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Comparing Cell Means Planned Comparisons & Post Hoc Tests

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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?

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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).

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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. ControlComp Tutor Comp Tutor+ Lab Comp Tutor + lab + quiz

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Planned Comparisons (1) Population Comparison: Weights are real numbers not all zero. Sum of weights must equal zero. Sample Comparison:

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Planned Comparison (2) A1A2A3A4 22262821 15273121 17242726 18232620 18252822 SourceSSdfMSMS F Cells (A1-A4) 21937312.17 Error72126 Total29115 (Data) (Summary Table) ComparisonA1A2A3A4 11/2 -1/2 2100 3001 (3 possible comparisons)

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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

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Significance Test A1A2A3A4 22262821 15273121 17242726 18232620 18252822 SourceSSdfMSMS F Cells (A1-A4) 21937312.17 Error72126 Total29115 df=N-J; 16-4=12=dfe. t(12) =-2.86, p <.05

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Significance Test A1A2A3A4 22262821 15273121 17242726 18232620 18252822 SourceSSdfMSMS F Cells (A1-A4) 21937312.17 Error72126 Total29115 df=N-J=

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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: TropicanaFlorida Fresh Pulpmaster 7.35.56.4 If my hypothesis is that Tropicana is better than all others, what are my contrast weights?

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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:

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Independence (2) ComparisonA1A2A3A4 1-1/31 2-1/20 1 31/2 -1/2 One and two are orthogonal; one and three are not. There are J-1 orthogonal comparisons. Use only what you need.

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Choosing Comparisons Usually done on basis of theory. But there are methods to generate all possible orthogonal comparisons. Group 12345 Comparison 14 203 3002 40001

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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:

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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.

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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.

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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?

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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.

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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-J df).

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Scheffé (2) SourceSSdfMSMS F Cells (A1-A4) 21937312.17 Error72126 Total29115 (Data from earlier problem.) The comparison is not significant because |-2.86|<3.24.

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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.

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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 q α. 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.

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HSD 2 Grp ->12345 M ->6382807770 SourceSSdfMSFp Grps2942.44725.64.13<.05 Error9801.055178.2 K = 5 groups; n=12 per group, v has 55 df. Tabled value of q with alpha =.05 is 3.98. Group15432 1630714 17*19* 570071012 477035 38002 2820

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Newman-Keuls Group12345 1 630714*17*19* 2 70071012 3 77035 4 8002 5 820 Layer 1 Layer 2 Layer 3 Layer 4 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. Layer refers to how many means apart.

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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.

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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).

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