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Milds Multivariate analysis of variance

In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is typically followed by significance tests involving individual dependent variables separately. It helps to answer:

1. Do changes in the independent variable(s) have significant effects on the dependent variables?
2. What are the relationships among the dependent variables?
3. What are the relationships among the independent variables?

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• ✪ 1 MANOVA - An Introduction
• ✪ Introduction to One-Way Multivariate Analysis of Variance (One-Way MANOVA)
• ✪ MANOVA - SPSS (part 1)
• ✪ Introduction to MANOVA, MANOVA vs ANOVA n MANOVA using R
• ✪ Mod-01 Lec-16 Multivariate Analysis of Variance (MANOVA)

Transcription

Hello, my friends. We're about to launch our inquiry into MANOVA. It's a beautiful morning here in Texas today. This old hot coffee sure is good. Mmm, mmm. Well, let's launch this little endeavor. MANOVA is about comparing one or more dependent variables across two or more groups. And MANOVA stands for multivariate analysis of variance. And ANOVA's a special case of MANOVA. And we've just gone through ANOVA. Now as I conduct the analyses in SPSS and later videos, I'm going to assume that you watched the video series for ANOVA. Do you remember the following? We started this out way back there in comparative analysis talking about t-Tests. And we said a t-Test has two groups, and compares them across the value of one dependent variable across two different groups. And of course, these values were normally distributed. Remember that we moved from t-Test to ANOVA and said ANOVA instead, can have two or more groups. And I drew this diagram with three, which compared across one dependent variable. And we said that t-Test is indeed a special case of ANOVA. That a t-Test is an ANOVA with only two groups. Well, now we move forward to MANOVA. MANOVA can have two or more groups. So here I've done three groups. And when we do MANOVA we can look at one or more dependent variables. In this diagram I have three groups, and we're looking at the values of two different dependent variables. See? ANOVA is a special case of MANOVA. ANOVA is a MANOVA with only one dependent variable. So MANOVA's pretty cool. Now let me tell you a little bit about how MANOVA works. We have the dependent variable one for each of these. We have the dependent variable two for each of these. So what MANOVA does is MANOVA does a curve fit of those variables. And it's important, again, that they have normality in those dependent variables. Because MANOVA then does a curve fit, and then MANOVA compares these unique curves. Do you see that? You have variable one, variable two for the first group. Variable one, value two for the second group. And of course, the same dependent variables for the third group. So it takes value one and value two and builds them into a curve fit, and then compares those curves. That's fairly profound. Now, the basic assumptions of MANOVA include the following. Certainly, you have independent random sampling, and you have a level of measurement of the variables. Now, you will control how the independent sampling is done. The level of the variables means simply that the groupings are nominal or ordinal. They're grouped out there. Male, female, different institutions, that type of things. And then the dependent variables need to either be at least scale-continuous type variables. Ratio variables make the best dependent variables in MANOVA, in my opinion. We have to have linearity of the dependent variables. Now that means that those dependent variables will be correlated. And we would establish that with a correlational analysis like a Pearson r. We must also have multivariate normality. That means that all of the dependent variables are normally distributed. And then we must have multivariate homogeneity of variance within groups, and multivariate homogeneity of variance between groups. Now, we'll establish some of these fairly easily. The test for the basic assumption of MANOVAs include the following. Now, I want to mention to you that the sampling and the data distributions are very easily under your control. You don't have to really do great statistical sampling there. But when we do the linearity of the dependent variable, we're going to run at Pearson r or some sort of correlational coefficient. When we look at the normality of the dependent variables, we'll do kurtosis and skewness as we did with ANOVA. We will do multivariate homogeneity of variance between the groups. So that'll be a Levene's test. And we'll do the multivariate homogeneity of covariance between the groups, which will be a Box M. That really is fairly cool. Now, let's move onto the research questions for MANOVA. And they are focused on differences. This is a causal comparative. We call it comparative analysis. So when we compare, we would be looking at differences. It's always in good form when you do a quantitative analysis, you ought to start by getting your descriptives. And this one is, what are the percentages of female, Hispanic, and African-American students in two-year, degree-granting, public for-profit and private not-for-profit colleges in Texas in 2011? Obviously, my groupings are my for-profit, my not-for-profit, and my public, and my public, for-profit and not-for-profit colleges. I have three groups. I will be looking at three variables. The percentages of female, percentages of Hispanic, and the percentage of African-American students. But all this does is just call for my descriptives. Now here's the real MANOVA research questions. Do differences exist in the percentages of female, Hispanic, and African-American students, between or among public for-profit, and private not-for-profit, two-year degree-granting colleges in Texas? Now, notice that I said between or among. If it had just been two, I would have said between. But there's two or more. There's actually three. So I said between or among. The question focuses on differences. Exist in the-- here are the dependent variables-- between or among the grouping variables. And it's in Texas in 2011. Now, the hypothesis should match the research questions, in my opinion. Now, I want to remind you that there are different ways of doing this. There are many ways of writing questions, many ways of writing hypotheses. There are many right ways of doing it. But however, there are some wrong ways. This is the way that I choose to do it. I have a descriptive question. I have a methodology question. And now I have my hypotheses for my methodology question. And you notice I have the plural hypotheses, because I'm going to include my null hypothesis and my alternative hypothesis. And my null hypothesis is that no differences exist in the percentage of female, Hispanic, and African-American students between or among public for-profit, and private not-for-profit two-year degree-granting colleges in Texas. Let's go back to the question. Do you see that my hypothesis exactly matches my question? The null is that there are no differences. The alternate is that differences exist. OK, the protocol for conducting MANOVA follows. Always, you provide your descriptives. You have to obtain those. I You examine the data level and the assumptions. You conduct the MANOVA. You can do a post hoc analysis as needed, if you find significant difference. Some folks disagree with doing post hoc analysis. You know, guys, there's researchers that agree with everything, and there's researchers that disagree with everything. You just have to make your own decision. And if you have significance, you do post hoc in my opinion. And then you evaluate the effect size and power, if the significance is found. Again, I want to thank you very much for your loyal support as we have pursued this mighty endeavour. I just can't tell you how much I appreciate your really getting this on for us. So, as always, your patronage keeps my family fed. Man, I need to make the money from teaching this course. Live long and prosper. And I think it may be more appropriate to say, may the odds be ever in your favor. Hope to meet you on the other side.

