Linear regression is one of the most popular
statistical techniques used by researchers. Despite its popularity,
interpretation of the regression coefficients of any but the simplest
models is sometimes difficult. This article explains how to interpret
the coefficients of continuous and categorical variables.
Although the example used here is a linear regression model with two
predictor variables, the same approach can be applied when interpreting
coefficients from any regression model without interactions, including
logistic and proportional hazards models.
A linear regression model with two
predictor variables can be expressed with the following equation:
Y = B0 + B1*X1 + B2*X2 + E.
The variables in the model are Y, the
response variable; X1, the first predictor variable; X2, the second
predictor variable; and E, the residual error, which is an unmeasured
variable. The parameters in the model are B0, the Y-intercept; B1, the
first regression coefficient; and B2, the second regression
coefficient.
One example would be a model of the height
of a shrub (Y) based on the amount of bacteria in the soil (X1) and
whether the plant is located in partial or full sun (X2 ). Height is
measured in cm, bacteria is measured in thousand per ml of soil, and
type of sun = 0 if the plant is in partial sun and type of sun = 1 if
the plant is in full sun. Let us say it turned out that the regression
equation was estimated as follows:
Y = 42 + 2.3*X1 + 11*X2
Interpreting the Intercept
B0, the Y-intercept, can be interpreted as
the value you would predict for Y if both X1 = 0 and X2 = 0. We would
expect an average height of 42 cm for shrubs in partial sun with no
bacteria in the soil. However, this is only a meaningful interpretation
if it is reasonable that both X1 and X2 can be 0, and if the dataset
actually included values for X1 and X2 that were near 0. If neither of
these conditions are true, then B0 really has no meaningful
interpretation. It just anchors the regression line in the right place.
In our case, it is easy to see that X2 sometimes is 0, but if X1, our
bacteria level, never comes close to 0, then our intercept has no real
interpretation.
Interpreting Coefficients of Continuous
Predictor Variables
Since X1 is a continuous variable, B1
represents the difference in the predicted value of Y for each one-unit
difference in X1, if X2 remains constant. This means that if X1
differed by one unit, and X2 did not differ, Y will differ by B1 units,
on average. In our example, shrubs with a 5000 bacteria count would, on
average, be 2.3 cm taller than those with a 4000/ml bacteria count,
which likewise would be about 2.3 cm taller than those with 3000/ml
bacteria, as long as they were in the same type of sun. Note that since
the bacteria count was measured in 1000 per ml of soil, 1000 bacteria
represent one unit of X1.
Interpreting Coefficients of Categorical
Predictor Variables
Similarly, B2 is interpreted as the
difference in the predicted value in Y for each one-unit difference in
X2, if X1 remains constant. However, since X2 is a categorical variable
coded as 0 or 1, a one unit difference represents switching from one
category to the other. B2 is then the average difference in Y between
the category for which X2 = 0 (the reference group) and the category
for which X2 = 1 (the comparison group). So compared to shrubs that
were in partial sun, we would expect shrubs in full sun to be 11 cm
taller, on average, at the same level of soil bacteria.
Interpreting Coefficients of Associated
Predictor Variables
It is important to keep in mind that each
coefficient is influenced by the other variables in a regression model.
Because predictor variables are nearly always associated, two or more
variables may explain the same variation in Y. Therefore, each
coefficient does not explain the total effect on Y of its corresponding
variable, as it would if it were the only variable in the model.
Rather, each coefficient represents the additional effect of
adding that variable to the model, if the effects of all other
variables in the model are already accounted for. Therefore, each
coefficient will change when other variables are added to or deleted
from the model.
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