normal_analysis: Analyzing Bayesian trial for normal mean data
In bayesCT: Simulation and Analysis of Adaptive Bayesian Clinical Trials

Function to analyze Bayesian trial for normal mean data which allows early stopping and incorporation of historical data using the discount function approach

normal_analysis(
  treatment,
  outcome,
  complete = NULL,
  mu0_treatment = NULL,
  sd0_treatment = NULL,
  N0_treatment = NULL,
  mu0_control = NULL,
  sd0_control = NULL,
  N0_control = NULL,
  alternative = "greater",
  N_impute = 100,
  h0 = 0,
  number_mcmc = 10000,
  prob_ha = 0.95,
  futility_prob = 0.1,
  expected_success_prob = 0.9,
  discount_function = "identity",
  fix_alpha = FALSE,
  alpha_max = 1,
  weibull_scale = 0.135,
  weibull_shape = 3,
  method = "fixed"
)

`treatment`	vector. treatment assignment for patients, 1 for treatment group and 0 for control group
`outcome`	vector. normal outcome of the trial.
`complete`	vector. similar length as treatment and outcome variable, 1 for complete outcome, 0 for loss to follow up. If complete is not provided, the dataset is assumed to be complete.
`mu0_treatment`	scalar. Mean of the historical treatment group.
`sd0_treatment`	scalar. Standard deviation of the historical treatment group.
`N0_treatment`	scalar. Number of observations of the historical treatment group.
`mu0_control`	scalar. Mean of the historical control group.
`sd0_control`	scalar. Standard deviation of the historical control group.
`N0_control`	scalar. Number of observations of the historical control group.
`alternative`	character. The string specifying the alternative hypothesis, must be one of `"greater"` (default), `"less"` or `"two.sided"`.
`N_impute`	scalar. Number of imputations for Monte Carlo simulation of missing data.
`h0`	scalar. Threshold for comparing two mean values. Default is `h0=0`.
`number_mcmc`	scalar. Number of Monte Carlo Markov Chain draws in sampling posterior.
`prob_ha`	scalar. Probability of alternative hypothesis.
`futility_prob`	scalar. Probability of stopping early for futility.
`expected_success_prob`	scalar. Probability of stopping early for success.
`discount_function`	character. If incorporating historical data, specify the discount function. Currently supports the Weibull function (`discount_function="weibull"`), the scaled-Weibull function (`discount_function="scaledweibull"`), and the identity function (`discount_function="identity"`). The scaled-Weibull discount function scales the output of the Weibull CDF to have a max value of 1. The identity discount function uses the posterior probability directly as the discount weight. Default value is `"identity"`. See `bdpnormal` for more details.
`fix_alpha`	logical. Fix alpha at alpha_max? Default value is FALSE.
`alpha_max`	scalar. Maximum weight the discount function can apply. Default is 1. For a two-arm trial, users may specify a vector of two values where the first value is used to weight the historical treatment group and the second value is used to weight the historical control group.
`weibull_scale`	scalar. Scale parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 0.135. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity".
`weibull_shape`	scalar. Shape parameter of the Weibull discount function used to compute alpha, the weight parameter of the historical data. Default value is 3. For a two-arm trial, users may specify a vector of two values where the first value is used to estimate the weight of the historical treatment group and the second value is used to estimate the weight of the historical control group. Not used when discount_function = "identity".
`method`	character. Analysis method with respect to estimation of the weight paramter alpha. Default method "`mc`" estimates alpha for each Monte Carlo iteration. Alternate value "`fixed`" estimates alpha once and holds it fixed throughout the analysis. See the the `bdpsurvival` vignette `vignette("bdpsurvival-vignette", package="bayesDP")` for more details.

a list of output for the analysis of Bayesian trial for normal mean.

prob_of_accepting_alternative: scalar. The input parameter of probability of accepting the alternative.
margin: scalar. The margin input value of difference between mean estimate of treatment and mean estimate of the control.
alternative: character. The input parameter of alternative hypothesis.
N_treatment: scalar. The number of patients enrolled in the experimental group for each simulation.
N_control: scalar. The number of patients enrolled in the control group for each simulation.
N_enrolled: vector. The number of patients enrolled in the trial (sum of control and experimental group for each simulation. )
N_complete: scalar. The number of patients who completed the trial and had no loss to follow-up.
post_prob_accept_alternative: vector. The final probability of accepting the alternative hypothesis after the analysis is done.
est_final: scalar. The final estimate of the difference in posterior estimate of treatment and posterior estimate of the control group.
stop_futility: scalar. Did the trial stop for futility during imputation of patient who had loss to follow up? 1 for yes and 0 for no.
stop_expected_success: scalar. Did the trial stop for early success during imputation of patient who had loss to follow up? 1 for yes and 0 for no.