Unit 4: Inference for numerical variables Lecture 3: ANOVA

Transcription

1 Unit 4: Inference for numerical variables Lecture 3: ANOVA Statistics 101 Thomas Leininger June 10, 2013

2 Announcements Announcements Proposals due tomorrow. Will be returned to you by Wednesday. You MUST complete the proposal process. A few things to watch out for: Data is plural, data set is singular. Avoid using population data - if you have population data, you might consider taking a random sample. Exploratory analysis: should include some summary statistics and some graphics AND interpretations. If using existing data, find out how your data were collected, and discuss the sampling method as well as any possible biases. Scope of inference: generalizability & causality. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

3 Aldrin in the Wolf River The Wolf River in Tennessee flows past an abandoned site once used by the pesticide industry for dumping wastes, including chlordane (pesticide), aldrin, and dieldrin (both insecticides). These highly toxic organic compounds can cause various cancers and birth defects. The standard methods to test whether these substances are present in a river is to take samples at six-tenths depth. But since these compounds are denser than water and their molecules tend to stick to particles of sediment, they are more likely to be found in higher concentrations near the bottom than near mid-depth. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

4 Aldrin in the Wolf River Data Aldrin concentration (nanograms per liter) at three levels of depth. aldrin depth bottom bottom bottom middepth middepth middepth surface surface surface Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

5 Aldrin in the Wolf River Exploratory analysis Aldrin concentration (nanograms per liter) at three levels of depth. surface middepth bottom n mean sd bottom middepth surface overall Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

6 Aldrin in the Wolf River Research question Is there a difference between the mean aldrin concentrations among the three levels? To compare means of 2 groups we use a Z or a T statistic. To compare means of 3+ groups we use a new test called ANOVA and a new statistic called F. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

7 Aldrin in the Wolf River Recap: 2-sample CIs and HTs n mean sd bottom middepth surface overall HT: T df = ( x 1 x 2 ) null value SE where SE = df = min(n 1 1, n 2 1) CI: ( x 1 x 2 ) ± t df SE s 2 1 n 1 + s2 2 n 2 Application exercise: Perform a HT and construct a CI for each difference. and Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

8 Aldrin in the Wolf River ANOVA ANOVA is used to assess whether the mean of the outcome variable is different for different levels of a categorical variable. H 0 : The mean outcome is the same across all categories, µ 1 = µ 2 = = µ k, where µ i represents the mean of the outcome for observations in category i. H A : At least one mean is different than others. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

9 Aldrin in the Wolf River Conditions 1 The observations should be independent within and between groups If the data are a simple random sample, this condition is satisfied. Carefully consider whether the between-group data is independent (e.g. no pairing). Always important, but sometimes difficult to check. 2 The observations within each group should be nearly normal. Especially important when the sample sizes are small. How do we check for normality? 3 The variability across the groups should be about equal. Especially important when the sample sizes differ between groups. How can we check this condition? Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

10 Aldrin in the Wolf River z/t test vs. ANOVA - Purpose z/t test ANOVA Compare means from two groups to see whether they are so far apart that the observed difference cannot reasonably be attributed to sampling variability. H 0 : µ 1 = µ 2 H A : µ 1 µ 2 H A : µ 1 < µ 2 Compare the means from two or more groups to see whether they are so far apart that the observed differences cannot all reasonably be attributed to sampling variability. H 0 : µ 1 = µ 2 = = µ k H A : At least one mean is different H A : µ 1 > µ 2 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

11 Aldrin in the Wolf River z/t test vs. ANOVA - Method z/t test Compute a test statistic (a ratio). z/t = ( x 1 x 2 ) (µ 1 µ 2 ) SE( x 1 x 2 ) ANOVA Compute a test statistic (a ratio). F = variability bet. groups variability w/in groups Large test statistics lead to small p-values. If the p-value is small enough H 0 is rejected, and we conclude that the population means are not equal. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

12 Aldrin in the Wolf River z/t test vs. ANOVA With only two groups t-test and ANOVA are equivalent, but only if we use a pooled standard variance in the denominator of the test statistic. With more than two groups, ANOVA compares the sample means to an overall grand mean. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

13 Aldrin in the Wolf River Hypotheses Question What are the correct hypotheses for testing for a difference between the mean aldrin concentrations among the three levels? (a) H 0 : µ B = µ M = µ S H A : µ B µ M µ S (b) H 0 : µ B µ M µ S H A : µ B = µ M = µ S (c) H 0 : µ B = µ M = µ S H A : At least one mean is different. (d) H 0 : µ B = µ M = µ S = 0 H A : At least one mean is different. (e) H 0 : µ B = µ M = µ S H A : µ B > µ M > µ S Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

14 ANOVA and the F test Test statistic Does there appear to be a lot of variability within groups? How about between groups? F = variability bet. groups variability w/in groups Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

15 ANOVA and the F test F distribution and p-value F = variability bet. groups variability w/in groups In order to be able to reject H 0, we need a small p-value, which requires a large F statistic. In order to obtain a large F statistic, variability between sample means needs to be greater than variability within sample means. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

16 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Degrees of freedom associated with ANOVA groups: df G = k 1, where k is the number of groups total: df T = n 1, where n is the total sample size error: df E = df T df G df G = k 1 = 3 1 = 2 df T = n 1 = 30 1 = 29 df E = 29 2 = 27 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

