Based on a paper by Geller, Johnston, and Madsen, 1997.
Eating disorders have become an important research problem for psychologists in the
last 20 years, and a variety of theories abound. Recently, people have begun looking at
eating disorders in the context of an individual's perception of themselves. Models such
as these argue that a person's concerns about weight and shape play a central role in
feelings of self-worth, which in turn play a role in eating disturbances.
Geller, Johnston, and Madsen (1997)
developed an inventory called the Shape and Weight Based Self-esteem Inventory (SAWBS),
and examined its psychometric properties. I do not have room here to go over all of their
work, but the paper is a nice example of what needs to be done in the development of any
new scale.
Geller et al. maintained that the SAWBS has an independent role in predicting
eating-disordered behavior over and above the role of more traditional variables such as
depression and general self-esteem. To investigate this hypothesis, they collected data on
84 female subjects on a number of different variables associated with eating disorders.
These measures are outlined below.
The intent of this investigation was to examine the psychometric properties of the
SAWBS. For our purposes the major question of interest will be the degree to which this
measure can provide additional predictability of eating disordered behaviors, after
we control for more traditional variables such as body mass, depression, and self-esteem.
I have chosen to use this example precisely because it looks at the additional
contribution of a variable. The general name for this approach is hierarchical regression,
which, as you will quickly discover, can be thought of as just plain old linear regression
with a difference. Hierarchical regression is not a new approach to regression, nor
is it a separate technique.
The data are available in the file named Geller.sav.[For those reading this from a web
page, I have made them available in Ascii form at Geller.dat.(The
variable names are SAWBS, WtPercep, ShPercep, Hiq, EDIComp, RSES, BDI, SES, and
SocDesir.) They closely mimic the data collected by Geller et al., both in terms of descriptive
statistics and the intercorrelation matrix. The variables were generated with a normal
random number generator, and were not truncated to integers, so fractional and negative
values are included. This does not in any way alter the interpretation.
Relationships Among Variables
The intercorrelation matrix for these variables is given in Table 1, and has been
pasted from SPSS printout. All correlations are based on a sample of N = .84.
SAWBS and Physical Characteristics
There are several correlations that are of interest here. In the first place, recall that the SAWBS deals with the extent to which self-worth is based on shape and weight, but is not a measure of actual shape and weight. As such, it should not be correlated with the Body Mass Index (BMI). In fact, that correlation is only .17, which is not significant.
SAWBS and Perceptions
It is apparent that the role that shape and weight play in determining an individual's
feeling of self-esteem is, in fact, related to that person's feelings of satisfaction with
their weight and shape, as can be seen by the significant correlations with WtPercep and
ShPercep. However, it is important to keep in mind that the WtPercep measure is not linear
in terms of desirability. For example, WtPercep runs from 1 = extremely overweight, to 4 =
neutral, to 7 = extremely underweight. Presumably the optimal level would be a 4, with
satisfaction falling off toward the extremes. This would have lead me to expect a
curvilinear relationship between that variable and others, and yet the relationship turns
out to be linear. The explanation for the linearity is apparent when you draw a
scatterplot of this relationship. Here you see that the highest value of WtPercep is
approximately 4.75, meaning that there are no data at the upper (underweight) end of the
scale. That, in itself, is interesting. (The scatterplot (Figure 1) follows the
intercorrelation matrix.)
Table 1
- - Correlation Coefficients - -
SAWBS WTPERCEP SHPERCEP HIQ EDICOMP RSES
SAWBS 1.0000 -.3930 -.3855 .6149 .6148 -.3783
P= . P= .000 P=.000 P= .000 P= .000 P= .000
WTPERCEP -.3930 1.0000 .5592 -.5647 -.6059 .3841
P= .000 P= . P=.000 P= .000 P= .000 P= .000
SHPERCEP -.3855 .5592 1.0000 -.5545 -.6666 .4173
P= .000 P= .000 P=. P= .000 P= .000 P= .000
HIQ .6149 -.5647 -.5545 1.0000 .8573 -.6092
P= .000 P= .000 P=.000 P= . P= .000 P= .000
EDICOMP .6148 -.6059 -.6666 .8573 1.0000 -.6812
P= .000 P= .000 P=.000 P= .000 P= . P= .000
RSES -.3783 .3841 .4173 -.6092 -.6812 1.0000
P= .000 P= .000 P=.000 P= .000 P= .000 P= .
