Survival and treatment effect estimation

Description

Provides functions to estimate the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using Kaplan-Meier estimation, landmark estimation for a randomized trial setting, inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation, and landmark estimation for an observational study setting. The landmark estimation approaches provide improved efficiency by incorporating intermediate event information and are robust to model misspecification. The IPTW Kaplan-Meier approach and landmark estimation in an observational study setting approach account for potential selection bias.

Author(s)

Layla Parast

References

Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481.

Xie, J., & Liu, C. (2005). Adjusted Kaplan-Meier estimator and log-rank test with inverse probability of treatment weighting for survival data. Statistics in Medicine, 24(20), 3089-3110.

Parast, L., Tian, L., & Cai, T. (2014). Landmark Estimation of Survival and Treatment Effect in a Randomized Clinical Trial. Journal of the American Statistical Association, 109(505), 384-394.

Parast, L. & Griffin B.A. (2017). Landmark Estimation of Survival and Treatment Effects in Observational Studies. Lifetime Data Analysis, 23:161-182.

Examples

data(example_rct)
delta.km(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2)
#executable but takes time
#delta.land.rct(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2, 
#landmark = 1, short = cbind(example_rct$TS,example_rct$DS), z.cov = as.matrix(example_rct$Z))
data(example_obs)
delta.iptw.km(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2,
cov.for.ps = as.matrix(example_obs$Z)) 
#executable but takes time
#delta.land.obs(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2, 
#landmark = 1, short = cbind(example_obs$TS,example_obs$DS), z.cov = as.matrix(example_obs$Z),
#cov.for.ps = as.matrix(example_obs$Z))

Nonparametric Nelson-Aalen estimate of survival

Description

Nonparametric Nelson-Aalen estimate of survival

Usage

Est.KM.FUN.weighted(xi, di, si, myt, weight.perturb = NULL, bw = NULL)

Arguments

Value

Smoothed survival estimate.

Author(s)

Layla Parast

Calculates kernel matrix

Description

Helper function; this calculates the kernel matrix

Usage

Kern.FUN(zz, zi, bw)

Arguments

Value

the kernel matrix

Author(s)

Layla Parast

Repeats a row.

Description

Helper function; this function creates a matrix that repeats vc, dm times where each row is equal to the vc vector.

Usage

VTM(vc, dm)

Arguments

Value

a matrix that repeats vc, dm times where each row is equal to the vc vector

Helper function

Description

Helper function; should not be called directly by user.

Usage

cumsum2(mydat)

Arguments

Value

Author(s)

Layla Parast

Helper function

Description

Helper function; should not be called directly by user.

Usage

helper.si(yy,FUN,Yi,Vi=NULL)

Arguments

Value

Author(s)

Layla Parast

Estimates survival and treatment effect using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Usage

delta.iptw.km(tl, dl, treat, tt, var = FALSE, conf.int = FALSE, ps.weights = NULL, 
weight.perturb = NULL, perturb.ps = FALSE, cov.for.ps = NULL)

Arguments

Details

Let T_{Li} denote the time of the primary event of interest for person i, C_i denote the censoring time, Z_{i} denote the vector of baseline (pretreatment) covariates, and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. Due to censoring, we observe X_{Li}= min(T_{Li}, C_{i}) and \delta_{Li} = I(T_{Li}\leq C_{i}). This function estimates survival at time t within each treatment group, S_j(t) = P(T_{L} > t | G = j) for j = 1,0 and the treatment effect defined as \Delta(t) = S_1(t) - S_0(t).

The inverse probability of treatment weighted (IPTW) Kaplan-Meier (KM) estimate of survival at time t for each treatment group is

\hat{S}_{IPTW,KM, j}(t) = \prod _{t_{kj} \leq t} \left [1-\frac{d_{kj}^w}{y_{kj}^w}\right ] \mbox{ if } t\geq t_{1j}, \mbox{ or } 1 \mbox{ if } t<t_{1j}

where t_{1j},...,t_{Dj} are the distinct observed event times of the primary outcome in treatment group j, d_{kj}^w = \sum_{i: X_{Li} = t_{kj}, \delta_{Li} = 1} {\hat{W}_j(Z_i)}^{-1}\delta_{Li} I(G_i = j) and y_{kj}^w = \sum_{i: X_{Li} \geq t_{kj}} {\hat{W}_j(Z_i)}^{-1} I(G_i = j), W_j(Z_i) = {P(G_{i} = j | Z_i)}, and \hat{W}_j(Z_i) is the estimated propensity score (see ps.wgt.fun for more information). The IPTW KM estimate of treatment effect at time t is \hat{\Delta}_{IPTW,KM}(t) = \hat{S}_{IPTW,KM, 1}(t) - \hat{S}_{IPTW,KM, 0}(t).

