Repeated Measurement ANOVA. Sungho Won, Ph. D. Graduate School of Public Health Seoul National University

Transcription

1 1 Repeated Measurement ANOVA Sungho Won, Ph. D. Graduate School of Public Health Seoul National University

2 2 Analysis of Variance (ANOVA) Main idea Evaluate the effect of treatment by analyzing the amount of variation among the subgroup sample means

3 Analysis of Variance (ANOVA) ANOVA tests mean differences between groups defined by categorical variables One-way ANOVA ONE factor with 2 or more levels Multi-way ANOVA 2 or more factors, each with 2 or more levels Fertirrigation: 4 levels Gender drug: 4 doses ASSUMPTIONS Independence of cases - this is a requirement of the design. Normality - the distributions in each cells are normal Homogeneity of variances - the variance of data in groups should be the same. 3

4 4 Preliminary Model for ANOVA Nonnormality F-test is very robust against non-normal data, especially in a fixedeffects model Large sample size will approximate normality by Central Limit Theorem (recommended sample size > 50) Simulations have shown unequal sample sizes between treatment groups magnify any departure from normality A large deviation from normality leads to hypothesis test conclusions that are too liberal and a decrease in power and efficiency Remedy for nonnormality Data transformation Modified F-tests Adjust the degrees of freedom Rank F-test Permutation (also called randomization) test on the F-ratio

5 5 Preliminary Model for ANOVA Transformation Transformation e.g. f(x) = x 1.5 e.g. f(x) = log(x) f(x) = atan(x)

6 6 Permutation test Permutation test Distribution of statistics is empirically estimated with the permuted samples under the null hypothesis. P-value can be obtained under the null hypothesis. It can be applied to any statistics. Example Group 1: 10, 15, 20 vs group 2: 3, 4, 5 We are interested in getting p-values for H 0 : μ 1 = μ 2 Group I : 10, 15, 20 Group II: 3, 4, 5 Number of possible permuted sample: 6 3 = 20

7 Permutation test Label Group I Group II test statistics

8 8 One-way ANOVA 1. Test normality 2. Test homogeneity of variance within each group 3. Run the ANOVA 4. Reject/accept the H 0 (all the means are equal) 2 approaches 5A. Multiple comparison to test differences between the level of factors 5B. Model simplification working with contrasts

9 9 One-way ANOVA Maize: 4 varieties (k) y: productivity (NORMAL CONTINUOS) x: variety (CATEGORICAL: four levels: x 1, x 2, x 3, x 4 ) H 0 : µ 1 = µ 2 = µ 3 = µ 4 H a : At least two means differ One-way ANOVA is used to test for differences among two or more independent groups ANOVA model y i = a + bx 2 + cx 3 + dx 4 a=µ1 b=µ1-µ2 c=µ1-µ3 d=µ1-µ4 y a b c d

10 10 One-way ANOVA Var 1 Var 2 Var 3 Var X 9.03 n i µ i Number of groups: k = 4 Number of observations N = 19 Grand mean = 7.80 Sum of squares (SS): deviance Degree of freedom (df) SS Total = Σ(y i grand mean) 2 SS Factor = Σ n i (group mean i grand mean) 2 SS Error (within group) = Σ(y i group mean i ) 2 Total: N 1 Group: k 1 Error: N k

11 11 One-way ANOVA SS Total SS Error Grand mean SS Factor mean 3 mean 4 SS Total = SS Factor + SS Error mean 1 mean 2

12 One-way ANOVA SSTotal=SSFactor SSTotal=SSFactor + SSError µ 3 µ4 µ 2 Grand mean µ 1 SS can be divided by the respective df to get a variance MS = SS /df Mean squared deviation F = MSFactor MSError 12

13 One-way ANOVA: F test (variance) F = Factor MS Error MS If the k means are not equal, then the Factor MS in the population will be greater than the population s Error MS How to define the correct F test can be a difficult task with complex design: This is a limitation of ANOVA!! If F calculated is large (e.g. P<0.05), then we can reject Ho All we conclude is that at least two means are different!!! A POSTERIORI MULTIPLE COMPARISONS WORKING WITH CONTRASTS 13

