Marginal estimates through regression standardization in competing - PowerPoint PPT Presentation

Marginal estimates through regression standardization in competing risks and relative survival models Paul C Lambert 1,2 1 Biostatistics Research Group, Department of Health Sciences, University of Leicester, UK 2 Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden 2019 Nordic and Baltic Stata Users Group meeting Stockholm, 30 August 2019 Paul C Lambert Standardization in competing risks 30 August 2019 1

Regression Standardization Fit a statistical model that contains exposure, X , and potential 1 confounders, ❩ . Predict outcome for all individuals assuming they are all exposed 2 (set X = 1). Take mean to give marginal estimate of outcome. 3 Repeat by assuming all are unexposed (set X = 0). 4 Take the difference/ratio in means to form contrasts. 5 Key point is the distribution of confounders, ❩ , is the same for the exposed and unexposed. If the model is sufficient for confounding control then such contrasts can be interpreted as causal effects. Also known as direct/model based standardization. G-formula (with no time-dependent confounders)[1]. Paul C Lambert Standardization in competing risks 30 August 2019 2

Why not margins? margins does regression standardization, so why not use this? It is an excellent command, but does not do what I wanted for survival data. In particular, extensions to competing risks and relative survival. Paul C Lambert Standardization in competing risks 30 August 2019 3

Marginal survival time With survival data X - is a binary exposure: 0 (unexposed) and 1 (exposed). T - is a survival time. T 0 - is the potential survival time if X is set to 0. T 1 - is the potential survival time if X is set to 1. The average causal difference in mean survival time E [ T 1 ] − E [ T 0 ] This is what stteffects can estimate. We often have limited follow-up and calculating the mean survival requires extrapolation and makes very strong distributional assumptions. Paul C Lambert Standardization in competing risks 30 August 2019 4

Marginal Survival functions Rather than use mean survival we can define our causal effect in terms of the marginal survival function. E [ T 1 > t ] − E [ T 0 > t ] We can limit t within observed follow-up time. For confounders, ❩ , we can write this as, E [ S ( t | X = 1 , ❩ )] − E [ S ( t | X = 0 , ❩ )] Note that this is the expectation over the distribution of ❩ . Paul C Lambert Standardization in competing risks 30 August 2019 5

Estimation Fit a survival model for exposure X and confounders ❩ . Predict survival function for each individual setting X = x and then average. Force everyone to be exposed and then unexposed. � N � N 1 S ( t | X = 1 , ❩ = ③ i ) − 1 � � S ( t | X = 0 , ❩ = ③ i ) N N i =1 i =1 Use their observed covariate pattern, ❩ = ③ i . We can standardize to an external (reference) population. � N 1 w i � S i ( t | X = x , ❩ = z i ) N i =1 standsurv will perform these calculation. Paul C Lambert Standardization in competing risks 30 August 2019 6

Competing risks h 1 ( t ) Cancer Alive Other h 2 ( t ) Separate models for each cause, e.g. h 1 ( t | ❩ ) = h 0 , 1 ( t ) exp ( β 1 ❩ ) h 2 ( t | ❩ ) = h 0 , 2 ( t ) exp ( β 2 ❩ ) Paul C Lambert Standardization in competing risks 30 August 2019 7

Two types of probability We may be interested in cause-specific survival/failure. (1) In the absence of other causes (net) � t F k ( t ) = 1 − S k ( t ) = P ( T k ≤ t ) = S k ( u ) h k ( u ) du 0 We may be interested in cumulative incidence functions. (2) In the presence of other causes (crude) � t CIF k ( t ) = P ( T ≤ t , event = k ) = S ( u ) h k ( u ) du 0 Both are of interest - depends on research question. (1) Needs conditional independence assumption to interpret as net probability of death. Paul C Lambert Standardization in competing risks 30 August 2019 8

Description of Example 102,062 patients with bladder cancer in England (2002-2013). Death due to cancer and other causes. Covariates age, sex and deprivation in five groups. Restrict here to most and least deprived. Models - Flexible parametric (Royston-Parmar) models[2] - Separate model for cancer and other causes. - Age modelled using splines (3 df) - 2-way interactions - Time-dependent effects for all covariates. Paul C Lambert Standardization in competing risks 30 August 2019 9

