Help for package pssmooth

Type:

Package

Title:

Flexible and Efficient Evaluation of Principal Surrogates/Treatment Effect Modifiers

Version:

1.0.3

BugReports:

https://github.com/mjuraska/pssmooth/issues

Description:

Implements estimation and testing procedures for evaluating an intermediate biomarker response as a principal surrogate of a clinical response to treatment (i.e., principal stratification effect modification analysis), as described in Juraska M, Huang Y, and Gilbert PB (2020), Inference on treatment effect modification by biomarker response in a three-phase sampling design, Biostatistics, 21(3): 545-560 <doi:10.1093/biostatistics/kxy074>. The methods avoid the restrictive 'placebo structural risk' modeling assumption common to past methods and further improve robustness by the use of nonparametric kernel smoothing for biomarker density estimation. A randomized controlled two-group clinical efficacy trial is assumed with an ordered categorical or continuous univariate biomarker response measured at a fixed timepoint post-randomization and with a univariate baseline surrogate measure allowed to be observed in only a subset of trial participants with an observed biomarker response (see the flexible three-phase sampling design in the paper for details). Bootstrap-based procedures are available for pointwise and simultaneous confidence intervals and testing of four relevant hypotheses. Summary and plotting functions are provided for estimation results.

License:

GPL-2

URL:

https://github.com/mjuraska/pssmooth

Encoding:

UTF-8

LazyData:

true

Imports:

graphics, stats, osDesign, np, chngpt, MASS

Suggests:

knitr, rmarkdown

RoxygenNote:

7.1.1

NeedsCompilation:

Packaged:

2020-11-18 01:01:21 UTC; mjuraska

Author:

Michal Juraska [aut, cre]

Maintainer:

Michal Juraska <mjuraska@fredhutch.org>

Repository:

CRAN

Date/Publication:

2020-11-18 13:20:03 UTC

Bootstrap Estimation of Conditional Clinical Endpoint Risk under Placebo and Treatment Given Biomarker Response to Treatment in a Baseline Surrogate Measure Three-Phase Sampling Design

Description

Estimates P\{Y(z)=1|S(1)=s_1\}, z=0,1, on a grid of s_1 values in bootstrap resamples (see riskCurve for notation introduction). Cases (Y=1) and controls (Y=0) are sampled separately yielding a fixed number of cases and controls in each bootstrap sample. Consequentially, the number of controls with available phase 2 data varies across bootstrap samples.

Usage

bootRiskCurve(
  formula,
  bsm,
  tx,
  data,
  pstype = c("continuous", "ordered"),
  bsmtype = c("continuous", "ordered"),
  bwtype = c("fixed", "generalized_nn", "adaptive_nn"),
  hinge = FALSE,
  weights = NULL,
  psGrid = NULL,
  iter,
  seed = NULL,
  saveFile = NULL,
  saveDir = NULL
)

Arguments

formula

a formula object with the binary clinical endpoint on the left of the ~ operator. The first listed variable on the right must be the biomarker response at t0 and all variables that follow, if any, are discrete baseline covariates specified in all fitted models that condition on them. Interactions and transformations of the baseline covariates are allowed. All terms in the formula must be evaluable in the data frame data.

bsm

a character string specifying the variable name in data representing the baseline surrogate measure

tx

a character string specifying the variable name in data representing the treatment group indicator

data

a data frame with one row per randomized participant endpoint-free at t_0 that contains at least the variables specified in formula, bsm and tx. Values of bsm and the biomarker at t_0 that are unavailable are represented as NA.

pstype

a character string specifying whether the biomarker response shall be treated as a continuous (default) or ordered categorical variable in the kernel density/probability estimation

bsmtype

a character string specifying whether the baseline surrogate measure shall be treated as a continuous (default) or ordered categorical variable in the kernel density/probability estimation

bwtype

a character string specifying the bandwidth type for continuous variables in the kernel density estimation. The options are fixed (default) for fixed bandwidths, generalized_nn for generalized nearest neighbors, and adaptive_nn for adaptive nearest neighbors. As noted in the documentation of the function npcdensbw in the np package: "Adaptive nearest-neighbor bandwidths change with each sample realization in the set when estimating the density at the point x. Generalized nearest-neighbor bandwidths change with the point at which the density is estimated, x. Fixed bandwidths are constant over the support of x."

