# Testing of the Null Hypothesis of Equal Marginal Causal Effect Predictiveness Curves for Two Biomarkers, Endpoints, or Baseline Covariate Subgroups 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 {c("$H_0^4: mCEP(s_1|X=0)=\n$", "$mCEP(s_1|X=1)$") 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). ```r 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 $s_1$. ## Returns A numeric value representing the two-sided p-value from the test of either $H_0^3$ or $H_0^4$. ## Examples ```r 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") ``` ## 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](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`

testEquality function