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Some regression methods, notably logistic regression and Cox proportional hazards regression, can produce degenerate estimates that are effectively infinite. (Note that zero is a degenerate estimate for odds ratios or hazard ratios, corresponding to an estimate of minus infinity for the log odds ratio or log hazard ratio.) This usually reflects a categorical predictor’s having a 0% or 100% rate of positive outcomes for one of its levels. In such cases, many software packages produce large estimated coefficients with even larger standard errors or with standard errors of zero. In such cases, the confidence intervals and P-values in the standard output are usually invalid.

When the effect that was estimated to be infinite is of interest, then other methods are needed to characterize the strength of evidence for the existence of the effect and its magnitude.

CLodds=PL option on the Model statement in SAS Proc Logistic.

Risklimits=PL option on the Model statement in SAS Proc PHreg.

LRCI option on the Model statement in Proc Genmod.

In Stata, the pllf command can produce a confidence bound.

While profile likelihood confidence bounds should in principle always be available, SAS Proc Genmod has been observed to produce an estimate, lower bound, and upper bound all equal to the same value in some challenging situations.

Example data

Outcome | Percent with | ||||||
---|---|---|---|---|---|---|---|

0 | 1 | Outcome=1 | |||||

Predictor | 0 | 10 | 10 | 50% | |||

1 | 0 | 20 | 100% |

A SAS program to analyze these data produces results that are summarized in the following table:

Results of Analysis of Example Data by Various Methods

Estimated | 95% Confidence Interval | |||||
---|---|---|---|---|---|---|

Method | Odds Ratio | Lower | Upper | P-value | ||

Usual default (Wald) | +∞ | 0 | +∞ | 0.95 | ||

Profile likelihood | +∞ | 8.4 | +∞ | <0.0001 | ||

"Exact" | 24.4 | 3.4 | +∞ | 0.0004 | ||

Firth - Wald | 41.0 | 2.0 | 829 | 0.015 | ||

Firth - profile likelihood | 41.0 | 4.5 | 5466 | 0.0001 |

The usual output, shown in the top row, inaccurately shows the study providing essentially no information. For the other methods, there is a more than 4-fold range in the lower confidence bounds.

The Firth results may seem somewhat confusing, because the estimate is usually the possible true value that is most supported by the study’s data, but the data clearly don’t support an odds ratio of 41.0 any more than they support any other large value. Also, the usual (informal) interpretation of a confidence interval is that the study provides strong evidence against values outside the confidence interval, but any evidence against values larger than the Firth upper bounds does not come from this data set. The estimate from the “exact” method is something called a median-unbiased estimate, but as with the Firth method it seems odd to pick this particular value based on the study’s data. One can instead report the infinite odds ratio estimate and still validly report the “exact” interval with it. The profile likelihood results do not have any counterintuitive features.

I | Attachment | Action | Size | Date | Who | Comment |
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lst | infinite_estimates.lst | manage | 7.0 K | 28 Jan 2013 - 17:53 | PeterBacchetti | |

sas | infinite_estimates.sas | manage | 0.5 K | 28 Jan 2013 - 17:52 | PeterBacchetti |

Topic revision: r4 - 10 Jun 2014 - 15:36:22 - PeterBacchetti

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