Package 'hds'

Estimate the time-varying coefficients from a local-in-time Cox model

Description

finda estimates the time-varying coefficients beta(t) at a single time from a local-in-time Cox model. Think of it as a Cox model where the the coefficients are allowed to vary with time. Further details can be found in Cai and Sun (2003) and Tian et al. (2005).

Usage

finda(tt, times, status, covars, start = rep(0, ncol(covars)), h = 400, ...)

Arguments

tt	Time to estimate beta(t) at
times	A vector of observed follow up times.
status	A vector of status indicators, usually 0=alive, 1=dead.
covars	A matrix or data frame of numeric covariate values, with a column for each covariate and each observation is on a separate row.
start	A vector of length p of starting values to be passed to optim for the numerical optimization procedure. p is the number of covariates. Defaults to all zeroes.
h	A single value on the time scale representing the bandwidth to use.
...	Additional parameters to pass to optim.

Details

The naming of the function finda stands for "find a(t)", where "a(t)" is the notation used in Cai and Sun (2003) to represent the time-varying Cox model coefficients. We also refer to "a(t)" as "beta(t)" through the documentation.

The user typically will not interact with this function, as finda is wrapped by hdslc.

Value

A vector of length p, where p is the number of covariates. The vector is the estimated beta(t) from the local-in-time Cox model at time tt.

References

Cai Z and Sun Y (2003). Local linear estimation for time-dependent coefficients in Cox's regression models. Scandinavian Journal of Statistics, 30: 93-111. doi:10.1111/1467-9469.00320

Tian L, Zucker D, and Wei LJ (2005). On the Cox model with time-varying regression coefficients. Journal of the American Statistical Association, 100(469):172-83. doi:10.1198/016214504000000845

---

Hazard discrimination summary estimator

Description

Returns hazard discimination summary (HDS) estimates at all specified evaluation times. See Liang and Heagerty (2016) for details on HDS.

Usage

hds(times, status, m, evaltimes = times[order(times)], se = TRUE)

Arguments

times	A vector of observed follow up times.
status	A vector of status indicators, usually 0=alive, 1=dead.
m	A matrix or data frame of covariate values, with a column for each covariate and each observation is on a separate row. Non-numeric values are acceptable, as the values will be transformed into a numeric model matrix through survival::coxph.
evaltimes	A vector of times at which to estimate HDS. Defaults to all the times specified by the times vector. If there are a lot of observations, then you may want to enter in a sparser vector of times for faster computation.
se	TRUE or FALSE. TRUE: calculate and return standard error estimates. FALSE: do not calculate standard errors estimates and return NAs. Defaults to TRUE. May want to set to FALSE to save computation time if using this function to compute bootstrap standard errors.

Details

A wrapper for hds_t. Since hds_t only estimates HDS at one time point, this function calls hds_t multiple times to estimate the entire HDS curve. hds and hdslc are the main functions the user will interact with in this package.

The covariate values m are centered for numerical stability. This is particularly relevant for the standard error calculations.

Value

A data frame with three columns: 1) the evaluation times, 2) the HDS estimates at each evaluation time, and 3) the standard error estimates at each evaluation time

References

Liang CJ and Heagerty PJ (2016). A risk-based measure of time-varying prognostic discrimination for survival models. Biometrics. doi:10.1111/biom.12628

See Also

hdslc

Examples

## Not run: 
head(hds(times = survival::pbc[1:312, 2],
         status = (survival::pbc[1:312, 3]==2)*1,
         m = survival::pbc[1:312, 5]))

hdsres   <- hds(times=pbc5[,1], status=pbc5[,2], m=pbc5[,3:7])
hdslcres <- hdslc(times = pbc5[,1], status=pbc5[,2], m = pbc5[,3:7], h = 730)
Survt    <- summary(survival::survfit(survival::Surv(pbc5[,1], pbc5[,2])~1))
Survtd   <- cbind(Survt$time, c(0,diff(1-Survt$surv)))
tden     <- density(x=Survtd[,1], weights=Survtd[,2], bw=100, kernel="epanechnikov")

