# Power/type I error calculation for data with two groups (treatment and control group, no covariates) with fixed a0 Power/type I error calculation for data with two groups (treatment and control group, no covariates) with fixed $a_0$ using power priors ```r power.two.grp.fixed.a0( data.type, n.t, n.c, historical = matrix(0, 1, 4), nullspace.ineq = ">", samp.prior.mu.t, samp.prior.mu.c, samp.prior.var.t, samp.prior.var.c, prior.mu.t.shape1 = 1, prior.mu.t.shape2 = 1, prior.mu.c.shape1 = 1, prior.mu.c.shape2 = 1, delta = 0, gamma = 0.95, nMC = 10000, nBI = 250, N = 10000 ) ``` ## Arguments - `data.type`: Character string specifying the type of response. The options are "Normal", "Bernoulli", "Poisson" and "Exponential". - `n.t`: Sample size of the treatment group for the simulated datasets. - `n.c`: Sample size of the control group for the simulated datasets. - `historical`: (Optional) matrix of historical dataset(s). If `data.type` is "Normal", `historical` is a matrix with four columns: * The first column contains the sum of responses for the control group. * The second column contains the sample size of the control group. * The third column contains the sample variance of responses for the control group. * The fourth column contains the discounting parameter value $a_0$ (between 0 and 1). For all other data types, `historical` is a matrix with three columns: * The first column contains the sum of responses for the control group. * The second column contains the sample size of the control group. * The third column contains the discounting parameter value $a_0$ (between 0 and 1). Each row represents a historical dataset. - `nullspace.ineq`: Character string specifying the inequality of the null hypothesis. The options are ">" and "<". If ">" is specified, the null hypothesis (for non-exponential data) is $H_0$: $\mu_t$ - $\mu_c$ $\ge$ $\delta$. If "<" is specified, the null hypothesis is $H_0$: $\mu_t$ - $\mu_c$ $\le$ $\delta$. The default choice is ">". - `samp.prior.mu.t`: Vector of possible values of $\mu_t$ to sample (with replacement) from. The vector contains realizations from the sampling prior (e.g. normal distribution) for $\mu_t$. - `samp.prior.mu.c`: Vector of possible values of $\mu_c$ to sample (with replacement) from. The vector contains realizations from the sampling prior (e.g. normal distribution) for $\mu_c$. - `samp.prior.var.t`: Vector of possible values of $\sigma^2_t$ to sample (with replacement) from. Only applies if `data.type` is "Normal". The vector contains realizations from the sampling prior (e.g. inverse-gamma distribution) for $\sigma^2_t$. - `samp.prior.var.c`: Vector of possible values of $\sigma^2_c$ to sample (with replacement) from. Only applies if `data.type` is "Normal". The vector contains realizations from the sampling prior (e.g. inverse-gamma distribution) for $\sigma^2_c$ - `prior.mu.t.shape1`: First hyperparameter of the initial prior for $\mu_t$. The default is 1. Does not apply if `data.type` is "Normal". - `prior.mu.t.shape2`: Second hyperparameter of the initial prior for $\mu_t$. The default is 1. Does not apply if `data.type` is "Normal". - `prior.mu.c.shape1`: First hyperparameter of the initial prior for $\mu_c$. The default is 1. Does not apply if `data.type` is "Normal". - `prior.mu.c.shape2`: Second hyperparameter of the initial prior for $\mu_c$. The default is 1. Does not apply if `data.type` is "Normal". - `delta`: Prespecified constant that defines the boundary of the null hypothesis. The default is zero. - `gamma`: Posterior probability threshold for rejecting the null. The null hypothesis is rejected if posterior probability is greater `gamma`. The default is 0.95. - `nMC`: Number of iterations (excluding burn-in samples) for the slice sampler or Gibbs sampler. The default is 10,000. - `nBI`: Number of