14 14 One-way ANOVA: F test (variance) Contrasts are the essence of hypothesis testing and model simplification in analysis of variance and analysis of covariance. They are used to compare means or groups of means with other means or groups of means We used contrasts to carry out t test AFTER having found out a significant effect with the F test - We can use contrasts in model simplification (merge similar factor levels) - Often we can avoid post-hoc multiple comparisons

15 Multi-way ANOVA: F test (variance) Multi-way ANOVA is used when the experimenter wants to study the effects of two or more treatment variables. ASSUMPTIONS Independence of cases - this is a requirement of the design Normality - the distributions in each of the groups are normal Homogeneity of variances - the variance of data in groups should be the same + Equal replication (BALANCED AND ORTHOGONAL DESIGN: this is required to estimate variance!!) Dose 1 Dose 2 Dose 3 Low temp - X 10 obs 10 obs High temp 10 obs 10 obs 8 obs X If you use traditional general linear models just one missing data can affect strongly the results 15

16 16 Fixed vs Random Effects If we consider more than one factor we have to distinguish two kinds of effects: Fixed effects: factors are specifically chosen and under control, they are informative (E.g. sex, treatments, wet vs. dry, doses, sprayed or not sprayed) Random effects: factors are chosen randomly within a large population, they are normally not informative (E.g. fields within a site, block within a field, split-plot within a plot, family, parent, brood, individuals within repeated measures) Random effects mainly occur in two contrasting kinds of circumstances 1. Observational studies with hierarchical structure 2. Designed experiments with different spatial or temporal dependence

17 17 Fixed vs Random Effects Random effect Imagine that we randomly select a of the possible levels of the factor of interest. In this case, we say that the factor is random if their effects can be assumed to follow the normal distribution. Typically random factors are categorical. While continuous covariates may be measured at random levels, we usually think of the effects as being systematic (such as linear, quadratic or even exponential) effects. Random effects are not systematic. If we have only random effect, then we are working on Random Effect MODEL: y ij = µ + r i (random) + ε ij, r i ~ N(0, σ 12 ), ε ij ~ N(0, σ 22 )

18 18 Fixed vs Random Effects Example From the text discusses a single random factor case about the difference of looms in a textile weaving company. Four looms have been chosen randomly from a population of looms within a weaving shed and four observations of fabric strength were made on each loom. The data obtained from the experiment are below. Loom Obs 1 Obs 2 Obs 3 Obs 4 row sum y ij = μ + α i + ε ij, α i ~N 0, σ α 2, ε ij ~N 0, σ ε 2

19 19 Fixed vs Random Effects Why is it so important to identify fixed vs. random effects? They affect the way to construct the F-test in a multifactorial ANOVA. Their misspecification leads to wrong conclusions You can find how to construct your F-test with different combinations of random and fixed effects and with different hierarchical structures (choose a well-known sampling design!!!) If we have both fixed and random effects, then we are working on MIXED MODELS y ij = µ + α i + r i + ε ij, r i ~ N(0, σ 12 ), ε ij ~ N(0, σ 22 )

20 20 Fixed vs Random Effects Why is it so important to identify fixed vs. random effects? They affect the way to construct the F-test in a multifactorial ANOVA. Their misspecification leads to wrong conclusions You can find how to construct your F-test with different combinations of random and fixed effects and with different hierarchical structures (choose a well-known sampling design!!!) If we have both fixed and random effects, then we are working on MIXED MODELS y i = µ + α i (fixed) + r i (random) + ε

21 21 Factorial ANOVA: two or more factors Factorial design: two or more factors are crossed. Each combination of the factors are equally replicated and each factor occurs in combination with every level of the other factors 3 levels of irrigation 4 fertilizer Orthogonal sampling

22 22 Factorial ANOVA: why? Why use a factorial ANOVA? Why not just use multiple one-way ANOVAs? With n factors, you d need to run n one-way ANOVA s, which would inflate your α-level However, this could be corrected with a Bonferroni correction The best reason is that a factorial ANOVA can detect interactions, something that multiple one-way ANOVA s cannot do. It should be noted that orthogonality is not often preserved in observational study. If they are not orthogonal, true effects of factor A can be estimated only if both A and B are included as covariates. Furthermore if A and B are highly correlated, their effects are not distinguishable because of multi-collinearity.