par(mar=c(2.25,2.25,0,0)+0.1, mgp=c(1.25,0.5,0))
plot(c(hdslcres[,1], hdslcres[,1]), c(hdslcres[,2] - 1.96*hdslcres[,3],
                                      hdslcres[,2] + 1.96*hdslcres[,3]),
     type="n", xlab="days", ylab="HDS(t)", cex.lab=.75, cex.axis=.75,
     ylim=c(-3,15), xlim=c(0,3650))
polygon(x=c(hdsres[,1], hdsres[312:1,1]), col=rgb(1,0,0,.25), border=NA,
        fillOddEven=TRUE, y=c(hdsres[,2]+1.96*hdsres[,3],
                              (hdsres[,2]-1.96*hdsres[,3])[312:1]))
polygon(x=c(hdslcres[,1], hdslcres[312:1, 1]), col=rgb(0,0,1,.25), border=NA,
        fillOddEven=TRUE, y=c(hdslcres[,2] + 1.96*hdslcres[,3],
                              (hdslcres[,2] - 1.96*hdslcres[,3])[312:1]))
lines(hdsres[,1], hdsres[,2], lwd=2, col="red")
lines(hdslcres[,1], hdslcres[,2], lwd=2, col="blue")
abline(h=1, lty=3)
legend(x=1200, y=14, legend=c("Proportional hazards",
                              "Local-in-time proportional hazards",
                              "Time density"), col=c("red", "blue", "gray"),
       lwd=2, bty="n", cex=0.75)
with(tden, polygon(c(x, x[length(x):1]),
                   c(y*3/max(y)-3.5, rep(-3.5, length(x))),
                   col="gray", border=NA, fillOddEven=TRUE))

## End(Not run)

---

Hazard discrimination summary estimate at one time point

Description

hds_t estimates HDS at time t under the PH assumption

Usage

hds_t(t, L0hat, betahat, m)

Arguments

t	The time at which to calculate HDS
L0hat	A data frame with variable names of hazard and time. Typically the object returned by basehaz.
betahat	A vector of coefficient estimates from the Cox model. Typically the coefficients value from the coxph.object object returned by coxph.
m	A numeric matrix of covariate values, with a column for each covariate and each observation is on a separate row.

Details

The user typically will not interact with this function. Rather, hds is a wrapper for this function and is what the user typically will use.

---

Hazard discrimination summary estimator

Description

Returns local constant HDS estimates at all specified evaluation times. See Liang and Heagerty (2016) for details on HDS.

Usage

hdslc(times, status, m, evaltimes = times[order(times)], h = 1.06 *
  sd(times) * (length(times)^(-0.2)), se = TRUE)

Arguments

times	A vector of observed follow up times.
status	A vector of status indicators, usually 0=alive, 1=dead.
m	A matrix or data frame of covariate values, with a column for each covariate and each observation is on a separate row. Non-numeric values are acceptable, as the values will be transformed into a numeric model matrix through survival::coxph.
evaltimes	A vector of times at which to estimate HDS. Defaults to all the times specified by the times vector. If there are a lot of observations, then you may want to enter in a sparser vector of times for faster computation.
h	A single numeric value representing the bandwdith to use, on the time scale. The default bandwidth is a very ad hoc estimate using Silverman's rule of thumb
se	TRUE or FALSE. TRUE: calculate and return standard error estimates. FALSE: do not calculate standard errors estimates and return NAs. Defaults to TRUE. May want to set to FALSE to save computation time if using this function to compute bootstrap standard errors.

Details

A local constant version of hds. While hds estimates HDS(t) assuming the Cox proportional hazards model, hdslc estimates HDS(t) using a relaxed, local-in-time Cox model. Specifically, the hazard ratios are allowed to vary with time. See Cai and Sun (2003) and Tian Zucker Wei (2005) for details on the local-in-time Cox model.

Point estimates use hdslc.fast and standard errors use hdslcse.fast. hdslc.fast requires an estimate of beta(t) (time-varying hazard ratio), which is estimated using finda(); and subject specific survival, which is estimated using sssf.fast(). hdslcse.fast requires the same and in addition standard error estimates of beta(t), which are estimated using betahatse.fast().

The covariate values m are centered for numerical stability. This is particularly relevant for the standard error calculations.

Value

A data frame with three columns: 1) the evaluation times, 2) the HDS estimates at each evaluation time, and 3) the standard error estimates at each evaluation time

References

Liang CJ and Heagerty PJ (2016). A risk-based measure of time-varying prognostic discrimination for survival models. Biometrics. doi:10.1111/biom.12628

Cai Z and Sun Y (2003). Local linear estimation for time-dependent coefficients in Cox's regression models. Scandinavian Journal of Statistics, 30: 93-111. doi:10.1111/1467-9469.00320

Tian L, Zucker D, and Wei LJ (2005). On the Cox model with time-varying regression coefficients. Journal of the American Statistical Association, 100(469):172-83. doi:10.1198/016214504000000845

