Stats/Methods
Bivariate Correlation & Regression
Correlation measures:
1.
Direction of relationship (+ or -)
2. Form of relationship (linear is most common)
3. Degree of relationship (measured by the numerical
value of the correlation. A value of
1.00 indicates a perfect relationship and a value of zero indicates no
relationship)
Examples of different values for linear correlations: (a) shows
a strong positive relationship, approximately +0.90; (b) shows a
relatively weak negative correlation, approximately
–0.40; (c) shows a perfect negative correlation, –1.00; (d)
shows no linear trend, 0.00. To compute a correlation
you need two scores, X and Y, for each individual in the sample.
The Pearson correlation
requires that the scores be numerical values from an interval or
ratio scale of measurement.
Other correlational methods
exist for other scales of measurement.
Conceptually, Pearson correlation coefficient computes
Regression--the statistical technique used to find the best-fitting straight
line (i.e., the regression line) for a set of data.
The total error between the data points and the line is obtained
by squaring each distance and then summing the squared values.
Total squared error =
Ʃ(Y
– Ŷ)2
The regression equation is designed to produce the minimum sum
of squared errors.
Best fitting line has the
smallest total squared error: the line is called the
least-squared-error solution and is expressed as:
Ŷ =
bX +
a
where the value of "b" is the
slope and the value of "a" is the y-intercept
Caution when interpreting
predicted values from regression equations:
1) The predicted value is not
perfect. There will be some error between predicted Y values and the actual
data...As absolute value of the correlation coefficient gets closer to zero, the
magnitude of the error will increase.
The unpredicted variability
can be used to compute the standard error of estimate, which is
a measure of the average distance between the actual
Y values and the predicted
Y
values.
The standard error of estimate (SEE) provides a measure of how
accurately the regression equation predicts the Y values. For example, SEE
of 2.16 would tell us that the standard or average distance between the actual
data points and the regression line is 2.16 units. There will be, on
average, 2.16 units of discrepancy between our predicted values we obtain using
the regression equation and the actual values in the data.
2) The regression equation
should not be used to make predictions for X values that fall outside of the
range of values covered by the sample data.
coefficient of
determination 
Indicates
the size or strength of the relationship between x and y. Measures the
proportion of variability in y that is associated with the x variable.
categories for
r2 (same as for t-test or ANOVA)
.01 for small
.09 for medium
.25 or larger for a large correlation
In regression,
r2 also provides a measure of the accuracy of the prediction
using the regression equation. Because r2 measures the
predicted portion of variability in the Y scores, we can use the expression (1 -
r2) to measure the unpredicted portion or residual
variability.
