Lecture 6: Single-classification multivariate ANOVA (k-group( MANOVA)

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1 Lecture 6: Single-classification multivariate ANOVA (k-group( MANOVA) Rationale and MANOVA test statistics underlying principles MANOVA assumptions Univariate ANOVA Planned and unplanned Multivariate ANOVA comparisons (MANOVA): principles and procedures L6. When to use ANOVA Tests for effect of discrete independent variables. Each independent variable is called a factor, and each factor may have two or more levels or treatments (e.g. crop yields with nitrogen (N) or nitrogen and phosphorous (N + P) added). ANOVA tests whether all group means are the same. Use when number of levels (groups) is greater than two. Frequency µ C µ N µ N+P Yield Control Experimental (N) Experimental (N+P) L6. Why not use multiple -sample tests? For k comparisons, the probability of accepting a true H 0 for all k is ( - α) k. For 4 means, ( - α) k = (0.95) 6 =.735. So α (for all comparisons) = So, when comparing the means of four samples from the same population, we would expect to detect significant differences among at least one pair 7% of the time. Frequency µ C : µ N+P µ c :µ N µ N :µ N+P µ C µ N µ N+P Control Yield Experimental (N) Experimental (N+P) L6.3

2 What ANOVA does/doesn t do Tells us whether all group means are equal (at a specified α level)......but if we reject H 0, the ANOVA does not tell us which pairs of means are different from one another. Frequency Frequency Control Experimental (N) Experimental (N+ P) µ C µ N µ N+P µ C µ N µ N+P Yield L6.4 Model I ANOVA: effects of temperature on trout growth 3 treatments determined (set) by investigator. 0.0 Dependent variable is 0.6 growth rate (λ), factor (X) is temperature. 0. Since X is controlled, we 0.08 can estimate the effect of 0.04 a unit increase in X (temperature) on λ (the 0.00 effect size) and can predict λ at Water temperature ( C) other temperatures. L6.5 Growth rate λ (cm/day) Model II ANOVA: geographical variation in body size of black bears 3 locations (groups) sampled from set of possible locations. Dependent variable is body size, factor (X) is location. Even if locations differ, we have no idea what factors are controlling this variability... so we cannot predict body size at other locations. Body size (kg) Riding Kluane Mountain Algonquin L6.6

3 Model differences In Model I, the putative causal factor(s) can be manipulated by the experimenter, whereas in Model II they cannot. In Model I, we can estimate the magnitude of treatment effects and make predictions, whereas in Model II we can do neither. In one-way (single classification) ANOVA, calculations are identical for both models but this is NOT so for multiple classification ANOVA! L6.7 How is it done? And why call it ANOVA? In ANOVA, the total variance in the dependent variable is partitioned into two components: among-groups: variance of means of different groups (treatments) within-groups (error): variance of individual observations within groups around the mean of the group L6.8 The general ANOVA model The general model is: Y ij = µ + α i+ ε ij ε 4 µ α Y µ ANOVA algorithms fit the above model (by least squares) to estimate the Y α i s. µ H 0 : all α i s = 0 Group Group Group 3 µ =µ = µ = µ 3 α =α =α 3 = 0 Group L6.9

4 Partitioning the total sums of squares µ Y µ µ 3 µ Total SS Model (Groups) SS Error SS Group Group Group 3 L6.0 The ANOVA table Source of Variation Sum of Squares Degrees of freedom (df) Mean Square F Total Error k ni (Yij Y) i= j = n - SS/df k Groups n i ( Y i Y ) k - SS/df i = k ni (Yi j Yi) i= j= n - k SS/df MS groups MS error L6. Use of single-classification MANOVA Data set consists of k groups ( treatments ), with n i observations per group, and p variables per observation. Question: do the groups differ with respect to their multivariate means? In single-classification ANOVA, we assume that a single factor is variable among groups, i.e., that all other factors which may possible affect the variables in question are randomized among groups. L6.

5 Examples Good(ish) 4 different concentrations of some suspected contaminant; 0 young fish randomly assigned to each treatment; at age months, a number of measurements taken on each surviving fish. Bad(ish) 0 young fish reared in 4 different treatments, each treatment consisting of water samples taken at different stages of treatment in a water treatment plant. L6.3 Multivariate variance: a geometric interpretation Univariate variance is a measure of the volume occupied by sample points in one dimension. Multivariate variance involving m variables is the volume occupied by sample points in an m -dimensional space. X Larger variance X Occupied volume X Smaller variance X L6.4 Multivariate variance: effects of correlations among variables X No correlation Correlations between pairs of variables reduce the volume occupied by sample points and hence, reduce the multivariate variance. Occupied volume X Positive correlation X Negative correlation X L6.5

6 C and the generalized multivariate variance L C = C N M O Q P = 3 4 c o r = = 05. = cos θ, θ = 60 The determinant of the ss sample covariance matrix C is a generalized multivariate variance because area of a h parallelogram with sides θ s given by the individual standard deviations and s angle determined by the correlation between opposite h variables equals the sin 60 = = ; h = 3. hypotenuse determinant of C. Area = Base Height = 3, Area = C L6.6 ANOVA vs MANOVA: procedure In ANOVA, the total sums of squares is partitioned into a within-groups (SS w ) and between-group SS b sums of squares: SST = SSb + SSw In MANOVA, the total sums of squares and cross-products (SSCP) matrix is partitioned into a within groups SSCP (W) and a between-groups SSCP (B) T= B+ W L6.7 ANOVA vs MANOVA: hypothesis testing In ANOVA, the null hypothesis is: H µ µ µ 0 : = = = k In MANOVA, the null hypothesis is H = = = 0 : µ µ µ k This is tested by means of the F statistic: MS MS b b F = = MS w MS e This is tested by (among other things) Wilk s lambda: W W Λ= =,0 Λ T B+ W L6.8

