In the case where the explanatory variable represents more than two groups, a significant ANOVA does not tell us which groups are different from the others. To determine which groups are different from the others, we would need to perform a post hoc test. A post hoc test conducts post hoc paired comparisons. Post hoc means after the fact, and these post hoc paired comparisons must be conducted in a particular way in order to prevent excessive Type 1 error. Type 1 error, as you'll recall, occurs when you make an incorrect decision about the null hypothesis, that is you reject the null hypothesis when the null hypothesis is true. Why can't we just perform multiple ANOVAs? That is, why can't we just subset our observations and take two at a time? That is, compare white versus black, white versus American Indian, Alaskan Native etc, etc, until all paired comparisons have been made. As you know, we accept significance and reject the null hypothesis at p ≤ 0.05. A 5% chance that we're wrong and have committed a Type 1 error. There's actually a 5% chance of making a Type 1 error for each analysis of variance that we conduct on this question. Therefore, performing multiple tests means that our overall chance of committing Type 1 error could be far greater than 5%. Here's how it works out. Using the formula displayed under this table, you can see that while one test has a Type 1 Error Rate of 0.05, by the time we've conducted 10 tests on this question, our chance of rejecting the null hypothesis, when the null hypothesis is true, is up to 40%. This increase in the Type 1 error rate is called the family-wise error rate and is the error rate for pair comparison. >> Post hoc tests are designed to evaluate the difference between pairs of means while protecting against inflation of Type 1 errors. And there are a lot of post talk tests to choose from when it comes to analysis of variance. There's the Sidak, and the Holm T test, and Fisher's Least Significant Difference test, Tukey's Honestly Significant Difference test, the Scheffee test, the Newman-Keuls test, Dunnett's Multiple Comparison test, the Duncan Multiple Range test, and the Bonferroni Procedure. It's enough to make your head swim. >> While there are certainly differences in how conservative each test is in terms of protecting against Type 1 error, in many cases it's far less important which post hoc test you conduct and far more important that you do conduct one. >> In order to conduct post hoc paired comparisons in the context of Mironova. Examining the association between ethnicity and number of cigarettes smoked per month among young adult smokers, I'm going to use the Duncan Test. To do this, all I need to do is add a slash and the word Duncan at the end of my MEANS statement. And then save and run my program. >> Here are the Proc ANOVA results with Duncan Post Hoc Test. The top of our results looks the same as in our original test. The F value or F statistic is 24.4 and it's significant at the P < 0.0001 level. However, if we scroll down, we see a new table displaying the results of the paired comparisons conducted by the Duncan Multiple Range Test. Here's how the table should be interpreted. Basically, means with the same capitol letter next to them are not significantly different. So you can see that ethnic group 1 and 3 are not significantly different because they both have A's. Groups 2, 4, and 5 are not significantly different because each has a C. Groups 3, 2, and 4, again are not significantly different from one another, each with a B next to the group number. So where are the significant differences? Group 1, which indicates White ethnicity, smoked significantly more cigarettes per month than ethnic groups 2, 4, and 5. That is Black, Asian, and Hispanic or Latino. Group 3, American Indian and Alaskan Native, smoked significantly more per month than group 5, which is Hispanic or Latino. Notice that some of the means have more than one letter next to them. So you need to be careful, and follow the rule that means that share even one letter in common are not significantly different from one another.