# Individual-level surrogate threshold effect for continuous normally distributed surrogate and true endpoints. Computes the individual-level surrogate threshold effect in the causal-inference single-trial setting where both the surrogate and the true endpoint are continuous normally distributed variables. For details, see paper in the references section. ```r ISTE.ContCont(Mean_T1, Mean_T0, Mean_S1, Mean_S0, N, Delta_S=c(-10, 0, 10), zeta.PI=0.05, PI.Bound=0, PI.Lower=TRUE, Show.Prediction.Plots=TRUE, Save.Plots="No", T0S0, T1S1, T0T0=1, T1T1=1, S0S0=1, S1S1=1, T0T1=seq(-1, 1, by=.001), T0S1=seq(-1, 1, by=.001), T1S0=seq(-1, 1, by=.001), S0S1=seq(-1, 1, by=.001), M.PosDef=500, Seed=123) ``` ## Arguments - `Mean_T1`: A scalar or vector that specifies the mean of the true endpoint in the experimental treatment condition (a vector is used to account for estimation uncertainty). - `Mean_T0`: A scalar or vector that specifies the mean of the true endpoint in the control condition (a vector is used to account for estimation uncertainty). - `Mean_S1`: A scalar or vector that specifies the mean of the surrogate endpoint in the experimental treatment condition (a vector is used to account for estimation uncertainty). - `Mean_S0`: A scalar or vector that specifies the mean of the surrogate endpoint in the control condition (a vector is used to account for estimation uncertainty). - `N`: The sample size of the clinical trial. - `Delta_S`: The vector or scalar of $\Delta S$ values for which the expected $\Delta T$ and its prediction error has to be computed. - `zeta.PI`: The alpha-level to be used in the computation of the prediction interval around $E(\Delta T)$. Default `zeta.PI=0.05`, i.e., the $95\%$ prediction interval. - `PI.Bound`: The ISTE is defined as the value of $\Delta S$ for which the lower (or upper) bound of the $(1-\alpha)\%$ prediction interval around $E(\Delta T)$ is 0. If another threshold value than 0 is desired, this can be requested by using the `PI.Bound` argument. For example, the argument `PI.Bound=5` can be used in the function call to obtain the values of $\Delta S$ for which the lower (or upper) bound of the $(1-\alpha)\%$ prediction intervals (in the different runs of the algorithm)around $\Delta T$ equal 5. - `PI.Lower`: Logical. Should a lower (`PI.Lower=TRUE`) or upper (`PI.Lower=FALSE`) prediction interval be used in the computation of ISTE? Default `PI.Lower=TRUE`. - `Show.Prediction.Plots`: Logical. Should plots that depict $E(\Delta T)$ against $\Delta S$ (prediction function), the prediction interval, and the ISTE for the different runs of the algorithm be shown? Default `Show.Prediction.Plots=TRUE`. - `Save.Plots`: Should the prediction plots (see previous item) be saved? If `Save.Plots="No"` is used (the default argument), the plots are not saved. If the plots have to be saved, replace `"No"` by the desired location, e.g., `Save.Plots="C:/Analysis directory/"` on a windows computer or `Save.Plots="/Users/wim/Desktop/Analysis directory/"` on macOS or Linux. - `T0S0`: A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the control treatment condition that should be considered in the computation of ISTE. - `T1S1`: A scalar or vector that specifies the correlation(s) between the surrogate and the true endpoint in the experimental treatment condition that should be considered in the computation of ISTE. - `T0T0`: A scalar that specifies the variance of the true endpoint in the control treatment condition that should be considered in the computation of ISTE. Default 1. - `T1T1`: A scalar that specifies the variance of the true endpoint in the experimental treatment condition that should be considered in the computation of ISTE. Default 1. - `S0S0`: A scalar that specifies the variance of the surrogate endpoint in the control treatment condition that should be considered in the computation of ISTE. Default 1. - `S1S1`: A scalar that specifies the variance of the surrogate endpoint in the experimental treatment condition that should be considered in the computation of ISTE. Default 1. - `T0T1`: A scalar or vector that contains the correlation(s) between the counterfactuals T0 and T1 that should be considered in the computation of ISTE. Default `seq(-1, 1, by=.001)`. - `T0S1`: A scalar or vector that contains the correlation(s) between the counterfactuals T0 and S1 that should be considered in the computation of ISTE. Default `seq(-1, 1, by=.001)`. - `T1S0`: A scalar or vector that contains the correlation(s) between the counterfactuals T1 and S0 that should be considered in the computation of ISTE. Default `seq(-1, 1, by=.001)`. - `S0S1`: A scalar or vector that contains the correlation(s) between the counterfactuals S0 and S1 that should be considered in the computation of ISTE. Default `seq(-1, 1, by=.001)`. - `M.PosDef`: The number of positive definite $\Sigma$ matrices that should be identified. This will also determine the amount of ISTE values that are identified. Default `M.PosDef=500`. - `Seed`: The seed to be used in the analysis (for reproducibility). Default `Seed=123`. ## Details See paper in the references section. ## Returns An object of class `ICA.ContCont` with