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Ann Thorac Surg 2005;80:1169
© 2005 The Society of Thoracic Surgeons
* Address reprint requests to Dr Anderson, 21672 Montbury Dr, Lake Forest, CA 92630 (Email: wnilesanderson{at}aol.com).
The article by Karkouti and associates [1] in this issue of The Annals of Thoracic Surgery makes effective use of two analytic methods that are often overlooked in analyzing clinical data series. Both involve assessment of the quality of the logistic regression model, which is central to the conclusions drawn in the article. The methodology of this article can serve as a good example for other articles submitted to The Annals.
Logistic regression is used in the article to study the relationship of hematocrit to stroke. The article presented both the c-index and Hosmer-Lemeshow tests to assess the accuracy of the logistic model, and the fit is certainly reasonable in this case. Both this use of a logistic model and the validation method are standard.
However, logistic regression depends on an important assumption that is frequently overlooked. The logistic model assumes that the logit of the probability for stroke is a linear function of the hematocrit. [The logit of a probability p is defined as log(p/1 – p), where log represents the natural logarithm.] Figure 2 of the article presents a visual verification of this linearity assumption: the logit of the probability is reasonably close to linear, and it fits comfortably within the confidence limits produced by the logistic model. In the article the graph serves as a useful complement to the formal goodness of fit statistics.
For a different data set, the plot might have not been close to linear. Then the graph would have suggested a transformation that would have produced a more useful logistic regression model. Use of the c-index and Hosmer-Lemeshow tests alone does not produce such information, because these tests do not distinguish between nonlinearity and random noise in checking the model fit.
Such an analysis is not as easy as it may sound, because it depends on the ability to use other tools to assess the relationship. The basis for deriving the relationship is the histogram of Figure 1; the histogram could be presented with more groups, with the logit transformation of the probabilities used for the graph. However, the histogram would be quite jumpy, and the article used cubic splines to smooth the curve. The exact details of how cubic splines work to smooth curves are not important, and other smoothing methods may work as well. Whatever method is used, the important point is that the central curve in Figure 2 was not derived from the logistic regression model; instead it is being used to validate the assumptions underlying the logistic regression model.
The analysis was aided by the large size of the data set; admittedly the situation would not be so simple with even a moderate size data set. Nevertheless the graphical analysis of the logistic regression model is a tool that all analysts should consider using when the logistic regression is crucial to the analysis of a clinical data series.
The article made effective use of another often overlooked validation method, which is to perform bootstrap repetitions of the analysis. Bootstrap samples will be somewhat different than the original data set, and the technique will determine how sensitive the conclusion is to small changes in the data. Here the conclusion proved to be quite robust, increasing the reader's confidence that the detected relationship is real.
Of course bootstrapping is not a cure-all, and the article still suffers from being a single center study, albeit on a very large series. Bootstrapping also cannot replace validation by a new study. In spite of these limitations, bootstrapping is a useful and easily implemented technique that should be considered by all analysts.
The two statistical techniques are described [2–5] and are also cited in the Karkouti and associates article [1]. The usefulness of these techniques is not limited to logistic regression, and their application in related situations is described by Katz [4].
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