To obtain variance estimates and construct confidence intervals, we use a perturbation-resampling method. Specifically, let \{V^{(b)}=(V_1^{(b)}, . . . ,V_n^{(b)})^{T}, b=1,...B\} be n\times B independent copies of a positive random variable U from a known distribution with unit mean and unit variance such as an Exp(1) distribution. To estimate the variance of our estimates, we appropriately weight the estimates using these perturbation weights to obtain perturbed values: \hat{S}_{IPTW,KM,0} (t)^{(b)}, \hat{S}_{IPTW,KM,1} (t)^{(b)}, and \hat{\Delta}_{IPTW,KM} (t)^{(b)}, b=1,...B. We then estimate the variance of each estimate as the empirical variance of the perturbed quantities. To construct confidence intervals, one can either use the empirical percentiles of the perturbed samples or a normal approximation.

Value

A list is returned:

Author(s)

Layla Parast

References

Xie, J., & Liu, C. (2005). Adjusted Kaplan-Meier estimator and log-rank test with inverse probability of treatment weighting for survival data. Statistics in Medicine, 24(20), 3089-3110.

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41-55.

Rosenbaum, P. R., & Rubin, D. B. (1984). Reducing bias in observational studies using subclassification on the propensity score. Journal of the American Statistical Association, 79(387), 516-524.

Examples

data(example_obs)
W.weight = ps.wgt.fun(treat = example_obs$treat, cov.for.ps = as.matrix(example_obs$Z))	
delta.iptw.km(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2, 
ps.weights = W.weight) 
delta.iptw.km(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2,
cov.for.ps = as.matrix(example_obs$Z)) 

Estimates survival and treatment effect using Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using Kaplan-Meier estimation

Usage

delta.km(tl, dl, treat, tt, var = FALSE, conf.int = FALSE, weight.perturb = NULL)

Arguments

Details

Let T_{Li} denote the time of the primary event of interest for person i, C_i denote the censoring time and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. Due to censoring, we observe X_{Li}= min(T_{Li}, C_{i}) and \delta_{Li} = I(T_{Li}\leq C_{i}). This function estimates survival at time t within each treatment group, S_j(t) = P(T_{L} > t | G = j) for j = 1,0 and the treatment effect defined as \Delta(t) = S_1(t) - S_0(t).

The Kaplan-Meier (KM) estimate of survival at time t for each treatment group is

\hat{S}_{KM, j}(t) = \prod _{t_{kj} \leq t} \left [1-\frac{d_{kj}}{y_{kj}}\right ] \mbox{ if } t\geq t_{1j}, \mbox{ or } 1 \mbox{ if } t<t_{1j}

where t_{1j},...,t_{Dj} are the distinct observed event times of the primary outcome in treatment group j, d_{kj} is the number of events at time t_{kj} in treatment group j, and y_{kj} is the number of patients at risk at t_{kj} in treatment group j. The Kaplan-Meier (KM) estimate of treatment effect at time t is \hat{\Delta}_{KM}(t) = \hat{S}_{KM, 1}(t) - \hat{S}_{KM, 0}(t).

To obtain variance estimates and construct confidence intervals, we use a perturbation-resampling method. Specifically, let \{V^{(b)}=(V_1^{(b)}, . . . ,V_n^{(b)})^{T}, b=1,...B\} be n\times B independent copies of a positive random variable U from a known distribution with unit mean and unit variance such as an Exp(1) distribution. To estimate the variance of our estimates, we appropriately weight the estimates using these perturbation weights to obtain perturbed values: \hat{S}_{KM,0} (t)^{(b)}, \hat{S}_{KM,1} (t)^{(b)}, and \hat{\Delta}_{KM} (t)^{(b)}, b=1,...B. We then estimate the variance of each estimate as the empirical variance of the perturbed quantities. To construct confidence intervals, one can either use the empirical percentiles of the perturbed samples or a normal approximation.