BDI .4237 -.4513 -.4732 .6561 .7112 -.7945
P= .000 P= .000 P=.000 P= .000 P= .000 P= .000
BMI .1656 -.6135 -.3238 .2188 .2373 -.0599
P= .132 P= .000 P=.003 P= .046 P= .030 P= .588
SES -.1338 .0040 .1444 -.1788 -.1794 .1762
P= .225 P= .971 P=.190 P= .104 P= .102 P= .109
SOCDESIR -.1259 .2294 -.0676 -.3564 -.1664 .2607
P= .254 P= .036 P=.541 P= .001 P= .130 P= .017
Figure 1 Scatterplot of SAWBS as a function of WtPercep.
Relationship between SAWBS and Eating Disorder Measures
If the SAWBS measures the degree to which shape and weight play a role in determining
the individual's feelings of self-esteem, then we would also expect that variable to be
related to symptoms of eating disorders. That would suggest that the SAWBS measure should
be correlated with both the Eating Disorders Inventory (EDI) and the Health Information
Questionnaire (HIQ). From Table 1 we can see that these correlations are both .61, and are
significant at p < .000. There is no way that we can establish a causal ordering
on these relationships, but it is apparent that women who show many symptoms of eating
disorders also report that their shape and weight play an important role in determining
their feelings of self-esteem.
SAWBS and Self-esteem
Another interesting relationship is that between the SAWBS and the Rosenberg
self-esteem Scale (RSES). The fact that someone's self-esteem is influenced by their
satisfaction with their shape and weight, does not in itself suggest that there needs to
be a correlation between those two variables. Remember that the SAWBS is not a measure of satisfaction,
but a measure of influence. However, from Table 1 we see that this correlation is
significant and negative (r = -.38).
Multiple Regression Analyses
Geller et al (1997) hypothesized that the SAWBS would be related to Eating Disorders
measures over and above other more traditional measures, such as the BMI, depression, and
actual self-esteem. There are at least two equivalent ways that we could look at this
question, but the approach we will take here is often referred to as "hierarchical
regression."
Hierarchical regression seems to be an "in prase" in the past few years, but
all that it really means is "Do these independent variables add anything new when
they are added to a mix of other independent variables?" We do this exactly the way
it sounds-we first use the "controlled for" variables to predict the dependent
variable, and then we add the extra variables to the model and see if prediction
improves reliably.
We will start with the EDI composite as our index of eating disorder symptoms. (It is
our dependent variable, and we would get similar results if we used HIQ instead.) First we
will predict EDIcomp from BMI, BDI, and RSES, which are traditional predictors of eating
disorders. The results of this regression are shown below. The printout is from
a previous version of SPSS, but I have used it because it saves you having to
download images. The results are exactly the same.
Table 2-The Reduced Model
* * * * M U L T I P L E R E G RE S S I O N * * * *
Equation Number 1 Dependent Variable.. EDICOMP
Variable(s) Entered on Step Number
1.. RSES
2.. BMI
3.. BDI
Multiple R .75812
R Square .57474
Adjusted R Square .55879
Standard Error 11.62412
Analysis of Variance
DF Sum ofSquares Mean Square
Regression 3 14609.13054 4869.71018
Residual 80 10809.61946 135.12024
F = 36.03983 Signif F = .0000
------------------ Variables in the Equation ------------------
Variable B SE B Beta T Sig T
BMI 1.019367 .413119 .180574 2.467 .0157
BDI 1.075868 .292390 .442643 3.680 .0004
RSES -.680090 .256260 -.318671 -2.654 .0096
(Constant) 13.566366 14.694770 .923 .3587
There are several interesting things to see in this printout, but I will skip over most
of them for the time being. One of the two most important things to notice is that these
three variables taken together have a multiple correlation with EDIcomp of .76, and
account for 57 percent of the variability in that measure. That is a substantial
percentage of the variability for measures that are not directly measuring feelings of
satisfaction with body image.
