ANOVA: Comparing More Than Two Means

Transcription

1 ANOVA: Comparing More Than Two Means Chapter 11 Cathy Poliak, Ph.D. Office Fleming 11c Department of Mathematics University of Houston Lecture Cathy Poliak, Ph.D. Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

2 Outline 1 Comparing More Than Two Means 2 ANOVA 3 Pairwise Tests Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

3 Popper #20 Questions We wish to test the hypotheses H 0 : µ = 13.5 versus H a : µ > 13.5 at α = 0.02 significance level. From a random sample of 40 the sample mean is Assume that the population standard deviation is Which test would be appropriate to use? a) z-test for proportions b) z-tests for means c) t-test for means d) t-tests for proportions 2. Calculate the test statistic. a) 2.4 b) c) 5.62 d) 0.89 e) Determine the p-value, approximately. 4. What is our decision of the test? a) 0 b) 0.02 c) d) 4.68 a) Reject H 0 b) Fail to reject H 0 c) Accept H 0 Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

4 Popper #20 Questions For each of the following scenarios, determine if it is a) paired t-test or b) two sample t-test 5. The weight of 14 patients before and after open-heart surgery. 6. The smoking rates of 14 men measured before and after a stroke. 7. The number of cigarettes smoked per day by 14 men who have had stokes compared with the number smoked by 14 men who have not had strokes. 8. The photosynthetic rates of 10 randomly chosen Douglas-fir trees compared with 10 randomly chosen western red cedar trees. 9. The photosynthetic rate measured on 10 randomly chosen Sitka spruce trees compared with the rate measured on the western red cedar growing next to each of the Sitka spruce trees. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

5 Weight Loss From: BS/BS704_HypothesisTesting-ANOVA/BS704_ HypothesisTesting-Anova_print.html Is there a difference in the mean weight loss among different programs? A clinical trial is run to compare weight loss programs and participants are randomly assigned to one of the comparison programs and are counseled on the details of the assigned program. Participants follow the assigned program for 8 weeks. Three popular weight loss programs are considered. Low calorie diet. Low fat diet Low carbohydrate diet Control group Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

6 Results Response variable = weight loss = weight at the end of 8 weeks - weight at beginning of the study The observed weight losses of twenty people in this study are as follows: Low Calorie Low Fat Low Carbohydrate Control Is there a statistically significant difference in the mean weight loss among the four diets? Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

7 Box Plots of Weight Loss calorie carb control fat Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

8 Hypotheses We want to know if there is a "statistically significant difference" in the mean weight loss among the four diets. Null hypothesis: mean weight loss is the same among the four diets H 0 : µ calorie = µ carb = µ control = µ fat Alternative hypothesis is that at least one of the mean weight loss among the four diets is different. Rejecting H0 is evidence that the mean of at least one group is different from the other means. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

9 Hypotheses We want to know if there is a "statistically significant difference" in the mean weight loss among the four diets. Null hypothesis: mean weight loss is the same among the four diets H 0 : µ calorie = µ carb = µ control = µ fat Alternative hypothesis is that at least one of the mean weight loss among the four diets is different. Rejecting H0 is evidence that the mean of at least one group is different from the other means. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

10 Hypotheses We want to know if there is a "statistically significant difference" in the mean weight loss among the four diets. Null hypothesis: mean weight loss is the same among the four diets H 0 : µ calorie = µ carb = µ control = µ fat Alternative hypothesis is that at least one of the mean weight loss among the four diets is different. Rejecting H0 is evidence that the mean of at least one group is different from the other means. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

11 Hypotheses We want to know if there is a "statistically significant difference" in the mean weight loss among the four diets. Null hypothesis: mean weight loss is the same among the four diets H 0 : µ calorie = µ carb = µ control = µ fat Alternative hypothesis is that at least one of the mean weight loss among the four diets is different. Rejecting H0 is evidence that the mean of at least one group is different from the other means. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

12 Assumptions The assumptions of analysis of variance are the same as those of the two sample t-test, but they must hold for all k groups. The measurements in every group is a SRS. We have a Normal distribution for each of the k populations. The variance is the same in all k populations. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture) / 23

13 Analysis: ANOVA ANalysis Of VAriance We can estimate how much variation among group means ought to be present from sampling error alone if the null hypothesis is true. ANOVA lets us determine whether there is more variance among the sample means than we would expect by chance alone. Cathy Poliak, Ph.D. Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

14 The Formulas Let X i. = group. ni j=1 X ij n i denote the average of the observations in the ith Let N = M i=1 n i, be the total number of observations in all the M groups. Let X M i=1.. = n i X i. N grand average) be the average of all the observations (the The treatment sum of squares (between groups) is SS(betw) = M i=1 n i( X i. X.. ) 2. The error sum of squares (residual) is SSE = M ni i=i n=1 (X ij X i. ) 2 = M i=1 (n 1)S2 i Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

