PRACTICE OF EPIDEMIOLOGY

Performance of Floating Absolute Risks

Patrick G. Arbogast

From the Department of Biostatistics, Vanderbilt University, Nashville, TN.

Received for publication November 17, 2004; accepted for publication April 29, 2005.

A recent investigation of hormone replacement therapy and breast cancer risk used a method called ‘‘floatingabsolute risks’’ (FARs) to compute confidence intervals for relative hazards. This method has been used in othermedical studies and has received controversy. This controversy stems from the correct implementation of thismethod. However, there has been no direct comparison of the FAR method, as it is sometimes incorrectly appliedand reported, with the conventional approach for computing confidence intervals from proportional hazards re-gression. In this paper, the author reports simulation results comparing these two methods and demonstrates thatthe FAR method, when applied incorrectly, can produce confidence intervals that are substantially too narrow.

confidence intervals; epidemiologic methods; hazard rate; proportional hazards models; regression; relative risk;survival analysis

Abbreviations: FAR, floating absolute risk; SEE, standard error estimate.

Recently, collaborators from the Million Women Studypublished results from an investigation of hormone replace-ment therapy and breast cancer risk (1). Relative hazards ofincident and fatal breast cancer by hormone replacementtherapy use were estimated from proportional hazards re-gression models. When hormone replacement therapygroups were expressed using more than two groups (e.g.,never, past, current users), confidence intervals were com-puted using a method called ‘‘floating absolute risks’’(FARs) (2). According to the article, the FAR method doesnot alter the estimated relative hazards, but it produces stan-dard errors and confidence intervals that enable valid com-parisons to be made between any two groups, even if neitheris the baseline group.

Easton et al. (2) originally proposed the FAR method andapplied it to two studies in which the exposure consisted ofmore than two categories. The first study (3) was an inves-tigation of prognostic factors in 333 patients with small-cellcarcinoma. Easton et al. examined the relation between per-formance status, defined by the World Health Organizationfive-point scale, and time to death from lung cancer using

proportional hazards regression. The second study (4) wasa matched case-control study consisting of 755 matchedpairs, in which the association between duration of oralcontraceptive use (never, 1–48, 49–96, �97 months) andbreast cancer risk was investigated. In both studies, Eastonand colleagues reported standard errors and confidence in-tervals using both the FAR method and the conventionalapproach, and the widths of the FAR confidence intervalswere consistently narrower than the corresponding confi-dence intervals computed using conventional methods. An-other interesting feature of the FAR method was that itincorporated sampling variation into the referent group,and a confidence interval for the referent group was reported.

In a 1997 letter, Peto (5) stated that the FAR methodshould be used in place of the conventional approach forcomputing confidence intervals. However, in a 1999 com-mentary, Greenland et al. (6) stated that the FAR methoddoes not yield valid confidence intervals for relative hazards,claiming that it can be proven mathematically that the 95percent confidence intervals constructed using the FARmethod will not cover any parameter with 95 percent

Reprint requests to Dr. Patrick G. Arbogast, Department of Biostatistics, S-2323 Medical Center North, Vanderbilt University, Nashville,

TN 37232-2158 (e-mail: patrick.arbogast@vanderbilt.edu).

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frequency. In addition, they stated that it did not make sensefor a fixed, reference relative hazard of 1.0 to have a confi-dence interval. Furthermore, they stated that conventionalconfidence intervals are asymptotically efficient and thatany intervals which are consistently narrower are invalid.Easton and Peto addressed these criticisms, stating that theclaim that the FAR method does not directly produce con-fidence intervals for relative hazards is ‘‘. . . obviouslytrue—they are not meant to be’’ (7, p. 393). However, thisis how the FAR method has been applied. Easton and Petoalso stated that the point of the FAR method is to allowcomputation of confidence intervals for relative hazardsfor any pair of categories. It appears that the problem is thatthe FAR method is not being applied correctly. The standarderrors derived from the FAR method, which correspond to‘‘absolute risks,’’ are applied as if they are the standarderrors of the relative hazards. If the FAR method is appliedas intended, then one gets the same standard errors as onegets using the conventional approach.

