Page 1Page 2Page 3Page 4Page 5Page 6Page 7Page 8Page 9Page 10Page 11Page 12Page 131Proportional hazards modelspurposes of modeling2causal; predictive; parsimonycausal questions: is some treatment or exposure harmful or beneficial? What is its effect?Examples: HIP trial: comparison of group randomized to receivescreening with group not receiving screeningdoes screening reduce mortality from breast cancer,lengthen life?3Randomized clinical trial for acute leukemia (1.2);comparison of 6-mercaptopurine (6-MP) vs. placebodoes 6-MP increase time to relapse?Infection in kidney dialysis patients (1.4)Comparison of subjects with surgically-placedcatheter, percutaneously-placed catheter; choice ofcatheter placement not made by randomizationDoes method of catheter placement increase time toinfection?4In this setting, have two sorts of predictor/explanatoryvariablesA / variable of primary interest, whose causal effect weare interested in (screening, 6-MP vs. placebo, catheterplacement, etc.)X / other variables (age, race, sex, other characteristics;baseline prognosis, etc.)Asymmetric roles: A is of primary interest, X may beuseful in learning effect of AInitially, will consider treatments A and other variables Xwhich do not vary with timetesting/estimation controlling for other variables Xpossible in nonparametric approach (stratification);limited as to # of factors which can be dealt withsimultaneously5Predictive:what is the survival experience of people who have givencharacteristics?What characteristics are associated with survival?Examples:well-publicized model for predicting risk of developingbreast cancer in womeninputs: age, breast cancer risk factors (age atmenarche, reproductive history (e.g., number ofchildren, age at first pregnancy))goal: determine a woman’s risk/probability ofdeveloping breast cancerall predictor variables play same role logically inmodel6laryngeal canceroutcome: time from first treatment of cancer untildeathpredictors: stage of cancer, agewant to know for patients of given age, stage, what isprognosis7parsimony:easier to summarize data on association of variablewith outcome with single hazard ratio than withsurvival curves (KM)greater efficiency in estimating effect of single factor(if model is correct); greater likelihood of misspecification of modelmodel also provides a unified approach to estimation andtesting treatment effects (up to a point)8Hazard models (review):multiplicative hazard modelsWhere is a baseline hazard functionmost popular: proportional hazards modelThus, Proportional hazards naturally restricts hazard to benonnegativefor other functions c(@) one may need to applyconstraint9proportional hazards: for all times t, the hazards givendifferent covariate levels X1 and X2 are proportional:i.e., ratio of hazards is the same at any time ttrue for vector covariatesagain, semiparametric model formulation is unspecified; or, equivalently, unknown parameter( in is infinite dimensional10inference may be about $ in a global sense ( is nuisance parameter)a subset of $ (other part of $, nuisance); localtests or inferencebaseline or conditional covariate-specific hazard or survival functions11there is a “correlation” between the types of inferenceoutlined above and our classification of causal vs.predictive inferencefor causal questions, one might take goal to makeinference about part of parameter vector $, $A (localinference)this subset of the parameter vector does not completelydescribe the effect of treatment more complete description may involve other part ofparameter vector $X, and baseline hazard full prediction involves both full parameter vector $ andrelative prediction/discrimination involves only parametervector $12in class, we will first follow book (chapter 8)first, global inference-(partial) likelihoods for data(8.2, 8.3)then, local inference for parts of $ (8.4)Neither requires consideration of purposes of fittingproportional hazards modelmodel-building (8.5)Properly involves consideration of inferentialpurpose/goalsestimation of the survival function (8.6)13Use as likelihood conditional probability that individualdies at ti given that 1 subject in risk set dies at this time:Derive “partial likelihood” from full