See Also

hds, finda

Examples

## Not run: 
head(hdslc(times = survival::pbc[1:312, 2],
           status = (survival::pbc[1:312, 3]==2)*1,
           m = survival::pbc[1:312, 5]))

hdsres   <- hds(times=pbc5[,1], status=pbc5[,2], m=pbc5[,3:7])
hdslcres <- hdslc(times = pbc5[,1], status=pbc5[,2], m = pbc5[,3:7], h = 730)
Survt    <- summary(survival::survfit(survival::Surv(pbc5[,1], pbc5[,2])~1))
Survtd   <- cbind(Survt$time, c(0,diff(1-Survt$surv)))
tden     <- density(x=Survtd[,1], weights=Survtd[,2], bw=100, kernel="epanechnikov")

par(mar=c(2.25,2.25,0,0)+0.1, mgp=c(1.25,0.5,0))
plot(c(hdslcres[,1], hdslcres[,1]), c(hdslcres[,2] - 1.96*hdslcres[,3],
                                      hdslcres[,2] + 1.96*hdslcres[,3]),
     type="n", xlab="days", ylab="HDS(t)", cex.lab=.75, cex.axis=.75,
     ylim=c(-3,15), xlim=c(0,3650))
polygon(x=c(hdsres[,1], hdsres[312:1,1]), col=rgb(1,0,0,.25), border=NA,
        fillOddEven=TRUE, y=c(hdsres[,2]+1.96*hdsres[,3],
                              (hdsres[,2]-1.96*hdsres[,3])[312:1]))
polygon(x=c(hdslcres[,1], hdslcres[312:1, 1]), col=rgb(0,0,1,.25), border=NA,
        fillOddEven=TRUE, y=c(hdslcres[,2] + 1.96*hdslcres[,3],
                              (hdslcres[,2] - 1.96*hdslcres[,3])[312:1]))
lines(hdsres[,1], hdsres[,2], lwd=2, col="red")
lines(hdslcres[,1], hdslcres[,2], lwd=2, col="blue")
abline(h=1, lty=3)
legend(x=1200, y=14, legend=c("Proportional hazards",
                              "Local-in-time proportional hazards",
                              "Time density"), col=c("red", "blue", "gray"),
       lwd=2, bty="n", cex=0.75)
with(tden, polygon(c(x, x[length(x):1]),
                   c(y*3/max(y)-3.5, rep(-3.5, length(x))),
                   col="gray", border=NA, fillOddEven=TRUE))

## End(Not run)

---

Hazard discrimination summary estimate (local constant) at one time point

Description

hdslc.fast estimates HDS at a single time using the local-in-time proportional hazards model. See Cai and Sun (2003, Scandinavian Journal of Statistics) for details on the local-in-time PH model.

Usage

hdslc.fast(S, betahat, m)

Arguments

S	A vector of length nrow(m) (which is typically the number of observations n), where each value is the subject-specific survival at time t where t is implied by the choice of betahat.
betahat	A p x 1 vector of coefficient estimates at time t of interest from the local-in-time Cox model. Vector length p is the number of covariates. Typically the output from hdslc::finda is passed here.
m	A numeric n x p matrix of covariate values, with a column for each covariate and each observation is on a separate row.

Details

The user typically will not interact with this function. Rather, hdslc wraps this function and is what the user typically will use.

Value

The HDS estimate at times t, where t is implied by choice of S and betahat passed to hdslc.fast.

---

Hazard discrimination summary (local constant) standard error estimate

Description

hdslcse.fast calculates an estimate of the variance for the local constant hazard discrimination summary estimator at a time t. The time t is implied by S, betahat, and betahatse

Usage

hdslcse.fast(S, betahat, m, betahatse)

Arguments

S	A vector of length nrow(m) (which is typically the number of observations n), where each value is the subject-specific survival at time t where t is implied by the choice of betahat.
betahat	A p x 1 vector of coefficient estimates at time t of interest from the local-in-time Cox model. Vector length p is the number of covariates. Typically the output from hdslc::finda is passed here.
m	A numeric n x p matrix of covariate values, with a column for each covariate and each observation is on a separate row.
betahatse	A p x p covariance matrix for betahat at time t

Details

The use will typically not interact with this function directly. Instead this function is wrapped by hdslc.

Value

Variance estimate that has not been normalized. To get a usable standard error estimate, divide the output of this function by the bandwidth and sample size, and then take the square root.

---

Cleaned up version of the Mayo PBC data.

Description

A cleaned up version of the Mayo PBC data from survival::pbc. Only the first 312 observations are used (the cases who participated in the randomized trial). Only five of the covariates (listed below) are used. Further, two of the covariates were log transformed.

Usage

pbc5

Format

A data frame with 312 rows and 7 variables:

time

follow up time in days

status

1=death, 0=censored

age

age in years

edema

0=no edema, 0.5=untreated or successfully treated, 1=edema despite diuretic therapy

bili

log serum bilirubin level (original value from survival::pbc is unlogged)

albumin

serum albumin

protime

log standardized blood clotting time (original value from survival::pbc is unlogged)

Source

Cleaned up version of survival::pbc

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