**Mediating Variables and Partial Correlation**

**Contents**

Mediating Variables

Partial Correlation

Which is the Mediator?

Mediating Pathways

Partial Correlation and Suppressor Variables

Residual Scores: Another Way to Understand Partial Correlation

Differential Research and Analysis of Covariance

Moderator Variables

Summary

Did you know that children with large big toes have better command of their language? One study found a strong correlation between language skills and size of the big toe in a sample of children. Of course, this should not be a surprise if you are told that the children who were tested ranged in age from 12 months to 12 years.

A number of states have laws requiring teenage drivers to log 30 hours of supervised driving after they obtain their license. Another recent study found that in these states 16-year-olds have 29% fewer fatal traffic accidents. Does this mean that fatal crashes in teenagers can be reduced by requiring parents to ride with teenagers? Or are there other differences between states with supervised driving laws and states without such laws? Perhaps teenagers in states with such laws simply drive less.

By now
you are well aware of the dangers inherent in interpreting correlational data,
whether the data come from correlational research or differential research. If
two variables, X and Y, are correlated, it is difficult or impossible to say why
this relationship exists. If a classification variable, X, is associated with
changes in a dependent variable, Y, it is difficult or impossible to say why the
changes in Y occur. Changes in X may be a cause of changes in Y, changes in Y
may be a cause of changes in X, some third variable, Z, may be producing changes
in both X and Y, or any combination of these possibilities may be true.

One
reason for the difficulty is the likely presence of "mediating", or
confounding, variables. A variable is a confounding variable if it is related
to both X and Y, so making causal interpretations impossible. We call it a mediating
variable if in some sense it "explains" the relationship between X and
Y. I put "explain" in quotation marks because, as we'll see, we have
to be very careful what we mean by that term.

In the study of language
skills, both language skills and body size (including the big toe) increase with
age. Presumably, if we were to control for age, the correlation of language skills
and big toe size would disappear. Age, then, is the mediating variable. In the
case of supervised driving laws and fatal traffic accidents, amount of driving
may be a mediating variable. We need to compare fatal accidents in the states
after controlling for driving time.

Note that in these examples we
still cannot say that changes in age "cause" changes in size of big
toe or changes in language skills, or that reduced time driving "causes"
the reduction in fatal accidents. Nevertheless, we can say that age or driving
time "account for" or "explain" the relationships, as long
as we are quite clear that "account for" and "explain" are
not used in a causal sense. It may turn out eventually that other mediating variables
are present, in which case our interpretation will have to change.

Remember, in science all conclusions are tentative. With correlational data this caution is doubly important.

Researchers have found a correlation between depression and the incidence of cancer. People with greater levels of depression are more likely to develop cancer. What do you think some of the mediating variables might be in this case? |

A common method for identifying
mediating variables is to use the statistical procedure called **multiple regression**.
For this course all you need to know about multiple regression is that it is a
method for examining the relationships among many variables. One component of
multiple regression is easy to use, however, and can be very useful for exploring
data obtained from correlational and differential research. In correlational research
it is called "partial correlation". In differential research it appears
as the "Analysis of Covariance".

You might find it
helpful first to look at the **Summary** and try to understand
the important principles that are involved, then return and read the details that
follow. Or, just plow ahead.

For the sake of generality, let X and Y be the two variables
we have found to be correlated. I'll use **r**(X,Y) to represent their correlation.
We'll call **r**(X,Y) the *simple* correlation.

We introduce
a third variable, Z, which may or may not mediate the relationship between X and
Y. We can find out if Z is a mediating variable by calculating the *partial*
correlation of X and Y, controlling for Z. I'll write the partial correlation
as **r**(X,Y|Z). The variable to the right of the vertical line is the control
variable.

