Analysis of variance (ANOVA) Comparing the means of more than two groups

Transcription

1 Analysis of variance (ANOVA) Comparing the means of more than two groups

2 Example: Cost of mating in male fruit flies Drosophila Treatments: place males with and without unmated (virgin) females

3 Five treatments ("groups"), randomly assigned

4 Response variable: longevity (days) The data (n = 25 males per group): GROUPS: 1preg virgin preg virgin none

5 Results: histograms of male longevity n = 25 males per group

6 Descriptive statistics Treatment i Y median s i n i 1preg virgin preg virgin none

7 Hypothesis testing with ANOVA Like a two-sample t-test, but to test differences between means of more than two groups H 0 : μ 1 = μ 2 = μ 3 = = μ k H A : At least one of the μ i is different

8 Hypothesis testing with ANOVA Rejecting H 0 in ANOVA indicates that at least one of the means μ i is different from the others.

9 Scenarios being compared μ 1 = μ 2 = μ 3 Frequency Y Not all μ's equal Frequency Y

10 Why we compare variances Under H 0, the variance among group means should be σ = σ / n, 2 2 Y or 2 2 σ = σ Y n 2 σ Y is the variance (squared SD) between group means Y i n is the sample size within groups σ 2 is the variance in Y within each group If the null hypothesis is not true, the variance among groups should be larger than this.

11 Variance within groups ("pooled sample variance") Error sum of squares SS error 2 = s i ( n 1) Error degrees of freedom df error = ( n 1) = N k where N is the total number of data points in all groups Mean Square Error (MS error ) MS = error SS df i i error error

12 where Y is the grand mean, Variance among groups Groups sum of squares SS groups = n i (Y i Y ) 2 Y n Y i i =. Degrees of freedom for groups df groups = k 1. Mean Square Groups (MS groups ) SSgroups MS groups = df. N groups

13 F = F-ratio MS MS groups error Under H 0, (except for chance) MS groups MS error So under H 0, (except for chance) F 1

14 F test Reject H 0 is rejected if F is significantly larger than 1 Compare F with critical value of the F distribution, F 0.05(1),dfgroups,dferror Reject H 0 F F 0.05(1),dfgroups,dferror

15 F distribution

16 ANOVA table for fruit fly experiment Source Sum of Squares df Mean Squares F P-value Groups <0.001 Error Total F 0.05(1),4,120 = Reject H 0 Conclusion: Mean longevity not equal in all groups

17 Cost of mating in male fruit flies Implication: sex is deadly (in males)

18 R 2 ("R-squared") R 2 measures the fraction of the variation in Y that is "explained" by group differences. Based on the fact that the total sum of squares is split into its two component parts, the sum of squares of error and the sum of squares of groups, SS = SS + total groups SS error 2 R = SS SS groups total.

19 Fruit flies R 2 = = Conclusion: 31% of the variation in longevity is "explained" by treatment

20 ANOVA vs two-sample t-test An ANOVA with k=2 is mathematically equivalent to a twotailed, 2-sample t-test of H 0 : μ 1 μ 2 = 0 H A : μ 1 μ 2 0

21 Assumptions of ANOVA The measurements in every group are a random sample from the corresponding population. The variable has a normal distribution in all k populations. The variance is the same in all k population.

22 The robustness of ANOVA The ANOVA is fairly robust to deviations from the assumption of normality, particularly when sample size is large. This robustness stems from a property of sample means described by the Central Limit Theorem: the sampling distribution of means is approximately normal when sample size is large, even when the variable itself does not have a normal distribution. ANOVA is also fairly robust to departures from the assumption of equal variance in the k populations if the samples sizes are all equal or nearly so.

23 Strategies if assumptions of ANOVA not met and we can't rely on robustness property Transformations (e.g., log-transformation, square root transformation, and arcsine transformation). Nonparametric alternatives

24 Nonparametric alternatives to ANOVA: Kruskal-Wallis test Analogous to the Mann-Whitney U-test for more than two groups. Based on ranks Test statistic H has an approximately χ 2 distribution H 0 : The population median is the same in all k groups. H A : At least one of the groups has a different median.

25 ANOVA: which means are different? H 0 : μ 1 = μ 2 = μ 3 = = μ k H A : At least one of the μ i is different (but which?)

26 ANOVA: which means are different? Two approaches to addressing this question: 1. Planned comparisons 2. Unplanned comparisons

27 Planned comparisons One or a very small number of focal comparisons that were planned at the time the study was designed.

28 Unplanned comparisons A comprehensive search for differences between groups

29 What's so important about the difference? 1. Planned comparisons: No need to correct for multiple comparisons Very few planned comparisons permitted 2. Unplanned comparisons Correction for multiple comparisons required because of rising Type 1 error rates

30 Planned confidence interval for the difference between two means Treatment i Y median s i n i 1preg virgin preg virgin none

31 Planned confidence interval for the difference between two means Formula for confidence interval similar to that for twosample case: ( Y Y ) SE t < μ μ < ( Y Y SE t i j 0.05(2), N k i j i j ) (2), N k except that we calculate SE using MS error instead of the pooled sample variance for just the two samples 1 SE = MS error + 1 n i n j and we use df error = N k for degrees of freedom.

32 Planned confidence interval for the difference between two means ( j Treatment Y i median s i n i 1preg virgin preg virgin none Yi Y ) = = SE = 4.188, df = 120, t 0.05(2),120 = % confidence interval: < μ μ < i j

33 Example of unplanned comparisons The Tukey-Kramer method for testing all pairs of means Comparison ( i Yj ) 1preg 8virgin reject 1preg 1virgin do not reject 1preg 8preg do not reject 1preg none do not reject none 8virgin reject none 1virgin do not reject none 8preg do not reject 8preg 8virgin reject 8preg 1virgin do not reject 1virgin 8virgin reject Y SE q critical q (0.05) conclusion 8virgin 1virgin 8preg none 1preg

34 The Tukey-Kramer method for testing all pairs of means With the Tukey-Kramer method, the probability of making at least one Type 1 error throughout the course of testing all pairs of means is no greater than the significance level α.

35 Fixed vs random effects ANOVA 1. Fixed effects: With fixed effects, the treatments are chosen by the experimenter. They are repeatable and of direct interest. 2. Random effects: With random effects, the treatments are a random sample from a "population" of treatments. For single-factor ANOVA, there is no difference in the F-test of the null and alternative hypotheses

36 Example of random effects ANOVA: Flycatcher patch height in 2 different years

37 Patch height measurements Bird Height 1 (mm) Height 2 (mm) Bird Height 1 (mm) Height 2 (mm)

38 Random effects ANOVA Each bird is a "group", randomly sampled from a population of groups (birds) n = 2 measurements per group We don't care so much about the individual birds, they are just a sample What we really want to know is whether the trait varies in the population (and by how much) H 0 : Patch height does not vary in the population H A : Patch height varies in the population

39 Random effects ANOVA Source Sum of Squares df Mean Squares F P-value Groups <0.001 Error Total F 0.05(1),29,30 = Reject H 0 Conclusion: patch height varies in the population

40 Repeatability Repeatability is the fraction of the variance in the trait that is among groups Repeatability = s 2 A 2 sa + MS error, where the variance among groups is 2 s A = MS groups n MS error.

41 Repeatability of patch height 2 s A = 2 = yielding Repeatability = =

42 Repeatability of patch height Conclusion: 88% of the variance in patch height in the population is among birds, whereas the remaining 12% is variance from measurement to measurement on the same birds.