11 Comparison of Several Multivariate Means Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

22 Paired Comparisons Measurements are recorded under different sets of conditions See if the responses differ significantly over these sets Two or more treatments can be administered to the same or similar experimental units Compare responses to assess the effects of the treatments

33 Example 6.1: Effluent Data from Two Labs

44 Single Response (Univariate) Case

55 Multivariate Extension: Notations

66 Result 6.1

77 Test of Hypotheses and Confidence Regions

88 Example 6.1: Check Measurements from Two Labs

99 Experiment Design for Paired Comparisons n Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random Treatments 1 and 2 assigned at random

1010 Alternative View

1111 Repeated Measures Design for Comparing Measurements q treatments are compared with respect to a single response variable Each subject or experimental unit receives each treatment once over successive periods of time

1212 Example 6.2: Treatments in an Anesthetics Experiment 19 dogs were initially given the drug pentobarbitol followed by four treatments Halothane Present Absent CO 2 pressure LowHigh 12 34

1313 Example 6.2: Sleeping-Dog Data

1414 Contrast Matrix

1515 Test for Equality of Treatments in a Repeated Measures Design

1616 Example 6.2: Contrast Matrix

1717 Example 6.2: Test of Hypotheses

1818 Example 6.2: Simultaneous Confidence Intervals

1919 Comparing Mean Vectors from Two Populations Populations: Sets of experiment settings Without explicitly controlling for unit- to-unit variability, as in the paired comparison case Experimental units are randomly assigned to populations Applicable to a more general collection of experimental units

2020 Assumptions Concerning the Structure of Data

2121 Pooled Estimate of Population Covariance Matrix

2222 Result 6.2

2323 Proof of Result 6.2

2424 Wishart Distribution

2525 Test of Hypothesis

2626 Example 6.3: Comparison of Soaps Manufactured in Two Ways

2727 Example 6.3

2828 Result 6.3: Simultaneous Confidence Intervals

2929 Example 6.4: Electrical Usage of Homeowners with and without ACs

3030 Example 6.4: Electrical Usage of Homeowners with and without ACs

3131 Example 6.4: 95% Confidence Ellipse

3232 Bonferroni Simultaneous Confidence Intervals

3333 Result 6.4

3434 Proof of Result 6.4

3535 Remark

3636 Example 6.5

37 Multivariate Behrens-Fisher Problem Test H 0 :  1 -  2 =0 Population covariance matrices are unequal Sample sizes are not large Populations are multivariate normal Both sizes are greater than the number of variables 37

38 Approximation of T 2 Distribution 38

39 Confidence Region 39

40 Example 6.6 Example 6.4 data 40

4141 Example 6.10: Nursing Home Data Nursing homes can be classified by the owners: private (271), non-profit (138), government (107) Costs: nursing labor, dietary labor, plant operation and maintenance labor, housekeeping and laundry labor To investigate the effects of ownership on costs

4242 One-Way MANOVA

4343 Assumptions about the Data

4444 Univariate ANOVA

4545 Univariate ANOVA

4646 Univariate ANOVA

4747 Univariate ANOVA

4848 Concept of Degrees of Freedom

4949 Concept of Degrees of Freedom

5050 Examples 6.7 & 6.8

5151 MANOVA

5252 MANOVA

5353 MANOVA

5454 Distribution of Wilk’s Lambda

5555 Test of Hypothesis for Large Size

5656 Popular MANOVA Statistics Used in Statistical Packages

5757 Example 6.9

5858 Example 6.8

5959 Example 6.9

6060 Example 6.9

6161 Example 6.10: Nursing Home Data Nursing homes can be classified by the owners: private (271), non-profit (138), government (107) Costs: nursing labor, dietary labor, plant operation and maintenance labor, housekeeping and laundry labor To investigate the effects of ownership on costs

6262 Example 6.10

6363 Example 6.10

6464 Example 6.10

6565 Bonferroni Intervals for Treatment Effects

6666 Result 6.5: Bonferroni Intervals for Treatment Effects

6767 Example 6.11: Example 6.10 Data

68 Test for Equality of Covariance Matrices With g populations, null hypothesis H 0 :  1 =  2 =... =  g =  Assume multivariate normal populations Likelihood ratio statistic for testing H 0

69 Box’s M-Test

70 Example 6.12 Example nursing home data

7171 Example 6.13: Plastic Film Data

7272 Two-Way ANOVA

7373 Effect of Interactions

7474 Two-Way ANOVA

7575 Two-Way ANOVA

7676 Two-Way MANOVA

7777 Two-Way MANOVA

7878 Two-Way MANOVA

7979 Two-Way MANOVA

8080 Bonferroni Confidence Intervals

8181 Example 6.13: MANOVA Table

8282 Example 6.13: Interaction

8383 Example 6.13: Effects of Factors 1 & 2

8484 Profile Analysis A battery of p treatments (tests, questions, etc.) are administered to two or more group of subjects The question of equality of mean vectors is divided into several specific possibilities –Are the profiles parallel? –Are the profiles coincident? –Are the profiles level?

8585 Example 6.14: Love and Marriage Data

8686 Population Profile

8787 Profile Analysis

8888 Test for Parallel Profiles

8989 Test for Coincident Profiles

9090 Test for Level Profiles

9191 Example 6.14

9292 Example 6.14: Test for Parallel Profiles

9393 Example 6.14: Sample Profiles

9494 Example 6.14: Test for Coincident Profiles

9595 Example 6.15: Ulna Data, Control Group

9696 Example 6.15: Ulna Data, Treatment Group

9797 Comparison of Growth Curves

9898 Comparison of Growth Curves

9999 Example 6.15

Example 6.16: Comparing Multivariate and Univariate Tests

Example 6.14: Comparing Multivariate and Univariate Tests

Strategy for Multivariate Comparison of Treatments Try to identify outliers –Perform calculations with and without the outliers Perform a multivariate test of hypothesis Calculate the Bonferroni simultaneous confidence intervals –For all pairs of groups or treatments, and all characteristics

Importance of Experimental Design Differences could appear in only one of the many characteristics or a few treatment combinations Differences may become lost among all the inactive ones Best preventative is a good experimental design –Do not include too many other variables that are not expected to show differences