hinge

a logical value (FALSE by default) indicating whether a hinge model (Fong et al., 2017) shall be used for modeling the effect of S(z) on the clinical endpoint risk. A hinge model specifies that variability in S(z) below the hinge point does not associate with the clinical endpoint risk. The hinge point is reestimated in each bootstrap sample.

weights

either a numeric vector of weights or a character string specifying the variable name in data representing weights applied to observations in the phase 2 subset in order to make inference about the target population of all randomized participants endpoint-free at t_0. The weights reflect that the case:control ratio in the phase 2 subset is different from that in the target population and are passed on to GLMs in the estimation of the hinge point. If NULL (default and recommended), weights for cases and controls are recalculated separately in each study group within each bootstrap sample; otherwise the same specified vector of weights is used in each bootstrap sample.

psGrid

a numeric vector of S(1) values at which the conditional clinical endpoint risk in each study group is estimated. If NULL (default), a grid of values spanning the range of observed values of the biomarker will be used.

iter

the number of bootstrap iterations

seed

a seed of the random number generator supplied to set.seed for reproducibility

saveFile

a character string specifying the name of an .RData file storing the output list. If NULL (default), the output list will only be returned.

saveDir

a character string specifying a path for the output directory. If NULL (default), the output list will only be returned; otherwise, if saveFile is specified, the output list will also be saved as an .RData file in the specified directory.

Value

If saveFile and saveDir are both specified, the output list (named bList) is saved as an .RData file; otherwise it is returned only. The output object is a list with the following components:

psGrid: a numeric vector of S(1) values at which the conditional clinical endpoint risk is estimated in the components plaRiskCurveBoot and txRiskCurveBoot
plaRiskCurveBoot: a length(psGrid)-by-iter matrix of estimates of P\{Y(0)=1|S(1)=s_1\} for s_1 in psGrid, with columns representing bootstrap samples
txRiskCurveBoot: a length(psGrid)-by-iter matrix of estimates of P\{Y(1)=1|S(1)=s_1\} for s_1 in psGrid, with columns representing bootstrap samples
cpointPboot: if hinge=TRUE, a numeric vector of estimates of the hinge point in the placebo group in each bootstrap sample
cpointTboot: if hinge=TRUE, a numeric vector of estimates of the hinge point in the treatment group in each bootstrap sample

References

Fong, Y., Huang, Y., Gilbert, P. B., and Permar, S. R. (2017), chngpt: threshold regression model estimation and inference, BMC Bioinformatics, 18.

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
X <- rbinom(n,1,0.5)
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.4)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(X,Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("X","Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=3)

out <- bootRiskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data,
                     psGrid=grid, iter=1, seed=10)

# alternatively, to save the .RData output file (no '<-' needed):
bootRiskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data,
              psGrid=grid, iter=1, seed=10, saveFile="out.RData", saveDir="./")

Plotting of the Estimated Marginal Causal Effect Predictiveness Curve

Description

Plots point estimates and, if available, pointwise and simultaneous Wald-type bootstrap confidence intervals for the specified marginal causal effect predictiveness (mCEP) curve.

Usage

plotMCEPcurve(
  object,
  confLevel = 0.95,
  hingePoint = NULL,
  title = NULL,
  xLab = NULL,
  yLab = NULL,
  yLim = NULL,
  pType = c("l", "p")
)

Arguments

object

an object returned by summary.riskCurve

confLevel

the confidence level (0.95 by default) of pointwise and simultaneous confidence intervals

hingePoint

the hinge point estimate (NULL by default)

title

a character string specifying the plot title

xLab

a character string specifying the x-axis label (NULL by default)

yLab

a character string specifying the y-axis label (NULL by default)

yLim

a numeric vector of length 2 specifying the y-axis range (NULL by default)

pType

a character string specifying the type of plot. Possible options are "l" for lines (default) and "p" for points.

Value

None. The function is called solely for plot generation.