Two separate cause-specific models Cancer Model stset dod, failure(status==1) exit(time min(dx+365.24*10,mdy(12,31,2013))) /// origin(dx) id(patid) scale(365.24) stpm2 dep5 male agercs* dep_agercs*, df(5) scale(hazard) /// tvc(agercs* male dep5) dftvc(3) estimates store cancer Other cause Model stset dod, failure(status==2) exit(time min(dx+365.24*10,mdy(12,31,2013))) /// origin(dx) id(patid) scale(365.24) stpm2 dep5 male agercs* dep_agercs*, df(5) scale(hazard) /// tvc(agercs* male dep5) dftvc(3) estimates store other Paul C Lambert Standardization in competing risks 30 August 2019 10

Conditional cause-specific CIFs (Females) Least Deprived 1.0 1.0 1.0 1.0 Age 50 Age 60 Age 70 Age 80 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 CIF CIF CIF CIF 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Years from diagnosis Years from diagnosis Years from diagnosis Years from diagnosis Most Deprived 1.0 1.0 1.0 1.0 Age 50 Age 60 Age 70 Age 80 0.8 0.8 0.8 0.8 0.6 0.6 0.6 0.6 CIF CIF CIF CIF 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.0 0.0 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Years from diagnosis Years from diagnosis Years from diagnosis Years from diagnosis Cancer Other Causes Paul C Lambert Standardization in competing risks 30 August 2019 11

Standardized cause-specific survival/failure Probability of death in the absence of other causes. Consider a single cause: standardize and form contrasts. Cancer specific survival/failure F 1 ( t ) = 1 − S 1 ( t ) E [ F 1 ( t ) | X = 1 , ❩ ] − E [ F 1 ( t ) | X = 0 , ❩ ] � N � N 1 F 1 ( t | X = 1 , ❩ = ③ i ) − 1 � � F 1 ( t | X = 0 , ❩ = ③ i ) N N i =1 i =1 Not a ‘real world’ probability, but comparisons between exposures where differential other cause mortality is removed is of interest. Paul C Lambert Standardization in competing risks 30 August 2019 12

Using standsurv Take mean of 102,062 survival functions where all individuals forced to be unexposed. Take mean of 102,062 survival functions where all individuals forced to be exposed. Paul C Lambert Standardization in competing risks 30 August 2019 13

Using standsurv Take mean of 102,062 survival functions where all individuals forced to be unexposed. Take mean of 102,062 survival functions where all individuals forced to be exposed. . estimates restore cancer . range tt 0 10 101 . standsurv, timevar(tt) failure ci /// at1(dep5 0 dep agercs1 0 dep agercs2 0 dep agercs3 0) /// at2(dep5 1 dep agercs1=agercs1 dep agercs2=agercs2 dep agercs3=agercs3) /// contrast(difference) /// atvar(F cancer s dep1 F cancer s dep5) /// contrastvar(F cancer diff) Paul C Lambert Standardization in competing risks 30 August 2019 13

Using standsurv Take mean of 102,062 survival functions where all individuals forced to be unexposed. Take mean of 102,062 survival functions where all individuals forced to be exposed. . estimates restore cancer . range tt 0 10 101 . standsurv, timevar(tt) failure ci /// at1(dep5 0 dep agercs1 0 dep agercs2 0 dep agercs3 0) /// at2(dep5 1 dep agercs1=agercs1 dep agercs2=agercs2 dep agercs3=agercs3) /// contrast(difference) /// atvar(F cancer s dep1 F cancer s dep5) /// contrastvar(F cancer diff) Paul C Lambert Standardization in competing risks 30 August 2019 13

Using standsurv Take mean of 102,062 survival functions where all individuals forced to be unexposed. Take mean of 102,062 survival functions where all individuals forced to be exposed. . estimates restore cancer . range tt 0 10 101 . standsurv, timevar(tt) failure ci /// at1(dep5 0 dep agercs1 0 dep agercs2 0 dep agercs3 0) /// at2(dep5 1 dep agercs1=agercs1 dep agercs2=agercs2 dep agercs3=agercs3) /// contrast(difference) /// atvar(F cancer s dep1 F cancer s dep5) /// contrastvar(F cancer diff) Paul C Lambert Standardization in competing risks 30 August 2019 13

Using standsurv Take mean of 102,062 survival functions where all individuals forced to be unexposed. Take mean of 102,062 survival functions where all individuals forced to be exposed. . estimates restore cancer . range tt 0 10 101 . standsurv, timevar(tt) failure ci /// at1(dep5 0 dep agercs1 0 dep agercs2 0 dep agercs3 0) /// at2(dep5 1 dep agercs1=agercs1 dep agercs2=agercs2 dep agercs3=agercs3) /// contrast(difference) /// atvar(F cancer s dep1 F cancer s dep5) /// contrastvar(F cancer diff) Paul C Lambert Standardization in competing risks 30 August 2019 13