If we have more than four groups, the number of pairwise comparisons we will want to look at will only increase even more. The following table illustrates how many pairwise comparisons are associated with each number of groups along with the family-wise error rate:. Notice that the family-wise error rate increases rapidly as the number of groups and consequently the number of pairwise comparisons increases.
This means we would have serious doubts about our results if we were to make this many pairwise comparisons, knowing that our family-wise error rate was so high.
Fortunately, post hoc tests provide us with a way to make multiple comparisons between groups while controlling the family-wise error rate. This means we have sufficient evidence to reject the null hypothesis that all of the group means are equal. Next, we can use a post hoc test to find which group means are different from each other. We will walk through examples of the following post hoc tests:.
R gives us two metrics to compare each pairwise difference:. Both the confidence interval and the p-value will lead to the same conclusion. In particular, we know that the difference is positive, since the lower bound of the confidence interval is greater than zero. Likewise, the p-value for the mean difference between group C and group A is 0. If the interval contains zero, then we know that the difference in group means is not statistically significant. In the example above, the differences for B-A and C-B are not statistically significant, but the differences for the other four pairwise comparisons are statistically significant.
This test provides a grid of p-values for each pairwise comparison. For example, the p-value for the difference between the group A and group B mean is 0. The p-value for this difference was. For example, using the code below we compare the group means of B, C, and D all to that of group A. Post hoc tests do a great job of controlling the family-wise error rate, but the tradeoff is that they reduce the statistical power of the comparisons.
This is because the only way to lower the family-wise error rate is to use a lower significance level for all of the individual comparisons. However, you should only run one post hoc test — do not run multiple post hoc tests. If your data met the assumption of homogeneity of variances, use Tukey's honestly significant difference HSD post hoc test. If your data did not meet the homogeneity of variances assumption, you should consider running the Games Howell post hoc test. First off, it is not essential that you present your results in a graphical form.
However, it can add a lot of clarity to your results. There are a few key points to producing a good graph. Firstly, you need to present error bars for each group mean. It is customary to use the standard deviation of each group, but standard error and confidence limits are also used in the literature.
A confidence interval not including zero means that a zero difference between these means in the population is unlikely. Obviously, , and result in the same conclusions. However, the tables we created don't come even close to APA standards. Honestly, I'm not sure how -or even if - it could be created from the menu but you can hopefully reuse it after just replacing the 2 variable names. First off, the capitals in this table A, B and so on indicate which means differ. SPSS also flags standard deviations and sample sizes.
This is utter stupidity because these are not compared. They always have the same flags as the means. So just ignore everything except the actual means here. Understanding this table starts with carefully reading its footnotes. Next, each statistically significant difference is indicated only once in this table. As indicated before, 4 means yield 6 unique pairs of means. Altogether, the table has 5 significance markers A, B and so on. After some puzzling these turn out to be homeopathic versus placebo.
So that's it for now. If you have any suggestions, please let me know by leaving a comment below. I want to thank for your perfect page! Am s statistics student and I am learning a lot of your page. THank you!
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