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A.S. Whittemore 395variate values. A risk model then uses these baseline covariates to assign eachsubject a risk of developing the outcome of interest during a specified subsequenttime period. For example, the Breast Cancer Family Registry (BCFR),a consortium of six institutions in the United States, Canada and Australia,has been monitoring the vital statuses and cancer occurrences of registry participantsfor more than ten years (John et al., 2004). The New York siteof the BCFR has used the baseline covariates of some 1900 female registryparticipants to assign each of them a ten-year probability of breast cancerdevelopment according to one of several risk models (Quante et al., 2012).These assigned risks are then compared to actual outcomes during follow-up.Subjects who die before outcome occurrence are classified as negative for theoutcome, so those with life-threatening co-morbidities may have low outcomerisk because they are likely to die before outcome development.Using cohort data to estimate outcome probabilities presents statisticalchallenges. For example, some subjects may not be followed for the full riskperiod; instead they are last observed alive and outcome-free after only a fractionof the period. An analysis that excludes these subjects may yield biasedestimates. Instead, censored time-to-failure analysis is needed, and the analysismust accommodate the competing risk of death (Kalbfleisch and Lawless,1998; Kalbfleisch and Prentice, 2002; Putter et al., 2007). Another challengearises when evaluating risk models that include biomarkers obtained fromblood collected at cohort entry. Budgetary constraints may prohibit costlybiomarker assessment for the entire cohort, and cost-efficient sampling designsare needed, such as a nested case-control design (Ernster, 1994), a casecohortdesign (Prentice, 1986), or a two-stage sampling design (Whittemoreand Halpern, 2013).Model calibration is often assessed by grouping subjects into quantiles ofassigned risk, and comparing estimated outcome probability to mean assignedrisk within each quantile. Results are plotted on a graph called an attributiondiagram (AD) (Hsu and Murphy, 1986). For example, the top two panelsof Figure 35.2 show ADs for subjects from the NY-BCFR cohort who havebeen grouped in quartiles of breast cancer risk as assigned by two breastcancer risk models, the BCRAT model and the International Breast CancerIntervention Study (IBIS) model (Tyrer et al., 2004). The null hypothesis ofequality between quantile-specific mean outcome probabilities and mean assignedrisks is commonly tested by classifying subjects into risk groups andapplying a chi-squared goodness-of-fit statistic. This approach has limitations:1) the quantile grouping is arbitrary and varies across cohorts sampled fromthe same population; 2) the averaging of risks over subjects in a quantile canobscure subsets of subjects with poor model fit; 3) when confidence intervalsfor estimated outcome probabilities exclude the diagonal line it is difficult totrouble-shoot the risk model; 4) assuming a chi-squared asymptotic distributionfor the goodness-of-fit statistic ignores the heterogeneity of risks withinquantiles.