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Multivariable Modeling Strategies

  • Chapter
Regression Modeling Strategies

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

Abstract

Chapter 2 dealt with aspects of modeling such as transformations of predictors, relaxing linearity assumptions, modeling interactions, and examining lack of fit. Chapter 3 dealt with missing data, focusing on utilization of incomplete predictor information. All of these areas are important in the overall scheme of model development, and they cannot be separated from what is to follow. In this chapter we concern ourselves with issues related to the whole model, with emphasis on deciding on the amount of complexity to allow in the model and on dealing with large numbers of predictors. The chapter concludes with three default modeling strategies depending on whether the goal is prediction, estimation, or hypothesis testing.

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Notes

  1. 1.

    Even then, the two blood pressures may need to be transformed to meet distributional assumptions.

  2. 2.

    Shrinkage (penalized estimation) is a general solution (see Section 4.5). One can always use complex models that are “penalized towards simplicity,” with the amount of penalization being greater for smaller sample sizes.

  3. 3.

    One can also perform a joint test of all parameters associated with nonlinear effects. This can be useful in demonstrating to the reader that some complexity was actually needed.

  4. 4.

    Lockhart et al. 425 provide an example with n = 100 and 10 orthogonal predictors where all true βs are zero. The test statistic for the first variable to enter has type I error of 0.39 when the nominal α is set to 0.05, in line with what one would expect with multiple testing using \(1 - 0.95^{10} = 0.40\).

  5. 5.

    AIC works successfully when the models being entertained are on a progression defined by a single parameter, e.g. a common shrinkage coefficient or the single number of knots to be used by all continuous predictors. AIC can also work when the model that is best by AIC is much better than the runner-up so that if the process were bootstrapped the same model would almost always be found. When used for one variable at a time variable selection. AIC is just a restatement of the P-value, and as such, doesn’t solve the severe problems with stepwise variable selection other than forcing us to use slightly more sensible α values. Burnham and Anderson 84 recommend selection based on AIC for a limited number of theoretically well-founded models. Some statisticians try to deal with multiplicity problems caused by stepwise variable selection by making α smaller than 0.05. This increases bias by giving variables whose effects are estimated with error a greater relative chance of being selected. Variable selection does not compete well with shrinkage methods that simultaneously model all potential predictors.

  6. 6.

    This is akin to doing a t-test to compare the two treatments (out of 10, say) that are apparently most different from each other.

  7. 7.

    These are situations where the true R 2 is low, unlike tightly controlled experiments and mechanistic models where signal:noise ratios can be quite high. In those situations, many parameters can be estimated from small samples, and the \(\frac{m} {15}\) rule of thumb can be significantly relaxed.

  8. 8.

    See [487]. If one considers the power of a two-sample binomial test compared with a Wilcoxon test if the response could be made continuous and the proportional odds assumption holds, the effective sample size for a binary response is 3n 1 n 2∕n ≈ 3min(n 1, n 2) if n 1∕n is near 0 or 1 [664, Eq. 10, 15]. Here n 1 and n 2 are the marginal frequencies of the two response levels.

  9. 9.

    Based on the power of a proportional odds model two-sample test when the marginal cell sizes for the response are n 1, …, n k , compared with all cell sizes equal to unity (response is continuous) [664, Eq, 3]. If all cell sizes are equal, the relative efficiency of having k response categories compared with a continuous response is \(1 - 1/k^{2}\) [664, Eq. 14]; for example, a five-level response is almost as efficient as a continuous one if proportional odds holds across category cutoffs.

  10. 10.

    This is approximate, as the effective sample size may sometimes be boosted somewhat by censored observations, especially for non-proportional hazards methods such as Wilcoxon-type tests. 49

  11. 11.

    An even more stringent assessment is obtained by stratifying calibration curves by predictor settings.

  12. 12.

    It is interesting that researchers are quite comfortable with adjusting P-values for post hoc selection of comparisons using, for example, the Bonferroni inequality, but they do not realize that post hoc selection of comparisons also biases point estimates.

  13. 13.

    There is an option to force continuous variables to be linear when they are being predicted.

  14. 14.

    If one were to estimate transformations without removing observations that had these constants inserted for the current Y -variable, the resulting transformations would likely have a spike at Y = imputation constant.

  15. 15.

    Study to Understand Prognoses Preferences Outcomes and Risks of Treatments

  16. 16.

    Whether this statistic should be used to change the model is problematic in view of model uncertainty.

  17. 17.

    The R function score.binary in the Hmisc package (see Section 6.2) assists in computing a summary variable from the series of binary conditions.

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    Frank E. Harrell Jr.

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Harrell, F.E. (2015). Multivariable Modeling Strategies. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_4

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