CRC Genetic Epidemiology Unit Strangeways Research Laboratory Worts Causeway Cambridge CB1 8RN, United Kingdom
Section of Epidemiology Institute of Cancer Research 15 Cotswold Road Belmont SM2 5NG, United Kingdom
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INTRODUCTION |
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The criticism that Greenland et al. raise of the FAR confidence intervals is that they are not, directly, confidence intervals for the relative risk estimates for each category relative to the baseline category. This is obviously truethey are not meant to be. The whole point of the FAR method is to allow computation of confidence limits for relative risks for any pair of categories. The variance of the log relative risk is obtained by adding the "floating variances" for the two categories, in exactly the same way as one does in, for example, an unpaired t test.
Standard confidence intervals do not provide the same information. This is clear in Greenland et al.'s example on birth weight and breast cancer. For example, one can combine the risk estimates for the first two and the last three categories in table 1 (1, p. 1078), by taking an inverse (FAR) variance weighted average of the corresponding log relative risks, and can deduce that the breast cancer risks associated with birth weights lower than 3.5 kg are significantly lower than those at or above 3.5 kg (relative risk = 0.77; 95 percent confidence interval: 0.63, 0.95). The "correct" confidence limits are uninformative for this comparison.
Greenland et al. suggest that the comparisons of other relative risk estimates would be possible if the average of the covariances among the log relative risk estimates were provided. This has the same rationale as the FAR method, providing information about the uncertainty associated with the baseline category which is lost in the standard analysis. Both methods are exact if there is no confounding due to other risk factors and approximate otherwise. The FAR method is aesthetically and perhaps formally referable, because it always gives estimates that are invariant to the choice of the baseline category, and because it minimizes the average covariance (and hence the discrepancy between the confidence limits derived and the "exact" confidence limits that would be obtained from the full covariance matrix). The main reason for referring the FAR confidence limits, however, is that they do provide a valid visual impression of the uncertainty associated with all of the estimates.
In the case of unmatched case-control (or cohort) analyses, an effect similar to that of the FAR method can also be achieved in logistic regression by fitting a separate parameter for each category, dispensing with the "intercept" parameter. Greenland et al. suggest reporting these results in terms of "case-control ratios" and describe the resulting curves as "floating trends." As their name suggests, they are exactly the same in concept as the floating absolute risks except for an arbitrary multiplicative factor. This factor, which describes the overall ratio of cases and controls, is of course a reflection of the study design and contains no additional information on relative risks. The FAR method is invariant to multiplying all risks by an arbitrary constant. However, from a presentational point of view, it seems simpler to assign one group a risk of 1.0, since then the point estimates are the same as the standard relative risk estimates relative to that category, which one would normally wish to present anyway. The method suggested by Greenland et al. requires an additional column of estimates of no intrinsic value.
While the two methods are essentially identical in the simplest case with no confounding factors, there could be differences if other factors are entered into the model. The approach of Greenland et al. requires that any confounding factors be defined to have a (weighted) mean of zero, which will approximately remove the covariance between the estimates in each category due to confounding variables. The same effect is achieved by the FAR method automatically, without any reparameterization.
The other major advantage of the FAR method is that it can be applied directly to conditional logistic regression analyses of matched case-control studies and to Cox regression, where there is no intercept term and therefore no analogy of the method suggested by Greenland et al.
Thus, the only apparently substantial criticism of FAR confidence intervals by Greenland et al. is that they "will never cover any parameter with 95 percent frequency" (1, p. 1083) and might therefore be misinterpreted. We chose the name "floating absolute risk" to describe the parameter for which they are correct confidence intervals, to draw attention to the fact that this is a new statistical concept. The FARs for a categorical variable are the logarithms of the actual values of the underlying rates in each group with the same unknown constant added to each. This explains the apparent paradox that this arbitrary constant can be chosen to make the estimate of the FAR in the reference group (as distinct from its actual value) equal to zero. It had not escaped our notice that it seems odd to assign a variance to zero; but that does not mean it is incorrect. We believe that the introduction of FAR unifies the mathematical structure as well as the analysis of data from case-control and cohort studies, and is thus conceptually as well as ractically useful.
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REFERENCES |
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Department of Epidemiology UCLA School of Public Health 22333 Swenson Drive Topanga, CA 90290
Strangeways Research Laboratory Worts Causeway Cambridge CB1 8RN, United Kingdom
Department of Epidemiology School of Public Health University of North Carolina Chapel Hill, NC 27599-7400
Channing Laboratory Harvard Medical School Boston, MA 02115
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INTRODUCTION |
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To avoid these problems, we proposed modification of the FAR method by anchoring the "floated" standard errors to the underlying rates or risks in a cohort study or to the underlying case-control ratios in a case-control study (8). Easton and Peto (1
) incorrectly label as "arbitrary" the multiplicative constant used in this anchoring. The constant is instead just the one needed to create valid intervals from the FAR standard errors. In a cohort study, the resulting additional column of estimates has intrinsic value because it displays the trend on an absolute incidence scale, with incidence calculated at the average covariate values (a procedure that roughly approximates standardization to the total sample) (9
, p. 1081).
The absolute scale is essential for evaluating the public health importance of the factor. In a case-control study, only relative values are estimable without external information. Nonetheless, plotting case-control ratios achieves the primary goal of giving a valid impression of trend uncertainty; furthermore, one can convert these ratios into absolute-incidence estimates if one divides them by the ratio of the case and control sampling fractions (9). Conversion to incidence can also be done if one knows the overall incidence in the source population (10
), as in a study that takes its cases from a population-based registry.
Easton and Peto state that the FAR method does not require an additional column. The absence of that extra column is exactly why the FAR method gives the incorrect impression that the intervals it generates apply to the relative risks. It is also no advantage that the FAR method can be applied directly to conditional logistic and Cox regression, because the intervals it generates in those applications are just as invalid as the FAR intervals in other settings. Nor does the FAR method have any computational advantage: It requires rather awkward covariance manipulations (2), whereas our modification requires only subtracting the covariate means from the covariates.
In summary, although we agree with Easton et al. (2) regarding the unsatisfactory properties of conventional confidence intervals, we maintain that their original FAR intervals (2
, 3
) are invalid and must be modified if they are not to mislead readers. We have suggested some computationally simpler ways to accomplish the same goals as the FAR method while avoiding its problems (8
). Our "floated trend" approach can be viewed as a direct correction of the FAR method. We believe that, without this correction, conventional intervals would be preferable to FAR intervals. Finally, we again caution that none of the methods under consideration (conventional, FAR, and our floated-trend approach) provide a satisfactory summary of trends compatible with the data under the analysis model (8
, p. 1085).
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