Letters to the Editor Re: “use of Two-segmented Logistic Regression to Estimate Change-points in Epidemiologic

We thank Ulm and Küchenhoff (1) for their valuable comments on our paper (2) about the estimation of changepoints in epidemiologic studies. We agree with them that several methods for change-point estimation in generalized linear models have already been described in the statistical literature (3–6). However, it is equally important to recognize that their epidemiologic application to dose-response assessment has been limited by the lack of an algorithm to simultaneously estimate the change-point and the other parameters of effect. In the above approaches, the maximum likelihood estimates are obtained from the profile loglikelihood of sequential models at different fixed changepoints, and hence, this estimation procedure does not provide appropriate error estimates for model parameters, except for the change-point (see the Discussion section of our paper (2) for a detailed description). To overcome this problem, the algorithm proposed in our article (2) provides simultaneous estimation of all model parameters and realistic measurements of statistical error. An alternative method recently proposed by Daniels et al. (7) is now also available. As stated in our paper (2), all inferences presented, such as confidence intervals or tests of hypothesis, were based on the assumption of existence of the change-point. Although we did not explicitly address the problem of testing for this assumption, we recognized, as stated in the Discussion section of our paper, that further research is needed in this area. Ulm (3) suggested a test procedure based on a quasi one-sided distribution. The justification of this method was supported by a simulation study, but it would be desirable to have a more rigorous theoretical justification. In our opinion, alternative methods to test formally for the existence of change-points are still needed. With respect to the example of alcohol intake and risk of myocardial infarction, several points need to be clarified. This example illustrates the applicability of two-segmented logistic models to estimate and provide inferences about the location of the change-point and the magnitude of other parameters of effect, when the change-point actually exists. If there is no change-point (i.e., β 2 = 0), the model reduces to a standard logistic regression, and hence, the change-point is not well defined. In such circumstances, the standard asymptotic properties of Wald and likelihood ratio statistics do not hold (8). Finally, we would like to stress that, although several models are used to display the dose-response relation of alcohol with the risk of myocardial infarction, only the quadratic-linear model provides a statistical estimation of the change-point. We concur with Ulm and Küchenhoff (1) on the importance of careful modeling and interpretation in assessing changeχ1 points and, as already discussed in our article (2), we suggest using nonparametric regression to check the appropriate parameterization of segmented models. Alternative parametric change-point models, which can accommodate many epidemiologic dose-response relations, are also described in the Discussion section of our paper. In conclusion, we believe that the two-segmented logistic regression model provides valuable inference procedures when threshold effects are anticipated and that it deserves wider use in epidemiologic doseresponse analyses.