23 Performance 23 Factorial ANOVA: Interactions Interaction: When the effects of one independent variable differ according to levels of another independent variable E.g. We are testing two factors, Gender (male and female) and Age (young, medium, and old) and their effect on performance Female If males performance differed as a function of age, i.e. males performed better or worse with age, but females performance was the same across ages, we would say that Age and Gender interact, or that we have an Age x Gender interaction Young Age Old Male It is necessary that the slopes differ from one another

24 Performance Performance Factorial ANOVA: Main Effects Main effects: the effect of a factor is independent from any other factors This is what we were looking at with one-way ANOVA s if we have a significant main effect of our factor, then we can say that the mean of at least one of the groups/levels of that factor is different than at least one of the other groups/levels Young Old Male Female It is necessary that the intercepts differ 24

25 Factorial ANOVA: Two-crossed fixed factor design Examples of good ANOVA results Effect Clone < 0.05 Treat < 0.05 C x T < 0.05 Effect Clone < 0.05 Treat < 0.05 C x T n.s. Effect Clone n.s. Treat < 0.05 C x T n.s. Effect Clone < 0.05 Treat n.s. C x T n.s. Mean y Mean y Mean y Mean y Treatment Treatment Treatment Treatment Worst case Effect Clone n.s. Treat n.s C x T n.s Mean y Treatment Two-crossed factor design CloneA CloneB Treatment: 3 levels 25

26 26 Factorial ANOVA: Two-crossed fixed factor design Two crossed fixed effects: every level of each factor occurs in combination with every level of the other factors Model 1: two fixed effects Model 2: two random effects (uncommon situation) Model 3: one random and one fixed effect We can test main effects and interaction: 1. The main effect of each factor is the effect of each factor independent of (pooling over) the other factors 2. The interaction between factors is a measure of how the effects of one factor depends on the levels of one or more additional factors (synergic and antagonist effect of the factors) Factor 1 x Factor 2 We can only measure interaction effects in factorial (crossed) designs

27 27 Factorial ANOVA: Two-crossed fixed factor design Two crossed fixed effects: Response variable: weight gain in six weeks Factor A: DIET (3 levels: barley, oats, wheat) Factor B: SUPPLEMENT (4 levels: S1, S2, S3, S4) DIET* SUPPLEMENT= 3 x 4 = 12 combinations We have 48 horses to test our two factors: 4 replicates barley+s1 barley+s2 oats+s1 oats+s2 wheat+s1 wheat+s2 barley+s3 barley+s4 oats+s3 oats+s4 wheat+s3 wheat+s4

28 Factorial ANOVA: Two-crossed fixed factor design The 48 horses must be independent units to be replicates DIET SUPPLEMENT DIET*SUPPLEMENT F test for main effects and interaction DIET SUPPLEMENT F D F S MS MS D error MS MS S error Barley Oats wheat Barley Oats Wheat S1 S2 S3 S4 DIET*SUPPLEMENT F DxS MS MS DxS error 28

29 29 Mixed MODELs: Fixed + Random Effects Pure fixed effect models REQUIRE INDEPENDENCE WHAT IF WE HAVE DEPENDENCE??? Mixed models can deal with spatial or temporal dependence The mixed models included the dependence in the data with appropriate random effects

30 30 Mixed MODELs: Split-Plot We can consider random factors to account for the variability related to the environment in which we carry out the experiment Mixed models can deal with spatial or temporal dependence The split-plot design is one of the most useful design in agricultural and ecological research The different treatment are applied to plot with different size organized in a hierarchical structure

31 Factorial ANOVA: Split-Plot (Mixed) N P P NP NP N P P NP N P NP NP NP N N P P Block A N P N P N P B C P NP P NP P NP D NP N NP N NP N P P P P P P NP N NP N NP N P NP P NP P NP NP NP NP NP NP NP N N N N N N P P P P P P Irrigation Density Fertilizer Response variable: Crop production in each cell Fixed effects: Irrigation (yes or no) Seed-sowing density sub-plots (low, med, high) Fertilizer (N, P or NP) Random effects: 4 blocks Irrigation within block Density within irrigation plots 31

32 Factorial ANOVA: Mixed Model The split-plot design i) 4 Blocks HIERARCHICAL STRUCTURE No water Water ii) irrigation plot (yes or no) N P NP P NP N iii) seed-sowing density sub-plots (low, med, high) P P NP N P NP N NP NP N iv) 3 fertilizer sub-sub-plot (N, P, o NP) P P Crop production 32