In other words, **r**(X,Y|Z) is a measure of the relationship
between X and Y if we control for Z, that is, if statistically we hold Z constant.
**If r(X,Y) is relatively large, but r(X,Y|Z) is much smaller, we can conclude
that Z is a mediating variable**. Z may explain, at least in part, the observed
relationship between X and Y. Be careful, though. Although we talk about "explaining"
the relationship, based on correlations alone we will never know what "causes"
the relationship.

In the big toe example, X is a measure of language
skills, Y is size of the big toe, and Z is a child's age. Age explains the relationship
between language skills and size of the big toe.

If you have three
variables, there are three simple correlations among them, **r**(X,Y), **r**(X,Z),
and **r**(Y,Z). Knowing these three correlations, it's very easy to calculate
the partial correlation **r**(X,Y|Z). You can use **this
calculator** for the purpose.

In the big toe example, suppose
**r**(X,Y) = 0.40, **r**(X,Z) = 0.55, and **r**(Y,Z) = 0.65. Then **r**(X,Y|Z)
= 0.07 (be sure you can duplicate that result). That is, the correlation between
X and Y is close to zero when Z is introduced as a mediator. Age explains the
relationship between language and big toe size.

You can also calculate partial
correlations in **SPSS** (use the **Correlations** command in the **Analyze**
menu and select **Partial Correlations**). Using SPSS, you can actually enter
several mediating variables at the same time, and find out if the X-Y relationship
might be explained by the combination of mediators.

Sometimes the partial
correlation, **r**(X,Y|Z), is smaller than the simple correlation, **r**(X,Y),
but it is still larger than zero. In this case we can say that the mediating variable
Z "partly explains" the correlation between X and Y. In other words,
there are other explanations for the relationship, including the possibility that
X and Y themselves are causally related.

In one study, the correlation between a child's school achievement and the number of hours the child watches TV was -0.33. The correlation between school achievement and teacher ratings of the child's aggressiveness was -0.48. Ratings of aggressiveness and number of hours watching TV correlated 0.55. Use partial correlations to interpret these results. Can we identify what causes what from these data? |

When you have three variables, any one
of them might be considered a mediator for the correlation between the other two.
Usually, however, it only makes theoretical sense to consider one of them to be
the mediator.

In the big toe example, using the data provided above, the partial correlation of language skills and age, controlling for big toe size, is 0.42. The partial correlation of big toe size and age, controlling for language skills, is 0.56. These partial correlations are both smaller than the simple correlations, so we might assume that big toe size partly explains the relationship between age and language skills, or that language skills partly explain the relationship between age and big toe size. Of course, these assumptions make no sense. This is a theoretical decision, though, based on what we know about the variables. The statistical analysis has little to say about which variable should be considered the mediator.

Asking which variable is the mediator leads to a consideration of "influence pathways" among the three variables. Consider Figure 1, which illustrates the big toe example. The presumed mediator, Age, is assumed to influence the X and Y variables, big toe size and language skills. We assume there is no direct connection between big toe size and language skills, apart from their joint dependence on age.

Figure 1. One way in which a mediating variable, Z, might influence X and Y.

Now consider a different kind of influence pathway. Let's assume we have found a correlation between students' SAT scores and their satisfaction with their college experience. Might there be a mediating variable in this case? What about the possibility that SAT scores influence the grades that students receive, and that their satisfaction depends only on their GPA? That is, GPA mediates the relationship between SAT scores and satisfaction.

This suggests a different kind of influence diagram, as in Figure 2. The X variable, SAT score, influences the mediator, GPA, which in turn influences the Y variable, Satisfaction. If this hypothesis is correct, the partial correlation of X and Y, controlling for Z, should be zero. (There's no direct connection between X and Y).

Figure 2. A second way in which a mediating variable, Z, might be related to X and Y.

Another example of Figure 2 might be the supervised driving laws and fatal traffic accidents example. It is possible that having these laws on the books (X) leads to less driving by teenagers (Z), which leads to fewer fatal accidents.

