A full analysis example Multiple correlations Partial correlations
New Dataset: Confidence
This is a dataset taken of the confidence scales of 41
employees some years ago using 4 facets of confidence
(Physical, Appearance, Emotional, and Problem Solving, as well as their gender and their citizenship status.
Example problem 1: Analyze the correlation between physical confidence and appearance confidence.
First question we should ask “Is Pearson correlation appropriate?”
Four requirements for correlation: 1. _____
2. _____ 3. _____ 4. _____
Example problem 1: Analyze the correlation between physical confidence and appearance confidence.
First question we should ask “Is Pearson correlation appropriate?”
Four requirements for correlation: 1. A straight-line relationship. 2. Interval data.
3. Random sampling (Will need to assume) 4. Normal distributed characteristics
Check for normality in each of the histograms. (Graphs Legacy Dialog Histogram)
The appearance variable is close enough to normal, although it has more on the upper and lower end than it should.
The physical variable has a negative skew, so that could be a problem.
There are at least two values that are far below the mean for confidence in physical. We should investigate them further.
Graphs Legacy Dialogs Boxplot
Use summaries of separate variables, and Options Exclude Variable-by-Variable
Boxplots identify outliers, from the boxplot we find that cases 31 and 37 are the outliers in physical confidence.
Looking at the data directly we find that neither of these cases even have a value for appearance.
The two outliers in ‘physical’ have no measured value for ‘appearance’
That means they will have no effect on a correlation between “physical” and “appearance”. Correlation can only consider cases where there are values for both variables (a point needs both an X and a Y to exist)
Next, we look at the scatterplot.
Graphs Legacy Dialog Scatter/Dot
No obvious signs of non-linear trends, but there doesn’t seem to be any strong trend at all.
We run the correlation to find it and see if it’s significant at alpha = 0.05.
Analyze Correlate Bivariate
Sig. (2-tailed) is .039, so the correlation is significant at alpha =.05. (Had we chosen the .01 level, this would not be the case)
We could also run a t-test by hand to verify the significance level we found. (r= .373, n=31)
t* = 2.045 at 0.05 level, 29 df t* = 2.756 at 0.01 level, 29 df
The
correlation matrix
is a table that shows the correlation between two variables.Physical Appearance
Physical 1.000 .373
Appearance .373 1
In this case, Physical is correlated with Appearance with r=.373 Likewise, Appearance is correlated with Physical with r=.373 Also, everything correlates with itself with r=1.000.
SPSS takes it a little farther by making a matrix of correlation coefficient, significance, and sample size.
Confidences are significantly correlated, there are 31 entries for each pair (not 41 because real data has blanks).
However, if we go to the correlations menu and select more than two variables of interest:
We get a 4x4 correlation matrix instead!
Cutting away all the sample size and significance stuff, I find:
Phys. Appear. Emot. Pr.Solve.
Physical 1 .373* .430** .730**
Appearance 1 .483** .527**
Emotional 1 .540**
Problem Solving 1
There is a positive correlation between every facet. That
means that any one facet of confidence increases, so do all the others.
* significant at 0.05 level * significant at 0.01 level
Phys. Appear. Emot. Pr.Solve.
Physical 1 .373* .430** .730**
Appearance 1 .483** .527**
Emotional 1 .540**
Problem Solving 1
Multiple correlation is useful as a first-look search for connections between variables, and to see broad trends between data.
If there were only a few variables connected to each other, it would help us identify which ones without having to look at all 6 pairs individually.
Pitfalls of multiple correlations:
1.
Multiple testing.
With 4 variables, there are 6correlations being tested for significance. At alpha =0.05, there’s a 26.5% chance that at least one correlation is
going to show as significant even if there are no correlations at all.
At 5 variables, there are 10 tests and a 40.1% chance of falsely rejecting at least one null. (Assuming no correlations)
At 6 variables, there are 15 tests and a 53.7% chance of falsely rejecting the null.
You don’t need to know how to handle multiple testing
problems in this class. However, be cautious when dealing with many variables.
Be suspicious of correlations that are significant, but just barely.
Example: The weakest correlation here is physical with appearance, a correlation of .373. That correlation being significant could be a fluke.
2. Diagnostics doesn’t get easier.
Doing correlations as a matrix allows you to do the math of a correlation much faster than checking them one at a time.
However, the diagnostic tests like histograms, scatterplots, and residual plots don’t get any faster.
Any correlation we’re interested in (even if it’s not showing as significant) still needs checks for normality and linearity before use in research.
One big advantage of correlating with multiple variables is that we can isolate the connections between different variables
where they might not be obvious otherwise.
Phys. Appear. Emot. Pr.Solve.
Physical 1 .373* .430** .730**
Appearance 1 .483** .527**
Emotional 1 .540**
Example: Is there really a correlation between appearance confidence and problem solving confidence SPECIFICALLY, or are they both attached to the same general confidence?
To isolate a correlation between two variables from a third variable, we want to only look at the part of that correlation that’s really between those two and not the third.
We want the
partial correlation
.Example: Ice cream sales increase when murder rates increase. These two variables have nothing logical to do with each other, however, they both increase when it’s hot out.
This is the
simple correlation
between these two variables.We want the relationship between murder and ice cream WITHOUT the
confounding variable
of heat.In the dataset “murderice.csv”, we can find run a partial correlation and find out.
First, a simple correlation reveals very significant correlations between everything.
But how much of that connection is truly between murder and ice cream?
Analyze
Correlate
Partial
From here, put the two variables of interest in the variable (you can put more than two if you wish).
The partial correlation between ice cream and murder is much
lower
than the simple correlation. It appears that heat (orsomething common to all three) was a major factor in both.
In fact, the correlation is no longer significant (we fail to reject the null that there is no correlation)
Also note: SPSS tells us in the output table that heat is a control variable, so we know from the output that this is a partial correlation (hint, hint).
We’re using three degrees of freedom, one for each variable involved, so the df is 57 even when n is 60 (for interest)
Key observation:
The partial correlation will be less than the simple correlation if both variables of interest are correlated to the confounding
variable in
the same way
.Here, both murder and ice cream are correlated to heat positively, so the partial correlation removes that common positive relationship murder and ice cream.
Removing a positive relationship makes the correlation less positive.
Likewise, if the correlation to the confounding variable is
opposing, then the partial correlation will be higher than the simple correlation.
If we’re only considering positive correlations, this means a confounding variable could be hiding or
masking
acorrelation hiding a correlation between two variables rather than creating a false correlation.
Example: Confidence.
Consider the correlation between types of confidence. Do the correlations between the other three still show after we
control for problem solving confidence?
Simple Correlations
Phys. Appear. Emot. Pr.Solve.
Physical 1 .373* .430** .730**
Appearance 1 .483** .527**
Emotional 1 .540**
The correlation between physical and anything is removed
entirely (that means that knowing problem solving confidence tells you as much about an employee’s physical confidence as knowing all three other facets)
With the heat behind murder and ice cream we had some
other non-math information to make the claim that heat was behind the other two variables.
It could have easily been something we didn’t measure, like the proportion of elderly in an area (retirees often migrate south for winter).
In the case of facets of confidence, we don’t have any reason why problem solving confidence would be the common
thread. The partial correlations shrink to nothing because after problem solving, the other variables we’re giving much info.
If we control for emotional confidence, we see there’s a connection between problem solving and physical when emotional is taken out of the picture.
Interestingly, controlling for appearance produces the same result. They all have a common thread and so increase
together, but the real connection is between problem solving and physical confidence.
