Lab 4 Fall ‘13
More Fun with Correlations
Last class we talked about correlations. Remember correlations do not allow us to make
conclusions about the causes of a phenomenon, they just tell us if we are able to better predict the value of
one variable based on knowledge of a persons’ score on another variable. Since correlations serve this
purpose it is valuable for us to know how to turn the correlation statistic into a prediction. Before, we get
into this, let’s look at predictions in general.
Assume that in past the mean final grade of my General Psychology courses has been 72%. A
student comes to me before the course begins. I know nothing whatsoever about them, yet they ask me to
predict what their final grade in the course will be. My best guess would be 72%. Why, because it is the
score which is most representative of scores obtained in the past. Clearly, I may not be correct. Not all the
students will receive a score of 72%, however, given that I have no information other than average past
performance of students in my General Psychology classes to base my prediction on, the class mean is my
best guess. If all of the students came to me beforehand and asked me to make the same prediction,
predicting, 72% for each of them will produce the best estimate over the long run.
Assume that I have kept records of student’s final grades in General Psychology and records of
students high school GPA’s for past years and have found that there is a positive correlation (r = 0.45)
between these two scores. GPA is not a perfect predictor of General Psychology grades, but I know from
this correlation statistic that people who have higher GPAs also tend to obtain higher grades in General
Psychology. If I ask the student what their High School GPA was, I should be able to better predict their
grade in General Psychology. If their GPA is higher than the mean GPA of my past students, I could
predict that they are likely to obtain a final grade higher than the mean of past students on the General
Psychology course. On the other hand, if they have a GPA that is lower than the mean of my past students,
my best prediction of their final General Psychology grade would be below the mean.
Regression Equations
The correlation and information about the means of the distributions I am predicting from (in this
case GPA, but lets call the score I am using to make my prediction from X) allows me to calculate my best
prediction of a person’s final grade score (we will call the score we are predicting Y). So, I am using a
person’s score on X (GPA) to predict their most likely score on Y (General Psychology Final Grade). The
formula for predicting X based on Y is not complicated. It is
Ypredicted  Y  b( X  X )
Where
Y is the mean of the distribution I am predicting (in this case General Psychology Final Grades)
And X is the mean of the distribution I am predicting from (in this case GPAs), and X is the score of the
person on distribution X. All we need to know now is what b is.
b is based on the correlation statistic between the two variables (r ).
br
sy
sx
Where sy is the standard deviation of Y, and sx is the standard deviation of X.
From my past classes I have found that the mean GPA of my students is 2.2 with a standard deviation of
0.5, and the mean Final Grade is 72% with a standard deviation of 5. If a student has a GPA of 3 I can
calculate my best estimate of their final grade by first calculating b and then putting into the formula above.
5
b  .45   .45  10  4.5
 .5 
Putting this into our regression equation, we can predict the most likely final grade for this student.
Ypredicted  72  4.5(3  2.2)  72  4.5(.8)  72  3.6  75.6
Lab 4 Fall ‘13
Not all students with GPAs of 3 will get 75.6 as their final grade, but 75.6 is my best estimate of the mean
final grade of students with a GPA of 3. Why? Because GPA is not a perfect predictor of General
Psychology Final Grades. We can determine how accurate this estimate is by determining the Standard
deviation of my distribution of estimated scores. The standard deviation of the predicted distributions
called the Standard Error of the Estimate.
sestimate  ( s y 1  rxy2 )
N 1
N 2
Where N is the number of pairs of scores I based my correlation on. Notice that the larger the sample that I
N 1
N  2 gets to one. With large sample sizes this term is
am basing my correlation on, the closer
essentially one and since multiplying a number by one does not change it, we can get rid of this term. Think
of it as a correction built into the formula to adjust the error term for smaller samples.
Assume I have based my correlation on GPAs and Final Grades from 500 students. We can get rid of the
N 1
N  2 part of the equation.
s estimate  (5 1  .45 2 )  5 1  .203  5(.89)  4.45
The standard error of the estimated (predicted) distribution is 4.45 (lets round to 4.5)
From our discussion of z-scores recall that approximately 95% of scores in a distribution fall within 2
standard deviations of the mean. The mean of my distribution of estimated Final Grades is 75.6. Therefore
95% of people with a GPA of 3 will fall between 75.6 -(2 x 4.5) and 75.6 +(2 x 4.5).
