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Regression Modeling Strategies pp 1–11Cite as

Introduction

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Part of the book series: Springer Series in Statistics ((SSS))

Abstract

Statistics comprises among other areas study design, hypothesis testing, estimation, and prediction. This text aims at the last area, by presenting methods that enable an analyst to develop models that will make accurate predictions of responses for future observations. Prediction could be considered a superset of hypothesis testing and estimation, so the methods presented here will also assist the analyst in those areas. It is worth pausing to explain how this is so.

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Notes

  1. 1.

    For example, unadjusted odds ratios from 2 × 2 tables are different from adjusted odds ratios when there is variation in subjects’ risk factors within each treatment group, even when the distribution of the risk factors is identical between the two groups.

  2. 2.

    Simple examples to the contrary are the less weight given to a false negative diagnosis of cancer in the elderly and the aversion of some subjects to surgery or chemotherapy.

  3. 3.

    To make an optimal decision you need to know all relevant data about an individual (used to estimate the probability of an outcome), and the utility (cost, loss function) of making each decision. Sensitivity and specificity do not provide this information. For example, if one estimated that the probability of a disease given age, sex, and symptoms is 0.1 and the “cost”of a false positive equaled the “cost” of a false negative, one would act as if the person does not have the disease. Given other utilities, one would make different decisions. If the utilities are unknown, one gives the best estimate of the probability of the outcome to the decision maker and let her incorporate her own unspoken utilities in making an optimum decision for her.

    Besides the fact that cutoffs that are not individualized do not apply to individuals, only to groups, individual decision making does not utilize sensitivity and specificity. For an individual we can compute \(\text{Prob}(Y = 1\vert X = x)\); we don’t care about Prob(Y = 1 | X > c), and an individual having X = x would be quite puzzled if she were given Prob(X > c | future unknown Y) when she already knows X = x so X is no longer a random variable.

    Even when group decision making is needed, sensitivity and specificity can be bypassed. For mass marketing, for example, one can rank order individuals by the estimated probability of buying the product, to create a lift curve . This is then used to target the k most likely buyers where k is chosen to meet total program cost constraints.

  4. 4.

    The ROC curve is a plot of sensitivity vs. one minus specificity as one varies a cutoff on a continuous predictor used to make a decision.

  5. 5.

    An exception may be sensitive variables such as income level. Subjects may be more willing to check a box corresponding to a wide interval containing their income. It is unlikely that a reduction in the probability that a subject will inflate her income will offset the loss of precision due to categorization of income, but there will be a decrease in the number of refusals. This reduction in missing data can more than offset the lack of precision.

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Authors and Affiliations

  1. Department of Biostatistics, School of Medicine Vanderbilt University, Nashville, TN, USA

    Frank E. Harrell Jr.

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  1. Frank E. Harrell Jr.
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Harrell, F.E. (2015). Introduction. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_1

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