The second thing to note is that all three variables make a significant contribution to
the prediction. That is interesting in light of the fact that BMI on its own is only
weakly correlated with EDIcomp (r = .24, p = .03). I would have expected
that this variable would have dropped to non-significance in the presence of two much more
powerful predictors.
The next step in the hierarchical regression is to add SAWBS to the predictor list and
see if it contributes anything additional to EDIcomp. Notice that we are not asking if
SAWBS is related to EDIcomp. We are asking if it is related to EDIcomp when we control
for BMI, BDI, and RSES. Another way of saying exactly the same thing is to ask if it
adds significantly to the accounted for variation over and above what was accounted for by
those three variables. (This is what we mean when we speak of
"hierarchical" regression--there is a hierarchy of predictors.) When we run this larger regression we obtain:
Table 3-The Full Model
Equation Number 1 Dependent Variable.. EDICOMP
Variable(s) Entered on Step Number
1.. SAWBS
2.. BMI
3.. RSES
4.. BDI
Multiple R .81895
R Square .67067
Adjusted R Square .65400
Standard Error 10.29382
Analysis of Variance
DF Sum of Squares Mean Square
Regression 4 17047.69469 4261.92367
Residual 79 8371.05531 105.96273
F = 40.22097 Signif F = .0000
------------------------- Variables in the Equation--------------
Variable B SE B Beta Tolerance VIF T
Sig T
BDI .805717 .264981 .331495 .350737 2.851 3.041
.0032
BMI .762357 .369742 .135046 .971748 1.029 2.062
.0425
RSES _.594573 .227632 -.278600 .366422 2.729 -2.612
.0108
SAWBS .130437 .027190 .346591 .798630 1.252 4.797
.0000
(Constant) 1.388258 13.020970 .875
.3844
I obtained this slightly expanded printout by choosing Collinearity Diagnostics from
the Statistics button in the regression dialog box. Notice the value of R2
= .67, which is an increase of .10 from the R2 in the previous model.
We will refer to the two models that we have here as the full and reduced
models. The full model has all of the predictors, while the reduced model has only some of
the predictors from the full model. (The reduced model cannot have any predictors that are
not included in the full model-it must be a proper subset of the full model.)
The added contribution of SAWBS, controlling for BMI, BDI, and RSES is .10. We can test
the significance of this increment by a standard formula presented in the text. (There is an easier way in this particular case, and I'll come back to that in a
minute.)
where f is the number of predictors in the full model, r is the number of
predictors in the reduced model, R2f is the squared
correlation from the full model, and R2r is the squared
correlation from the reduced model. With an F of nearly 24, this increment (the
increase in predictability due to adding SAWBS) is clearly
significant.
It is apparent from these results that SAWBS has a significant and important
contribution to make over and above the contributions of the other variables. We refer to
this as the unique contribution of SAWBS.
Because our full model contains only one additional variable, we actually knew whether
its additional contribution was significant without calculating the F statistic
above. The t test for the significance of the slope of SAWBS is exactly the same
test, because it tests whether the slope for SAWBS is significant in the presence of
the other variables. (If I had not rounded so severely in the calculation of F, the square
of t would be exactly equal to the F that I calculated.) This scheme only
works when I add one predictor. If I had added two predictors, the F would be a
test on whether those two predictors contributed significant new accountable variation,
whereas the t tests on their slopes would deal with the variables individually.
These results tell us that even when we control for body mass, depression, and
self-esteem, the role that shape and weight play in forming an individual's self-esteem
play an important role in predicting eating disorders. This is potentially useful
information.
