Analysis of Variance
Chapter 11 Two-Way ANOVA
An analysis method for a quantitative outcome and two categorical explanatory variables.
If an experiment has a quantitative outcome and two categorical explanatory variables that are defined in such a way that each experimental unit (subject) can be exposed to any combination of one level of one explanatory variable and one level of the other explanatory variable, then the most common analysis method is two-way ANOVA. Because there are two different explanatory variables the effects on the outcome of a change in one variable may either not depend on the level of the other variable (additive model) or it may depend on the level of the other variable (interaction model). One common naming convention for a model incorporating a k-level categorical explanatory variable and an m-level categorical explanatory variable is “k by m ANOVA” or “k x m ANOVA”. ANOVA with more that two explanatory variables is often called multi-way ANOVA. If a quantitative explanatory variable is also included, that variable is usually called a covariate. In two-way ANOVA, the error model is the usual one of Normal distribution with equal variance for all subjects that share levels of both (all) of the explanatory variables. Again, we will call that common variance σ 2 . And we assume independent errors.
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CHAPTER 11. TWO-WAY ANOVA
Two-way (or multi-way) ANOVA is an appropriate analysis method for a study with a quantitative outcome and two (or more) categorical explanatory variables. The usual assumptions of Normality, equal variance, and independent errors apply.
The structural model for two-way ANOVA with interaction is that each combination of levels of the explanatory variables has its own population mean with no restrictions on the patterns. One common notation is to call the population mean of the outcome for subjects with level a of the first explanatory variable and level b of the second explanatory variable as µab . The interaction model
An analysis method for a quantitative outcome and two categorical explanatory variables.
If an experiment has a quantitative outcome and two categorical explanatory variables that are defined in such a way that each experimental unit (subject) can be exposed to any combination of one level of one explanatory variable and one level of the other explanatory variable, then the most common analysis method is two-way ANOVA. Because there are two different explanatory variables the effects on the outcome of a change in one variable may either not depend on the level of the other variable (additive model) or it may depend on the level of the other variable (interaction model). One common naming convention for a model incorporating a k-level categorical explanatory variable and an m-level categorical explanatory variable is “k by m ANOVA” or “k x m ANOVA”. ANOVA with more that two explanatory variables is often called multi-way ANOVA. If a quantitative explanatory variable is also included, that variable is usually called a covariate. In two-way ANOVA, the error model is the usual one of Normal distribution with equal variance for all subjects that share levels of both (all) of the explanatory variables. Again, we will call that common variance σ 2 . And we assume independent errors.
267
268
CHAPTER 11. TWO-WAY ANOVA
Two-way (or multi-way) ANOVA is an appropriate analysis method for a study with a quantitative outcome and two (or more) categorical explanatory variables. The usual assumptions of Normality, equal variance, and independent errors apply.
The structural model for two-way ANOVA with interaction is that each combination of levels of the explanatory variables has its own population mean with no restrictions on the patterns. One common notation is to call the population mean of the outcome for subjects with level a of the first explanatory variable and level b of the second explanatory variable as µab . The interaction model