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
X <- rbinom(n,1,0.5)
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.3)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(X,Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("X","Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=3)

out <- riskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data, psGrid=grid)
boot <- bootRiskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data,
                      psGrid=grid, iter=2, seed=10)
sout <- summary(out, boot, contrast="te")
plotMCEPcurve(sout)

Estimation of Conditional Clinical Endpoint Risk under Placebo and Treatment Given Biomarker Response to Treatment in a Baseline Surrogate Measure Three-Phase Sampling Design

Description

Estimates P\{Y(z)=1|S(1)=s_1\}, z=0,1, on a grid of s_1 values following the estimation method of Juraska, Huang, and Gilbert (2018), where Z is the treatment group indicator (Z=1, treatment; Z=0, placebo), S(z) is a continuous or ordered categorical univariate biomarker under assignment to Z=z measured at fixed time t_0 after randomization, and Y is a binary clinical endpoint (Y=1, disease; Y=0, no disease) measured after t_0. The estimator employs the generalized product kernel density/probability estimation method of Hall, Racine, and Li (2004) implemented in the np package. The risks P\{Y(z)=1|S(z)=s_1,X=x\}, z=0,1, where X is a vector of discrete baseline covariates, are estimated by fitting inverse probability-weighted logistic regression models using the osDesign package.

Usage

riskCurve(
  formula,
  bsm,
  tx,
  data,
  pstype = c("continuous", "ordered"),
  bsmtype = c("continuous", "ordered"),
  bwtype = c("fixed", "generalized_nn", "adaptive_nn"),
  hinge = FALSE,
  weights = NULL,
  psGrid = NULL,
  saveFile = NULL,
  saveDir = NULL
)

Arguments

formula

bsm

a character string specifying the variable name in data representing the baseline surrogate measure

tx

a character string specifying the variable name in data representing the treatment group indicator

data

pstype

a character string specifying whether the biomarker response shall be treated as a continuous (default) or ordered categorical variable in the kernel density/probability estimation

bsmtype

a character string specifying whether the baseline surrogate measure shall be treated as a continuous (default) or ordered categorical variable in the kernel density/probability estimation

bwtype

hinge

weights

either a numeric vector of weights or a character string specifying the variable name in data representing weights applied to observations in the phase 2 subset in order to make inference about the target population of all randomized participants endpoint-free at t_0. The weights reflect that the case:control ratio in the phase 2 subset is different from that in the target population and are passed on to GLMs in the estimation of the hinge point. If NULL (default), weights for cases and controls are calculated separately in each study group.

psGrid

saveFile

a character string specifying the name of an .RData file storing the output list. If NULL (default), the output list will only be returned.

saveDir

Value

If saveFile and saveDir are both specified, the output list (named oList) is saved as an .RData file; otherwise it is returned only. The output object (of class riskCurve) is a list with the following components:

psGrid: a numeric vector of S(1) values at which the conditional clinical endpoint risk is estimated in the components plaRiskCurve and txRiskCurve
plaRiskCurve: a numeric vector of estimates of P\{Y(0)=1|S(1)=s_1\} for s_1 in psGrid
txRiskCurve: a numeric vector of estimates of P\{Y(1)=1|S(1)=s_1\} for s_1 in psGrid
fOptBandwidths: a conbandwidth object returned by the call of the function npcdensbw containing the optimal bandwidths, selected by likelihood cross-validation, in the kernel estimation of the conditional density of S(1) given the baseline surrogate measure and any other specified baseline covariates
gOptBandwidths: a conbandwidth object returned by the call of the function npcdensbw or npudensbw containing the optimal bandwidths, selected by likelihood cross-validation, in the kernel estimation of the conditional density of S(0) given any specified baseline covariates or the marginal density of S(0) if no baseline covariates are specified in formula
cpointP: if hinge=TRUE, the estimate of the hinge point in the placebo group
cpointT: if hinge=TRUE, the estimate of the hinge point in the treatment group

References

Fong, Y., Huang, Y., Gilbert, P. B., and Permar, S. R. (2017), chngpt: threshold regression model estimation and inference, BMC Bioinformatics, 18.

Hall, P., Racine, J., and Li, Q. (2004), Cross-validation and the estimation of conditional probability densities, JASA 99(468), 1015-1026.

Juraska, M., Huang, Y., and Gilbert, P. B. (2020), Inference on treatment effect modification by biomarker response in a three-phase sampling design, Biostatistics, 21(3): 545-560, https://doi.org/10.1093/biostatistics/kxy074.

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
X <- rbinom(n,1,0.5)
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.4)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(X,Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("X","Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=3)

out <- riskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data, psGrid=grid)

# alternatively, to save the .RData output file (no '<-' needed):
riskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data, saveFile="out.RData",
          saveDir="./")

Summary of Point and Interval Estimation of a Marginal Causal Effect Predictiveness Curve

Description

Summarizes point estimates and pointwise and simultaneous Wald-type bootstrap confidence intervals for a specified marginal causal effect predictiveness (mCEP) curve (see, e.g., Juraska, Huang, and Gilbert (2018) for the definition).