Value

A list is returned:

Author(s)

Layla Parast

References

Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481.

Examples

data(example_rct)
delta.km(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2)

Estimates survival and treatment effect using landmark estimation

Description

Estimates the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using landmark estimation for an observational study setting

Usage

delta.land.obs(tl, dl, treat, tt, landmark, short = NULL, z.cov = NULL, 
var = FALSE, conf.int = FALSE, ps.weights = NULL, weight.perturb = NULL, 
perturb.ps = FALSE, cov.for.ps = NULL, bw = NULL)

Arguments

Details

Let T_{Li} denote the time of the primary event of interest for person i, T_{Si} denote the time of the available intermediate event(s), C_i denote the censoring time, Z_{i} denote the vector of baseline (pretreatment) covariates, and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. Due to censoring, we observe X_{Li}= min(T_{Li}, C_{i}) and \delta_{Li} = I(T_{Li}\leq C_{i}) and X_{Si}= min(T_{Si}, C_{i}) and \delta_{Si} = I(T_{Si}\leq C_{i}). This function estimates survival at time t within each treatment group, S_j(t) = P(T_{L} > t | G = j) for j = 1,0 and the treatment effect defined as \Delta(t) = S_1(t) - S_0(t).

To derive these estimates using landmark estimation for an observational study setting, we first decompose the quantity into two components S_j (t)= S_j(t|t_0) S_j(t_0) using a landmark time t_0 and estimate each component separately incorporating inverse probability of treatment weights (IPTW) to account for potential selection bias. Let W_j(Z_i) = {P(G_{i} = j | Z_i)}, and \hat{W}_j(Z_i) be the estimated propensity score (or probability of treatment, see ps.wgt.fun for more information). In this presentation, we assume Z_i indicates the vector of baseline (pretreatment) covariates and that Z_i is used to estimate the propensity scores and incorporated into the survival and treatment effect estimation. However, the function allows one to use different subsets of Z_i for the propensity score estimation vs. survival estimation, as is appropriate in the setting of interest. Intermediate event information is used in estimation of the conditional component S_j(t|t_0),

S_j(t|t_0)= P(T_L>t |T_L> t_0,G=j)=E[E[I(T_L>t | T_L> t_0,G=j,H)]]=E[S_{j,H} (t|t_0)]

where S_{j,H}(t|t_0) = P(T_L>t | T_L> t_0,G=j,H) and H = \{Z, I(T_S \leq t_0), min(T_S, t_0) \}. Then S_{j,H}(t|t_0) is estimated in two stages. The first stage involves fitting a weighted Cox proportional hazards model among individuals with X_L> t_0 to obtain an estimate of \beta, denoted as \hat{\beta},

S_{j,H}(t|t_0)=\exp \{-\Lambda_{j,0} (t|t_0) \exp(\beta^{T} H) \}

where \Lambda_{j,0} (t|t_0) is the cumulative baseline hazard in group j. Specifically, \hat{\beta} is the solution to the weighted Cox partial likelhoodand, with weights \hat{W}_j(Z_i)^{-1}. The second stage uses a weighted nonparametric kernel Nelson-Aalen estimator to obtain a local constant estimator for the conditional hazard \Lambda_{j,u}(t|t_0) = -\log [S_{j,u}(t|t_0)] as

\hat{\Lambda}_{j,u}(t|t_0) = \int_{t_0}^t \frac{\sum_i \hat{W}_j(Z_i)^{-1} K_h(\hat{U}_i - u) dN_i(z)}{\sum_i \hat{W}_j(Z_i)^{-1} K_h(\hat{U}_i - u) Y_i(z)}

where S_{j,u}(t|t_0)=P(T_L>t | T_L> t_0,G=j,\hat{U}=u), \hat{U} = \hat{\beta}^{T} H, Y_i(t)=I(T_L \geq t),N_i (t)=I(T_L\leq t)I(T_L<C),K(\cdot) is a smooth symmetric density function, K_h (x/h)/h, h=O(n^{-v}) is a bandwidth with 1/2 > v > 1/4, and the summation is over all individuals with G=j and X_L>t_0. The resulting estimate for S_{j,u}(t|t_0) is \hat{S}_{j,u}(t|t_0) = \exp \{-\hat{\Lambda}_{j,u}(t|t_0)\}, and the final estimate

\hat{S}_j(t|t_0) = \frac{n^{-1} \sum_{i =1}^n \hat{W}_j(Z_i)^{-1} \hat{S}_j(t|t_0, H_i) I(G_i=1)I(X_{Li} > t_0)}{n^{-1} \sum_{i =1}^n \hat{W}_j(Z_i)^{-1} I(G_i=1)I(X_{Li} > t_0) }

is a consistent estimate of S_j(t|t_0).

