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3. Analysis of Variance (page 169) Method to compare means of two or more normal populations based on independent random samples when population variances assumed to be equal. ANALYSIS OF VARIANCE (ANOVA) Extension of two indep samples POOLED t-test Recall two independent samples t-test: H 0 : _____________________ Assumptions … t statistic with _______________ df.

4. ANOVA Assumptions Assumptions: Each sample is a... random sample. The k random samples are... independent. For each popul, the model for response is a... normal model. The k population variances are.... Equal. Popul 1 Popul 2 Popul k

5. ANOVA Hypotheses page 170 H 0 : _______________________________________ H a : _______________________________________ One possible H a picture

6. ANOVA Hypotheses Question: Why call a technique for testing the equality of the means “analysis of VARIANCE”? Answer: We are going to compare two estimators of  2, the common population variance. MS Groups (Mean Square between Groups) MSE (Mean Square Within or due to Error):

7. F Test Statistic These two estimates are used to form the F statistic: Think about this form … which MS is in numerator? which MS is in the denominator? If this F ratio is too __________________ we would reject the null hypothesis. How will you fill in the blank? Read through the Logic of ANOVA on your own!

8. Computing the F Test Statistic page 173 Step 1: Calculate the mean and variance for each sample: Step 2: Calculate the overall sample mean (using all N observations):

9. Computing the F Test Statistic Step 3: Calculate the sum of squares between groups: Step 4: Calculate the sum of squares within groups (due to error): Step 5: Optional - Calculate the total sum of squares: SSTotal = SSGroups + SSE

10. Computing the F Test Statistic Step 6: Fill in the ANOVA table: SourceSum of SquaresDFMean SquareF GroupsSS Groups Error (Within) SSE TotalSS Total

11. The p-value for an F Test If H 0, is true, then the F statistic has an F(k-1, N-k) distribution.  Use SPSS to provide p-value  Know how ANOVA table is constructed  Be able to sketch picture of p-value for an F-test

12. Yellow Card on ANOVA

13. Try It! Comparing 3 Drugs page 175 Quantitative response (smaller  more favorable) N = 19 patients, randomly assigned to one of three drugs What should we do first? Data from Drug 1Data from Drug 2Data from Drug 3 7.37.15.8 8.210.66.5 10.111.28.8 6.09.04.9 9.58.57.9 10.98.5 7.85.2

14. What assumption is best checked with this graph? A) That each sample is random B) That the 3 samples are independent C) That each population has a normal model D) That the 3 populations have equal variance

15. State the hypotheses for ANOVA bottom pg 175 H 0 : _______________________________________ H a : _______________________________________

16. Computing the F Test Statistic Step 1: Calculate the mean and variance for each sample:

17. Computing the F Test Statistic Step 2: Calculate the overall sample mean (using all N observations):

18. Computing the F Test Statistic Step 3: Calculate the sum of squares between groups:

19. Computing the F Test Statistic Step 4: Calculate the sum of squares within groups (due to error):

20. Computing the F Test Statistic Step 5: Optional - Calculate the total sum of squares: No thank you … But we do know … SSTotal = SSGroups + SSE

21. Computing the F Test Statistic Step 6: Fill in the ANOVA table: SourceSum of SquaresDFMean SquareF Groups Error (Within) Total