Usage

## S3 method for class 'riskCurve'
summary(
  object,
  boot = NULL,
  contrast = c("te", "rr", "logrr", "rd"),
  confLevel = 0.95,
  ...
)

Arguments

object

an object of class riskCurve, typically returned by riskCurve

boot

an object returned by bootRiskCurve. If NULL (default), only point estimates are reported.

contrast

a character string specifying the mCEP curve. It must be one of te (treatment efficacy), rr (relative risk), logrr (log relative risk), and rd (risk difference [placebo minus treatment]).

confLevel

the confidence level of pointwise and simultaneous confidence intervals

...

for other methods

Value

A data frame containing point and possibly interval estimates of the specified mCEP curve.

References

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.4)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=2)

out <- riskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data, psGrid=grid)
boot <- bootRiskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data,
                      psGrid=grid, iter=2, seed=10)
summary(out, boot, contrast="te")

Testing of the Null Hypotheses of a Flat and a Constant Marginal Causal Effect Predictiveness Curve

Description

Computes a two-sided p-value either from the test of {H_0^1: mCEP(s_1)=CE for all s_1}, where CE is the overall causal treatment effect on the clinical endpoint, or from the test of {H_0^2: mCEP(s_1)=c for all s_1 in the interval limS1 and a specified constant c}, each against a general alternative hypothesis. The testing procedures are described in Juraska, Huang, and Gilbert (2018) and are based on the simultaneous estimation method of Roy and Bose (1953).

Usage

testConstancy(
  object,
  boot,
  contrast = c("te", "rr", "logrr", "rd"),
  null = c("H01", "H02"),
  overallPlaRisk = NULL,
  overallTxRisk = NULL,
  MCEPconstantH02 = NULL,
  limS1 = NULL
)

Arguments

object

an object returned by riskCurve

boot

an object returned by bootRiskCurve

contrast

a character string specifying the mCEP curve. It must be one of te (treatment efficacy), rr (relative risk), logrr (log relative risk), and rd (risk difference [placebo minus treatment]).

null

a character string specifying the null hypothesis to be tested; one of H01 and H02 as introduced above

overallPlaRisk

a numeric value of the estimated overall clinical endpoint risk in the placebo group. It is required when null equals H01.

overallTxRisk

a numeric value of the estimated overall clinical endpoint risk in the treatment group. It is required when null equals H01.

MCEPconstantH02

the constant c in the null hypothesis H_0^2. It is required when null equals H02.

limS1

a numeric vector of length 2 specifying an interval that is a subset of the support of S(1) and that is used in the evaluation of the null hypothesis H_0^2. If NULL (default), then H_0^2 is evaluated for all s_1.

Value

A numeric value representing the two-sided p-value from the test of either H_0^1 or H_0^2.

References

Roy, S. N. and Bose, R. C. (1953), Simultaneous condence interval estimation, The Annals of Mathematical Statistics, 24, 513-536.

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
X <- rbinom(n,1,0.5)
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.4)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(X,Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("X","Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=3)

out <- riskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data, psGrid=grid)
boot <- bootRiskCurve(formula=Y ~ S + factor(X), bsm="Sb", tx="Z", data=data,
                      psGrid=grid, iter=2, seed=10)
fit <- glm(Y ~ Z, data=data, family=binomial)
prob <- predict(fit, newdata=data.frame(Z=0:1), type="response")

testConstancy(out, boot, contrast="te", null="H01", overallPlaRisk=prob[1],
              overallTxRisk=prob[2])
testConstancy(out, boot, contrast="te", null="H02", MCEPconstantH02=0, limS1=c(qS[1],1.5))

Testing of the Null Hypothesis of Equal Marginal Causal Effect Predictiveness Curves for Two Biomarkers, Endpoints, or Baseline Covariate Subgroups

Description

Computes a two-sided p-value either from the test of {H_0^3: mCEP_1(s_1)=mCEP_2(s_1) for all s_1 in limS1}, where mCEP_1 and mCEP_2 are each associated with either a different biomarker (measured in the same units) or a different endpoint or both, or from the test of {H_0^4: mCEP(s_1|X=0)= mCEP(s_1|X=1) for all s_1 in limS1}, where X is a baseline dichotomous phase 1 covariate of interest, each against a general alternative hypothesis. The testing procedures are described in Juraska, Huang, and Gilbert (2018) and are based on the simultaneous estimation method of Roy and Bose (1953).

