Multivariate Analysis of Variance MANOVA Multivariate Analysis of Variance

One way Multivariate Analysis of Variance (MANOVA) Comparing k p-variate Normal Populations

Comparing k mean vectors Situation We have k normal populations Let denote the mean vector and covariance matrix of population i. i = 1, 2, 3, … k. Note: we assume that the covariance matrix for each population is the same.

We want to test against

The data Assume we have collected data from each of k populations Let denote the n observations from population i. i = 1, 2, 3, … k.

The summary statistics Sample mean vectors Sample covariance matrices S1, S2, etc.

Computing Formulae: Compute 1) 2) 3)

4) 5)

Let = the Between SS and SP matrix

Let = the Within SS and SP matrix

The Manova Table Source SS and SP matrix Between Within

There are several test statistics for testing against

1. Roy’s largest root 2. Wilk’s lambda (L) This test statistic is derived using Roy’s union intersection principle 2. Wilk’s lambda (L) This test statistic is derived using the generalized Likelihood ratio principle

3. Lawley-Hotelling trace statistic 4. Pillai trace statistic (V)

Example In the following study, n = 15 first year university students from three different School regions (A, B and C) who were each taking the following four courses (Math, biology, English and Sociology) were observed: The marks on these courses is tabulated on the following slide:

The data

Summary Statistics

Computations : 1) 2) 3)

4) =

5) =

Now = the Between SS and SP matrix

Let = the Within SS and SP matrix

Using SPSS to perform MANOVA

Selecting the variables and the Factors

The output

Univariate Tests

Profile Analysis

Repeated Measures Designs

In a Repeated Measures Design We have experimental units that may be grouped according to one or several factors (the grouping factors) Then on each experimental unit we have not a single measurement but a group of measurements (the repeated measures) The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors)

Example In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. The enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for n = 15 cardiac surgical patients.

The data is given in the table below. Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

The subjects are not grouped (single group). There is one repeated measures factor -Time – with levels Day 0, Day 1, Day 2, Day 7 This design is the same as a randomized block design with Blocks = subjects

The Anova Table for Enzyme Experiment The Subject Source of variability is modelling the variability between subjects The ERROR Source of variability is modelling the variability within subjects

Example : (Repeated Measures Design - Grouping Factor) In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

The 24 patients were randomly divided into three groups of n= 8 patients. The first group of patients were left untreated as a control group while the second and third group were given drug treatments A and B respectively. Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery for three treatment groups (control, Drug A, Drug B)

The subjects are grouped by treatment control, Drug A, Drug B There is one repeated measures factor -Time – with levels Day 0, Day 1, Day 2, Day 7

The Anova Table There are two sources of Error in a repeated measures design: The between subject error – Error1 and the within subject error – Error2

Tables of means Drug Day 0 Day 1 Day 2 Day 7 Overall Control 118.63 77.88 60.50 55.75 78.19 A 103.25 68.25 52.00 51.50 68.75 B 103.38 69.38 54.13 51.50 69.59 Overall 108.42 71.83 55.54 52.92 72.18

Example : Repeated Measures Design - Two Grouping Factors In the following example , the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task. In addition the researcher was interested in how the error rates also changed over time. Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.

The number of errors committed at each instance is tabulated below.

The Anova Table