Interpreting Bias Corrected bootstrapping confidence intervals in logistic regression

Question

I have a question about interpreting and using the bias corrected confidence intervals for logistic regression as produced by SPSS. I understand the rationale for using bootstrapping, but want confirmation that the BCa confidence intervals produced by the bootstrapping cannot be used as is but need to be exponentiated in order to obtain the actual confidence intervals. The screen shot below indicates this issue. The original CI is 2.263 - 3.378. In the table below that the estimates from the bootstrapping are provide. The variable remains significant, which means the CI cannot contain the neutral value of 1. The BCa CI indicated is 0.813 - 1.226, which at face value suggests the variable is not significant. However, when I exponentiate the BCa CI I arrive at 2.254-3.407, which excludes the value of 1 and is also similar to the original CI estimates. Am I thus correct in finding the exponent of the BCA supplied values for the bootstrapping CI? Many thanks.

Answer 1

You have to be careful to distinguish coefficients in their original scales from coefficients in transformed scales.

Logistic and survival regression models usually do their work in a coefficient scale in which the null hypothesis for a coefficient is a value of 0. That's the scale in which the results for "B," "Bias," and "Std.Error" are reported in your printouts. The BCa CI of 0.813 - 1.226 are well outside the null value of 0 on that scale.

Many find it easier to think about the results in terms of odds ratios (logistic) or hazard ratios (survival). For that you exponentiate the coefficient in its original scale. Thus the null hypothesis of 0 for a coefficient in the original scale becomes a null hypothesis of exp(0) = 1 in the odds-ratio or hazard-ratio scale.

What's confusing in this SPSS output is that the non-bootstrapped results show CI already in the exponentiated scale where the null hypothesis is that the exponentiated coefficient value is 1, while the bootstrapped results report the CI in the original scale, where the null hypothesis is that the original-scale coefficient equals 0.