Efficient Estimators for Survival and Ordinal Outcomes in RCTs Without Proportional Hazards and Odds Assumptions with Variable Selection
Nick Williams and Iván Díaz
The development version can be installed from GitHub with:
devtools::install_github("nt-williams/adjrct")adjrct implements efficient estimators for the restricted mean survival time (RMST) and survival probability for time-to-event outcomes (Díaz et al., 2019), and the average log odds ratio (Díaz et al., 2016) and Mann-Whitney estimand (Vermeulen et al., 2014) for ordinal outcomes in randomized controlled trials (RCT) without the proportional hazards or odds assumptions. Prognostic baseline variables should be incorporated to obtain equal or better asymptotic precision compared to un-adjusted estimators. Under random censoring, the primary estimator (TMLE) is doubly robust–it is consistent if either the outcome or censoring model is correctly specified. For survival outcomes, estimators are implemented using a formula interface based on that of the survival package for familiar users.
To allow for estimation of multiple estimands without having to
re-estimate nuisance parameters we first create a Survival metadata
object using the survrct() function. We specify the model
parameterization using a typical R formula with Surv() (based on the
survival package)
specifying the left-hand side of the formula. We also specify a formula
for the propensity.
library(adjrct)
data("c19.tte")
surv <- survrct(Surv(days, event) ~ A + age + sex + bmi + dyspnea,
A ~ 1, data = c19.tte)
surv
#> survrct metadata
#>
#> Outcome regression: Surv(days, event) ~ A + age + sex + bmi + dyspnea
#> Propensity: A ~ 1
#>
#> • Estimate RMST with `rmst()`
#> • Estimate survival probability with `survprob()`
#> • Inspect nuisance parameter models with `get_fits()`
#>
#> Estimator: TMLE
#> Target variable: A
#> Status Indicator: event
#> Max coarsened time: 15Using the metadata from the previous step we can now estimate the restricted mean survival time and survival probability for a single or multiple time horizons. If multiple times are evaluated, two confidence bands are returned: 95% point-wise intervals as well as 95% uniform confidence bands based on the multiplier-bootstrap from Kennedy (2019).
rmst(surv, 14)
#> RMST Estimator: tmle
#>
#> Marginal RMST: E(min[T, 14] | A = a)
#> Treatment Arm
#> Estimate: 12.43
#> Std. error: 0.12
#> 95% CI: (12.2, 12.66)
#> Control Arm
#> Estimate: 11.37
#> Std. error: 0.17
#> 95% CI: (11.04, 11.7)
#>
#> Treatment Effect: E(min[T, 14] | A = 1) - E(min[T, 14] | A = 0)
#> Additive effect
#> Estimate: 1.06
#> Std. error: 0.2
#> 95% CI: (0.66, 1.46)
survprob(surv, 14)
#> Survival Probability Estimator: tmle
#>
#> Marginal Survival Probability: Pr(T > 14 | A = a)
#> Treatment Arm
#> Estimate: 0.76
#> Std. error: 0.02
#> 95% CI: (0.73, 0.79)
#> Control Arm
#> Estimate: 0.73
#> Std. error: 0.02
#> 95% CI: (0.7, 0.76)
#>
#> Treatment Effect: Pr(T > 14 | A = 1) - Pr(T > 14 | A = 0)
#> Additive effect
#> Estimate: 0.03
#> Std. error: 0.02
#> 95% CI: (-0.02, 0.07)We can similarly create an Ordinal metadata object using the
ordinalrct() function.
data("c19.ordinal")
ord <- ordinalrct(state_ordinal ~ A + age + dyspnea + sex,
A ~ 1, data = c19.ordinal)
ord
#> ordinalrct metadata
#>
#> Outcome regression: state_ordinal ~ A + age + dyspnea + sex
#> Propensity: A ~ 1
#>
#> • Estimate log odds ratio with `log_or()`
#> • Estimate Mann-Whitney with `mannwhitney()`
#> • Estimate with `cdf()`
#> • Estimate with `pmf()`
#> • Inspect nuisance parameter models with `get_fits()`
#>
#> Estimator: tmle
#> Target variable: A
#> Outcome variable: state_ordinalThe average log odds ratio, Mann-Whitney statistic, CDF, and PMF can then be estimated using the metadata.
