David A. Kenny

December 26, 2015

Multiple Regression

View Multiple Regression webinars (small charge click here) or powerpoints (small charge click here)

The
example equation:

Y = a + bX + cZ + e

Y criterion variable

X predictor
variable

a intercept:
the predicted value of Y when all the predictors are zero

b regression
coefficient: how much of a difference in Y results from a one unit difference
in X

e residual

predicted Y given X and Z or equivalently a + bX + cZ (often
called "Y hat")

R multiple correlation: the correlation between Y and

The coefficients (a, b, and c) are chosen so that the sum of squared errors is
minimized. The estimation technique is then called least squares or ordinary
least squares (OLS). Given the criterion of least squares, the mean of the
errors is zero and the errors correlate zero with each predictor.

If the predictor and
criterion variables are all standardized, the regression coefficients are
called **beta weights**. A beta weight equals the correlation when there is
a single predictor. If there are two or predictors, a beta weights can be
larger than +1 or smaller than -1, but this is due to multicollinearity.

The predictors in a regression equation have no order and one cannot be said to
enter before the other.

Generally in interpreting a regression equation, it makes no scientific sense
to speak of the variance due to a given predictor. Measures of variance depend
on the order of entry in step-wise regression and on the correlation between
the predictors. Also the semi-partial correlation or unique variance has little
interpretative utility.

The standard significance test of whether a specified regression coefficient is equal to zero is to determine if the
multiple correlation significantly declines when the
predictor variable is removed from the equation and the other predictor
variables remain. In most computer programs this is test is given by the *t* or *F* next to the coefficient.

**Multicollinearity**

If two predictors are highly correlated or if one predictor has a large
multiple correlation with the other predictors, there is said to be *multicollinearity*. With perfect multicollinearity (correlations of plus or minus one), estimation
of regression coefficients is impossible. Multicollinearity results in large
standard errors for coefficient, and so a statistically significant regression
coefficient is difficult (power is low).

Multicollinearity for a given predictor is typically measured by what is called *tolerance*. It is defined as 1 - R^{2} where R^{2} is the
multiple correlation where the predictor now becomes
the criterion and the other predictors are the predictors. Generally tolerance values below .20 are considered
potentially problematic. Another measure
is the variance inflation factor which is defined as 1/(1
- R^{2}). Values above 5 are considered to
be potentially problematic.

**Suppression**

It can occur that a predictor may have little or
correlation with the criterion, but have a moderate to large regression
coefficient. For this to happen, two
conditions must co-occur: 1) the
predictor must be co-linear with one or more other predictor and 2) these
predictors have non-trivial coefficients. With suppression, because the suppressor is correlated with a predictor
that has large effect on the criterion, the suppressor
should correlate with the criterion. To explain this, the suppressor is assumed to have an effect that
compensates for the lack of correlation.

**Diagnostics**

It is advisable to examine the errors in the regression equation to determine if they have a normal distribution and that their variance does not change as a function any predictors. Moreover, to evaluate violation of the linearity assumption for a given predictors, one graph the predictor against the errors.

**Example**

Consider the hypothetical regression equation in which Age (in years) and
Gender (1 = Male and -1 = Female) predict weight (in pounds):

Weight = 12 + 22(Gender) + 3(Age) + Error

We interpret the unstandardized coefficients as follows:

intercept:
the predicted weight for people who are zero years of age and half way between
male and female is 12 pounds

gender: a
difference between men and women on the gender variable equals 2 and so there
is a 44 (2 times 22) pound difference between the two groups

age: a
difference of one year in age results in a difference of 3 pounds

It is advisable to center the Age variable. To center Age, we would subtract the mean age from Age. Doing so, would change the intercept to the predicted score for persons of average age in the study.

Note that if we recoded gender to be 1 = Male and 0 = Female, the new equation would be:

Weight = -10 + 44(Gender) + 3(Age) + Error

intercept:
the predicted weight for women who are zero years of age and is ‑10
pounds

gender: men
weigh on average 44 more pounds than
women, controlling for age

age: a
difference of one year in age results in a difference of 3 pounds

Another site that more extensively describes multiple regression: Statsoft