Clearly, there is controversy regarding the FARmethod asit continues to be used. Although Greenland et al. analyti-cally addressed the FAR method, there has been no directcomparison of the FAR method, as it is incorrectly applied,with the conventional approach for relative hazards to assessthe impact of this. Hence, I conducted a series of simulationstudies comparing confidence intervals computed usingthese two approaches. In the next section, the FAR methodis briefly described, followed by simulation study results.After that, the findings are discussed.

FLOATING ABSOLUTE RISK

Easton et al. (2) derived the FAR method for proportionalhazards regression and conditional logistic regression, usingan easy-to-implement heuristic formulation and a formalderivation of an augmented likelihood. Both derivationsyield virtually identical results.

Consider a categorical exposure z consisting of s catego-ries. In data analyses, z typically is expressed as z ¼ (z1, . . .,zs–1) denoting (s – 1) indicator variables; the category nothaving a designated indicator is the referent group. Under theproportional hazards model, the hazard function at time t is

kðt;z;bÞ¼ k0ðtÞexpðb#zÞ;where k0(t) is the baseline hazard, and b ¼ (b1, . . ., bs–1)denotes the logarithm of the hazards of each category rela-tive to the referent group. Let bj denote the estimated bjs(j ¼ 1, . . ., s – 1) from the proportional hazards regression.The bjs are, of course, correlated. Easton and colleaguesdefined independent parameters a0, . . ., as–1 such that bj ¼aj – a0, and they interpreted the ajs as the logarithms ofthe ‘‘absolute risks’’ associated with each category. Hence,subtraction of a0 produces the logarithm of the relative haz-ards bjs. However, when the FAR method has been imple-mented, the variances of the estimated ajs (j ¼ 1, . . ., s – 1)are used to compute confidence intervals for relative hazardsinstead of the variances of aj � a0: Easton and colleaguesdeveloped two methods to estimate the variances of the ajs;a heuristic formulation and an augmented likelihood.

Under the heuristic formulation, the covariance matrix forthe ajs is calculated from the covariance matrix for the bjsbased on the relation bj ¼ aj � a0; that is, covðbi; bjÞ ¼covðai; ajÞ � covðai; a0Þ � covða0; ajÞ þ covða0; a0Þ (i, j ¼1, . . ., s – 1). For a detailed description of this formulation,refer to the article by Easton et al. (2).

The augmented likelihood is a formal derivation of the ajsby adding a term to the log-partial likelihood of the pro-portional hazards model, which is used to estimate the logrelative hazards bjs. Easton and colleagues stated that theajs can be used to estimate the bjs and that the variances andcovariances are precisely the same as those obtained fromthe original partial likelihood. For a detailed description ofthe augmented likelihood, refer to Easton et al. (2).

SIMULATION STUDIES

Extensive simulation studies were conducted to comparethe confidence intervals based on the FAR method withthose based on the conventional approach. The FAR methodwas implemented using the heuristic formulation. The ex-posure z was a simulated categorical variable, and a numberof different configurations of categories were considered.Survival was generated using the hazard function k(t, z; b)¼exp(b#z), where the baseline hazard k0(t) ¼ 1 and variousbs were considered to assess the effect of different relativehazards. Survival was censored such that approximately25 percent of each sample was censored to reflect real datasettings. For each set of simulation parameters, 1,000 sam-ples were generated. For each simulated sample, the datawere fit using proportional hazards regression, and b; as wellas the covariance matrices for b and a; was recorded. Then,95 percent confidence intervals were computed using theconventional approach and the FAR method, in which theFAR method was applied as described in the previous sec-tion. In addition, the mean standard errors of b and a; as wellas their standard error estimates (SEEs), were computedfrom the 1,000 simulated samples. The SEE is the standarddeviation of the bs from the simulated samples. The meanstandard errors should approximate the SEEs. Coverageprobabilities, the proportion of the 95 percent confidenceintervals that includes the true relative hazard, were alsocomputed for each method. If a method has proper cover-age, then for a 95 percent confidence interval, the coverageshould be very close to 0.95.