(2 x 4.5 = 9) therefore I predict that 95% of the students who have GPAs of 3 will obtain a final grade
between 66.6 and 84.6. I can predict with 95% confidence that a student with a grade point average of 3
will obtain a final grade between 67 and 85.
Okay, too much math. On your lab I will ask you to calculate these, but I will not make you do this for your
in-class exam. What you need to know for the exam.
1. If you know the mean of the predictor distribution and the mean of the distribution you are predicting
and the correlation between the two variables you can use a regression equation to estimate the mean of
the distribution of scores for any given X score.
2. The higher the correlation between the two variables, the less variance there will be in your predicted
distribution. At the extremes if X perfectly predicts Y, (r = 1 or r = -1) then the regression equation will
give you a perfect prediction of Y. If r = 0, then your best prediction of Y is the mean of the distribution
of Y. A correlation of zero gives you no predictive power.
Partial Correlations
Suppose you are interested in the effect of counselor’s touching their clients during counseling
sessions. You observe a large number of counselors during counseling sessions and count the number of
times each counselor touched his or her client. At the end of the session, you ask each client to fill out a
client satisfaction measure. You have two measures for each client (how often they are touched -- we will
call variable X) and their satisfaction rating (we will call this variable Y). You enter your data into SPSS
Lab 4 Fall ‘13
and find a significant positive correlation -- hurrah!. When you announce to your fellow counselors that
your data indicate that frequent counselor touch contributed to greater client satisfaction, a skeptic could
argue that the correlation is not due to counselor touch at all but rather to counselor empathy. They argue
that perhaps more empathetic counselors also tend to touch more -- clients give higher satisfaction ratings
not because of being touched, but because their counselor is more empathic. They argue, therefore, that the
correlation between touch and satisfaction is just an irrelevant side effect of the relationship between
touching and empathy.
To answer this criticism, you might well choose to use the statistical technique known as partial
correlation. Partial correlation allows you to measure the degree of relationship between two variables (X
and Y) with the effect of a third variable (Z) “partial led out” or “controlled for” or “statistically removed
from the equation“. In order to do this you of course would have needed to measure counselor empathy.
Although the formula for calculating a partial correlation by hand is not very difficult, SPSS will calculate
these for you, so we will not go through the math. Conceptually, what a partial correlation does is
statistically remove variation in your client satisfaction scores that can be attributed to variation in empathy
scores. The partial correlation is denoted rxy.z.
From our example, we found that the correlation between counselor touch (X) and client
satisfaction (Y) is rxy = .36. The correlation between empathy (Z) and counselor touch (X) is rzx = .65 and
the relationship between counselor empathy (Z) and client satisfaction (Y) is r yz is .60. These are called
zero order correlations and they are the same as Pearson correlations we talked about last week.. They are
called zero order because they are correlations between two variables with zero other variables controlled
for. The partial correlation program of SPSS tells you that r xy.z. (rtouch satisfaction. controlling for empathy) = -.05
(essentially zero). This result would tend to support the hypothesis of your skeptical colleague: There does
not seem to be a very strong relationship, if any between counselor touch and client satisfaction when
empathy is partial led out. When one variable is partial led out the statistic is referred to as a first order
correlation. (I.e., one variable has been controlled for). It is possible to control for several variables at the
same time. If two variables are partialled out it would be called a second order correlation (and so on….).
How does this fit in with your research projects? Several of you have run correlational studies,
and when we have put together your design you might have noted that there may be factors that would affect
your results, or may be alternative explanations for any relationship you might find between two variables.
I have suggested that you might want to measure these variables. For example, in a study I previously
conducted with a BRII student we were looking at the relationship between Year in College and Attitudes
Toward Learning. Past research has suggested that as students progress through their college careers, their
attitudes toward the role of their professors, and their peers change. A possible confound here is that as
students progress through their college careers they also age. Grade level and age are positively correlated.