33 33 Factorial ANOVA: Mixed Model Model formulation Y~ fixed effects + error terms y ~ a*b*c + Error(a/b/c) Here you can specify your sampling hierarchy Yield Block Irrigation Density Fertilizer Uninformative Informative!!! Yield ~ irrigation*density*fertilizer+ Error(block/irrigation/density))

34 34 Factorial ANOVA: Mixed Model Mixed models using traditional ANOVA requires perfect orthogonal and balanced design (THEY WORK WELL WITH THE PROPER SAMPLING) avoid to work with F-test for multi-way ANOVA in non-orthogonal sampling designs If something has gone wrong with the sampling You can use linear mixed models with the REML estimation

35 35 ANOVA with Repeated Measures Statistical Model Plot of means by time for each of two treatments Sub 1 t 1 y 11 y 1t n y n1 y nt A standard ANOVA model Model y hij = hj + hij hj the expected response of a subject on treatment h at time t j, hij the independent errors, N(0, 2 ). h = 1,, g (# of groups); i =1,, n h (# of subjects in the hth group); j =1,, k (# of measurements).

36 ANOVA with Repeated Measures Mixed effects models The random effect for an individual is a random displacement term that applies to all measurements for that individual. Time Point whole Plot Sub 1 j t Trt = 1 1 y 11 y 1j y 1t i y i1 y ij y it n y n1 y nj y nt Trt = 2 U h1 : Split Plot 1 U hi : Split Plot i U hn : Split Plot n Subjects are not nested for treatment 36

37 37 ANOVA with Repeated Measures A mixed effects model The random/mixed effects model is alternately referred to a split plot, hierarchical, two-stage, or variance components model, or the random intercept model y hij = hj + U hi + hij hj the expected response of a subject on trt h at time t j U hi the unobserved random effect for subject i in group h; h =1,, g; j =1,,k; i =1,,n h ; U hi N(0, 2 ) for all h, i ; hij ~ N(0, 2 ) are residual errors; U hi, hij are independent

38 38 ANOVA with Repeated Measures A mixed effects model Decomposition of Variance of Covariance Var (y hij ) = : the population variance of effect 2 : the measurement variance This assumes the equal covariance between y hij and y hij'. However it is not always true and should be confirmed with statistical inference. Hierarchical or two-stage model is applicable to the mixed-effects model 1. E (y hij U hi ) = hj + U hi = U hi *, V(y hij U hi ) = 2 2. U hi ~ N(0, 2 ) The covariance/correlation between measurements within a subject?

39 39 ANOVA with Repeated Measures A mixed effects model Why not use subject as a fixed effect? May require a lot of additional parameters (depending on number of subjects). Their effects are small and can be assumed to follow the normal distribution. Implies interest in estimating a separate effect for each person. Comments Model fitting does not require estimation of subject effects. Instead of using a separate parameter (fixed effect) for each subject, we assume subject effects are random realization from a population. The only additional parameter is the random effect variance, v 2.

40 40 ANOVA with Repeated Measures Repeated Measures ANOVA It represents an univariate approach to analyzing repeated measures data with particular focus on hypothesis testing. It can be thought of as an extension of the paired-sample t-test to include comparison between more than two repeated.

41 ANOVA with Repeated Measures Between-subject and within-subject effects Between-subject effects Each subject is exposed to one of the treatments being tested Estimate E(Y E = 1) - E(Y E = 0) Example: one-way ANOVA Within-subject effects Each subject is exposed to more than one of the treatments being tested Estimate E(Y i E = 1 - Y i E = 0 ), i: subject Example: Paired t-test Mixed (within and between-subjects) design There is at least one within-subjects factor and at least one between-subjects factor in the same experiment 41

42 42 ANOVA with Repeated Measures Between-subject and within-subject effects Remarks Within-subjects allow observation of change over time. The main disadvantage is possible confounding, which is generated by learning effect and can often be overcome by using counterbalancing. For between-subject designs, as long as group assignment is random, causal estimates are obtained by comparing the behavior of those in one experimental condition with the behavior of those in another. However it is hardly achieved for observational studies