Estimation of S_j(t_0) uses a similar two-stage approach but using only baseline covariates, to obtain \hat{S}_j(t_0). The final overall estimate of survival at time t is, \hat{S}_{LM,j} (t)= \hat{S}_j(t|t_0) \hat{S}_j(t_0). The treatment effect in terms of the difference in survival at time t is estimated as \hat{\Delta}_{LM}(t) = \hat{S}_{LM,1}(t) - \hat{S}_{LM,0}(t). To obtain an appropriate h we first use the bandwidth selection procedure given by Scott(1992) to obtain h_{opt}; and then we let h = h_{opt}n_j^{-0.10}.

To obtain variance estimates and construct confidence intervals, we use a perturbation-resampling method. Specifically, let \{V^{(b)}=(V_1^{(b)}, . . . ,V_n^{(b)})^{T}, b=1,...B\} be n\times B independent copies of a positive random variable U from a known distribution with unit mean and unit variance such as an Exp(1) distribution. To estimate the variance of our estimates, we appropriately weight the estimates using these perturbation weights to obtain perturbed values: \hat{S}_{LM,0} (t)^{(b)}, \hat{S}_{LM,1} (t)^{(b)}, and \hat{\Delta}_{LM} (t)^{(b)}, b=1,...B. We then estimate the variance of each estimate as the empirical variance of the perturbed quantities. To construct confidence intervals, one can either use the empirical percentiles of the perturbed samples or a normal approximation.

Value

A list is returned:

Author(s)

Layla Parast

References

Parast, L. & Griffin B.A. (2017). Landmark Estimation of Survival and Treatment Effects in Observational Studies. Lifetime Data Analysis, 23:161-182.

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41-55.

Rosenbaum, P. R., & Rubin, D. B. (1984). Reducing bias in observational studies using subclassification on the propensity score. Journal of the American Statistical Association, 79(387), 516-524.

Examples

data(example_obs)
W.weight = ps.wgt.fun(treat = example_obs$treat, cov.for.ps = as.matrix(example_obs$Z))	
#executable but takes time
#delta.land.obs(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2, 
#landmark = 1, short = cbind(example_obs$TS,example_obs$DS), z.cov = as.matrix(example_obs$Z),
#ps.weights = W.weight)
#delta.land.obs(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2, 
#landmark = 1, short = cbind(example_obs$TS,example_obs$DS), z.cov = as.matrix(example_obs$Z),
#cov.for.ps = as.matrix(example_obs$Z))

Estimates survival and treatment effect using landmark estimation

Description

Estimates the probability of survival past some specified time and the treatment effect, defined as the difference in survival at the specified time, using landmark estimation for a randomized trial setting

Usage

delta.land.rct(tl, dl, treat, tt, landmark, short = NULL, z.cov = NULL, 
var = FALSE, conf.int = FALSE, weight.perturb = NULL, bw = NULL)

Arguments

Details

Let T_{Li} denote the time of the primary event of interest for person i, T_{Si} denote the time of the available intermediate event(s), C_i denote the censoring time, Z_{i} denote the vector of baseline (pretreatment) covariates, and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. Due to censoring, we observe X_{Li}= min(T_{Li}, C_{i}) and \delta_{Li} = I(T_{Li}\leq C_{i}) and X_{Si}= min(T_{Si}, C_{i}) and \delta_{Si} = I(T_{Si}\leq C_{i}). This function estimates survival at time t within each treatment group, S_j(t) = P(T_{L} > t | G = j) for j = 1,0 and the treatment effect defined as \Delta(t) = S_1(t) - S_0(t).

