13.2 Intepreting regression coefficients

• Nonlinearity of logistic function -> impact depends on where you evaluate the function.

• The averages of the predictors is a useful start.

Intercept is not interesting at ‘0’ income, so evaluate at mean:

invlogit(coef(fit_1)[1] + coef(fit_1)[2]*mean(nes$income) )

## (Intercept) 
##   0.4014185

• The coefficient for income is 0.33, so aA difference of income = 0.33 on the logit scale. To understand this on the probability scale, as an example, we can find the difference in probability near the mean income (about 3):

invlogit(coef(fit_1)[1] + coef(fit_1)[2]*3 ) -
   invlogit(coef(fit_1)[1] + coef(fit_1)[2]*2 )

## (Intercept) 
##  0.07451764

So near the average income, a change in 1 income class gives about 0.07 increase in probability of voting for bush

• Divide-by-4: The steapest slope (upper bound on effect on probability) occurs for p=0.5, and is equal to \(\beta/4\) . The can be used as a rule of convenience. For our case, the probabilities are near 0.5, and rule of 4 gives:

coef(fit_1)[2]/4

##     income 
## 0.08124866

So the upper bound on the effect of increasing income category here is 0.08

What about confidence interval in that:

posterior_interval(fit_1)['income',]/4

##         5%        95% 
## 0.05741692 0.10585723