43 43 ANOVA with Repeated Measures Partition of SS A x (B x S): A - between, B within, S - subjects Each level of factor A contains a different group of e.g. randomly assigned subjects On the other hand, each level of factor B at any given level of factor A contains the same subjects SStotal SS b/t subjects SS w/in subjects SS A SS subj w/in groups SS B SS error

44 44 Sources of Variance Partition of SS SS total : Deviation of each individual score from the grand mean SS b/t subjects Deviation of subjects' individual means (across treatments) from the grand mean. This is largely uninteresting, as we can pretty much assume that subjects differ SS w/in subjects : How SS vary about their own mean, breaks down into: SS B 1 As in between subjects ANOVA, it is the comparison of treatment means to each other (by examining their deviations from the grand mean) 2 However this is now a partition of the within subjects variation SS error 1 Variability of individuals scores about their treatment mean

45 45 ANOVA with Repeated Measures Example : The Sitka spruce 1989 Goal : to investigate the effect of ozone on tree growth 79 trees are grown in four controlled environment chambers Observations: 103, 130, 162, 190, 213, 247,273, 308 (days) 27 trees, ozone 70 ppb 27 trees, ozone 70 ppb 12 trees, No ozone 12 trees, No ozone

46 ANOVA with Repeated Measures Repeated Measures ANOVA Linear model y hij = + h + j + hj + U hi + hij y hij the jth measurement (response) (j =1,, n) for the ith subject (i =1,, m h ) in group h (h =1,,g); the unknown intercept; h the effect of treatment h, h h = 0; j the effect of time j, j j =0 for all h; hj the effect of treatment h at time j; j hj = 0 for all h, h hj = 0 for j =1,, n; U hi the random effect of the ith subject in the hth group, U hi IN(0, 2 ); hij ~ IN(0, 2 ) are residual errors; U hi, hij are independent. m h m h F 1 : To test for difference in expected response between treatments H 0 : 1 = 1 = = g vs H 1 : not H 0. F 2 : To test for lack of parallel group mean profile H 0 : 11 = 12 = = gn vs H 1 : not H 0. 46

47 47 ANOVA with Repeated Measures ANOVA Table Source of variation Sum of squares df Between treatments Whole plot residual Whole plot total Between times Treatment by time interaction Split-plot residual Split-plot total g BTSS m mh( yh y) 1 h1 RSS TSS BTSS g m TSS m h ( y y ) 1 h1 n i1 hi 2 BTSS ( ) 2 n y 1 j y j n g 2 2 m ( ) 1 h 1 h y j hj 1 2 ISS y BTSS BTSS RSS TSS ISS BTSS TSS TSS g 2 h1 m h n i1 j1 2 ( y y ) hi j 2 2 whole plot split plot g 1 m g n 1 ( g1) ( n1) ( m g) ( n 1) nm 1 F 1 : To test for difference in expected response between treatments: F { BTSS / ( g 1)} / { RSS / ( m g)}, F 2 : To test for lack of parallel group mean profile: F { ISS / [( g 1)( n 1)]} / { RSS / [( m g)( n 1)]}, 2 2 2

48 48 ANOVA with Repeated Measures ANOVA Table Source of Var Sum of Squares df Expected Mean Square Between Trts BTSS 1 = m h m h (y h.. y... ) 2 g 1 m h m h h2 /(h-1) + n Whole plot residual RSS 1 = TSS 1 BTSS 1 m g n Whole plot total TSS 1 = m h i (y hi. y... ) 2 m 1 Between Times BTSS 2 = n j (y..j y... ) 2 n 1 g j n h j2 /(n-1) + 2 Trt x Time Interaction ISS 2 = j h n h (y h.j y... ) 2 (g 1)(n 1) j h n h hj2 /((g-1)(n-1)) + 2 BTSS 1 BTSS 2 Error RSS 2 = TSS 2 ISS 2 BTSS 2 (m-g)(n-1) 2 TSS 1 Total TSS 2 = h i j (y hij y... ) 2 nm - 1 Under the null hypotheses, α h = 0, τ j = 0, and hj = 0. v 2 = 0 can be 0, which means no covariances between repeated measurements for each subject.

