Two-Way ANOVA：双向方差分析

Various Estimates in Two-Way ANOVA
Estimates for the main effects of the two independent variables The “between estimate,” or between mean square for the row variable (for example, ethnicity) is based on the deviation of each row mean of the DV (mean for Hispanic, mean for African-American) from the overall or grand mean of the DV Similarly, the “between estimate,” or between mean square for the column variable, gender, is based on the deviation of each column mean of the DV (mean for females, mean for males) from the overall or grand mean of the DV Each of these estimates is calculated as if the other factor did not exist
Estimates in Anova
The estimate or mean square for the interaction effect of gender and ethnicity is based on the deviation of the cell means (mean on the DV from each of these combinations: Hispanic/female; Hispanic/male; African-American/female; African-American/male) from the grand mean, after differences due to the two factors (gender, ethnicity) acting independently and the error variance (individual variability within the cells) have been accounted for or “removed”


The third will tell you if the interaction of the two independent variables has a significant effect on the DV
What is the combined effect of gender and ethnicity on income that could not be detected by considering the two IVs separately? (e.g., what is the interaction of gender and ethnicity with respect to income; is the effect of gender different for different categories of ethnicity?

The Three Effects in a Two-Way ANOVA
Let’s consider an example: What is the impact of gender, ethnicity, and their interaction on annual income?
One of these will tell you if your first independent variable has a significant main effect on the DV A second will tell you if your second independent variable has a significant main effect on the DV
Two-Way ANOVA
Two-way Analysis of Variance
Two-way ANOVA is applied to a situation in which you have two independent nominal-level variables and one interval or better dependent variable Each of the independent variables may have any number of levels or conditions (e.g., Treatment 1, Treatment 2, Treatment 3…… No Treatment) In a two-way ANOVA you will obtain 3 F ratios One of these will tell you if your first independent variable has a significant main effect on the DV A second will tell you if your second independent variable has a significant main effect on the DV The third will tell you if the interaction of the two independent variables has a significant effect on the DV, that is, if the impact of one IV depends on the level of the other
More Conventions to Know
An independent variable is called a factor, and its separate impact on the DV is called a main effect The term between effect or between-groups effect in ANOVA language refers to the differences in the DV between or among levels of a factor and is the same thing as the variable’s main effect (e.g., differences in the DV between men and women, or between African Americans and Hispanics) The term within effect or within-groups effect in ANOVA language refers to the differences in the DV within a level of the factor (e.g., differences among the individuals within the “female” category or the “African-American” category
The Null Hypotheses in a Two-Way ANOVA
The null hypotheses in a two-way ANOVA are these:
The population means for the DV are equal across levels of the first factor The population means for the DV are equal across levels of the second factor The effects of the first and second factors on the DV are independent of one another
For convenience purposes, one factor or IV is usually called the “column” variable and the other the “row” variable When describing your design in the opening statement of a Method section you will refer to it as a 2 X 2 design, or a 3 X 3 design, where the first number refers to the number of levels of the row variable and the second number refers to the number of levels of the column variable. When there are more than two factors involved, in a multiple factor ANOVA, you will see 4 X 2 X 4, which means that there are three factors in the design, the first with four, the second with two, and the third with four levels of the factor. The order is usually arbitrary
What is the main effect of ethnicity on income, regardless of (across all levels of) gender What is the main effect of gender on income, regardless of (across all levels of) ethnicity?
An Interaction Effect in Two-Way Analysis of Variance
What is the impact of gender and ethnicity on annual salary, and how do they interact? In this example, there may not be much of a main effect either for gender or ethicnity, but there may be an interaction effect: for example, are females who are Hispanic paid more than males who are Hispanic, while females who are African-American are paid less than males who are African-American?
Female Hispanic Salary Average is High Salary Averagerage is Low Salary Average is High
AfricanAmerican
Some Conventions to Know