7 SSCP matrices: within, between, and total The total (T) SSCP matrix (based on p variables X, X,, X p ) in a sample of objects belonging to m groups G, G,, G m with sizes n, n,, n m can be partitioned into withingroups (W) and betweengroups (B) SSCP matrices: T = B+ W x ijk x jk x k n j t = ( x x )( x x ) t m Value of variable X k for ith observation in group j Mean of variable X k for group j Overall mean of variable X k rc, w Element in row r and rc column c of total (T, t) and within (W, w) SSCP rc ijr r ijc c j= i= m n j rc = ijr jr ijc jc j= i= w ( x x )( x x ) L6.9 The distribution of Λ Unlike F, Λ has a very complicated distribution but, given certain assumptions it can be approximated b as Bartlett s χ (for moderate to large samples) or Rao s F (for small samples) χ = [( N ) 0.5( p+ k)]ln Λ df = p( k ) F / s = Λ ms p( k )/+ Λ / s pk ( ) m= N ( p+ k)/ p ( k ) 4 s = p + ( k ) 5 df= pk ( ), ms pk ( )/+ L6.0 Assumptions All observations are independent (residuals are uncorrelated) Within each sample (group), variables (residuals) are multivariate normally distributed Each sample (group) has the same covariance matrix (compound symmetry) L6.

8 Effect of violation of assumptions Assumption Effect on α Effect on power Independence of observations Normality Equality of covariance matrices Very large, actual α much larger than nominal α Small to negligible Small to negligible if group Ns similar, if Ns very unequal, actual α larger than nominal α Large, power much reduced Reduced power for platykurtotic distributions, skewness has little effect Power reduced, reduction greater for unequal Ns. L6. Checking assumptions in MANOVA Independence (intraclass correlation, ACF) No Use group means as unit of analysis Assess MV normality Yes N i > 0 Check group sizes N i < 0 MVN graph test Check Univariate normality L6.3 Checking assumptions in MANOVA (cont d) MV normal? Most variables normal? No Transform offending variables Yes Yes Check homogeneity of covariance matrices No Group sizes more or less equal (R <.5)? Yes No Yes END Yes Groups reasonably large (> 5)? Transform variables, or adjust α L6.4

9 Then what? Question Procedure What variables are responsible for detected differences among groups? Do certain groups (determined beforehand) differ from one another? Which pairs of groups differ from one another (groups not specified beforehand)? Check univariate F tests as a guide; use another multivariate procedure (e.g. discriminant function analysis) Planned multiple comparisons Unplanned multiple comparisons L6.5 What are multiple comparisons? Pair-wise comparisons of different treatments These comparisons may involve group means, medians, variances, etc. for means, done after ANOVA In all cases, H 0 is that the groups in question do not differ. Frequency µ C : µ N+P µ c :µ N µ N :µ N+P µ C µ N µ N+P Control Yield Experimental (N) Experimental (N+P) L6.6 Types of comparisons Y planned (a priori): independent of ANOVA results; theory predicts Planned which treatments should be different. X X X 3 X 4 X 5 unplanned (a posteriori): unplanned depend on ANOVA results; unclear which Y treatments should be different. Test of significance are very different between the X X X 3 X 4 X 5 two! L6.7

10 Planned comparisons (a( a priori contrasts): catecholamine levels in stressed fish Comparisons of interest are 0.7 determined by experimenter 0.6 beforehand based on theory 0.5 and do not depend on 0.4 ANOVA results. 0.3 Prediction from theory: 0. catecholamine levels 0. increase above basal levels 0.0 only after threshold PA O = torr is reached. PA O (torr) So, compare only treatments 50 above and below 30 torr (N T = Predicted threshold ). L6.8 [Catecholamine] Unplanned comparisons (a( a posteriori contrasts): catecholamine levels in stressed fish Comparisons are determined by ANOVA results. Prediction from theory: catecholamine levels increase with increasing PA O. So, comparisons between any pairs of treatments may be warranted (N T = ). [Catecholamine] PA O (torr) Predicted relationship L6.9 The problem: controlling experiment-wise α error For k comparisons, the probability of accepting H 0 (no difference) is ( - α) k. For 4 treatments, ( - α) k = (0.95) 6 =.735, so experiment-wise α (α e ) = Thus we would expect to 0.0 reject H 0 for at least one paired comparison about Number of treatments 7% of the time, even if all four treatments are Nominal α =.05 identical. L6.30 Experiment-wise α (α e )

11 Unplanned comparisons: Hotelling T and univariate F tests Follow rejection of null Then use univariate t- in original MANOVA by tests to determine all pairwise multivariate which variables are tests using Hotelling T contributing to the to determine which detected pairwise groups are different differences but test at modified α opinion is divided as to maintain overall to whether these nominal type I error should be done at a rate (e.g. Bonferroni modified α. correction) L6.3 How many different variables for a MANOVA? In general, try to use a Measurement error is small number of variables multiplicative among because: variables: the larger the In MANOVA, power number of variables, generally declines with the larger the increasing number of measurement noise variables. Interpretation is easier If a number of variables with a smaller number are included that do not of variables differ among groups, this will obscure differences on a few variables L6.3 How many different variables for a MANOVA : recommendation Choose variables carefully, attempting to keep them to a minimum Try to reduce the number of variables by using multivariate procedures (e.g. PCA) to generate composite, uncorrelated variables which can then be used as input. Use multivariate procedures (such as discriminant function analysis) to optimize set of variables. L6.33

Chapter 7, continued: MANOVA

Chapter 7, continued: MANOVA The Multivariate Analysis of Variance (MANOVA) technique extends Hotelling T 2 test that compares two mean vectors to the setting in which there are m 2 groups. We wish to