3-way interaction- analysing and plotting results
james
can someone please advise on this:
In using hierarchical multiple regression, i've tested for 2 and 3 way
interactions in my models. What I've discovered is a significant 3 way
interaction of a*b*c. This is fine, but the problem lies in
deconstructing this relationship. Specifically, in following Cohen and
Cohen's method of rearranging the regression equations to produce 1SD+,
M, 1SD- (though I take out the M), how does one begin to plot the
relationship?
I know how to rearrange the equation in a 2 way interaction, but how do
i do this with a 3 way interaction? I've seen it where a median split
on variable 'c' is performed to separate the dataset into a 'low' and
'high' c group. I can understand this and this is the method I plan to
use (I know it has problems using a median split, but this is the
easiest way I know how to do it). Therefore, using this method, after
splitting the data set, am I right in saying I need to rerun the
analysis, centering 'a' 'b' and 'c' to produce standardised interaction
terms AGAIN based on the data in the 'low' c group, then use the new
unstandardised beta weights to produce the graph from that? (and then
repeat on the 'high' c group to produce the second graph?).
Hope I've explained my problem enough, and I look forward to some help
for my poor statistical knowledge.
thanks,
J
Ray Koopman
Are all three variables -- a, b, c -- continuous (or scored
continuously)?
james
yes, they are all continuously scored variables (i.e., interval/ratio
level).
j
Bruce Weaver
If it was an ANOVA model (i.e., all variables categorical), you would no
doubt look at a plot of the cell means to work out the nature of the
interaction. For 3 continuous variables, a plot of the predicted (or
fitted) value of Y at various combinations of the 3 variables will give
you an analogous plot.
An easy way to get those fitted values is as follows:
1. Generate a little data set that has all of the explanatory variables
in your regression model, but no value for Y (the outcome). It is
useful to include a flag variable = 1 to help you filter out these
records later.
E.g.,
data list list /a b c flag(3f8.2).
begin data
1 1 1 1
1 1 2 1
1 2 1 1
1 2 2 1
2 1 1 1
2 1 2 1
2 2 1 1
2 2 2 1
end data.
The values of a, b, and c should be desired (and realistic) combinations
of the 3 variables.
2. Use ADD FILES to add the data from the original file.
add files
file = * /
file = "C:\path\mydata.sav"
.
exe.
3. Run the regression model again, saving the predicted/fitted values of
Y. Fitted values will be saved for ALL records in the file, including
the records you created with the DATA LIST command. Filter on FLAG,
then produce descriptive stats by A by B by C; or even better, create a
plot of the fitted values of Y at the desired combinations of A, B, and C.
--
Bruce Weaver
bwe...@lakeheadu.ca
www.angelfire.com/wv/bwhomedir
Ray Koopman
> yes, they are all continuously scored variables (i.e., interval/ratio
> level).
You have a regression function that is something like
w0 + w1*A + w2*B + w3*C + w12*A*B + w13*A*C + w23*B*C + w123*A*B*C,
where the w's are the regression coefficients, and A, B, & C are
scores on the predictors. Reorganize the function as
(w0 + w1*A + w2*B + w12*A*B) + (w3 + w13*A + w23*B + w123*A*B)*C,
where the parenthesized expressions are the intercept and slope of
the regression on C for particular values of A & B. You want to show
how the slope changes as a function of A & B.
Make a 3 x 3 array of boxes, like a Sudoku but without the little
squares. Each box corresponds to a particular combination of A & B.
In each box, plot the regression on C for those values of A & B.
The middle box is for A & B at their means; the right (left) column
is for A at its mean plus (minus) 1 s.d.; the upper (lower) row is
for B at its mean plus (minus) 1 s.d. (Or try 2 s.d.s, instead of 1.)
The bottom edge of each box is in units of C; the left edge of each
box is in units of the d.v. All 9 boxes should be on the same scale,
so the plots can be compared. The 3-way interaction will be evidenced
by the pattern of changes in the slope of the regression line.
It may help to put axes in the boxes. And if you're up to it,
you could also put confidence bounds on each regression line.
Do the whole thing in the three possible ways, showing the regression
on C for particular A & B combinations, on B for particular A & C
combinations, and on A for particular B & C combinations. One of them
should give you what you want.