4.3 Interpreting the logistical regression
We turn now to the interpretation of the coefficients in the logit equation. The estimated values of each of the coefficients describe the amount by which Z, the log odds, changes in response to a one unit change in the corresponding Xi, where Xi is continuous. In the event that Xj is itself a binary variable describing a particular category (eg, male or female) and taking only the values of zero or one, the value of the estimated coefficient on Xj (denoted 
) describes the change in the log odds of moving from the category coded 0 to the alternative category coded (1).
However, this interpretation is not especially intuitive. A preferable approach is to consider the impact of a unit change in a particular Xj on the odds rather than the log odds. The odds ratio is defined as
(6)
ie, it is the ratio of two odds. Consider the following which involves finding the odds ratio for a one unit change in say X1:
(7)
This expression reduces to 
as all other terms cancel out. This result simply states that the ratio of the odds for a one unit increase in X1 is given by ; ie, a one unit change in X1 results in a change in the odds ratio. It is constant; specifically it does not depend on the values of the other variables (Xj). Note that while the odds ratio is constant, this does not imply that the odds themselves are constant at various values of the Xj. In fact owing to the multiplicative effect, the actual change in the odds depends on the starting point. An odds ratio of two would increase odds of one to two, and odds of two to four.
Table 4-2 shows the direction of change in the odds of an event occurring for a one unit change in a variable, based on the sign of the corresponding coefficient.
| If the estimate of β1 is: | then the odds of the event will: |
|---|---|
| Positive | Increase |
| Negative | Decrease |
| Zero | Unchanged |
It is worth noting that while the distribution of the coefficients is symmetrical about zero, the odds have a lower bound of zero and so are not symmetrical (as shown in Table 4-1).
