How troublesome this problem is depends on two things:
(1) the question(s) you really want to ask of the data;
(2) whether or not the design is in fact balanced.
(Although your Subject: lines refers to "nested or unbalanced",
your post describes an incomplete kind of nesting, and gives no
information about balance -- that is, about numbers of observations
in each of the four Subject groups.)
First, the simplest combination of possibilities: the data are balanced
(equal N in each group A, B, C, D), and your interest lies in the
difference in performance between time 1 and time 2.
For this case, compute the change for each subject. (I would take
<change> = <time 2> - <time 1>, so that positive values of <change>
imply increases in the dependent variable between time 1 and time 2.)
Now you have a complete crossed design, with the four Subject groups
nested within the four [<condition> vs <condition>] comparisons. For
this your standard 2-way ANOVA will give correct results. Here is the
design as you described it, and the design for the <change> variable
beside it:
Original data Change data
Time 1 2 1st cond 1 2
Cond 1 2 1 2 2nd cond 1 2 1 2
Sub A X x Subj group A B C D
Sub B X x
Sub C X x
Sub D X x
If the design is unbalanced, this still works, so long as you can
carry out the 2-way ANOVA. This cannot be much of a problem, since you
report having analyzed some of your data by ANOVA.
If your interest (also?) lies in the actual performance score for each
person at both times and in both conditions, you can carry out the same
ANOVA separately on the <time 1> data for each person, and again on the
<time 2> data. (I suspect you've already done one or both of these.)
What these analyses will tell you is how the pretest [or posttest]
performance score varies according to the <condition> vs <condition>
table. Rearranging the one-line table above into 2x2 form:
First condition 1 2
2nd cond = 1 A C
2nd cond = 2 B D
This is a perfectly legitimate analysis; it just isn't a <time> by
<condition> analysis, which may be how you were trying to interpret
your results. What you can find out from it is whether the pretest
score depends on the condition imposed at pretest, on the condition
imposed at posttest, and on the interaction between pretest and posttest
conditions; and of course you can find out the same information for the
posttest score. What you would hope to find, I suppose, is that there
are NO effects of condition on either score; and in particular you
would probably expect that there is no significant interaction for the
pretest scores, and no significant effect of <2nd condition> on the
pretest scores.
To the extent that you find any significant effects in these analyses,
at least in the analysis of pretest scores, you may need to re-interpret
the results of your change score analysis, since <change> may now depend
not just on the <condition>:<condition> comparison but also on the
pretest score.
In fact, it would be sensible to treat your data as a 2x2 ANCOVA, with
<posttest> (or <change>, it doesn't much matter) as the dependent
variable and <pretest> as the covariate. I would carry out THAT
analysis in a multiple-regression (= MR) program rather than an ANCOVA
module, because ANCOVA programs often impose the constraint that the
slope of <posttest> (or <change>) on <pretest> be the same in all cells
of the design; I'd rather DISCOVER that to be the case than ASSUME it.
In MR, the predictors would be <1st cond>, <2nd cond>, their interaction
(usually constructed as <1st cond>*<2nd cond>, <pretest>, and the
interactions of <pretest> with <1st cond>, <2nd cond>, and
<1st cond>*<2nd cond> (usually constructed as <pretest>*<1st cond>,
<pretest>*<2nd cond>, and <pretest>*<1st cond>*<2nd cond>). You might
want to recode the <condition> variables to (+1/-1) rather than (1/2) or
(2/1), so that if your design IS balanced (or nearly so) these
predictors will be mutually orthogonal (or nearly so).
Good luck! -- DFB.
On 4 Jul 2003, Felix Bach wrote:
> Hello,
> I did get a very tricky nested(?) anova design from a friend: A
> subject had to do two experiments. The first is done at time 1, the
> other afterwards (time 2). There were two kinds of conditions in the
> experiment 1 and 2. Each person did experiment 1 and 2, so that there
> are the following four possible combinations:
>
> Time 1 2
> Cond 1 2 1 2
>
> Sub A X x
> Sub B X x
> Sub C X x
> Sub D X x
>
> I was thinking of some nesting methods but none seem to work. Another
> possibility is to analyse it as an ANOVA as a Type4 SS Model. This
> works (well, I get an output) as long as I have got only one
> interaction in my model. Does anyone has got an idea how to correctly
> analyse it? (I know planning a good design beforehand is the best
> advice)
I think your main problem was that you did not correctly identify the
factors in the design. Viewed as <1st cond>, <2nd cond>, and <time>,
you have in fact a three-way ANOVA with repeated measures on <time>. If
you stack your pretest data on top of your posttest data, so that you
have two observations (pre and post) for each subject, the design looks
like this (in the tabular form you used above), where I use <A,B,C,D>
for pretest score and <a,b,c,d> for posttest:
First condition 1 2
Second condition 1 2 1 2
Time 1 A B C D
Time 2 a b c d
and an ANOVA should produce the following sources of variation:
<1st cond>, <2nd cond>, <1st cond>*<2nd cond>, <time>,
<time>*<1st cond>, <time>*<2nd cond>, <time>*<1st cond>*<2nd cond>.
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
.
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