# Multivariate differences between groups controlling for one factor (MANOVA)

I have a sample of 100 participants who have scores on 5 different variables (V1-V5). Some participants took part in a workshop, others did not. I am interested in investigating the influence of workshop participation and gender on the scores of variables. For gender and workshop participation significant differences where shown (see below). My data looks the following:

  V1 V2 V3 V4 V5 gender workshop
1  2  2  5  5  4   male      yes
2  4  3  4  3  3 female      yes
3  3  5  5  3  1 female      yes
4  1  5  5  5  1 female      yes
5  5  2  2  4  5 female       no
6  5  4  5  1  7   male      yes
...


Here is what I did:

MANOVA

library (car)

DV <- as.matrix(x[, 1:5])
output = lm(DV ~ gender*workshop, data = x, contrasts=list(gender = contr.sum, workshop = contr.sum))
manova_out = Manova(output, type = "III")
summary (manova_out, multivariate = T)

Multivariate Tests: gender
Df test stat approx F num Df den Df  Pr(>F)
Pillai            1 0.1336421 2.838336      5     92 0.01983 *
Wilks             1 0.8663579 2.838336      5     92 0.01983 *
Hotelling-Lawley  1 0.1542574 2.838336      5     92 0.01983 *
Roy               1 0.1542574 2.838336      5     92 0.01983 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Multivariate Tests: workshop
Df test stat approx F num Df den Df   Pr(>F)
Pillai            1 0.0967188 1.970179      5     92 0.090415 .
Wilks             1 0.9032812 1.970179      5     92 0.090415 .
Hotelling-Lawley  1 0.1070749 1.970179      5     92 0.090415 .
Roy               1 0.1070749 1.970179      5     92 0.090415 .
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Multivariate Tests: gender:workshop
Df test stat  approx F num Df den Df  Pr(>F)
Pillai            1 0.0242859 0.4579838      5     92 0.80649
Wilks             1 0.9757141 0.4579838      5     92 0.80649
Hotelling-Lawley  1 0.0248904 0.4579838      5     92 0.80649
Roy               1 0.0248904 0.4579838      5     92 0.80649


Workshop participation appears to have a significant effect on the scores of variables.

### My questions:

Is this effect maybe just due to the larger amount of females in the sample? How can I make sure that my significant results for workshop participation are only due to workshop participation and not gender? Is there a way of controlling for gender?

Would checking for an interaction effect between gender and workshop be useful? Is a MANOVA be the right method in that case? But what would the MANOVA results tell me? And to what extend are my results influenced by the different amount of females/males and workshop participants/not workshop participants?

### For replication

My data

library (dplyr)
set.seed(2)
n <- 100
x <- replicate(5, sample(1:5, n, rep=T))
colnames(x) <- paste0("V", 1:5)
x <- as.data.frame(x)
x$gender <- sample(c("male", "female"), n, prob=c(.4, .6), rep=T) x$workshop <- sample(c("yes", "no"), n, rep=T)
x[x$workshop == "yes", 1:5] = x[x$workshop == "yes", 1:5] + sample(0:1, 100, rep=T, prob=c(.9, .1))
x[x$gender == "male", 1:5] = x[x$gender == "male", 1:5] + sample(0:1, 100, rep=T, prob=c(.9, .1))

x$workshop <- as.factor(x$workshop)
x$gender <- as.factor(x$gender)

• (1) The imbalance between females and males does not affect the conclusions. (2) You are already "controlling for gender" by including it into the model. The workshop participation effect is not due to gender. (3) MANOVA is a reasonable approach in your situation. (4) You do not have a "significant effect" of workshop participation ($p=0.09$), unless you are willing to work with $0.1$ significance threshold which is a VERY WEAK standard of evidence. Commented Aug 8, 2016 at 11:31
• gender effect is the difference between the genders, averaged across the workshops. workshop effect is the difference between the workshops, averaged across the genders. So these are controlled, not marginal, effects. Additional interaction effect estimates whether the workshop effect varies for the two genders (and vice versa). If a 2-factor design is balanced interaction effect cannot contaminate the two main effects. Commented Aug 8, 2016 at 11:51
• Note also you are using type III SS. Under it, only non-contaminated portions of the three effects are being drawn, this is the most "universal" approach as it tests for the same hypotheses irrespective of (un)balancedness of a design. Commented Aug 8, 2016 at 12:02
• Would there be any difference between a MANOVA as outlined above and a MANCOVA when a factor (here gender) is regarded as the covariate and one is interested in the effect of workshop only? Commented Aug 9, 2016 at 9:15
• @Mark A covariate in MANCOVA is usually understood to be continuous, not categorical. If it is categorical (like here), then it is MANOVA. Commented Aug 10, 2016 at 16:16