# [S] Singly and doubly multivariate test with S-PLUS

Gabriel Baud-Bovy (baudbovy@fpshp1.unige.ch)
Thu, 19 Feb 98 18:00:55 +0100

First, I would like to apologize if I am taking too much
bandwith with this problem. Besides, this posting is
a little bit long. I would appreciate some help all the more.

Short story: I have a problem in performing a "singly multivariate"
test in SPLUS of data that should have been analysed as doubly
multivariate (I cannot do so for a reason explained below).

Long story: I have a doubly multivariate repeated measure design
with 2 dependent variables (DVs) and 2 within-subject factors.
The first factor has 12 levels while the second factor
has 3 factors. I measured the 2 DVs on the 10 subjects
for all combinations of within factor levels only
once (factorial design, no replication). Thus I have
a 11 by 72 response matrix Y.

I want to test the main effects. Unfortunately, I cannot
perform a doubly multivariate analysis for the 12-levels
factor (no problem with the 3-levels factor).
The reason is certainly that the design matrix T to test this effect has
72 rows and 22 = 2*(12-1) columns (i.e. the transformation
matrix in SPSS jargon). Thus I have actually 22 DVs measured
11 times (please, let me know if I am wrong!):

manova( Y %*% T ~ 1)

Tabachnik (p. 381) writes: "when using MANOVA it is
necessary to have more cases than DVs in every cell
[...] There are two reasons for the requirement. First,
the power of the analysis is lowered [...] because
of reduced degrees of freedom for error [...] The
second reason is associated with the assumption of
homogeneity of variance-covariance matrices".

For a totally within design, the second reason seems
not to be pertinent because there is only one
cell (no between-subject factor). Thus, the
problem must lie into the reduced DF for error.
As a matter of fact, my own calculation shows that
the DF of the denominator for Rao's F approximation
of Wilk's lamba is -11 (!).

I would like to have more insight about this
problem (discussion in a book or a paper) as well as
a few pointers on what I should do from here to
analyze these data.

An example in Tabachnik (p. 477) provides some clues.
The idea is to do a "singly multivariate test" of this
effect, i.e. multivariate in the DVs but univariate
for the within-subject factors. In this case, the
within factors must satisfy assumption of homogeneity
of covariance (see p. 476). In Tabachnik's example (data at the
end of the posting), the SPLUS command to test the effect of
the within-subject factor (month):

manova(cbind(wtloss,esteem)~month*cbw+Error(su/month),data=tabach)

where wtloss and esteen are the two DVs, month the within-subject
factor and cbw the between-subject factor. The summary
gives the same result as the one printed in his book though
it may appear strange to mixt multivariate and univariate
analyses. The error term is included because this is
a repeated measure design.

In my design, I have two DVs (ex and ey) and two within-subject
factors (hp and ang). The logical command to test the main effect
is:

manova(cbind(ex,ey)~ang+hp+Error(su/(ang+hp)),data=mydata)

Unfortunately, the summary command says that there is an
invalid value for dimension 1. The same problem arises when
ignoring the between-subject factor in Tabachnik's example:

manova(cbind(wtloss,esteem)~month+Error(su/month),data=tabach)

Does that mean that it is impossible to do an "univariate multivariate
test" for a mathematical reason when there is only within-subject
variables? Or is my SPLUS command wrong?

Of course, I could neglect the suppress the error term in the
formula:

manova(cbind(wtloss,esteem)~month,data=tabach)
manova(cbind(wtloss,esteem)~month+su,data=tabach)

but I don't think either would be correct since month is a
within variable (repeated measure design).

Thank you.

Gabriel Baud-Bovy

# data frame for Tabachnik's example.
d1_ c(4,4,4,3,5,6,6,5,5,3,4,5, 6,5,7,6,3,5,4,4,6,7,4,7,
8,3,7,4,9,2,3,6,6,9,7,8)
d2_ c(3,4,3,2,3,5,5,4,4,3,2,2, 3,4,6,4,2,5,3,2,5,6,3,4,
4,6,7,7,7,4,5,5,6,5,9,6)
d3_ c(3,3,1,1,2,4,4,1,1,2,2,1, 2,1,3,2,1,4,1,1,3,4,2,3,
2,3,4,1,3,1,1,2,3,2,4,1)
d4_10+c(4,3,7,1,6,7,7,3,4,4,6,5, 2,3,7,6,6,3,2,2,7,9,5,6,
6,9,5,6,3,6,3,5,5,6,6,7)
d5_10+c(3,4,2,1,5,8,6,5,4,5,6,3, 1,4,1,5,7,1,1,1,6,9,5,4,
2,9,1,2,2,3,3,2,3,4,6,7)
d6_10+c(5,7,6,2,4,8,9,5,5,3,1,6, 4,5,8,8,5,5,4,1,9,9,5,8,
6,6,9,8,7,7,6,8,8,7,9,7)
tabach_cbind.data.frame( su=rep( paste("s",as.character(1:36), sep=""), 3),
cbw=rep( rep( c("ctl","diet","diex"),rep(12,3) ), 3 ),
month=rep(1:3,rep(36,3)),
wtloss=c(d1,d2,d3), esteem=c(d4,d5,d6) )
tabach\$month_factor(tabach\$month)

-------------------------------------------------------------
Gabriel Baud-Bovy baudbovy@fpshp1.unige.ch
Université de Genève, FAPSE tel. +41 22 705 97 67
9, route de Drize fax +41 22 300 14 82
1227 Carouge, Switzerland home tel. +41 22 320 21 38
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