testEquality {pssmooth}

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

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.

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

See Also

riskCurve, bootRiskCurve and testConstancy

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")