The influence diagrams in Figures 1 and 2 may suggest "causal" relationships, but strictly speaking we cannot show, using correlations alone, that the relationships are causal. Nor is there any way to tell from the correlations and partial correlations which of the two influence diagrams is correct. Both are possible once we show that the partial correlation of X and Y, controlling for Z, is effectively zero. However, extensions of the partial correlation method, called "Path Analysis", are often used to provide weak support for causal models such as these, and to differentiate among them.

Consider the previous example concerning school achievement and the number of hours a child watches TV, with teacher ratings of aggressiveness as the mediator. Would the influence pathway be like that in Figure 1 or like that in Figure 2? Can you think of another mediator where the other kind of influence diagram would be appropriate? |

**Partial
Correlation and Suppressor Variables**

Sometimes you will observe
a very strange result when calculating partial correlations. It is possible for
the simple correlation between X and Y to be close to zero, but for the partial
correlation, **r**(X,Y|Z), to be large. In this case Z is *suppressing*,
rather than mediating, the relationship between X and Y.

Here's an
example. Suppose we ask women to judge the attractiveness of several men, and
we want to find out if judgments of attractiveness are related to a man's weight.
We observe a correlation, **r**(X,Y), of 0.05. There seems to be no relationship.

We introduce as a mediating variable a measure of a man's body fat. We
find that the body fat index correlates 0.55 with weight (which seems plausible),
and correlates -0.45 with judgments of attractiveness (it's a negative correlation:
the women do not find obese men attractive). Plugging these numbers into the calculator,
we find that the partial correlation of weight with judgments of attractiveness
is 0.40. In other words, when we control for body fat, heavier men are indeed
seen as more attractive.

We might call body fat index a "suppressor
variable". That is, if it is not controlled, it suppresses the relationship
between weight and attractiveness, because it is positively correlated with one
and negatively correlated with the other.

The term "suppressor variable" has a number of different meanings in statistics, and there is some debate surrounding the term. You can ignore all of these complexities. From time to time you will see partial correlations that are larger than the simple correlations, and it may be difficult to understand why that would happen. This example may help to make it clearer. Probably, the mediator is suppressing the relationship by having opposite correlations with the first two variables.

A study measured adult patients' time to recover from an illness. The correlation between recovery time and a measure of family support (support received from family members during the illness) was found to be 0.03. That is, they were unrelated. However, the correlation between recovery time and the patient's age was 0.52, while family support and age correlated -0.49. What do you make of these results? |

**Residual
Scores: Another Way to Understand Partial Correlation**

Let's go back to the language skills example. We are pretty sure that the important predictor of language skills is age, not big toe size. Here's another way to think about the issue that may help you understand partial correlation better.

Suppose we want to predict scores on a test of language skills from a child's age. Then we could use a regression equation,

Predicted Language Score = A + B * Age,

where A and B are parameters obtained from a regression analysis (look it up if you don't remember regression analysis).

These
predictions will probably be pretty close, but they won't be exact. So for each
child we can calculate a **Residual Score**,

Residual = Language Score - Predicted Language Score.

The residual, then, is the
error of the prediction, or the degree to which the language skills score is *not*
accounted for by age. We call this "regressing language skills on age".
Think of the residuals as what's left over in the language scores after we account
for Age.

To find out if big toe size has any relationship with language skills, after accounting for age, we could now find the correlation between big toe size and the residual scores. This new correlation will, in fact, be the partial correlation of big toe size and language skills, with age held constant. It's the correlation of big toe size with "what's left over" in language skills after accounting for age. And, of course, after accounting for age, there's nothing left over for big toe size to explain, so the correlation of big toe size with residuals will be close to zero.

If we have two variables, X and Y,
and a possible mediator, Z, the partial correlation ** r(X,Y|Z)** is the same
thing as the correlation between X and the residual scores after we regress Y
on Z. We first find out if the mediator is related to one of the variables, then
see if the residuals are related to the other variable. That is, after taking
account of the mediator Z, we find out if the variable X can be used to predict
the remaining part of Y .