15 Diets Example Low Calorie Low Fat Low Carbohydrate Control Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

16 The F Test The mean square for treatments is MSTr = SSTr M 1. The mean square for error is MSE = SSE N M. The test statistic is F = MSTr MSE. This test statistic has an F distribution with parameters "numerator degrees of freedom" = M - 1 and "denominator degrees of freedom" = N - M. Where N is the total number of observations and M is the number of groups. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

17 The ANOVA Table Source of degrees of Sum of Mean F Variation freedom Squares Square MSTr Treatments M - 1 SSTr MSTr MSE Error N - M SSE MSE Total N - 1 SST Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

18 ANOVA for Diets Source of degrees of Sum of Mean F Variation freedom Squares Square Diets Error Total p-value = 1 - pf(f,m - 1, N - M) Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

19 R Code > diet.lm=lm(loss~diet,data=diet) > anova(diet.lm) Analysis of Variance Table Response: Loss Df Sum Sq Mean Sq F value Pr(>F) Diet ** Residuals Signif. codes: 0 *** ** 0.01 * Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

20 Tukey s Method (The T Method) The T-method is used to determine which pair (or pairs) of means differs significanctly. 1. Select α, determine Q α,k,n k in R it is qtukey(1 α, k, N k) where, k = number of groups and N = total sample size. 2. Calculate w = Q α,k,n k MSE/j. Where j = the number of elements in each group. 3. List the sample means in increasing order and underline those pairs that differ by less than w. 4. Any pair of sample means not underscored by the same line corresponds to a pair of population or treatment means that are judged significantly different. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

21 Tukey s Method for Diet Example Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

22 Multiple Comparisons If our p-value is small for the ANOVA F test, this implies that at least one of the means is different from the other. Which one(s) are different? We could do a t-test for each pair of means. Problem: when we do multiple t-tests our P(Type 1 error) becomes greater than α. Solution: There are methods of adjustments to reduce the significance level of the pairwise test enough so that the probability of one or more type I errors in the whole set of comparisons is less than α. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

23 Multiple Comparisons If our p-value is small for the ANOVA F test, this implies that at least one of the means is different from the other. Which one(s) are different? We could do a t-test for each pair of means. Problem: when we do multiple t-tests our P(Type 1 error) becomes greater than α. Solution: There are methods of adjustments to reduce the significance level of the pairwise test enough so that the probability of one or more type I errors in the whole set of comparisons is less than α. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

24 Multiple Comparisons If our p-value is small for the ANOVA F test, this implies that at least one of the means is different from the other. Which one(s) are different? We could do a t-test for each pair of means. Problem: when we do multiple t-tests our P(Type 1 error) becomes greater than α. Solution: There are methods of adjustments to reduce the significance level of the pairwise test enough so that the probability of one or more type I errors in the whole set of comparisons is less than α. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

25 Multiple Comparisons If our p-value is small for the ANOVA F test, this implies that at least one of the means is different from the other. Which one(s) are different? We could do a t-test for each pair of means. Problem: when we do multiple t-tests our P(Type 1 error) becomes greater than α. Solution: There are methods of adjustments to reduce the significance level of the pairwise test enough so that the probability of one or more type I errors in the whole set of comparisons is less than α. Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

26 The Bonferroni Method The Bonferroni Method of adjustments reduces the significance level for the pairwise test to α/k, where k is the number of comparisons. R Code: > attach(diet) > pairwise.t.test(loss,diet,"bonferroni") Pairwise comparisons using t tests with pooled SD data: Loss and Diet calorie carb control carb control fat P value adjustment method: bonferroni Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

27 The Bonferroni Method The Bonferroni Method of adjustments reduces the significance level for the pairwise test to α/k, where k is the number of comparisons. R Code: > attach(diet) > pairwise.t.test(loss,diet,"bonferroni") Pairwise comparisons using t tests with pooled SD data: Loss and Diet calorie carb control carb control fat P value adjustment method: bonferroni Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

28 Example MPG Is there a difference in the average miles per gallon for different makes of automobiles? The following table shows the mean mpg of three different makes of automobiles. The data is on the data sets list called mpg. Make n X S Honda Toyota Nissan Cathy Poliak, Ph.D. cathy@math.uh.edu Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

29 Cathy Poliak, Ph.D. Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23

30 Cathy Poliak, Ph.D. Office Fleming 11c (Department Chapter 11of Mathematics University of Houston Lecture ) / 23