In one series of studies, I considered three categories anddefined z ¼ (z1, z2) in which z1 and z2 denoted indicatorvariables. I simulated samples of 50, 100, and 1,000 pa-tients. Different frequency distributions of the exposurewere considered (table 1). In particular, I considered cate-gorical exposures in which the referent group was commonas well as rare and in which one of the exposure categorieswas common as well as rare. The log relative hazards, b ¼(b1, b2), were specified such that the corresponding relativehazards were 1.5 and 2.0. Simulation results are summa-rized in table 1. The mean standard errors from the conven-tional approach approximate the SEE well, and the 95percent confidence intervals have proper coverage. Themean standard errors from the FAR method are consistently

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and substantially smaller than the SEE, and the correspond-ing confidence intervals have poor coverage. For the FARmethod, coverage is particularly poor when the referentgroup is rare. When two groups other than the referent groupare compared, that is, z1 against z2, the FAR and conven-tional methods are identical (results not reported).

Another series of simulations mimicked the lung cancerstudy Easton et al. (2) reported in their paper. For the sim-ulation study, I sampled 300 persons, where performancestatus was simulated to mimic the frequency distributionreported by Easton et al. (table 2). Time to death was gen-

erated, with the hazard function mimicking the relative haz-ards reported by Easton et al. (table 2). Simulation resultsare summarized in table 2. The mean standard errors fromthe conventional approach approximate the SEE well, andthe 95 percent confidence intervals have proper coverage.The mean standard errors from the FAR method are consis-tently and substantially smaller than the SEE, and the cor-responding confidence intervals have poor coverage.

Another series of simulation studies mimicked the resultreported from the hormone replacement therapy and breastcancer analysis of the Million Women Study (1). The time to

TABLE 1. Simulation results for an exposure with three categories

Sample size andfrequency (%)

Parameter SEE*Conventional FAR*

Mean SE* CP* Mean SE CP

50 patients

(40, 30, 30) b1 0.47 0.44 0.94 0.32 0.83

b2 0.47 0.44 0.94 0.32 0.82

(20, 40, 40) b1 0.54 0.50 0.95 0.27 0.70

b2 0.56 0.51 0.93 0.28 0.70

(20, 60, 20) b1 0.52 0.48 0.95 0.22 0.63

b2 0.65 0.59 0.93 0.40 0.79

100 patients

(40, 30, 30) b1 0.30 0.30 0.95 0.22 0.82

b2 0.30 0.30 0.96 0.22 0.84

(10, 45, 45) b1 0.48 0.45 0.95 0.17 0.57

b2 0.49 0.45 0.94 0.18 0.55

(10, 80, 10) b1 0.47 0.43 0.94 0.13 0.44

b2 0.61 0.58 0.94 0.40 0.82

1,000 patients

(40, 30, 30) b1 0.09 0.09 0.95 0.07 0.85

b2 0.09 0.09 0.95 0.07 0.84

(1, 50, 49) b1 0.41 0.39 0.93 0.05 0.21

b2 0.41 0.39 0.94 0.05 0.21

(1, 98, 1) b1 0.44 0.39 0.93 0.04 0.13

b2 0.63 0.56 0.93 0.39 0.78

* SEE, standard error estimate; FAR, floating absolute risk; SE, standard error; CP, coverage probability.

TABLE 2. Simulation results mimicking small cell lung cancer study (Eur J Cancer Clin Oncol 1987;

23:1598–9) (3)

Performancestatus

Frequency(%)

Relativehazard

SEE*Conventional FAR*

Mean SE* CP* Mean SE CP

0 13 1.0 0.19

1 51 1.93 0.21 0.20 0.95 0.10 0.62

2 22 4.15 0.26 0.25 0.95 0.15 0.76

3 11 5.31 0.31 0.31 0.94 0.23 0.85

4 3 5.06 0.60 0.55 0.93 0.50 0.91

* SEE, standard error estimate; FAR, floating absolute risk; SE, standard error; CP, coverage probability.