A college looked at data for its students during the last 10 years. They found a strong correlation between students' college GPA and their high school GPA. They also found strong correlations between both GPAs and students' combined SAT scores. They used SAT scores to predict high school GPA, then took residual scores from the regression analysis and correlated the residuals with college GPAs. The correlation of college GPA with residual scores was almost zero. What do you make of these findings? |

**Differential
Research and Analysis of Covariance**

In differential research, one or more of the variables is a classification variable, with a small, fixed number of levels. Examples include the study of sex differences (male versus female), class year, psychiatric diagnosis, etc. The supervised driving example is an example of differential research, where States is the classification variable.

The problem in interpreting the results of differential research is exactly the same as the problem in using correlational research. Since no variable has been manipulated, no causal conclusions can be derived. Thus, for example, if we observe a difference in skills between males and females, we cannot say why the difference occurred. There may be one or more confounding variables that are correlated with sex. For example, males and females may have had systematically different educational experiences. Any confounding variable is a potential mediating variable, i.e., a variable that explains the differences among levels of the classification variable.

Differential
research often uses the **Analysis of Variance** (ANOVA) to test for differences
among levels of the classification variable. In these situations, there is a procedure
similar to the use of partial correlations that can be used to identify mediating
variables. It is called the **Analysis of Covariance** (ANCOVA). Statistically,
ANCOVA is quite complex, but by using SPSS it is very easy to do the calculations.

Assume that you use Sex as a classification variable. This factor would vary between groups, because you have a group of males and a group of females. You will have one or more dependent variables, and you can use a straight forward ANOVA to find out if there are any differences between males and females on these dependent variables. (You could also use a t-test in this case, which would give you equivalent results)

Any potential mediator may be treated as a **covariate**. You
should measure each covariate (each potential mediator) for each of the subjects.
For example, you might record the number of math courses and language courses
taken by each person. You then repeat the analysis using ANCOVA, in which the
covariates are introduced as control variables.

If you find that the previously
significant differences in the ANOVA are no longer significant in the ANCOVA,
this means that the covariates "explain" the originally observed differences.
If you still find significant differences, it means that the covariates do not
explain the differences, at least not completely. **If the effect of the classification
variable is significant, but the effect disappears when we introduce a covariate,
we can conclude that covariate is a mediating variable**.

Consider the supervised driving laws example again. There is a significant difference in fatal accidents between states with and without such laws. Would the difference still be significant if we added amount of teenage driving as a covariate?

ANCOVA
can also be thought of in terms of **residuals** (see above).
After regressing the dependent variable on the covariate, we find out if different
levels of the classification variable are associated with different residuals.
That is, after finding out what effect the covariate has, we look to see if the
classification variable has any effect on what is left over.

When using SPSS you will need to use the General Linear Model (GLM) analysis. See the separate notes on How to use SPSS, especially the sections dealing with the use of GLM for ANOVA and ANCOVA. As long as it is clear which are your classification (between groups) factors, which are your dependent variables, and which are your covariates, it should not be difficult.

A comparison of patients diagnosed with manic depressive disorder and schizophrenia found a large, significant difference between the two groups on a test of rational decision making, F(1, 26) = 14.7, p < .01. Schizophrenic patients scored lower. An IQ test was given to each patient, and IQ was used as a covariate. In the analysis of covariance, the difference between the groups was reduced, although it was still significant, F(1, 25) = 7.6, p < .01. How would you interpret these results? |

We saw above that with partial correlations an extraneous variable
is sometimes a **suppressor variable** rather than a
mediating variable. The same thing can happen with Analysis of Covariance.
An ANOVA may show that a classification variable has no effect on the dependent
variable. When a covariate is introduced, however, the ANCOVA shows a significant
effect. The covariate in this case is a suppressor variable. When we control
for it, an otherwise hidden effect becomes apparent.

