How to Lie with Statistics

How to Lie with Statistics

Confidence intervals for the predicted values - logistic regression

Using predict
after logistic
to get predicted probabilities and confidence intervals is somewhat tricky. The
following two commands will give you predicted probabilities:

        . logistic ...
        . predict phat

The following does not give you the standard error of the predicted
probabilities:

        . logistic ...
        . predict se_phat, stdp

Despite the name we chose, se_phat does not contain the
standard error of phat. What does it contain? The standard error
of the predicted index. The index is the linear combination of the estimated
coefficients and the values of the independent variable for each observation
in the dataset. Suppose we fit the following logistic
regression model:

        . logistic y x 

This model estimates b₀ and b₁ of the following model:


P(y = 1) = exp(b₀+b₁*x)/(1 + exp 0+b₁*x))
Here the index is b₀ + b₁*x. We could get
predicted values of the index and its standard error as follows:

        . logistic y x
        . predict lr_index, xb
        . predict se_index, stdp

We could transform our predicted value of the index into a predicted
probability as follows:

. gen p_hat = exp(lr_index)/(1+exp(lr_index))

This is just what predict does by default after a logistic regression
if no options are specified. Using a similar procedure, we can get a 95%
confidence interval for our predicted probabilities by first generating the
lower and upper bounds of a 95% confidence interval for the index and then
converting these to probabilities:

. gen lb = lr_index - invnorm(0.975)*se_index
. gen ub = lr_index + invnorm(0.975)*se_index
. gen plb = exp(lb)/(1+exp(lb))
. gen pub = exp(ub)/(1+exp(ub))

Generating the confidence intervals for the index and then
converting them to probabilities to get confidence intervals for the predicted
probabilities is better than estimating the standard error of the predicted
probabilities and then generating the confidence intervals directly from that
standard error. The distribution of the predicted index is
closer to normality than the predicted probability.

Confidence intervals for the predicted values - logistic regression-stata