Introduction

This landpred package provides nonparametric models for landmark prediction of long-term survival outcomes, incorporating covariate and short-term event information. The package supports the construction of flexible varying-coefficient models that use discrete covariates, as well as multiple continuous covariates. The goal is to improve prediction accuracy when censored short-term events are available as predictors, using robust nonparametric procedures that don’t require correct model specification and avoid restrictive parametric assumptions found in existing multistate survival methods. More information on these models can be found in Parast et al. (2012, Journal of the American Statistical Association, doi:10.1080/01621459.2012.721281), and Parast et al. (2011, Biometrical Journal, https://doi.org/10.1002/bimj.201000150).

Tutorial

Generating Data

We will generate a dataframe with two continuous covariates, $Z_1,Z_2$ , and one discrete covariate, $B$ . We will also have a censored short event $X_S$ and a censored long event, our outcome, $X_L$ .

n <- 500
# Generate covariates
Z1 <- rnorm(n)
Z2 <- rnorm(n)
B <- rbinom(n, 1, 0.5)  # Binary discrete covariate

# Generate event times based on exponential model
X_L_raw <- rexp(n, rate = exp(-0.5 * Z1 + 0.3 * Z2 + 0.4 * B))
X_S_raw <- rexp(n, rate = exp(0.2 * Z1 - 0.3 * Z2 - 0.3 * B))

# Generate censoring times
C_L <- runif(n, 2, 8)  # censoring for long event
C_S <- runif(n, 1, 4)  # censoring for short event

# Apply censoring
X_L <- pmin(X_L_raw, C_L)
D_L <- as.numeric(X_L_raw <= C_L)  # 1 if event, 0 if censored

X_S <- pmin(X_S_raw, C_S)
D_S <- as.numeric(X_S_raw <= C_S)  # 1 if event, 0 if censored

# Return simple data frame
df <- data.frame(
  X_L = X_L,       # long event time
  D_L = D_L,       # long event indicator
  X_S = X_S,       # short event time
  D_S = D_S,       # short event indicator
  Z1 = Z1,         # continuous covariate 1
  Z2 = Z2,         # continuous covariate 2
  B = B            # discrete covariate
)

Workflow

The package exports the landpred function, which will construct a landpred object given a formula and data. We can supply any number of continuous covariates to this, or a singular discrete covariate, as demonstrated below. Note that short term information needs to be wrapped in a Surv from the survival package.

library(landpred)
library(survival)

# Create model with continuous covariates and short term information
obj1 <- landpred(Surv(X_L, D_L) ~ Surv(X_S, D_S) + Z1 + Z2, data=df)

# Create model with discrete marker and short term information
# Notice the discrete flag is now enabled, which is false by default.
obj2 <- landpred(Surv(X_L, D_L) ~ Surv(X_S, D_S) + B, data=df, discrete=TRUE)

# Create model with discrete marker and no short term information
# Notice the discrete flag is now enabled, which is false by default.
obj3 <- landpred(Surv(X_L, D_L) ~  B, data=df, discrete=TRUE)

We will proceed with the object with continuous covariates, and short-term information. We can call summary on this landpred object to get more information from it. Note this landpred object returned by the landpred call serves as a baseline object. To get a model for a specific time ande delta, we must call the get_model function on this object as specified in the summary function. Workflows / functions will be the same in the discrete and continuous case.

summary(obj1)
#> 
#> Landpred Object Summary
#> Call get_model() to get time-specific model for t0 + tau
#> 
#> Call:
#> landpred(formula = Surv(X_L, D_L) ~ Surv(X_S, D_S) + Z1 + Z2)
#> 
#> Discrete: FALSE    Short Covariate: TRUE     N: 500

Our get_model function accepts a landpred object as input, and three required parameters, t0, tau, and bw. This returns a landmark prediction model for that specific time and delta. For the coefficients and standard errors of the coefficeints, we call summary on the returned model to get coefficients when no information in the short covariate is provided (has not occured yet). We can also provide t_s to this call to get coefficients for if the short event has occured yet.

We must first choose our bandwidth value. We provide a utility to select this optimal value using cross-validation:

t0 <- 0.5
tau <- 1.5
bw <- optimize_bandwidth(obj1, t0=t0, tau=tau, lower=0.01, upper=1, transform=identity)
bw
#> [1] 0.1333657

The optimal bandwidth is 0.133, which controls the smoothness of the local regression used to incorporate short-term event information. A smaller bandwidth gives more weight to nearby observations, while a larger bandwidth smooths over more data points.

# Get model using the optimized bandwidth
model <- get_model(obj1, t0=t0, tau=tau, bw=bw, transform=identity)

# Summary with no short covariate information
summary(model)
#> 
#> Continuous Landpred Model:
#> 
#> Coefficients (No short covariate):
#>             Estimate Std. Error
#> (Intercept)  1.50236     0.1934
#> Z1          -0.83170     0.1699
#> Z2           0.47622     0.1519
#> ---
#> Fit on n=261 observations.
#> 
#> t0: 0.500      tau: 1.500

The model summary shows coefficients for predicting survival beyond time t0 + tau, given survival to the landmark time t0 = 0.5. Calling summary without t_s returns the coefficients for the model when no short event information is provided.

# Summary with short covariate information (if short event occurred at time 0.3)
summary(model, t_s=0.3)
#> 
#> Continuous Landpred Model:
#> 
#> Coefficients (t_s=0.300000):
#>             Estimate Std. Error
#> (Intercept)  1.55723     0.5463
#> Z1          -0.89340     0.4170
#> Z2           0.49641     0.5249
#> ---
#> Fit on n=105 observations.
#> 
#> t0: 0.500      tau: 1.500

When we condition on the short event occurring at a specific time (here 0.3), the model provides locally-weighted coefficients that incorporate this timing information for more accurate predictions.

Prediction

We can also do prediction with this model to get survival probabilities for a given observation:

probs <- predict(model, newdata=df[1:5, , drop=FALSE])
probs
#> [1] 0.86239039 0.95018224 0.09478818 0.85729249 0.78854048

These probabilities represent the estimated risk of the long-term event occurring by time t0 + tau, conditional on surviving to the landmark time t0. Higher values indicate greater risk. Here, newdata is a dataframe also containing the short event information for each observation. If this short event time is less than the value of t_0, then the short event time is incorporated into the probability, giving P(X_L > t_0 + tau | X=x, X_s=t_s). If the short event time is greater than t_0, then the short event time is not incorporated into the probability, giving instead P(X_L > t_0 + tau | X=x). ```