If for each Subject you have 4 Measures in each of the 3 Conditions, then
both Conditions and Measures are repeated-measures factors: you design
may be symbolized as S x C x M -- that is, Subjects (5 levels) are
crossed with both Conditions and Measures. This design is equivalent to
R(SxCxM) where R (Replications) is "the ubiquitous nested factor" as
one author has put it, random with one level. (And since it only has one
level, it has zero degrees of freedom and zero sum of squares; but using
it formally often helps one to see what the proper error mean square
would be for each effect modelled in the design, even if no such mean
square is actually available in the data.)
Your choices then are, for each of the three factors, whether to
treat it as fixed or random. Conditions are presumably fixed -- they
usually are, because they usually represent all the conditions one is
interested in considering. (I can imagine wanting to treat them as a
random sample of 3 drawn randomly from a population of possible
experimental conditions, but that seems to me very unlikely.)
Measures might go either way. If what they represent is a series
of opportunities to observe the subjects' response to each condition, one
might treat the factor as fixed, the levels representing the sequence
(1st, 2nd, 3rd, 4th) in which the opportunities are presented. This
would permit examining differences among the 4 levels as possibly
reflecting learning (one becomes a little more skilled each time one is
asked to respond to a condition, perhaps?), or fatigue (after one has
done it once, the action starts to become boring or otherwise wearisome),
or a kind of resultant between learning and fatigue. Or, if you really
think it reasonable to model each encounter as equivalent to each other
encounter (in the same Condition), and the only variation among levels of
Measure is random replication variance, Measure might be treated as
random.
Subjects are usually treated as random, because one usually wants
to generalize to a population of subjects "like these", and one may even
have selected the Ss randomly from a pool of potential Ss for the
experiment. But you haven't very many Subjects, and perhaps you want to
model individuial differences between them of some kind or other; or,
for some as yet unspecified reason, you are interested only in these
particular Ss and not in a population of Ss which they might be argued to
represent; in either of which cases you may wish to treat Ss as fixed.
Of course, to carry out _any_ tests of hypotheses, at least one
of the three factors must be declared random, or you will have no
legitimate error mean square against which to test the hypothesis mean
square for any of the possible effects.
In terms of your three possibilities:
(a) has C and S fixed, M random;
(b) has C and M fixed, S random (although I don't think it correct to
describe S as a "repeated-measure" factor: in my lexicon, a "repeated
measure" factor is any factor in a design that is _crossed with_ S);
(c) has C fixed, S and M random.
It may be informative to carry out more than one formal analysis,
using different fixed/random choices. This would tell you what results
are robust with respect to those choices, and what results depend on how
you choose to treat one or another of the formal factors. In case it's
useful, here is a table of the proper error mean squares for each effect:
Error mean square under
Source (a) (b) (c)
C CM CS (CS + CM - CSM)
S SM -- SM
M -- SM SM
CS CSM -- CSM
CM -- CSM CSM
SM -- -- CSM
CSM -- -- --
(Where the entry is "--", the proper error mean square would be R(SCM),
if it were available. In its absence, one could use the mean square for
CSM, making the assumption that there is no 3-way interaction -- that may
or may not be a reasonable assumption to make.)
-- DFB.
On Fri, 9 Feb 2001, Sylvain Clément wrote:
> We have data from an experiment in psychology of hearing. There are 3
> experimental conditions (factor C). We have collected data from 5
> subjects (factor S). For each subject we get 4 measures of performance
> (M for Measure factor) in each condition. What is the best way to
> analyse these data?
>
> We've seen these possibilities :
>
> a) ANOVA with repeated measures with 2 fixed factors : subjects &
> conditions and the different measures as the repeated measure factor
> (random factor).
>
> b) ANOVA with two fixed factor (condition & measure) and a random
> factor (repeated measure-> subject factor).
>
> c) ANOVA with one fixed factor (condition) and the other two as
> random.
< snip, arguments in favor of one or another of these
choices >
> I have only little theoretical background in stats and I like to know
> what exactly imply these possible designs.
The main differences I see are, what error mean squares apply to what
effects (main effects and interactions) are of interest. In any case,
you cannot test hypotheses about the 3-way SCM interaction, which will
be the error mean square for some of the effects of interest.
> Thanks in advance for your help
>
> Sylvain Clement
> "Auditory function team"
> Bordeaux, France
>
>
>
>
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Donald F. Burrill [EMAIL PROTECTED]
348 Hyde Hall, Plymouth State College, [EMAIL PROTECTED]
MSC #29, Plymouth, NH 03264 (603) 535-2597
Department of Mathematics, Boston University [EMAIL PROTECTED]
111 Cummington Street, room 261, Boston, MA 02215 (617) 353-5288
184 Nashua Road, Bedford, NH 03110 (603) 471-7128
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