You will meet ANCOVA again when considering the design of true experiments, where covariates can be used for quite different purposes.

When using ANCOVA, it is necessary to make some critical statistical assumptions. Understanding these assumptions is not required for this course, but if you use ANCOVA in serious research, you will need to explore this topic further.

The term "moderator variable" sounds a lot
like "mediating variable", and the two are easily confused. There is
a very important difference, however. The difference is well summarized by the
two terms themselves. A mediating variable "mediates" a relationship
(i.e., serves as a facilitator, makes the relationship possible). A *moderating
variable* "moderates" a relationship (i.e., produces changes in the
relationship, or modifies the relationship).

The concept of moderator
variable is most easily explained if the moderator is dichotomous (i.e., has only
two levels). Sex might be a good example. Suppose we study the correlation between
anxiety and threat in a social setting. Men and women read scenarios describing
varying levels of threat to their safety or property, and rate their reactions
to the threat on several dimensions. We find that the overall correlation between
threat and anxiety is positive but small.

We now calculate the correlation
separately for men and for women. For women we find that the correlation is fairly
large; for men it is essentially zero. If X is anxiety and Y is threat, we can
write the simple correlation as, say, **r**(X,Y) = 0.24. However, **r**(X,Y|men)
= 0.04, **r**(X,Y|women) = 0.46.

Notice that if Z is a third variable,
I write **r**(X,Y|some level for Z). Z is a moderator if **r**(X,Y|some
level for Z) is different for different levels of Z. Thus, Sex is a moderator
variable. The correlation between threat and anxiety is different for different
sexes.

The concept of a moderator variable is not restricted to dichotomous
variables. For example, another moderator might be age: the correlation might
change as we look at people of different ages.

Most important, you should
be able to recognize that a moderator variable is nothing like a mediating variable.
In the example here, gender does not "account for" the relationship
between threat and anxiety (nor does it "suppress" it). Rather, the
relationship between threat and anxiety changes, depending on whether we are talking
about men or about women.

The concept of moderator variables is closely related to the concept of interaction, which is discussed at length when we talk about factorial experimental designs (see the notes on interactions).

A study found a small but significant correlation between a person's willingness to help others and his or her score on a test of intelligence. People with higher intelligence scores expressed greater willingness to help others. Can you
think of any Can
you think of any |

When we find a relationship between two dependent variables, or between
a classification variable and a dependent variable, it is usually impossible to
provide a causal explanation for this relationship. The reason is that there is
a potentially infinite number of confounding variables that might be involved.
In some cases, however, there is a theoretical reason for believing that a third,
mediating variable can explain the relationship. We can test this hypothesis by
examining **partial correlations** or by using the **Analysis
of Covariance**.

Suppose X and Y are the two variables involved in the original relationship, and that Z is the possible mediating variable. To find out if Z is indeed a mediator, we use a three step procedure:

1. Examine the relationship between Z and one of the original variables, say, Y.

2. Find that part of Y that is

notpredicted by Z. We call this a "residual".3. Find out if X is related to the residuals for Y.

If there is
no relationship between X and the residuals, then Z completely accounts for the
relationship between X and Y. If the relationship between X and the residuals
is just as strong as the original relationship, then Z is **not** a mediating
variable - it is not involved in the relationship. On rare occasions the relationship
between X and the residuals is stronger than the original relationship, in which
case Z is called a **suppressor variable**.

In the
case of correlational research, this three step procedure is carried out by examining
**partial correlations**. In the case of differential research,
we use the **Analysis of Covariance**.

In none of these cases can we unambiguously identify any variable as the cause of any other variable, but the procedures can help us develop and test theories to explain relationships among the variables.

Note finally that **moderator
variables** are not related to partial correlation and the Analysis of Covariance.
Rather, they are closely related to the concept of **interaction**.