Semi-partial Correlation
We know that SAWBS accounts for additional variation, but how much additional
variation? Well, we know that also. For the reduced model, R2r
= .57, and for the full model R2f = .67. This increase in the
multiple correlation of .10 is the amount that SAWBS accounts for over and above the other
variables. This is known as the squared semi-partial correlation for SAWBS. All squared
semi-partial correlations are of this type-they are the increase in R2
when we add in just that variable. In the book I show how you can calculate these for each
variable in the model from the individual t tests on the slope. In other words, you
could look at the increment in R2 when each variable is added
after the others, or you could just look at the t values for the full model and
make the calculation directly. One important thing to note is that if you order the
variables in the full model by the magnitude of their t statistics, you are
essentially ordering them on the basis of the size of their squared semi-partial
correlations, given the other variables in the set.
Tolerance
Earlier I noted that it was interesting that BMI was a significant predictor in the reduced model (and the full model) even though it had a relatively low correlation with SAWBS when looked at alone. Usually, variables that are only weakly (even if significantly) correlated with a dependent variable tend to drop out in more complex models because their role is taken over by the other variables in the model. So why didn't this happen to BMI? At least a partial answer can be found in the column headed Tolerance in the full model. The tolerance of a variable is a measure of how that variable correlates with the other independent variables in the model. The more highly it is correlated, the lower its tolerance will be (and the more able the other variables are to carry its weight). Actually,
tolerance = 1 - R2predictor. Other predictors
If you were to calculate the multiple correlation of BMI with BDI, RSES, and SAWBS, you
would find
R2BMI.BDI, RSES, SAWBS = .02825. If we subtract this from
1.00 we get .97175, which is the tolerance given in Table 3.
Because BMI is nearly independent of the other variables in the model, they do not
carry any of its information, and therefore they cannot take its place. Hence there is
still a role for BMI to play in the regression, which is why it is not eliminated from the
model. I must admit that this is the most extreme example of this situation that I have
ever seen in real data.
The logical question for people to ask next would be "How small does the tolerance
have to be before a variable really doesn't have anything new to add?" The answer is
probably found by looking at the column of t. If a variable is significant, it is
adding important information. However in practice, when the tolerance of a variable falls
below about .10 (it shares 90% of its variation with other dependent variables), it is
unlikely to remain significant in a model. Moreover, tolerances below .10 can make your
model very unstable, leading to noticeably different regression equations from sample to
sample. I would suggest that when you have variables with a tolerance of .10 or
below, you should drop either that variable or others with which it correlates
highly.)
Interaction Models-Moderation Effects
One of the more interesting problems in multiple regression (as well as one of the most
frustrating) is the treatment of interactions. I say "frustrating" because it is
often very difficult to find significant interactions, even when you have good reason to
expect them. I say "interesting" because the use of interaction terms gives us
considerable flexibility and explanatory power in modeling behavior.
We will start with the situation in which we want to predict eating disorders (as
measured this time by HIQ) on the basis of self-esteem (RSES) and Shape and Weight based
self-esteem (SAWBS). Remember that SAWBS is not a measure of self-esteem, it is just a
measure of how important those variables are in controlling one's sense of self-esteem.
It seems logical (at least to me), that if someone already has a pretty good sense of
self-esteem, the SAWBS measure might be largely irrelevant when it comes to predicting
eating disorders. On the other hand, someone with a very poor self-esteem might show
eating disorders if shape and weight are important to them, and show some other disorder
if those variables are not important. If you think of this from an analysis of variance
perspective, it sounds like I'm talking about an interaction. I predict a simple effect of
SAWBS under one level of RSES, and no effect of SAWBS under another level of RSES. That's
in fact exactly what I'm doing, but how do we get at an interaction in linear regression?
Remember that in Anova we wrote our interaction terms as AXB. That "X" in the
middle wasn't just there to fill up some space-it actually represents a multiplicative
effect. The same is true in linear regression; our interaction term will be the product of
our two main effects. We will create a new variable which is the product of the two
independent variables under consideration (SAWBS and RSES), and then use all three
variables as predictors. If the interaction term is significant, we have our interaction.
Why can't it ever be simple?
