A regression coefficient describes the size and direction of the
relationship between a predictor and the response variable. Coefficients are
the numbers by which the values of the term are multiplied in a regression
Use the coefficient to determine whether a change in a predictor variable
makes the event more likely or less likely. The coefficient for a term
represents the change in the link function associated with an increase of one
coded unit in that term, while the other terms are held constant.
The size of the effect is usually a good way to assess the practical
significance of the effect that a term has on the response variable. The size
of the effect does not indicate whether a term is statistically significant
because the calculations for significance also consider the variation in the
response data. To determine statistical significance, examine the p-value for
The relationship between the coefficient and the probability depends on
several aspects of the analysis, including the link function, the reference
event for the response, and the reference levels for categorical predictors
that are in the model. Generally, positive coefficients make the event more
likely and negative coefficients make the event less likely. An estimated
coefficient near 0 implies that the effect of the predictor is small.
Terms that are not factors, such as the covariate, block, and center point
term, do not have high and low levels. These terms do not have effects but do
- The coefficient for a covariate is in the same units as the
covariate. The coefficient represents the change in the link function for a one
unit increase in the covariate. If the coefficient is negative, as the
covariate increases, the probability decreases. If the coefficient is positive,
as the covariate increases, the probability increases. Because covariates are
not coded and are not usually orthogonal to the factors, the presence of
covariates usually increases VIF values. For more information, go to the
section on VIF.
- Blocks are categorical variables with a (−1, 0, +1) coding scheme.
Each coefficient represents the difference between the link function for the
block and the average value.
- The center point term is coded as 1 for a center point and as 0 for
other points. Usually, you do not interpret the coefficient of the CenterPt
term because the term represents as many aliased quadratic effects as factors
in the design. Usually, you use the p-value to determine the value of further
data collection to estimate quadratic effects of the factors.
- If the coefficient for a center point is statistically significant,
you can conclude that at least one of the factors has a curved relationship
with the response.
Interpretation for the logit link function
The logit link provides the most natural interpretation of the estimated
coefficients and is therefore the default link in Minitab. The interpretation
uses the fact that the odds of a reference event are P(event)/P(not event) and
assumes that the other predictors remain constant. The greater the log odds,
the more likely the reference event is. Therefore, positive coefficients
indicate that the event becomes more likely and negative coefficients indicate
that the event becomes less likely. A summary of interpretations for different
types of factors follows.
- Continuous factors
- The coefficient of a continuous factor is the estimated change in
the natural log of the odds for the reference event for each increase of one
coded unit in the factor. For example, if each coded unit of a time factor
represents a change of 30 seconds, and the coefficient for time is 1.4, then
the natural log of the odds increases by 1.4 if you increase the time by 30
- Estimated coefficients can also be used to calculate the odds ratios,
or the ratio between two odds.
- Categorical factors
- For a 2-level factor, the interpretation of the coefficient of a
categorical factor can be between the two levels. The coefficient of a
categorical factor is the estimated change in the natural log of the odds of
the event for a change of one coded unit. The difference between the low and
high levels of a categorical factor is 2 coded units. For example, a
categorical variable has the levels Fast and Slow. Slow is the low level, coded
as -1. Fast is the high level, coded as +1. If the coefficient for the variable
is 1.3, then a change from Slow to Fast increases the natural log of the odds
of the event by 2.6.
- For a categorical factor with more than 2 levels, the coefficient of
a categorical factor is the estimated change in the natural log of the odds of
the event from the average value. For example, the coefficient for level A of a
factor is 1.3. When the factor level is A, the prediction for the natural log
of the odds increases by 1.3 compared to the equation averaged over the factor.
- Estimated coefficients can also be used to calculate the odds ratio,
or the ratio between two odds.