To derive these estimates using landmark estimation, we first decompose the quantity into two components S_j (t)= S_j(t|t_0) S_j(t_0) using a landmark time t_0 and estimate each component separately. Intermediate event information is used in estimation of the conditional component S_j(t|t_0),

S_j(t|t_0)= P(T_L>t |T_L> t_0,G=j)=E[E[I(T_L>t | T_L> t_0,G=j,H)]]=E[S_{j,H} (t|t_0)]

where S_{j,H}(t|t_0) = P(T_L>t | T_L> t_0,G=j,H) and H = \{Z, I(T_S \leq t_0), min(T_S, t_0) \}. Then S_{j,H}(t|t_0) is estimated in two stages: 1) fitting the Cox proportional hazards model among individuals with X_L> t_0 to obtain an estimate of \beta, denoted as \hat{\beta},

S_{j,H}(t|t_0)=\exp \{-\Lambda_{j,0} (t|t_0) \exp(\beta^{T} H) \}

where \Lambda_{j,0} (t|t_0) is the cumulative baseline hazard in group j and then 2) using a nonparametric kernel Nelson-Aalen estimator to obtain a local constant estimator for the conditional hazard \Lambda_{j,u}(t|t_0) = -\log [S_{j,u}(t|t_0)] as

\hat{\Lambda}_{j,u}(t|t_0) = \int_{t_0}^t \frac{\sum_i K_h(\hat{U}_i - u) dN_i(z)}{\sum_i K_h(\hat{U}_i - u) Y_i(z)}

where S_{j,u}(t|t_0)=P(T_L>t | T_L> t_0,G=j,\hat{U}=u), \hat{U} = \hat{\beta}^{T} H, Y_i(t)=I(T_L \geq t),N_i (t)=I(T_L\leq t)I(T_L<C),K(\cdot) is a smooth symmetric density function, K_h (x/h)/h, h=O(n^{-v}) is a bandwidth with 1/2 > v > 1/4, and the summation is over all individuals with G=j and X_L>t_0. The resulting estimate for S_{j,u}(t|t_0) is \hat{S}_{j,u}(t|t_0) = \exp \{-\hat{\Lambda}_{j,u}(t|t_0)\}, and the final estimate

\hat{S}_j(t|t_0) =\frac{n^{-1} \sum_{i =1}^n \hat{S}_j(t|t_0, H_i) I(G_i=1)I(X_{Li} > t_0)}{n^{-1} \sum_{i =1}^n I(G_i=1)I(X_{Li} > t_0) }

is a consistent estimate of S_j(t|t_0).

Estimation of S_j(t_0) uses a similar two-stage approach but using only baseline covariates, to obtain \hat{S}_j(t_0). The final overall estimate of survival at time t is, \hat{S}_{LM,j} (t)= \hat{S}_j(t|t_0) \hat{S}_j(t_0). The treatment effect in terms of the difference in survival at time t is estimated as \hat{\Delta}_{LM}(t) = \hat{S}_{LM,1}(t) - \hat{S}_{LM,0}(t). To obtain an appropriate h we first use the bandwidth selection procedure given by Scott(1992) to obtain h_{opt}; and then we let h = h_{opt}n^{-0.10}.

To obtain variance estimates and construct confidence intervals, we use a perturbation-resampling method. Specifically, let \{V^{(b)}=(V_1^{(b)}, . . . ,V_n^{(b)})^{T}, b=1,...B\} be n\times B independent copies of a positive random variable U from a known distribution with unit mean and unit variance such as an Exp(1) distribution. To estimate the variance of our estimates, we appropriately weight the estimates using these perturbation weights to obtain perturbed values: \hat{S}_{LM,0} (t)^{(b)}, \hat{S}_{LM,1} (t)^{(b)}, and \hat{\Delta}_{LM} (t)^{(b)}, b=1,...B. We then estimate the variance of each estimate as the empirical variance of the perturbed quantities. To construct confidence intervals, one can either use the empirical percentiles of the perturbed samples or a normal approximation.

Value

A list is returned:

Author(s)

Layla Parast

References

Parast, L., Tian, L., & Cai, T. (2014). Landmark Estimation of Survival and Treatment Effect in a Randomized Clinical Trial. Journal of the American Statistical Association, 109(505), 384-394.

Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical report, University of California Berkeley.

Scott, D. (1992). Multivariate density estimation. Wiley.