Unfortunately, the world is not quite as simple as that last paragraph implies. If we
do what I called for there, you should be able to see the problem. First, I will use SPSS
(or whatever program I have at hand) to create a variable that is the product of SAWBS and
RSES. Call it SAWXRSES. The run the regression of HIQ on SAWBS, RSES, and SAWXRSES, in the
process asking for the intercorrelation matrix. The result is shown below:
Table 4 -Interactions
Variable(s) Entered on Step Number
1.. SAWXRSES
2.. RSES
3.. SAWBS
Multiple R .73810
R Square .54479
Adjusted R Square .52772
Standard Error 6.25376
Analysis of Variance
DF Sum of Squares Mean Square
Regression 3 3744.47265 1248.15755
Residual 80 3128.75735 39.10947
F = 31.91446 Signif F = .0000
------------------ Variables in the Equation ------------------
Variable B SE B Beta T Sig T
RSES -.536468 .139465 -.483411 -3.847 .0002
SAWBS .062231 .057902 .317992 1.075 .2857
SAWXRSES 7.34027E-04 .001598 .125947 .459 .6472
(Constant) 31.110904 5.451364 5.707 .0000
- - Correlation Coefficients - -
HIQ SAWBS RSES SAWXRSES
HIQ 1.0000 .6149 -.6092 .4349
P= . P= .000 P=.000 P= .000
SAWBS .6149 1.0000 -.3783 .9055
P= .000 P= . P=.000 P= .000
RSES -.6092 -.3783 1.0000 -.0435)
P= .000 P= .000 P=. P= .695
SAWXRSES .4349 .9055 -.0435 1.0000
P= .000 P= .000 P=.695 P= .
First of all, notice that the multiple correlation is .73810. Write that down on the
back of your hand so that you don't forget it. You'll want it in a minute. Notice also
that the only variable that is significant is RSES. SAWBS isn't even close to significant,
even though it does have a significant first-order correlation with HIQ. Finally, note
that the interaction term isn't significant either, but that it's regression coefficient
is 7.34027E-04. (To read this value, the E-04 says to move the decimal point four places
to the left, giving .000734.) Well, that's a bummer! We have much less than we expected,
as well as no interaction. What went wrong?
If you look at the intercorrelation matrix which follows the regression, you will see
that the interaction term is very highly correlated with the SAWBS main effect. And you
should remember that when you have highly correlated independent variables, the solution
is both unsatisfactory and unstable. So we have to find a way to get rid of that high
correlation.
One simple way to break up the high correlation is to use what are called "centered
variables." A centered variable is created by subtracting the variable's mean
from every observation. (Centered variables are really just deviation scores, to use a
different language.) If you center both SAWBS and RSES, to create CENSAWBS and CENRSES,
you will not change the simple correlation of either of those variables with each other or
with the dependent variable-centering is just a linear transformation, such as changing
feet to inches, or Fahrenheit to Celsus. You will, however, drastically reduce the
correlation between either of those variables and their product (CENSAWRS). If you create
those variables and run the regression, you get:
Table 5-Regression with centered variables
HIQ CENRSES CENSAWBS CENSAWRS
HIQ 1.0000 -.6092 .6149 -.1414
CENRSES -.6092 1.0000 -.3783 .2102
CENSAWBS .6149 -.3783 1.0000 -.1842
CENSAWRS -.1414 .2102 -.1842 1.0000
*** MULTIPLE REGRESSION***
Variable(s) Entered on Step Number
1.. CENSAWRS
2.. CENSAWBS
3.. CENRSES
Multiple R .73810
R Square .54479
Adjusted R Square .52772
Standard Error 6.25376
Analysis of Variance
DF Sum of Squares Mean Square
Regression 3 3744.47265 1248.15755
Residual 80 3128.75735 39.10947
F = 31.91446 Signif F = .0000
------------------ Variables in the Equation ------------------
Variable B SE B Beta T Sig T
CENSAWBS .088656 .016055 .453021 5.522 .0000
CENRSES -.494188 .091531 -.445313 -5.399 .0000
CENSAWRS 7.34027E-04 .001598 .035687 .459 .6472
(Constant) 16.904613 .719333 23.500 .0000
Notice that the two main effects still have the same correlation with the criterion
(HIQ), but that the correlation between the product term and the dependent variable has
been changed. This is to be expected. The most important thing to be seen here is that the
interaction is no longer very highly correlated with SAWBS. That means we have a much
better change of getting at the independent contributions of these variables. (In other
language, we can say that its Tolerance is high.)