49 49 ANOVA with Repeated Measures Results Growth of Sitka spruce with and without ozone (continued) The following table presents a split-plot ANOVA for the 1989 sitka spruce data. We ignore possible chamber effects to ease comparison with the results from the robust analysis. Source of variation Sum of squares df Mean square Between treatments Whole plot residuals Whole plot total Between times Treatment by time interaction Split-plot residual Split-plot total What are the test statistics for F 1, F 2?

50 50 ANOVA with Repeated Measures Assumptions in the Sitka Spruce Example U IN v IN 2 2 hi ~ (0, ), hij ~ (0, ) 2 2 v if i i ', j j ' 2 cov( yhij, yh' i' j ') v if i i ', j j ' 0 if i i ', j j ' Homogeneity of variances Sphericity Independence

51 51 ANOVA with Repeated Measures Assumptions in the Sitka Spruce Example Between subjects When analysing data from different participants, you have to assume homogeneity of variances, i.e., same variances in all groups Homogeneity of variances is tested with Levene's test Within subjects Since your subjects are always the same, there will necessarily be a relation between the measurements in the various conditions in a within subj design. While the relations cannot be independent they should be the same between all pairs. variance A-B variance A-C variance B-C

52 ANOVA with Repeated Measures Sphericity The validity of (some of the) F-tests rests on assumption of the splitplot model in particular, type H matrix including compound symmetry (the equal correlation) Type H matrix: Sphericity assume that variances in the differences between conditions is equal (basically it means that the correlation between treatment levels is the same). Sphericity test: perform a statistical test for the structure, type H matrix (Type H matrix satisfies sphericity). 52

53 53 ANOVA with Repeated Measures Sphericity Start 1 Month 2 Months Start 1 Month Start 2 Months 1 Month 2Months Variance

54 54 ANOVA with Repeated Measures Mauchly Sphericity Test Likelihood ratio test: H 0 : type H matrix is preserved vs H 1 : not H 0 sup H Lw ( ) sup H ( ) 0H L log ~ ( df ) When the significance level of the Mauchly s test is < 0.05, then sphericity cannot be assumed

55 55 Examples Latin square / split-plot design Recall the repeated-measures set-up: Treatment A1 A2 A3 S1 S1 S1 S2 S2 S2 S3 S3 S3 S4 S4 S4 S1, S2, and S3 indicate different subject.

56 56 Examples Four assumptions need to be tested/satisfied Compound Symmetry Homogeneity of variance in each column σ A12 = σ A2 2 = σ A3 2 Homogeneity of covariance between columns σ A1A2 = σ A2A3 = σ A3A1 No A x S Interaction (Additivity) Sphericity Variance of difference scores between pairs are equal σ YA1-YA2 = σ YA1-YA3 = σ YA2-YA3 The variance of each cell cannot be computed.

57 57 Examples Usually, testing sphericity will suffice Sphericity can be tested using the Mauchly test in SAS SAS code proc glm data=temp; class sub; model a1 a2 a3 = sub / nouni; repeated as 3 (1 2 3) polynomial / summary printe; run; quit; Results Sphericity Tests Mauchly's Variables DF Criterion Chi-Square Pr > ChiSq Transformed Variates 2 Det = Orthogonal Components 2 Det = In addition, the PRINTE option provides sphericity tests for each set of transformed variables. If the requested transformations are not orthogonal, the PRINTE option also provides a sphericity test for a set of orthogonal contrasts.

58 58 ANOVA with Repeated Measures Remarks Repeated measures ANOVA is a somewhat ad hoc approach to analysis. For each model we need to find appropriate sums of squares, F-test. If sphericity is violated, Use modified tests (Huynh-Feldt or Greenhouse-eisser). Use MANOVA However, for longitudinal data this covariance model typically has limited scientific appeal. The method assumes balanced data (the same number and timing of measurements across subject). F-test validity is not guaranteed for unbalanced data.

59 59 ANOVA with Repeated Measures Limitation ANOVA/MANOVA assume categorical predictors. ANOVA/MANOVA do not handle time-dependent covariates (predictors measured over time). They assume everyone is measured at the same time (time is categorical) and at equally spaced time intervals. You don t get parameter estimates (just p-values) Missing data must be imputed. They require restrictive assumptions about the correlation structure.

60 60 Analysis Procedure Procedure with PROC GLM

61 Questions?? 61