ANOVA Example

We look at hypothetical data about the effect of drug treatment on the amount of time (in seconds) a stimulus is endured. We do an ANOVA following the formal hypothesis testing steps. Note that the books steps are augmented here to reflect current thinking about using visualizations to investigate the assumptions underlying the analysis.

1. State the Hypotheses:
   The hypotheses, for ANOVA, are:
   
   
2. Set the Decision Criterion
   We arbitrarily set
   

3. Gather the Data:
   The data are obtained from 60 subjects, 20 in each of 3 different experimental conditions. The conditions are a Placebo condition, and two different drug conditions. The independent (classification) variable is the experimental condition (Placebo, DrugA, DrugB). The dependent variable is the time the stimulus is endured.

   Here are the data as shown in ViSta's data report:

   

   and here they are in ViSta's data object:

   
   

   The data may be gotten from the ViSta Data Applet. Then, you can do the analysis that is shown below yourself.

4. Visualize the Data
   We visualize the data and the model in order to see if the assumptions underlying the independent-measures F-test are met. The assumptions are:
   1. The observations within each sample must be independent (this assumption is satisfied by the nature of the experimental design).
   2. The populations from which the samples are selected must be normal (the data and model visualizations can inform us about this).
   3. The populations from which the samples are selected must have equal variance (the data and model visualizations can inform us about this also). This is called homogeneity of variance.

   The data visualization is shown below. The boxplot shows that there is somewhat more variance in the "DrugA" group, and that there is an outlier in the "DrugB" group. The Q plots (only the "DrugB" Q-Plot is shown here) and the Q-Q plot show that the data are normal, except for the outlier in the "DrugB" group.
   

5. Evaluate the Null Hypothesis
   We use ViSta to calculate the observed F-ratio, and the observed probability level. The report produced by ViSta is shown below. The information we want is near the bottom:
   
   We note that F=4.37 and p=.01721. Since the observed p < .05, we reject the null hypothesis and conclude that it is not the case that all group means are the same. That is, at least one group mean is different than the others.

   Here is the F distribution for df=2,57 (3 groups, 60 observations). Note the location of the the observed F=4.37.
   

6. Visualize the Model
   Finally, we also visualize the ANOVA model to see if the assumptions underlying the independent-measures F-test are met. The boxplots are the same as those for the data. The partial regression plot shows that the model is significant at the .05 level of significance, since the curved lines cross the horizontal line. The residual plot shows the outline in the "DrugB" group, and shows that the "DrugA" group is not as well fit by the ANOVA model as the other groups. Here is the model visualization:
   

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