In the multiple regression in Table 5, notice that the overall multiple correlation
coefficient is .73810. If that number doesn't seem familiar, look at the back of your
hand. We have not explained any more or less of the variation, we have just shoved the
pieces around.
Also notice the regression coefficient for the interaction effect (CENSAWRS =
.000734027). Doesn't that look familiar? Again, we haven't disrupted the test of the
relationship between the criterion and the interaction variable, we have just diddled with
the relationship between the product term and the two main effects.
Finally, notice the regression equations for the other variables. Both CENSAWBS and
CESRSES have significant regression coefficients, even though they didn't have them a few
minutes ago. The point here is that these variables are highly correlated with the
interaction term, which they create. Breaking up that correlation lets each variable speak
for itself.
Now we can say that HIQ is linearly connected to both self-esteem and SAWBS, although
there is no interaction between those two variables. You might wonder why I went through
all of this when there was no interaction. Well, the simple answer is that I had to do
something, and these were the data at hand. It is very easy to talk about interactions,
and to explain why they should be there, but it is quite a different thing to find good
ones in linear regression. There are at least two reasons for this:
In the first place, interactions may not be as common as we would like to think, and
they can easily be hidden by strong main effects. In the second place, the experiments
that we design in a laboratory, and the data that we collect in a natural setting, have
tremendous differences in power. If you want to find interactions in natural settings, be
prepared to invest a lot of effort. This point is very well made in a paper by McClelland & Judd, C. M. (1993).
Having written that last paragraph, I fear that I have left the reader wondering why
anyone would ever look for interactions with regression if they are so hard to find.
First, I may have overstated the case for how difficult it is to find them. Moreover, when
we do find them they add some very nice explanatory mechanisms for theorists. I strongly
advising looking for them, but you won't find them too often.
Moderating and Mediating Effects
This paper is already much too long, but I can't leave it without making at least
passing reference to a critically important, and often cited, paper by Baron and Kenny (1986). They distinguish between a moderator
effect, in which we look for a variable which partitions the data into separate
subgroups with different relationships between the independent and dependent variables,
and mediator effects, in which the independent variable works through some other
variable to influence the dependent variable.
I like to think of a moderator effect as being similar to an interaction in Anova,
where simple effects are different for different levels of one independent variable. In
our example here, we have already looked at one sort of interaction. In a slightly
different study we might find that there is a relationship between SAWBS and HIQ for
females, but no relationship between those variables for males. In this case, SEX would be
a moderator variable.
If, however, we believed that body image had no direct effect on depression, but
that it lowered self-esteem, which in turn produced depression, then self-esteem would be
a mediating variable, because it mediated the apparent relationship between body
image and depression.
After reading the Baron and Kenny paper, you might try to look at the relationship
between ShPercep, RSES, and depression. A case could be made that the relationship between
ShPercep and BDI is mediation by RSES.
Finally!!
This document has gone on much too long, and I will end it here-at least for today. I
may add more later.
Baron, R. M. & Kenny, D. A. (1986) The moderator-mediator
variable distinction in social psychological research: Conceptual, strategic, and
statistical considerations. Journal of Personality and Social Psychology, 51,
1173-1182.
Return
Geller, J., Johnston, C, & Madsen, K. (1997) The role of shape and weight in self-concept The shape and weight based
self-esteem inventory. Cognitive Therapy and Research, 21, 5-24.
Return
McClelland, G. H.,& Judd, C. M. (1993) Statistical
difficulties in detecting statistical interactions and moderator effects. Psychological
Bulletin, 114 376-390.
Return
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Last Revised: 7/13/98