components, - **ISTE_Low_PI**: The vector of individual surrogate threshold effect (ISTE) values, i.e., the values of $\Delta S$ for which the lower bound of the $(1-\alpha)\%$ prediction interval around $\Delta T$ is 0 (or another threshold value, which can be requested by using the `PI.Bound` argument in the function call). - **ISTE_Up_PI**: Same as `ISTE_Low_PI`, but using the upper bound of the $(1-\alpha)\%$ prediction interval. - **MSE**: The vector of mean squared error values that are obtained in the prediction of $\Delta T$ based on $\Delta S$. - **gamma0**: The vector of intercepts that are obtained in the prediction of $\Delta T$ based on $\Delta S$. - **gamma1**: The vector of slope that are obtained in the prediction of $\Delta T$ based on $\Delta S$. - **Delta_S_For_Which_Delta_T_equal_0**: The vector of $\Delta S$ values for which $E(\Delta T = 0)$. - **S_squared_pred**: The vector of variances of the prediction errors for $\Delta T$. - **Predicted_Delta_T**: The vector/matrix of predicted values of $\Delta T$ for the $\Delta S$ values that were requested in the function call (argument `Delta_S`). - **PI_Interval_Low**: The vector/matrix of lower bound values of the $(1-\alpha)\%$ prediction interval around $\Delta T$ for the $\Delta S$ values that were requested in the function call (argument `Delta_S`). - **PI_Interval_Up**: The vector/matrix of upper bound values of the $(1-\alpha)\%$ prediction interval around $\Delta T$ for the $\Delta S$ values that were requested in the function call (argument `Delta_S`). - **T0T0**: The vector of variances of T0 (true endpoint in the control treatment) that are used in the computation (this is a constant if the variance is fixed in the function call). - **T1T1**: The vector of variances of T1 (true endpoint in the experimental treatment) that are used in the computations (this is a constant if the variance is fixed in the function call). - **S0S0**: The vector of variances of S0 (surrogate endpoint in the control treatment) that are used in the computations (this is a constant if the variance is fixed in the function call). - **S1S1**: The vector of variances of S1 (surrogate endpoint in the experimental treatment) that are used in the computations (this is a constant if the variance is fixed in the function call). - **Mean_DeltaT**: The vector of treatment effect values on the true endpoint that are used in the computations (this is a constant if the means of T0 and T1 are fixed in the function call). - **Mean_DeltaS**: The vector of treatment effect values on the surrogate endpoint that are used in the computations (this is a constant if the means of S0 and S1 are fixed in the function call). - **Total.Num.Matrices**: An object of class `numeric` that contains the total number of matrices that can be formed as based on the user-specified correlations in the function call. - **Pos.Def**: A `data.frame` that contains the positive definite matrices that can be formed based on the user-specified correlations. These matrices are used to compute the vector of the ISTE values. - **ICA**: Apart from ISTE, ICA is also computed (the individual causal association). For details, see `ICA.ContCont`. - **zeta.PI**: The `zeta.PI` value specified in the function call. - **PI.Bound**: The `PI.Bound` value specified in the function call. - **PI.Lower**: The `PI.Lower` value specified in the function call. - **Delta_S**: The `Delta_S` value(s) specified in the function call. ## References Van der Elst, W., Alonso, A. A., and Molenberghs, G. (submitted). The individual-level surrogate threshold effect in a causal-inference setting. ## Author(s) Wim Van der Elst, Ariel Alonso, & Geert Molenberghs ## See Also `ICA.ContCont` ## Examples ```r # Define input for analysis using the Schizo dataset, # with S=BPRS and T = PANSS. # For each of the identifiable quantities, # uncertainty is accounted for by specifying a uniform # distribution with min, max values corresponding to # the 95% confidence interval of the quantity. T0S0 <- runif(min = 0.9524, max = 0.9659, n = 1000) T1S1 <- runif(min = 0.9608, max = 0.9677, n = 1000) S0S0 <- runif(min=160.811, max=204.5009, n=1000) S1S1 <- runif(min=168.989, max = 194.219, n=1000) T0T0 <- runif(min=484.462, max = 616.082, n=1000) T1T1 <- runif(min=514.279, max = 591.062, n=1000) Mean_T0 <- runif(min=-13.455, max=-9.489, n=1000) Mean_T1 <- runif(min=-17.17, max=-14.86, n=1000) Mean_S0 <- runif(min=-7.789, max=-5.503, n=1000) Mean_S1 <- runif(min=-9.600, max=-8.276, n=1000) # Do the ISTE analysis ## Not run: ISTE <- ISTE.ContCont(Mean_T1=Mean_T1, Mean_T0=Mean_T0, Mean_S1=Mean_S1, Mean_S0=Mean_S0, N=2128, Delta_S=c(-50:50), zeta.PI=0.05, PI.Bound=0, Show.Prediction.Plots=TRUE, Save.Plots="No", T0S0=T0S0, T1S1=T1S1, T0T0=T0T0, T1T1=T1T1, S0S0=S0S0, S1S1=S1S1) # Examine results: summary(ISTE) # Plots of results. # Plot ISTE plot(ISTE) # Other plots, see plot.ISTE.ContCont for details plot(ISTE, Outcome="MSE") plot(ISTE, Outcome="gamma0") plot(ISTE, Outcome="gamma1") plot(ISTE, Outcome="Exp.DeltaT") plot(ISTE, Outcome="Exp.DeltaT.Low.PI") plot(ISTE, Outcome="Exp.DeltaT.Up.PI") ## End(Not run) ```

ISTE.ContCont function