Someone is bound to suggest this as a possible explanation of any correlation we find between grade level
and attitudes. We can explore this alternative explanation by calculating a partial correlation. Essentially,
what SPSS does is remove any variance in her attitude scores that can be attributed to age differences
between her subjects. The partial correlation between attitude and grade level with age partial led out will
give us an estimate of how related grade level and attitude would be if all the subjects at the different grade
levels were the same age (a sample that would be very difficult to find in real life).
When we partial out variables, we may reduce a correlation to a non-significant level (indicating
that the variable that has been partialled out (Z) can account for the correlation between X and Y. If X was
not related to Z, then it would not be correlated with Y. In Renee’s example, if the partial correlation
between Attitudes and Grade Level with Age partialled out is reduced to a non-significant level, Renee
would have to conclude that the correlation she obtained between Grade Level and Attitudes was a side
effect of the relationship between Age and Attitudes. Relationships which are due to a third variable (recall
the third variable problem) are called spurious.
Sometimes when a variable is partialled out, it increases the correlation between X and Y. In other
words rxy.z is actually larger than rxy. Variables that increase a correlation when they are partialled out are
called suppressor variables. They tend to have a very low correlation with X, but a significant correlation
with Y. Partialling such a variable out suppresses, or controls for irrelevant variance, or what is sometimes
referred to as noise.
Lab 4 Fall ‘13
Multiple Correlations
Many of you have likely run into a statistic called a multiple correlation when you have been
reading the literature in your topic area. A multiple correlation (denoted R) allows you to look at more than
one predictor at time. Let’s return to Renee’s study. Suppose she finds a positive correlation between
attitudes and grade level, and she also finds a positive correlation between age and attitudes. She partials
out age and finds the correlation between attitudes and grade level is still significant. So she does the
partial correlation the other way around and partials out grade level from the correlation between attitudes
and age and again finds there is a significant partial correlation. What she has found is that age is a
predictor of attitudes (even when grade level is controlled for) and that grade level is a predictor of attitudes
(even when age is controlled for). In other words, age and grade level give us independent information that
could help us better predict attitudes towards learning. So, if we wanted to get a good estimate of attitudes,
our best prediction would be based on both age and grade level. We would get a better prediction if we
used more than one predictor variable. This is what a multiple correlation does. Multiple Correlations
allow you to address the following research questions.



How well can a set of variables predict a particular outcome?
Which variable in a set of variables is the best predictor of an outcome? and
Whether a particular predictor variable is still able to predict an outcome when the effects of another
variable (or set of variables) are controlled for. (e.g., can grade level still predict attitudes after age has
been controlled for).?
When telling SPSS to run this analysis you need to indicate to the program which variable you are
predicting (SPSS calls it the dependant variable). Since the Attitudes Towards Learning study we wanted to
know if age and grade level predict attitude, her dependant variable is Attitudes. You also need to
indicate which variables you are using as predictor variables (SPSS calls these independent variables). In
the Attitudes study these would be age and grade level.
There are different ways to conduct a Multiple Regression. For this course we will use only one, called a
standard multiple regression. It enters all the variables at the same time and will provide the following
output.
1. Zero order correlations. The table containing zero order correlations is titled correlations. This is the
same type of output you would obtain if you did a Pearson correlation. It I did a multiple correlation
analysis using attitudes the variable I am predicting, and age and Grade level as my predictor variable,
the zero order correlations that would be reported would be
Correlation between attitudes and attitudes (remember this will always be +1.0)
Correlation between attitudes and age
Correlation between attitudes and grade level.
2. The next table is labeled Model Summary. In the column labeled R you will find the multiple
correlation. This is the correlation between the set of independent variables you specified and your
dependant variable. The next column presents R2. This is the same as the coefficient of determination
we discussed with correlations, however , it is based on the set of predictor variables. The next column
is the adjusted R2. When the sample size is small R is somewhat overestimated. The adjusted R2
corrects for this. When you report R2 in your results section, you should report this adjusted value. The
final column of this table gives you the Standard Error of the Estimate. This is the standard deviation
of the predicted distribution.
3. The next table is titled ANOVA. It is just like the ANOVA’s we discussed in Intro to Experimental. It
is the test of significance. SPSS compares the R value you obtained to see if it is significantly higher
than 0. In other words, if the value in the sig. column of this table is less than .05, your R value is
significant. You can say that this correlation is high enough that it is unlikely to be due to chance.
