1. Lecture 13: Tues., Feb. 24 Comparisons Among Several Groups – Introduction (Case Study 5.1.1) Comparing Any Two of the Several Means (Chapter 5.2) The One-Way Analysis of Variance F-test (Chapter 5.3) Robustness to Assumptions (5.5.1) Thursday: Multiple Comparisons (6.3-6.4)

2. Comparing Several Groups Chapter 5 and 6: Compare the means of I groups (I>=2). Examples: –Compare the effect of three different teaching methods on test scores. –Compare the effect of four different therapies on how long a cancer patient lives. –Compare the effect of using different amounts of fertilizer on the yield of a crop. –Compare the amount of time that ten different tire brands last. As in Ch. 1-4, studies can either seek to compare treatments (causal inferences) or population means

3. Case Study 5.1.1 Female mice randomly assigned to one of six treatment groups –NP: Mice in this group ate as much as they pleased of nonpurified, standard diet –N/N85: Fed normally both before and after weaning. After weaning, ration controlled at 85 kcal/wk –N/R50: Fed normal diet before weaning and reduced calorie diet of 50 kcal/wk after weaning –R/R50: Fed reduced calorie diet of 50 kcal/wk both before and after weaning –N/R50 lopro: Fed normal diet before weaning, a restricted diet of 50 kcal/wk after weaning and dietary protein content decreased with advancing age –N/R40: Fed normally before weaning and given severely reduced diet of 40 kcal/wk after weaning.

4. Questions of Interest Specific comparisons of treatments, see Display 5.3 (section 5.2) Are all of the treatments the same? (F-test, Section 5.3). Multiple comparisons (Chapter 6) Terminology for several group problem: one-way classification problem, one-way layout Setup in JMP: One column for response (e.g., lifetime), a second column for group label.

5. Ideal Model for Several Samples Ideal model: –The populations 1,2,…,I have normal distributions with means –Each population has the same standard deviation –Observations within each sample are independent –Observations in any one sample are independent of observations in other samples Sample sizes. Total sample size

6. Randomized Experiments Terminology of samples from multiple populations used but methods also apply to data from randomized experiments in which response of Y 1 on treatment 1 would produce response of on treatment 2 and on treatment 3, etc Can think of as equivalent to and as equivalent to (additive treatment effect of treatment 3 compared to treatment 2) Phrase concluding statements in terms of treatment effects or population means depending on type of study.

7. Comparing any two of several means Compare mean of mice on N/R50 diet to mean of N/N85 diet, (i.e., what is the additive treatment effect of N/N85 diet?) What’s different from two group problem? We have additional information about the variability in the populations from the additional groups. We use this information in constructing a more accurate estimate of the population variance.

8. Comparing any two means Comparison of and Use usual t-test but estimate from weighted average of sample standard deviations in all groups, use df=n-I. 95% CI for : (Note: Multiplier for degree of confidence equals, the multiplier if there were only two groups) See handout for implementation in JMP

9. Note about CIs and hyp. tests Suppose we form a 95% confidence interval for a parameter, e.g., The 95% confidence interval will contain 0 if and only if the p-value of the two sided test that the parameter equals 0 (e.g., vs. ) has p-value >=0.05. In other words the test will only give a “statistically significant” result if the confidence interval does not contain 0.

10. One-Way ANOVA F-test Basic Question: Is there any difference between any of the means? H 0 : H A : At least two of the means and are not equal Could do t-tests of all pairs of means but this has difficulties (Chapter 6 – multiple comparisons) and is not the best test. Test statistic: Analysis of Variance F-test.

11. ANOVA F-test in JMP Convincing evidence that the means (treatment effects) are not all the same

12. Rationale behind the test statistic If the null hypothesis is true, we would expect all the sample means to be close to one another (and as a result, close to the grand mean). If the alternative hypothesis is true, at least some of the sample means would differ. Thus, we measure variability between sample means. Large variability within the samples weakens the “ability” of the sample means to represent their corresponding population means. Therefore, even though sample means may markedly differ from one another, variability between sample means must be judged relative to the “within samples variability”.

13. F Test Statistic Notation: = jth observation in ith group, = sample mean of ith group, = grand mean (sample mean of all observations) F-test statistic: Test statistic is essentially (Variability of the sample means)/(Variability within samples). Large values of F are implausible under H 0. F statistic follows F(I-1,n-I) distribution under H 0. Reject H 0 if F>F( ) [See Table A.4]

14. ANOVA F-test in JMP F=57.1043, p-value <0.0001 Convincing evidence that the means (treatment effects) are not all the same

15. Robustness to Assumptions Robustness of t-tests and F-tests for comparing several groups are similar to robustness for two group problem. –Normality is not critical. Extremely long-tailed or skewed distributions only cause problems if sample size in a group is <30 –The assumptions of independence within and across groups are critical. –The assumption of equal standard deviations in the population is crucial. Rule of thumb: Check if largest sample standard deviation divided by smallest sample standard deviation is <2 –Tools are not resistant to severely outlying observations. Use outlier examination strategy in Display 3.6