Analysis of Variance
Analysis of Variance
Lecture 11 April 26th, 2011
A. Introduction
When you have more than two groups, a t-test (or the nonparametric equivalent) is no longer applicable. Instead, we use a technique called analysis of variance. This chapter covers analysis of variance designs with one or more independent variables, as well as more advanced topics such as interpreting significant interactions, and unbalanced designs.
B. One-Way Analysis of Variance
The method used today for comparisons of three or more groups is called analysis of variance (ANOVA). This method has the advantage of testing whether there are any differences between the groups with a single probability associated with the test. The hypothesis tested is that all groups have the same mean. Before we present an example, notice that there are several assumptions that should be met before an analysis of variance is used.
Essentially, we must have independence between groups (unless a repeated measures design is used); the sampling distributions of sample means must be normally distributed; and the groups should come from populations with equal variances (called homogeneity of variance).
Example:
15 Subjects in three treatment groups X,Y and Z. X Y Z
700 480 500
850 460 550
820 500 480
640 570 600
920 580 610 The null hypothesis is that the mean(X)=mean(Y)=mean(Z). The alternative hypothesis is that the means are not all equal. How do we know if the means obtained are different because of difference in the reading programs(X,Y,Z) or because of random sampling error? By chance, the five subjects we choose for group X might be faster readers than those chosen for groups Y and Z.
We might now ask the question, “What causes scores to vary from the grand mean?” In this example, there are two possible sources of variation, the first source is the training method (X,Y or Z). The second source of variation is due to the
Lecture 11 April 26th, 2011
A. Introduction
When you have more than two groups, a t-test (or the nonparametric equivalent) is no longer applicable. Instead, we use a technique called analysis of variance. This chapter covers analysis of variance designs with one or more independent variables, as well as more advanced topics such as interpreting significant interactions, and unbalanced designs.
B. One-Way Analysis of Variance
The method used today for comparisons of three or more groups is called analysis of variance (ANOVA). This method has the advantage of testing whether there are any differences between the groups with a single probability associated with the test. The hypothesis tested is that all groups have the same mean. Before we present an example, notice that there are several assumptions that should be met before an analysis of variance is used.
Essentially, we must have independence between groups (unless a repeated measures design is used); the sampling distributions of sample means must be normally distributed; and the groups should come from populations with equal variances (called homogeneity of variance).
Example:
15 Subjects in three treatment groups X,Y and Z. X Y Z
700 480 500
850 460 550
820 500 480
640 570 600
920 580 610 The null hypothesis is that the mean(X)=mean(Y)=mean(Z). The alternative hypothesis is that the means are not all equal. How do we know if the means obtained are different because of difference in the reading programs(X,Y,Z) or because of random sampling error? By chance, the five subjects we choose for group X might be faster readers than those chosen for groups Y and Z.
We might now ask the question, “What causes scores to vary from the grand mean?” In this example, there are two possible sources of variation, the first source is the training method (X,Y or Z). The second source of variation is due to the