Comparing Cell Means Planned Comparisons & Post Hoc Tests
Questions • What is the main difference between planned comparisons and post hoc tests? • Generate numbers (like 0 1, -1 or 1 –1/2, -1/2) to create a contrast appropriate for a given problem. • How many independent comparisons can be made in a given design? • What is the difference between a per comparison and a familywise error rate? • How does Bonferroni deal with familywise error rate problems? • What is the studentized range statistic? How is it used?
Questions (2) • What is the difference between the Tukey HSD and the Newman-Keuls? • What are the considerations when choosing a post hoc test (what do you need to trade-off)? • Describe (make up) a concrete example where you would use planned comparisons instead of an overall F test. Explain why the planned comparison is the proper analysis. • Describe (make up) a concrete example where you would use a post hoc test. Explain why the post hoc test is needed (not the specific choice of post hoc test, but rather why post hoc test at all).
Planned vs. Post Hoc • Planned Comparisons or Contrasts • Use instead of overall F test. Planned before the study. • Post Hoc or Incidental tests. • Use after significant overall F test to investigate specific means. No specific plan before study.
Planned Comparisons (1) Population Comparison: Weights are real numbers not all zero. Sum of weights must equal zero. Sample Comparison:
Planned Comparison (2) (3 possible comparisons) (Data) (Summary Table)
Sampling Variance of Planned Comparisons The sample comparison is an unbiased estimate of the population comparison. The variance of the sampling distribution of the comparison: Sampling variance will be large when within cells variance is large, the weights are large, and the number of people in each cell is small. Estimated by: We substitute for
Significance Test df=N-J; 16-4=12=dfe. t(12) =-2.86, p < .05
Significance Test df=N-J=
Review • What is the main difference between planned comparisons and post hoc tests? • Suppose I do a blind orange juice taste test and discover that my means are: If my hypothesis is that Tropicana is better than all others, what are my contrast weights?
Independence of Planned Comparisons You can make several planned comparisons on the same data. Some of these comparisons are independent; some are dependent. We want them independent. Two comparisons from a normal population with equal sample sizes in each cell are independent if the sum of the products of weights is zero. With unequal sample sizes, it’s:
Independence (2) One and two are orthogonal; one and three are not. There are J-1 orthogonal comparisons. Use only what you need.
Choosing Comparisons Usually done on basis of theory. But there are methods to generate all possible orthogonal comparisons.
Error Rates • With 1 test, we set alpha = Type I error rate. • With multiple tests, original (nominal) alpha is called the per comparison error rate ( ). • With comparisons, we have a family of tests on the same data. Want to know the probability of at least 1 Type I error in the family of tests. Such a probability is called familywise error rate ( ). • For independent tests, • E.g., 10 tests:
Bonferroni Tests • Familywise error depends on the number of tests (K) and the nominal alpha, . • Bonferroni’s solution is to set: • Suppose we want FW error to be .05 and we will have 4 comparisons. Then Where is an aspiration level. We use the adjusted alpha (.0125) for each of the 4 tests.
Bonferroni Test (2) • Use the adjusted alpha (e.g., .0125) for each comparison. • Look at the p value on the printout (use .0125 instead of .05). • Use a statistical function (e.g., Excel, SAS) if you want to find the critical value. • E.g., Excel function TINV says with p=.0125 and df=12, t is 2.93.
Review • How many independent comparisons can be made in a given design? • What is the difference between a per comparison and a familywise error rate? • How does Bonferroni deal with familywise error rate problems?
Post Hoc Tests • Given a significant F, where are the mean differences? • Often do not have planned comparisons. • Usually compare pairs of means. • There are many methods of post hoc (after the fact) tests.
Scheffé • Can use for any contrast. Follows same calculations, but uses different critical values. • Instead of comparing the test statistic to a critical value of t, use: Where the F comes from the overall F test (J-1 and N-Jdf).
Scheffé (2) (Data from earlier problem.) The comparison is not significant because |-2.86|<3.24.
Paired comparisons Newman Keuls and Tukey HSD are two (of many) choices. Both depend on q, the studentized range statistic. Suppose we have J independent sample means and we find the largest and the smallest. MSerror comes from the ANOVA we did to get the J means. The n refers to sample size per cell. If two cells are unequal, use 2n1n2/(n1+n2). The sampling distribution of q depends on k, the number of means covered by the range (max-min), and on v, the degrees of freedom for MSerror.
Tukey HSD HSD = honestly significant difference. For HSD, use k = J, the number of groups in the study. Choose alpha, and find the df for error. Look up the value qα. Then find the value: Compare HSD to the absolute value of the difference between all pairs of means. Any difference larger than HSD is significant.
HSD 2 K = 5 groups; n=12 per group, v has 55 df. Tabled value of q with alpha =.05 is 3.98.
Newman-Keuls Layer refers to how many means apart. Layer 4 Layer 3 Layer 2 Layer 1 Same as HSD except the value of q changes with layers. For layer k-1 (here 4), use HSD. For each layer down, subtract 1 from the value of k for the tabled value of q.
Comparing Post Hoc Tests The Newman-Keuls found 3 significant differences in our example. The HSD found 2 differences. If we had used the Bonferroni approach,we would have found an interval of 15.91 required for significance (and therefore the same two significant as HSD). Thus, power descends from the Newman-Keuls to the HSD to the Bonferroni. The type I error rates go just the opposite, the lowest to Bonferroni, then HSD and finally Newman-Keuls. Do you want to be liberal or conservative in your choice of tests? Type I error vs Power.
Review • What is the studentized range statistic? How is it used? • What is the difference between the Tukey HSD and the Newman-Keuls? • What are the considerations when choosing a post hoc test (what do you need to trade-off)? • Describe (make up) a concrete example where you would use planned comparisons instead of an overall F test. Explain why the planned comparison is the proper analysis. • Describe (make up) a concrete example where you would use a post hoc test. Explain why the post hoc test is needed (not the specific choice of post hoc test, but rather why post hoc test at all).