Slope versus Correlation Coefficient
It is quite possible for two different bivariate data sets to have identical slopes for
predicting Y from X but different correlation coefficients (aka standardized slopes) for
predicting Y from X. Consider the data set Identical-Slopes_Different-Rs. Notice that
Y2 has a larger standard deviation than does Y1.
Descriptive Statistics
N
X1
Y1
Y2
Valid N
(listwise)
Mean
10
10
10
10
Std.
Deviation
14.00
24.00
24.00
2.981
3.651
8.944
For predicting Y1 from X1, the slope is 1. On average, Y1 increases 1 point for
each 1 point increase in X1. The standardized slope (aka , and, for a bivariate
regression, also known as Pearson r) is .816. With an r this high, there will be little error
in the prediction of Y. The coefficient of alienation is a useful index of the amount of
error in prediction. Here it is (1 – r2) = .334. The linear model explains all but 33.4% of
the variance in Y.
Unstandardized
Coefficients
Model
1
B
(Constant)
X1
Standardized
Coefficients
Std. Error
10.000
3.571
1.000
.250
Beta
.816
For predicting Y2 from X1, the slope is also 1, but r drops to .333. There is
considerably more error in prediction here. The coefficient of alienation has increased
to .889.
Unstandardized
Coefficients
Model
1
B
(Constant)
X2
Std. Error
10.000
14.283
1.000
1.000
Standardized
Coefficients
Beta
.333
Slope_vs_R_SPSS.docx
2
Looking at scatter plots might help. In the plot on the left, notice how close the
actual Y scores are to the predicted Y scores (the regression line). In the plot on the
right there are greater deviations between actual Y and predicted Y.
Y1 predicted from X1
Y2 predicted from X1
So, now you have seen that the regression for predicting Y from X in two different
populations may have identical slopes but different correlation coefficients. Now I shall
show you that that the populations can differ with respect to slopes but not correlation
coefficients.
Y1
Pearson Correlation
X1
Y3
.816**
.816**
.004
.004
10
10
Sig. (2-tailed)
N
Predicting Y1 from X1
Unstandardized
Coefficients
Model
1
B
(Constant)
X1
Standardized
Coefficients
Std. Error
10.000
3.571
1.000
.250
Beta
.816
3
Predicting Y3 from X1
Model
Unstandardized Coefficients
Standardized
t
Sig.
Coefficients
B
(Constant)
1
X1
Std. Error
10.000
7.141
2.000
.500
Beta
.816
1.400
.199
4.000
.004
For both regressions, the r = .816, but one has a slope of 1 and the other a slope
of 2.
Y1 predicted from X1
Y3 predicted from X1
Karl L. Wuensch, Dept. of Psychology, East Carolina University, December, 2014.