Relationship with ANOVA

MANOVA is a generalized form of univariate analysis of variance (ANOVA), although, unlike univariate ANOVA, it uses the covariance between outcome variables in testing the statistical significance of the mean differences.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear. The diagonal entries are the same kinds of sums of squares that appear in univariate ANOVA. The off-diagonal entries are corresponding sums of products. Under normality assumptions about error distributions, the counterpart of the sum of squares due to error has a Wishart distribution.

MANOVA is based on the product of model variance matrix, $\Sigma _{\text{model}}$ and inverse of the error variance matrix, $\Sigma _{\text{res}}^{-1}$ , or $A=\Sigma _{\text{model}}\times \Sigma _{\text{res}}^{-1}$ . The hypothesis that $\Sigma _{\text{model}}=\Sigma _{\text{residual}}$ implies that the product $A\sim I$ . Invariance considerations imply the MANOVA statistic should be a measure of magnitude of the singular value decomposition of this matrix product, but there is no unique choice owing to the multi-dimensional nature of the alternative hypothesis.

The most common statistics are summaries based on the roots (or eigenvalues) $\lambda _{p}$ of the $A$ matrix:

• Samuel Stanley Wilks' $\Lambda _{\text{Wilks}}=\prod _{1...p}(1/(1+\lambda _{p}))=\det(I+A)^{-1}=\det(\Sigma _{\text{res}})/\det(\Sigma _{\text{res}}+\Sigma _{\text{model}})$ distributed as lambda (Λ)
• the Pillai-M. S. Bartlett trace, $\Lambda _{\text{Pillai}}=\sum _{1...p}(\lambda _{p}/(1+\lambda _{p}))=\mathrm {tr} (A(I+A)^{-1})$ • the Lawley-Hotelling trace, $\Lambda _{\text{LH}}=\sum _{1...p}(\lambda _{p})=\mathrm {tr} (A)$ • Roy's greatest root (also called Roy's largest root), $\Lambda _{\text{Roy}}=\max _{p}(\lambda _{p})=\|A\|_{\infty }$ Discussion continues over the merits of each, although the greatest root leads only to a bound on significance which is not generally of practical interest. A further complication is that, except for the Roy's greatest root, the distribution of these statistics under the null hypothesis is not straightforward and can only be approximated except in a few low-dimensional cases. An algorithm for the distribution of the Roy's largest root under the null hypothesis was derived in  while the distribution under the alternative is studied in.

The best-known approximation for Wilks' lambda was derived by C. R. Rao.

In the case of two groups, all the statistics are equivalent and the test reduces to Hotelling's T-square.

Correlation of dependent variables

MANOVA's power is affected by the correlations of the dependent variables and by the effect sizes associated with those variables. For example, when there are two groups and two dependent variables, MANOVA's power is lowest when the correlation equals the ratio of the smaller to the larger standardized effect size.