each of these variables has a small number of levels or categories (usually two to four).” (p 188)

“ANCOVA typically is used to adjust or control for differences between the groups based on another, typically interval-level variable, called the covariate. For example, imagine that we found that boys and girls differ on math achievement. However, this could be due to the fact that boys take more math courses in high school. ANCOVA allows us to adjust the math achievement scores based on the relationship between number of math courses taken and math achievement. We can then determine if boys and girls still have different math achievement scores after making the adjustment. ANCOVA can also be used if one wants to use one or more discrete or nominal variables and one or two continuous variables to predict differences in one dependent variable.” (p 188)

[“Assumptions of Factorial ANOVA and ANCOVA” (p 188) …]

“The assumptions for factorial ANOVA and ANCOVA include that the observations are independent, the variances of the groups are equal (homogeneity of variances), and the dependent variable is normally distributed for each group.” (p 188)

[“Additional Assumptions for ANCOVA” (p 188) …]

“For ANCOVA there is a fourth assumption that there is a linear relationship between the covariates and the dependent variable. … one of the most important assumptions, and it can be checked with an _F_ test on the interaction of the independent variables with the covariate.” (p 188)

[“Problem 9.1: Factorial (Two-Way) ANOVA” (p 189) …]

“We would use a _t_ test or one-way ANOVA to examine differences between two or more groups (comprising the levels of _one_ independent variable or factor) on a continuous dependent variable. These designs, in which there is only one independent variable and it is a discrete or categorical variable, are called single-factor designs. In this problem, we will compare groups formed by combining _two_ independent variables. The appropriate statistic for this type of problem is called a

two-factor,two-way, orfactorial ANOVA.” (p 189)

[Adapted from SPSS output for Problem 9.1 (p 191) …]

[Notes to above table …]

“This [the Sig. value] indicates that the assumption of homogeneity of variances has been violated. Because Levene’ s test is significant, we know that the variances are significantly different. … this violation should be considered when deciding which post hoc test to use.” (p 191)

[“Interpretation of Output 9.1” (p 192) …]

“… the interaction is statistically significant, _F_(2,67) = 3.97, _p_ = .024. … When the interaction is statistically significant, you should analyze the

‘simple effects’(differences between means for one variable at each particular level of the other variable).” (p 193)

“

Eta, the correlation ratio, is used when the independent variable is _nominal_ and the dependent variable (_math achievement_ in this problem) is _scale_. Eta is an indicator of the proportion of variance that is due to between-groups differences.Partial eta squaredis the ratio of the variance associated with a particular between-groups ‘effect” to the sum of that same number (variance associated with that ‘effect’) and error variance.” (p 193)

“Effect size measures, which are more independent of sample size, tell how strong the relationship is and, thus, give you some indication of its importance.” (p 194)

[“Problem 9.2: Post Hoc Analyses of a Significant Interaction” (p 194) …]

[Writing about assumptions …]

“The assumptions of independent observations was met, and assumptions of homogeneity of variances and normal distributions of the dependent variable for each group were checked. The assumption of homogeneity of variances was violated; thus, results should be viewed with caution. The assumption of normal distributions of the dependent variable for each group was not violated.” (p 200)

[“9.2c. Computation of Contrasts” (p 201)]

[“Problem 9.3: Analyses of Covariance (ANCOVA)” (p 205) …]

“ANCOVA is an extension of ANOVA that typically provides a way of statistically controlling for the effects of continuous or scale variables that you are concerned about but that are not the focal point or independent variable(s) in the study. These continuous variables are called

covariates(or sometimes, control variables). Covariates usually are variables that may cause you to draw incorrect inferences about the prediction of the dependent variable from the independent variable, if not controlled (then are possible confounds). It is also possible to use ANCOVA when you are interested in examining a combination of a categorical (nominal) variable and a continuous (scale) variable as predictors of the dependent variable.” (p 205)

[“Interpretation of Output 9.3” (p 210) …]

“The ANCOVA (

Tests of Between-Subject Effects) table is interpreted in much the same way as ANOVA tables in earlier outputs. The covariate (_mathcrs_) has a highly significant “effect” on math achievement, as should be the case. However, the “effect” of gender is no longer significant, _F_(1, 72) = .36, _p_ = .55. You can see from theEstimated Marginal Meanstable that the statistically adjusted math achievement means for boys and girls are quite similar once differences in the number of _math courses taken_ were accounted for. This suggests that the fact that males took more math courses may have been the reason for their higher math achievement.” (p 210)

“The

Observed Powerfor _math courses taken_ was 1.0, which indicates extremely high power. For gender the observed power was .091. This is very low power. The effect size for gender is also very small (partial eta= .071), so it may be that we have overlooked an important gender difference because of this low power. It is possible that once the strong relation between the number of math courses taken and math achievement was taken into account, there was no longer an important “effect” of gender on math achievement.” (p 210)

[“Example of How to Write About Problems 9.3” (p 211)]