4. Finally, in the fourth table, titled coefficients, the coefficients for each of your predictor variables will
be given. The first Column of this table identifies which predictor variable the rows refer to. The first
Lab 4 Fall ‘13
row is labeled constant. The constant (c ) is used instead of the mean of X in a regression equation that
includes multiple predictor variables.
In the second column, labeled as Unstandardized Coefficients, and there are two columns. The first is
labeled B and the second standard error. These are the b values we discussed with the regression
equation. If you wanted to predict a value for any set of predictor variable scores you cold simply use
the formula.
Predicted = b1 X1 + b2 X2 + …+ c
If, for example, I wanted to predict the attitude score or a subject with a grade level score of 3, who was
24 years old, I could plug in the B values from this column into the formula
B1(3)+ B2(24) + c = predicted attitude score.
The next column reports Standardized Coefficients. or Beta values. These values could also be used in a
regression equation to predict the z-score of an individual. Unstandardized B values can not be compared
meaningfully to each other because they are not on the same metric. Because Beta values are in
standardized form (transformed to z-scores) they are all on the same metric. We can compare them. If one
Beta value is higher than the others, we can meaningfully say that that beta value is a better individual
predictor of the dependant variable (when all other predictor variables are partialled out).
The next two columns give t-statistics and sig values for the coefficients. If the value in the sig column is
less than .05, the partial correlation is significant. It is higher than would be expected based on chance
alone.
Lab 5
1. One the data disk this week you will find variables defined and labeled as the following.
ID – Subject ID
Sex
Age
tmastery – total scores from the Total Mastery Scale which measures peoples perceived control over events
and circumstances in their lives. Higher values indicate higher levels of perceived control
tpstress - total scores from a perceived stress scale. High scores indicate higher levels of stress.
tmarlow – total scores on the Marlow-Crowne Social Desirability scale which measures the degree to which
people try to present themselves in a positive light.
Tpcontrol - scores from a scale which measures the degree to which people feel they have control over their
internal states (emotions, thoughts, and physical reactions). Higher scores indicate higher levels
of control
For the first part of the lab you are going to find the partial correlation between Perceived Control of
Internal States and Perceived Stress score, while controlling for social Desirability. Social Desirability
refers to people’s tendency to present themselves in a positive or “socially desirable” way (also known as
faking good) when completing questionnaires. Social desirability scales therefore provide a measure of
Demand Characteristics.
Lab 4 Fall ‘13
A. SPSS Procedure for Partial Correlations
1. From the menu at the top of the screen click on Analyze, then click on Correlate, then on Partial.
2. Click on the on the two variable you want to correlate Click on the arrow to move these into the
Variables box.
3. Click on the variable you wish to control for. (partial out) Move this into the controlling for box
4. Choose Two tailed significance and display actual significance levels
5. On the options menu, select Means and Standard Deviations, zero order correlations and under missing
values select exclude cases pairwise.
5. Click on continue and then on Ok.
This will produce a table of descriptive statistics and two correlation matrixes. The first correlation matrix
contains zero order Pearson Correlations between the variables.
Q1. Using this correlation matrix, the descriptive statistics and the regression formula we discussed
in class, predict your best estimate of perceived stress for a person who has a tpcontrol score of
64.
Q2. Within what range could you be 95% confident that this person’s tpstress score would fall?
The second matrix provided the first order correlations. Does Social desirability appear to be a viable third
variable.
B. Multiple Correlation (R)
Run a Multiple Correlation predicting Total Perceived stress using Total Mastery and Total Perceived
control.
1. From the menu at the top of the screen click on Analyze, then click on Regression, then on Linear.
2. Enter the appropriate Independent and dependant variables using the arrow keys.
3. Be sure that the selection ENTER is in the box for method.
4. On the statistics menu be sure the only option you select is Descriptives (otherwise you will get a whole
wack of tables and stuff you will not understand.
5. On the options menu be sure to select pairwise under missing values .
Based on the print out.
Q3. Predict your best estimate of Perceived Stress, for a subject who has a perceive Master score of
18 and a Perceive Control score of 40.
Q4. Which independent variable is the better predictor of perceived stress? What values are you
basing your answer on, and why?