Examples

data(example_rct)
#executable but takes time
#delta.land.rct(tl=example_rct$TL, dl = example_rct$DL, treat = example_rct$treat, tt=2, 
#landmark = 1, short = cbind(example_rct$TS,example_rct$DS), z.cov = as.matrix(example_rct$Z))

Hypothetical data from an observational study

Description

Hypothetical data from an observational study to be used in examples.

Usage

data(example_obs)

Format

A data frame with 4000 observations on the following 6 variables.

TL

the observed event or censoring time for the primary outcome, equal to min(T, C) where T is the time of the primary outcome and C is the censoring time.

DL

the indicator telling whether the individual was observed to have the event or was censored, equal to 1*(T<C) where T is the time of the primary outcome and C is the censoring time.

TS

the observed event or censoring time for the intermediate event, equal to min(TS, C) where TS is the time of the intermediate event and C is the censoring time.

DS

the indicator telling whether the individual was observed to have the intermediate event or was censored, equal to 1*(TS<C) where TS is the time of the primary outcome and C is the censoring time.

Z

a baseline covariate vector

treat

treatment indicator

Examples

data(example_obs)
names(example_obs)

Hypothetical data from a randomized trial

Description

Hypothetical data from a randomized trial to be used in examples.

Usage

data(example_rct)

Format

A data frame with 3000 observations on the following 6 variables.

TL

the observed event or censoring time for the primary outcome, equal to min(T, C) where T is the time of the primary outcome and C is the censoring time.

DL

the indicator telling whether the individual was observed to have the event or was censored, equal to 1*(T<C) where T is the time of the primary outcome and C is the censoring time.

TS

the observed event or censoring time for the intermediate event, equal to min(TS, C) where TS is the time of the intermediate event and C is the censoring time.

DS

the indicator telling whether the individual was observed to have the intermediate event or was censored, equal to 1*(TS<C) where TS is the time of the primary outcome and C is the censoring time.

Z

a baseline covariate vector

treat

treatment indicator

Examples

data(example_rct)
names(example_rct)

Calculates propensity score weights

Description

Calculates propensity score (or inverse probability of treatment) weights given the treatment indicator and available baseline (pretreatment) covariates.

Usage

ps.wgt.fun(treat, cov.for.ps, weight = NULL)

Arguments

Details

Let Z_{i} denote the matrix of baseline (pretreatment) covariates and G_i be the treatment group indicator such that G_i = 1 indicates treatment and G_i = 0 indicates control. This function estimates P = P(G_i = 1 | Z_i) using logistic regression. The propensity score (or inverse probability of treatment) weights are then equal to 1/\hat{P} for those in treatment group 1 and 1/(1-\hat{P}) for those in treatment group 0. These weights reflect the situation where the average treatment effect (ATE) is of interest, not average treatment effect in the treated (ATT).

Value

propensity score (or inverse probability of treatment) weights

Author(s)

Layla Parast

References

Rosenbaum, P. R., & Rubin, D. B. (1983). The central role of the propensity score in observational studies for causal effects. Biometrika, 70(1), 41-55.

Rosenbaum, P. R., & Rubin, D. B. (1984). Reducing bias in observational studies using subclassification on the propensity score. Journal of the American Statistical Association, 79(387), 516-524.

Examples

data(example_obs)
W.weight = ps.wgt.fun(treat = example_obs$treat, cov.for.ps = as.matrix(example_obs$Z))	
delta.iptw.km(tl=example_obs$TL, dl = example_obs$DL, treat = example_obs$treat, tt=2, 
ps.weights = W.weight) 

Estimates survival using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Usage

surv.iptw.km(tl, dl, tt, var = FALSE, conf.int = FALSE, ps.weights, 
weight.perturb = NULL,perturb.ps = FALSE, perturb.vector = FALSE)

Arguments

Details

See documentation for delta.iptw.km for details.

Value

A list is returned:

Author(s)

Layla Parast

References

Xie, J., & Liu, C. (2005). Adjusted Kaplan-Meier estimator and log-rank test with inverse probability of treatment weighting for survival data. Statistics in Medicine, 24(20), 3089-3110.

