Anonymous:
Specific Question:
At the end of the Tuesday’s class, by using the clinical data set we found the LINC as below
-0.38695
-0.056443
0.8660428
0.2587766
-0.221139
We use the piece of code “linc=inv(Sp)*diff” to get the LINC. And then you put all these numbers into mslc in SAS as mslc=-0.38695t1-0.056443t2+0.8660428t3+0.2587766t4-0.221139t5. After ran the code, compared with other t values under t0~t4, the one under mslc is the biggest. You also said linear combination gives the most significant result and in this case it gives us the most significant difference between the two groups. Now I have two questions. First, why can we get the LINC though inv(Sp)*diff? How does it make sense? Is that a definition or needed to be proved? Second, after we get the LINC, how and where to use it, for instance, in this case? I still don’t understand after review the class recording.
As far as the definition of the MSLC, it is indeed the result of a mathematical optimization problem. The problem is,
"Consider all possible univariate linear combinations Y=c'X. Consider a hypothesis involving the data Y (eg H0: the means of Y are the same for all groups). Choose a vector c such that the univariate ANOVA F statistic using Y=c'X is a maximum."
The solution is the MSLC, and the optimization theory tells us that there is no other linear combination that is more significant (ie, that has a higher F statistic).
Note that the F statistic is the same whether we use, eg, Y or 10*Y, so the linear combination coefficients obtained from the solution to the optimization problem are not completely unique - they are unique only up to a constant of proportionality. For example,
-3.8695
-0.56443
8.660428
2.587766
-2.21139
is also an MSLC.
In the class example, we used it to state that the largest difference between BA and non-BA students was in differentiation of teaching and non-teaching TTU functions. The measure "differentiation of teaching and non-teaching TTU functions" was suggested by the MSLC.
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General Question:
Recently, I read an essay named Multivariate Analysis Versus Multiple Univariate Analyses from http://management.uta.edu/Casper/MultiStat/Huberty%20&%20Morris%201989%20-%20MANOVA%20vs.%20ANOVA.pdf. It talks about ANOVA and MANOVA. It says, “with multiple outcome variables, the typical analysis approach used in the group-comparison context, at least in the behavioral science, is to either conduct multiple ANOVA or conduct a MANOVA followed by multiple ANOVAS”. In this article we can see from a table of “frequencies of alternative analyses with multiple outcome variables in 1986 journal issues” that the Multiple ANOVAS and MANOVA plus ANOVAS methods are much more popular than MANOVA method. I have no idea about the Multiple ANOVAS before seeing this essay and I am wondering how it works and whether this method is still widely used or not in nowadays. Also, in this essay, “in six articles in which Multiple ANOVAS were used, three justifications for not doing a MANOVA were given: (a) low outcome variable, (b) small number of outcome variables, (c) small design cell frequencies”. Are they really the right reasons? Another confusion is that from the essay it states “the MANOVA-ANOVA approach is seldom, if ever, appropriate”.
Nice article. Everywhere they mention "LDF", simply substitute "MSLC" and you have more answers to your first question. The reason I call it "MSLC" rather than "LDF" as in the article is that "MSLC" is much more general. "LDF" is but a special case of "MSLC."
Recall that the MANOVA tests leave you wanting more. When you reject the MANOVA test, it tells you there is a difference between some groups, for some variables. It does not tell you which groups differ, and it does not tell you which variables are involved in the difference.
If there are p=5 variables in the MANOVA test, you can run 5 univariate ANOVAs, one for each variable. At least that will tell you which variables re involved, but it still won;t tell you which groups are involved.
PROC GLM gives all the univariate ANOVAs first, then the MANOVA test at the end of the output.
These are the multiple ANOVAs they refer to in the paper.
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I think you are missing their main issues when you cite
... three justifications for not doing a MANOVA were given: (a) low outcome variable, (b) small number of outcome variables, (c) small design cell frequencies”. Are they really the right reasons? Another confusion is that from the essay it states “the MANOVA-ANOVA approach is seldom, if ever, appropriate”.
The main issues are given in their "discussion" section at the end, which re-state and emphasize what I have said in class: (1) The MANOVA leaves you wanting more; in fact, you may not even want to do the MANOVA if your interest is in the univariate tests, (2) The most interesting thing about MANOVA might be the MSLC (which they call LDF), as it gets at underlying structure.
Nice paper! Highly recommended for all to distinguish univariate and multivariate, and to better understand MSLC (which they call LDF).
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