Usage

testEquality(
  object1,
  object2,
  boot1,
  boot2,
  contrast = c("te", "rr", "logrr", "rd"),
  null = c("H03", "H04"),
  limS1 = NULL
)

Arguments

object1

an object returned by riskCurve pertaining to either mCEP_1(s_1) in H_0^3 or mCEP(s1|X=0) in H_0^4

object2

an object returned by riskCurve pertaining to either mCEP_2(s_1) in H_0^3 or mCEP(s1|X=1) in H_0^4

boot1

an object returned by bootRiskCurve pertaining to either mCEP_1(s_1) in H_0^3 or mCEP(s1|X=0) in H_0^4

boot2

an object returned by bootRiskCurve pertaining to either mCEP_2(s_1) in H_0^3 or mCEP(s1|X=1) in H_0^4

contrast

a character string specifying the mCEP curve. It must be one of te (treatment efficacy), rr (relative risk), logrr (log relative risk), and rd (risk difference [placebo minus treatment]).

null

a character string specifying the null hypothesis to be tested; one of H03 and H04 as introduced above

limS1

a numeric vector of length 2 specifying an interval that is a subset of the support of S(1). If NULL (default), then the specified null hypothesis is evaluated for all s_1.

Value

A numeric value representing the two-sided p-value from the test of either H_0^3 or H_0^4.

References

Roy, S. N. and Bose, R. C. (1953), Simultaneous condence interval estimation, The Annals of Mathematical Statistics, 24, 513-536.

Examples

n <- 500
Z <- rep(0:1, each=n/2)
S <- MASS::mvrnorm(n, mu=c(2,2,3), Sigma=matrix(c(1,0.9,0.7,0.9,1,0.7,0.7,0.7,1), nrow=3))
p <- pnorm(drop(cbind(1,Z,(1-Z)*S[,2],Z*S[,3]) %*% c(-1.2,0.2,-0.02,-0.2)))
Y <- sapply(p, function(risk){ rbinom(1,1,risk) })
X <- rbinom(n,1,0.5)
# delete S(1) in placebo recipients
S[Z==0,3] <- NA
# delete S(0) in treatment recipients
S[Z==1,2] <- NA
# generate the indicator of being sampled into the phase 2 subset
phase2 <- rbinom(n,1,0.4)
# delete Sb, S(0) and S(1) in controls not included in the phase 2 subset
S[Y==0 & phase2==0,] <- c(NA,NA,NA)
# delete Sb in cases not included in the phase 2 subset
S[Y==1 & phase2==0,1] <- NA
data <- data.frame(X,Z,S[,1],ifelse(Z==0,S[,2],S[,3]),Y)
colnames(data) <- c("X","Z","Sb","S","Y")
qS <- quantile(data$S, probs=c(0.05,0.95), na.rm=TRUE)
grid <- seq(qS[1], qS[2], length.out=3)
out0 <- riskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data[data$X==0,], psGrid=grid)
out1 <- riskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data[data$X==1,], psGrid=grid)
boot0 <- bootRiskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data[data$X==0,],
                       psGrid=grid, iter=2, seed=10)
boot1 <- bootRiskCurve(formula=Y ~ S, bsm="Sb", tx="Z", data=data[data$X==1,],
                       psGrid=grid, iter=2, seed=15)

testEquality(out0, out1, boot0, boot1, contrast="te", null="H04")

Bootstrap Estimation of Conditional Clinical Endpoint Risk under Placebo and Treatment Given Biomarker Response to Treatment in a Baseline Surrogate Measure Three-Phase Sampling Design

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Plotting of the Estimated Marginal Causal Effect Predictiveness Curve

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Estimation of Conditional Clinical Endpoint Risk under Placebo and Treatment Given Biomarker Response to Treatment in a Baseline Surrogate Measure Three-Phase Sampling Design

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Summary of Point and Interval Estimation of a Marginal Causal Effect Predictiveness Curve

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Testing of the Null Hypotheses of a Flat and a Constant Marginal Causal Effect Predictiveness Curve

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Testing of the Null Hypothesis of Equal Marginal Causal Effect Predictiveness Curves for Two Biomarkers, Endpoints, or Baseline Covariate Subgroups

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