Examples

data(example_obs)
W.weight = ps.wgt.fun(treat = example_obs$treat, cov.for.ps = as.matrix(example_obs$Z))	
example_obs.treat = example_obs[example_obs$treat == 1,]
surv.iptw.km(tl=example_obs.treat$TL, dl = example_obs.treat$DL, tt=2, ps.weights = 
W.weight[example_obs$treat == 1]) 

Estimates survival using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time using inverse probability of treatment weighted (IPTW) Kaplan-Meier estimation

Usage

surv.iptw.km.base(tl, dl, tt, ps.weights, weight.perturb = NULL, perturb.ps = FALSE)

Arguments

Value

estimate of survival at the specified time

Author(s)

Layla Parast

Estimates survival using Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time using Kaplan-Meier estimation

Usage

surv.km(tl, dl, tt, var = FALSE, conf.int = FALSE, weight.perturb = NULL, 
perturb.vector = FALSE)

Arguments

Details

See documentation for delta.km for details.

Value

A list is returned:

Author(s)

Layla Parast

References

Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53(282), 457-481.

Examples

data(example_rct)
example_rct.treat = example_rct[example_rct$treat == 1,]
surv.km(tl=example_rct.treat$TL, dl = example_rct.treat$DL, tt=2)

Estimates survival using Kaplan-Meier estimation

Description

Estimates the probability of survival past some specified time using Kaplan-Meier estimation

Usage

surv.km.base(tl, dl, tt, weight.perturb = NULL)

Arguments

Value

estimate of survival at the specified time

Author(s)

Layla Parast

Estimates survival using landmark estimation

Description

Estimates the probability of survival past some specified time using landmark estimation for an observational study setting

Usage

surv.land.obs(tl, dl, tt, landmark, short = NULL, z.cov = NULL, var = FALSE, 
conf.int = FALSE, ps.weights, weight.perturb = NULL, perturb.ps = FALSE, 
perturb.vector = FALSE, bw = NULL)

Arguments

Details

See documentation for delta.land.obs for details.

Value

A list is returned:

Author(s)

Layla Parast

References

Parast, L. & Griffin B.A. (2017). Landmark Estimation of Survival and Treatment Effects in Observational Studies. Lifetime Data Analysis, 23:161-182.

Examples

data(example_obs)
W.weight = ps.wgt.fun(treat = example_obs$treat, cov.for.ps = as.matrix(example_obs$Z))	
example_obs.treat = example_obs[example_obs$treat == 1,]
#executable but takes time
#surv.land.obs(tl=example_obs.treat$TL, dl = example_obs.treat$DL, tt=2, landmark = 1, 
#short = cbind(example_obs.treat$TS,example_obs.treat$DS), z.cov = example_obs.treat$Z, 
#ps.weights = W.weight[example_obs$treat == 1])

Estimates survival using landmark estimation

Description

Estimates the probability of survival past some specified time using landmark estimation for an observational study setting

Usage

surv.land.obs.base(tl, dl, tt, landmark, short = NULL, z.cov = NULL, 
ps.weights, weight.perturb = NULL, perturb.ps = FALSE, bw = NULL)

Arguments

Value

estimate of survival at the specified time

Author(s)

Layla Parast

Estimates survival using landmark estimation

Description

Estimates the probability of survival past some specified time using landmark estimation for a randomized trial setting

Usage

surv.land.rct(tl, dl, tt, landmark, short = NULL, z.cov = NULL, var = FALSE, 
conf.int = FALSE, weight.perturb = NULL, perturb.vector = FALSE, bw = NULL)

Arguments

Details

See documentation for delta.land.rct for details.

Value

A list is returned:

Author(s)

Layla Parast

References

Parast, L., Tian, L., & Cai, T. (2014). Landmark Estimation of Survival and Treatment Effect in a Randomized Clinical Trial. Journal of the American Statistical Association, 109(505), 384-394.

Examples

data(example_rct)
example_rct.treat = example_rct[example_rct$treat == 1,]
#executable but takes time
#surv.land.rct(tl=example_rct.treat$TL, dl = example_rct.treat$DL, tt=2, landmark = 1, 
#short = cbind(example_rct.treat$TS,example_rct.treat$DS), z.cov = example_rct.treat$Z)

Estimates survival using landmark estimation

Description

Estimates the probability of survival past some specified time using landmark estimation for a randomized trial setting

Usage

surv.land.rct.base(tl, dl, tt, landmark, short = NULL, z.cov = NULL, 
weight.perturb = NULL, bw = NULL)

Arguments

Details

See documentation for delta.land.rct for details.

Value

estimate of survival at the specified time

Author(s)

Layla Parast