log_or(ord)
#> Log OR Estimator: tmle
#>
#> Arm-specific log odds:
#> Treatment Arm
#> Estimate: 1.62
#> Std. error: 0.09
#> 95% CI: (1.44, 1.8)
#> Control Arm
#> Estimate: 0.87
#> Std. error: 0.07
#> 95% CI: (0.73, 1.02)
#>
#> Average log odds ratio:
#> Estimate: 0.75
#> Std. error: 0.11
#> 95% CI: (0.52, 0.97)
mannwhitney(ord)
#> Mann-Whitney Estimand
#>
#> Estimator: tmle
#> Estimate: 0.45
#> Std. error: 0.01
#> 95% CI: (0.42, 0.47)
cdf(ord)
#> CDF Estimator: tmle
#>
#> Arm-specific CDF: Pr(K <= k | A = a)
#> Treatment Arm
#> k Estimate Std. error 95% CI Uniform 95% CI
#> 1 0 0.602 0.018 (0.57 to 0.64) (0.56 to 0.65)
#> 2 1 0.685 0.017 (0.65 to 0.72) (0.64 to 0.73)
#> 3 2 0.742 0.016 (0.71 to 0.77) (0.70 to 0.78)
#> 4 3 0.903 0.011 (0.88 to 0.92) (0.88 to 0.93)
#> 5 4 0.975 0.006 (0.96 to 0.99) (0.96 to 0.99)
#> 6 5 1.000 - - -
#>
#> Control Arm
#> k Estimate Std. error 95% CI Uniform 95% CI
#> 1 0 0.554 0.018 (0.52 to 0.59) (0.51 to 0.60)
#> 2 1 0.610 0.017 (0.58 to 0.64) (0.57 to 0.65)
#> 3 2 0.686 0.017 (0.65 to 0.72) (0.65 to 0.73)
#> 4 3 0.714 0.016 (0.68 to 0.75) (0.68 to 0.75)
#> 5 4 0.877 0.011 (0.86 to 0.90) (0.85 to 0.90)
#> 6 5 1.000 - - -
pmf(ord)
#> PMF Estimator: tmle
#>
#> Arm-specific PMF: Pr(K = k | A = a)
#> Treatment Arm
#> k Estimate Std. error 95% CI Uniform 95% CI
#> 1 0 0.602 0.018 (0.57 to 0.64) (0.56 to 0.65)
#> 2 1 0.083 0.010 (0.06 to 0.10) (0.06 to 0.11)
#> 3 2 0.057 0.008 (0.04 to 0.07) (0.04 to 0.08)
#> 4 3 0.161 0.013 (0.13 to 0.19) (0.13 to 0.19)
#> 5 4 0.072 0.010 (0.05 to 0.09) (0.05 to 0.10)
#> 6 5 0.025 0.006 (0.01 to 0.04) (0.01 to 0.04)
#>
#> Control Arm
#> k Estimate Std. error 95% CI Uniform 95% CI
#> 1 0 0.554 0.018 (0.52 to 0.59) (0.51 to 0.60)
#> 2 1 0.056 0.008 (0.04 to 0.07) (0.04 to 0.08)
#> 3 2 0.075 0.010 (0.06 to 0.09) (0.05 to 0.10)
#> 4 3 0.028 0.006 (0.02 to 0.04) (0.01 to 0.04)
#> 5 4 0.163 0.014 (0.14 to 0.19) (0.13 to 0.20)
#> 6 5 0.123 0.011 (0.10 to 0.14) (0.10 to 0.15)Díaz, I., E. Colantuoni, D. F. Hanley, and M. Rosenblum (2019). Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards. Lifetime Data Analysis 25 (3), 439–468.
Díaz, I., Colantuoni, E. and Rosenblum, M. (2016), Enhanced precision in the analysis of randomized trials with ordinal outcomes. Biom, 72: 422-431. https://doi.org/10.1111/biom.12450
Vermeulen, K., Thas, O., and Vansteelandt, S. (2015), Increasing the power of the Mann‐Whitney test in randomized experiments through flexible covariate adjustment. Statist. Med., 34, pages 1012– 1030. doi: 10.1002/sim.6386
Edward H. Kennedy (2019) Nonparametric Causal Effects Based on Incremental Propensity Score Interventions, Journal of the American Statistical Association, 114:526, 645-656, DOI: 10.1080/01621459.2017.1422737