Multivariate analysis of variance and covariance

Transcription

1 Introduction Multivariate analysis of variance and covariance Univariate ANOVA: have observations from several groups, numerical dependent variable. Ask whether dependent variable has same mean for each group, or different mean. (Followup: ask which means significantly different from others.) Mathematically: one numerical dependent, one or more categorical independent. ANCOVA: also have a numerical independent variable ANCOVA example Two drugs, A and B, are supposed to improve people s moods. An experiment is carried out on 20 volunteers, randomly assigned to the two drugs. Each subject is measured at 8:00 am, given appropriate drug, then measured again at 10:00 am. Dependent is score at 10:00 am ( after ), independent are drug (categorical), score at 8:00 am (numerical) ( before ). SAS code: class drug; model after = drug before / solution; lsmeans drug; Model as whole is significant: Dependent Variable: AFTER Source DF Sum of Squares F Value Pr > F Model Error Corrected Total Look at type III SS s: Source DF Type III SS F Value Pr > F DRUG BEFORE After score depends on before score; after scores different for drugs after allowing for differences in before scores

2 Parameter estimates: T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT B DRUG a B b B... BEFORE Mean score on drug A 5.15 higher than for drug B. Can draw regression lines predicting after from before for each group. Parallel here (drug A line always 5.15 above drug B line). Relax parallelism assumption by adding interaction to model. Add before*drug to model line, get test: Source DF Type III SS F Value Pr > F DRUG BEFORE BEFORE*DRUG Interaction not significant. Parallel lines (and previous analysis) OK Multiple analysis of variance In discriminant analysis, wanted to test that group means really different. In general, have several outcome variables for each group in experiment. Could do sequence of one-way ANOVAs. But (as with Hotelling s T-squared) may be deceived by correlation between variables. So want single test for all variables at once. Mathematics of ANOVA: observations y ij from group i, means ȳ i for group i, ȳ overall, n obs. per group. Total SS equals group SS plus error SS. (y ij ȳ) 2 = n i (ȳ i ȳ) 2 + (y ij ȳ i ) 2. Depends on comparison between SS s around overall mean and around group means. MANOVA: now have sum of squares matrices (like var/cov). Decomposition now as below (all y s vectors, length number of variables): (y ij ȳ)(y ij ȳ) = n (ȳ i ȳ)(ȳ i ȳ) + (y ij ȳ i )(y ij ȳ i ). i More concisely, S T = S G + S E (total is group plus error). Cannot now divide to get F -statistic, because S s matrices. Idea: use feature of matrices to mimic F -statistic:

3 Example: advertising messages Wilks s Lambda: Λ = S E / S T. Pillai s Trace: tr(s G S 1 T ). Hotelling-Lawley Trace: tr(s 1 E S G) Roy s Maximum Root: largest eigenvalue of S 1 E S G. Note ways of getting number from matrix: determinant, sum of diagonal elements, largest eigenvalue. All above tests have F approximations. No consensus on which test best, so look for consistency between all. Text example: comparing two advertisements. Ad 1: hard sell, Ad 2: humorous. Each subject then asked to rate how much they liked the product, and how likely they were to buy it (1 7 scale). 2 outcome variables, 2 groups (ads). Plot (p. 414) shows little difference in means of likability or buy intention (relative to SD). SAS PROC GLM does MANOVA. Code as below for variables ad, like, buy : class ad; model like buy = ad; manova h=_all_; Output: individual ANOVAs for each variable (not significant) plus MANOVA: Wilks Lambda Pillai s Trace Hotelling-Lawley Trace Roy s Greatest Root Strongly significant difference between groups. Is difference between effects of ads. Look again at Figure (page 414): difference in groups is that for ad 1, consumers found it less likable and had greater intention to buy. Is a multivariate difference, needs multivariate method to find it. In general, when null hypothesis (no difference) rejected, can do discriminant analysis to describe differences between groups. Which groups are different? In ANOVA, follow up rejection of F -test with procedure to determine which groups significantly different. Problem: with g groups, compare g(g 1)/2 pairs of groups do this many tests simultaneously. If not careful, reject some H 0 s by chance. Solution: Bonferroni, do each of k simultaneous tests at level α/k. Apply Bonferroni here by doing two-sample tests for each pair of groups on each dependent variable. Use S E divided by error df as pooled var/cov matrix

4 Illustrate with iris data. 3 groups: setosa, versicolor, virginica; 4 variables (sepal/petal length/width) so total of 12 tests. Use α = MANOVA: Wilks Lambda Pillai s Trace Hotelling-Lawley Trace Roy s Greatest Root No doubt that means different. Add following code to PROC GLM: contrast set-ver species 1-1 0; contrast set-vir species 1 0-1; contrast ver-vir species 0 1-1; Idea: contrast between setosa and versicolor is contrast between first and second species, etc. Output: Dependent Variable: SEPALLEN set-ver set-vir ver-vir Dependent Variable: SEPALWID set-ver set-vir ver-vir Dependent Variable: PETALLEN set-ver set-vir ver-vir Dependent Variable: PETALWID set-ver set-vir ver-vir Repeated measures Common to use standard ANOVA-type experimental design, but to collect output at several time points. Look for differences among At α = 0.01, assess significance of each at level 0.01/12 = All species are significantly different on all variables except for versicolor and virginica on sepal width. groups that are consistent over time. Several approaches to analysis. One way is to use MANOVA and take advantage of relatedness of dependent variables. Illustrate with NEWFOOD data from text. Simple model: test for effect of price (only) on sales at 3 time points. Code: class price adv; model sales1 sales2 sales3 = price; manova h=_all_; repeated time;

5 Output of MANOVA tests effect of price on overall sales: Wilks Lambda Pillai s Trace Hotelling-Lawley Trace Roy s Greatest Root Price definitely affects sales (over all times). Output from repeated assesses effect of time on sales: Wilks Lambda Pillai s Trace Hotelling-Lawley Trace Roy s Greatest Root Also time/price interaction effect on sales: Wilks Lambda Pillai s Trace Hotelling-Lawley Trace Roy s Greatest Root Not significant. Effect of price on sales consistent over time; effect of time on sales consistent for each price group. Understand test results by looking at means (edited): Sales patterns change over time PRICE= SALES SALES SALES PRICE= SALES SALES SALES PRICE= SALES SALES SALES For each price level, sales highest after 2 months, sales for 4 and 6 months similar. For all times, sales highest when price is 1 (low) and lowest when price is 3 (high). Consistency of pattern supports lack of interaction between price and time