17 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Sum of squares between groups, SSG Measures the variability between groups k SSG = n i ( x i x) 2 i=1 where n i is each group size, x i is the average for each group, x is the overall (grand) mean. n mean bottom middepth surface overall SSG = ( 10 ( ) 2) + ( 10 ( ) 2) + ( 10 ( ) 2) = Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

18 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Sum of squares total, SST Measures the total variability n SST = (x i x) i=1 where x i represents each observation in the dataset. SST = ( ) 2 + ( ) 2 + ( ) ( ) 2 = ( 1.3) 2 + ( 0.3) 2 + ( 0.2) (0.1) 2 = = Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

19 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Sum of squares error, SSE Measures the variability within groups: SSE = SST SSG SSE = = Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

20 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Mean square error Mean square error is calculated as sum of squares divided by the degrees of freedom. MSG = 16.96/2 = 8.48 MSE = 37.33/27 = 1.38 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

21 ANOVA output, deconstructed Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total Test statistic, F value The F statistic is the ratio of the between group and within group variability. F = MSG MSE F = = 6.14 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

22 ANOVA output, deconstructed p-value Df Sum Sq Mean Sq F value Pr(>F) (Group) depth (Error) Residuals Total p-value is the probability of at least as large a ratio between the between group and within group variability, if in fact the means of all groups are equal. It s calculated as the area under the F curve, with degrees of freedom df G and df E, above the observed F statistic. dfg = 2 ; dfe = Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

23 ANOVA output, deconstructed Conclusion If p-value is small (less than α), reject H 0. The data provide convincing evidence that at least one mean is different from (but we can t tell which one). If p-value is large, fail to reject H 0. The data do not provide convincing evidence that at least one pair of means are different from each other, the observed differences in sample means are attributable to sampling variability (or chance). Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

24 ANOVA output, deconstructed Conclusion - in context Question What is the conclusion of the hypothesis test for α = 0.05? (p-value = ) The data provide convincing evidence that the average aldrin concentration (a) is different for all groups. (b) on the surface is lower than the other levels. (c) is different for at least one group. (d) is the same for all groups. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

25 Checking conditions (1) independence Does this condition appear to be satisfied? Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

26 Checking conditions (2) approximately normal Does this condition appear to be satisfied? 9 8 bottom middepth 5.0 surface Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

27 Checking conditions (3) constant variance Does this condition appear to be satisfied? bottom sd=1.58 middepth sd=1.10 surface sd=0.66 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

28 Multiple comparisons & Type 1 error rate Which means differ? Earlier we concluded that at least one pair of means differ. The natural question that follows is which ones? We can do two sample t tests for differences in each possible pair of groups. Can you see any pitfalls with this approach? When we run too many tests, the Type 1 Error rate increases. This issue is resolved by using a modified significance level. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

29 Multiple comparisons & Type 1 error rate Multiple comparisons The scenario of testing many pairs of groups is called multiple comparisons. The Bonferroni correction suggests that a more stringent significance level is more appropriate for these tests: α = α/k where K is the number of comparisons being considered. If there are k groups, then usually all possible pairs are compared and K = k(k 1) 2. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

30 Multiple comparisons & Type 1 error rate Determining the modified α Question In the aldrin data set depth has 3 levels: bottom, mid-depth, and surface. If α = 0.05, what should be the modified significance level for two sample t tests for determining which pairs of groups have significantly different means? (a) α = 0.05 (b) α = 0.05/2 = (c) α = 0.05/3 = (d) α = 0.05/6 = Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

31 Multiple comparisons & Type 1 error rate Which means differ? Question Based on the box plots below, which means would you expect to be significantly different? 9 (a) bottom & surface 8 (b) bottom & mid-depth 7 (c) mid-depth & surface 6 5 (d) bottom & mid-depth; mid-depth & surface 4 3 bottom sd=1.58 middepth sd=1.10 surface sd=0.66 (e) bottom & mid-depth; bottom & surface; mid-depth & surface Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

32 Multiple comparisons & Type 1 error rate Which means differ? (cont.) If the ANOVA assumption of equal variability across groups is satisfied, we can use the data from all groups to estimate variability: Estimate any within-group standard deviation with MSE, which is s pooled Use the error degrees of freedom, n k, for t-distributions Difference in two means: after ANOVA σ 2 1 SE = + σ2 2 MSE n 1 n 2 n 1 + MSE n 2 Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

33 Multiple comparisons & Type 1 error rate Is there a difference between the average aldrin concentration at the bottom and at mid depth? n mean sd bottom middepth surface overall Df Sum Sq Mean Sq F value Pr(>F) depth Residuals Total T dfe = ( x bottom x middepth ) MSE n bottom + MSE n middepth T 27 = ( ) = = < p value < 0.10 (two-sided) α = 0.05/3 = Fail to reject H 0, the data do not provide convincing evidence of a difference between the average aldrin concentrations at bottom and mid depth. Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34

34 Multiple comparisons & Type 1 error rate Application exercise: Post-hoc comparison Is there evidence of a difference between the average aldrin concentration at the bottom and at surface? (a) yes (b) no Statistics 101 (Thomas Leininger) U4 - L3: ANOVA June 10, / 34