Understanding MANOVA in case of a single predictor

Question

I'm trying to understand the statistical analysis I saw in a clinical study. They measured performance of 3 groups of subjects with a series of performance measures (A, B, C, ..., N). The objective was to find out if there was a difference between the performance of the 3 groups, and if yes, then in which ways they differed.

Their analysis consisted of doing a MANOVA like this:

set.seed(100)
group <- rep(c(0,1), each=40)
A <- rnorm(80, 5, .5) + .1 * group
B <- rnorm(80, 9, .3) + .2 * group + .5 * A
C <- rnorm(80, 12, .3) + .2 * group + .7 * B
d.1 <- data.frame(A = A, B = B, C = C, group = group)

fit.manova <- manova(cbind(A, B, C) ~ group, d.1)
summary(fit.manova, test="Pillai")

          Df  Pillai approx F num Df den Df    Pr(>F)    
group      1 0.19669   6.2027      3     76 0.0007949 ***
Residuals 78

When they showed they see significant differences, they continued to perform a series of ANOVA tests for each DV, i.e.:

summary(lm(A ~ group, d.1))
summary(lm(B ~ group, d.1))
summary(lm(C ~ group, d.1))

to find out which performance measures where different between the groups.

What I'd like to know is:

1. Is this approach (MANOVA followed by series of ANOVA) justified? Are there strict assumptions before we could take this path?
2. If yes, should there be some kind of correction for the second step, i.e. series of ANOVAs for individual DVs (multiple-comparisons)? What kid of correction?
3. What is the recommended approach for problems like this with multiple DVs?

EDIT 1: changed the text to include an example code.

EDIT 2: updated the example. DVs are now correlated.

EDIT 3: this is actually a very common situation in studies involving objective measures. Devices typically just spit out an array of measures, even if you don't specifically ask for them. For example, you have two groups, a control and patient group, and do a laboratory gait analysis. The gait analysis systems gives you 50 different gait measures. Your research question might be: do the two groups have similar gait? If not, in which ways do they differ?

Answer 1

1) Is this approach (MANOVA followed by series of ANOVA) justified? Are there strict assumptions before we could take this path?

In the example data you provide, there is no correlation between A, B, and C. Hence, a MANOVA seems beside the point. Unless you are interested in the relationship between A, B, and C, or have some reason to think that the three will be somehow correlated, just skip to the ANOVA.

2) If yes, should there be some kind of correction for the second step, i.e. series of ANOVAs for individual DVs (multiple-comparisons)? What kind of correction?

No. Likely not.

3) What is the recommended approach for problems like this with multiple DVs?

Well, if you know a relationship between them, or know that they will all be influenced by, say, subject, you have two possible options. If you know the relationship between them, try something like Structural Equation Modeling. If there is some reason to suspect that each metric will be influenced in the same way by subject, then you need to control for this.

Might I recommend you see the following paper, as it addresses most of your questions: H. J. Keselman, Carl J. Huberty, Lisa M. Lix, Stephen Olejnik, Robert A. Cribbie, Barbara Donahue, Rhonda K. Kowalchuk, Laureen L. Lowman, Martha D. Petoskey, Joanne C. Keselman and Joel R. Levin. 1998. Statistical Practices of Educational Researchers: An Analysis of their ANOVA, MANOVA, and ANCOVA Analyses. REVIEW OF EDUCATIONAL RESEARCH. 68; 350-386 DOI: 10.3102/00346543068003350

Answer 2

The answer is in how you have simulated your data. It defines the stochastic process that you are assuming, and is enlightening on how you ought to draw inference.

This

set.seed(100)
group <- rep(c(0,1), each=40)
A <- rnorm(80, 5, .5) + .1 * group
B <- rnorm(80, 9, .3) + .2 * group + .5 * A
C <- rnorm(80, 12, .3) + .2 * group + .7 * B
d.1 <- data.frame(A = A, B = B, C = C, group = group)

sets up a multivariate normal distribution.

If you know the dependency structure and it is a directed acyclic graph--like your example where A depends on group, B depends on A and group, C depends on B and group--just do a series of linear regressions and do inference on the coefficient of the group term.

If the dependency structure is more complicated, you should have a look at structural equations models.

Either way, the superior method is to write out your likelihood function (which is the product of multivariate normals) and figure out how to estimate the parameters of your link function (in this case, linear).