# Testing of the Null Hypotheses of a Flat and a Constant Marginal Causal Effect Predictiveness Curve 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). ```r 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 $s_1$. ## Returns A numeric value representing the two-sided p-value from the test of either $H_0^1$ or $H_0^2$. ## 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) 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)) ``` ## 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 `testEquality`

testConstancy function