burn-in samples for the slice sampler or Gibbs sampler. The default is 250. - `N`: Number of simulated datasets to generate. The default is 10,000. ## Returns The function returns a S3 object with a `summary` method. Power or type I error is returned, depending on the sampling prior used. The posterior probabilities of the alternative hypothesis are returned. Average posterior means of $\mu_t$ and $\mu_c$ and their corresponding biases are returned. If `data.type` is "Normal", average posterior means of $\tau$ and $\tau_k$'s (if historical data is given) are also returned. ## Details If `data.type` is "Bernoulli", "Poisson" or "Exponential", a single response from the treatment group is assumed to follow Bern($\mu_t$), Pois($\mu_t$) or Exp(rate=$\mu_t$), respectively, where $\mu_t$ is the mean of responses for the treatment group. If `data.type` is "Normal", a single response from the treatment group is assumed to follow $N(\mu_t, \tau^{-1})$ where $\tau$ is the precision parameter. The distributional assumptions for the control group data are analogous. `samp.prior.mu.t` and `samp.prior.mu.c` can be generated using the sampling priors (see example). If `data.type` is "Bernoulli", the initial prior for $\mu_t$ is beta(`prior.mu.t.shape1`, `prior.mu.t.shape2`). If `data.type` is "Poisson", the initial prior for $\mu_t$ is Gamma(`prior.mu.t.shape1`, rate=`prior.mu.t.shape2`). If `data.type` is "Exponential", the initial prior for $\mu_t$ is Gamma(`prior.mu.t.shape1`, rate=`prior.mu.t.shape2`). The initial priors used for the control group data are analogous. If `data.type` is "Normal", each historical dataset $D_{0k}$ is assumed to have a different precision parameter $\tau_k$. The initial prior for $\tau$ is the Jeffery's prior, $\tau^{-1}$, and the initial prior for $\tau_k$ is $\tau_k^{-1}$. The initial prior for the $\mu_c$ is the uniform improper prior. If a sampling prior with support in the null space is used, the value returned is a Bayesian type I error rate. If a sampling prior with support in the alternative space is used, the value returned is a Bayesian power. If `data.type` is "Normal", Gibbs sampling is used for model fitting. For all other data types, numerical integration is used for modeling fitting. ## Examples ```r data.type <- "Bernoulli" n.t <- 100 n.c <- 100 # Simulate three historical datasets historical <- matrix(0, ncol=3, nrow=3) historical[1,] <- c(70, 100, 0.3) historical[2,] <- c(60, 100, 0.5) historical[3,] <- c(50, 100, 0.7) # Generate sampling priors set.seed(1) b_st1 <- b_st2 <- 1 b_sc1 <- b_sc2 <- 1 samp.prior.mu.t <- rbeta(50000, b_st1, b_st2) samp.prior.mu.c <- rbeta(50000, b_st1, b_st2) # The null hypothesis here is H0: mu_t - mu_c >= 0. To calculate power, # we can provide samples of mu.t and mu.c such that the mass of mu_t - mu_c < 0. # To calculate type I error, we can provide samples of mu.t and mu.c such that # the mass of mu_t - mu_c >= 0. sub_ind <- which(samp.prior.mu.t < samp.prior.mu.c) # Here, mass is put on the alternative region, so power is calculated. samp.prior.mu.t <- samp.prior.mu.t[sub_ind] samp.prior.mu.c <- samp.prior.mu.c[sub_ind] N <- 1000 # N should be larger in practice result <- power.two.grp.fixed.a0(data.type=data.type, n.t=n.t, n.c=n.t, historical=historical, samp.prior.mu.t=samp.prior.mu.t, samp.prior.mu.c=samp.prior.mu.c, delta=0, N=N) summary(result) ``` ## References Yixuan Qiu, Sreekumar Balan, Matt Beall, Mark Sauder, Naoaki Okazaki and Thomas Hahn (2019). RcppNumerical: 'Rcpp' Integration for Numerical Computing Libraries. R package version 0.4-0. https://CRAN.R-project.org/package=RcppNumerical Chen, Ming-Hui, et al. "Bayesian design of noninferiority trials for medical devices using historical data." Biometrics 67.3 (2011): 1163-1170. ## See Also `two.grp.fixed.a0`

power.two.grp.fixed.a0 function