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invasive and fatal breast cancer and distribution of hormonereplacement therapy use were simulated to mimic the resultsreported in the article, specifically figures 1 and 6. Althoughthe hormone replacement therapy and breast cancer analysisincluded over 800,000 patients, because of computationalconsiderations, samples of 1,000 patients were generated forthe simulation study. Simulation results are summarized intable 3. As observed in the previous simulation studies, theconventional approach produces appropriate standard errorsand confidence intervals. For the FAR method applied toinvasive breast cancer, the current users have poor coverage.For the other hormone replacement therapy groups, themean standard errors and coverage probabilities are reason-able. This may be due to a couple of factors, such as a prev-alent referent group or relative hazards of former hormonereplacement therapy users being close to one. For fatalbreast cancer, the conventional analysis produces appropri-ate results, whereas the FAR method underestimates thestandard errors and has poor coverage.

DISCUSSION

In this report, simulation studies have demonstrated thatconfidence intervals for relative hazards based on an incor-rect application of the FAR method, which sometimes oc-curs in practice, can be too narrow and potentiallymisleading. Conventional analyses provide correct confi-dence intervals for relative hazards, even when comparingtwo categories in which neither is the referent group. How-ever, to do so requires the investigators either to provide allof the potential associations of interest in their report or,which is rarely if ever done, to provide the covariance ma-trix so that the reader can make his/her own calculations.When the FAR method is implemented as designed, theresulting confidence intervals are identical to those obtainedfrom the conventional approach.

The purpose of the FAR method is to allow the reader tocompute confidence intervals of any exposure level relative

to another, even if neither is the referent group. However,investigators who choose to report confidence intervalsbased on the FAR method need to exercise great care as toits correct implementation. Failure to do so can lead topotentially misleading results, specifically associations thatappear to be statistically significant but in fact are not, whichcan have a substantial public health impact.

ACKNOWLEDGMENTS

The author wishes to thank Dr. Frank Harrell, Dr. WayneRay, and Dr. Marie Griffin for their valuable comments thatgreatly enhanced the quality of this manuscript.

Conflict of interest: none declared.

REFERENCES

1. Beral V, Million Women Study Collaborators. Breast cancerand hormone-replacement therapy in the Million WomenStudy. Lancet 2003;362:419–27.

2. Easton DF, Peto J, Babiker AG. Floating absolute risk: analternative to relative risk in survival and case-control anal-ysis avoiding an arbitrary reference group. Stat Med 1991;10:1025–35.

3. Vincent MD, Ashley SE, Smith IE. Prognostic factors insmall cell lung cancer: a simple prognostic index is betterthan conventional staging. Eur J Cancer Clin Oncol 1987;23:1598–9.

4. Oral contraceptive use and breast cancer risk in young women.UKNationalCase-Control StudyGroup.Lancet 1989;1:973–82.

5. Peto R. Birthweight as risk factor for breast cancer. (Letter).Lancet 1997;349:501.

6. Greenland S, Michels KB, Robins JM, et al. Presenting sta-tistical uncertainty in trends and dose-response relations. Am JEpidemiol 1999;149:1077–86.

7. Easton D, Peto J. Re: ‘‘Presenting statistical uncertainty intrends and dose-response relations.’’ (Letter). Am J Epidemiol2000;152:393–4.

TABLE 3. Simulation results mimicking the Million Women Study (Lancet 2003;362:419–27) (1)

Hormone replacement therapyuse at baseline

Frequency(%)

Relativehazard

SEE*Conventional FAR*

Mean SE* CP* Mean SE CP

Invasive breast cancer (figure 1)

Never 49 1.0 0.05

Current 36 1.66 0.08 0.80 0.95 0.06 0.86

Last use: <5 years ago 10 1.04 0.12 0.12 0.95 0.11 0.92

Last use: 5–9 years ago 4 1.01 0.17 0.18 0.97 0.17 0.95

Last use: �10 years ago 2 0.90 0.37 0.35 0.94 0.35 0.94

Fatal breast cancer (figure 6)

Never 47 1.0 0.05

Current 34 1.22 0.08 0.08 0.95 0.06 0.87

Past 18 1.05 0.09 0.09 0.95 0.08 0.90

* SEE, standard error estimate; FAR, floating absolute risk; SE, standard error; CP, coverage probability.

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