---
title: "Normal Mean Estimation with bayesDP"
author: "Donnie Musgrove"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: yes
    fig_caption: yes
vignette: >
  %\VignetteIndexEntry{Normal Mean Estimation}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, SETTINGS-knitr, include=FALSE}
library(bayesDP)
stopifnot(require(knitr))
opts_chunk$set(
  dev = "png",
  dpi = 150,
  fig.asp = 0.8,
  fig.width = 5,
  out.width = "60%",
  fig.align = "center"
  )

# Run two models to document the discount function plots
set.seed(42)  
fit01 <- bdpbinomial(y_t=10, N_t=500, y0_t=25, N0_t=250, method="fixed")
fit02 <- bdpbinomial(y_t=10, N_t=500, y0_t=10, N0_t=250, method="fixed",
                     discount_function="weibull")

fit_scaledweibull <- bdpbinomial(y_t=10, N_t=500, y0_t=25, N0_t=250, 
                                 discount_function="scaledweibull",
                                 method="fixed")
fit_identity <- bdpbinomial(y_t=10, N_t=500, y0_t=10, N0_t=250,
                            method="fixed")
```
# Introduction
The purpose of this vignette is to introduce the `bdpnormal` function. `bdpnormal` is used for estimating posterior samples from a Gaussian mean outcome for clinical trials where an informative prior is used. In the parlance of clinical trials, the informative prior is derived from historical data. The weight given to the historical data is determined using what we refer to as a discount function. There are three steps in carrying out estimation:

1. Estimation of the historical data weight, denoted $\hat{\alpha}$, via the discount function

2. Estimation of the posterior distribution of the current data, conditional on the historical data weighted by $\hat{\alpha}$

3. If a two-arm clinical trial, estimation of the posterior treatment effect, i.e., treatment versus control

Throughout this vignette, we use the terms `current`, `historical`, `treatment`, and `control`. These terms are used because the model was envisioned in the context of clinical trials where historical data may be present. Because of this terminology, there are 4 potential sources of data:

1. Current treatment data: treatment data from a current study

2. Current control data: control (or other treatment) data from a current study

3. Historical treatment data: treatment data from a previous study

4. Historical control data: control (or other treatment) data from a previous study

If only treatment data is input, the function considers the analysis a one-arm trial. If treatment data + control data is input, then it is considered a two-arm trial.


## Estimation of the historical data weight
In the first estimation step, the historical data weight $\hat{\alpha}$ is estimated. In the case of a two-arm trial, where both treatment and control data are available, an $\hat{\alpha}$ value is estimated separately for each of the treatment and control arms. Of course, historical treatment or historical control data must be present, otherwise $\hat{\alpha}$ is not estimated for the corresponding arm.

When historical data are available, estimation of $\hat{\alpha}$ is carried out as follows. Let $\bar{y}$, $s$, and $N$ denote the sample mean, sample standard deviation, and sample size of the current data, respectively. Similarly, let $\bar{y}_0$, $s_0$, and $N_0$ denote the sample mean, sample standard deviation, and sample size of the historical data, respectively. Then, the posterior distribution of the mean for current data, under vague (flat) priors is

$$ \begin{array}{rcl}
\tilde{\sigma}^2\mid\bar{y},s,N & \sim & InverseGamma\left(\frac{N-1}{2},\,\frac{N-1}{2}s^2 \right),\\
\\
\tilde{\mu}\mid\bar{y},N,\tilde{\sigma}^2 & \sim & \mathcal{N}ormal\left(\bar{y},\, \frac{1}{N}\tilde{\sigma}^2  \right).
\end{array}
$$

Similarly, the posterior distribution of the mean for historical data, under vague (flat) priors is 
$$ \begin{array}{rcl}
\sigma^2_0 & \sim & InverseGamma\left(\frac{N_0-1}{2},\,\frac{N_0-1}{2}s_0^2 \right),\\
\\
\mu_0 \mid \bar{y}_0, N_0, \sigma^2_0 &  \sim  & \mathcal{N}ormal\left(\bar{y}_0,\, \frac{1}{N_0}\sigma^2_0  \right).
\end{array}
$$

We next compute the posterior probability $p = Pr\left(\tilde{\mu} < \mu_0\mid\bar{y},s,N,\bar{y}_0,s_0,N_0\right)$. Finally, for a discount function, denoted $W$, $\hat{\alpha}$ is computed as 
$$
\hat{\alpha} = \alpha_{max}\cdot W\left(p, \,w\right),\,0\le p\le1,
$$
where $w$ may be one or more parameters associated with the discount function and $\alpha_{max}$ scales the weight $\hat{\alpha}$ by a user-input maximum value. More details on the discount functions are given in the discount function section below.

There are several model inputs at this first stage. First, the user can select the discount function type via the `discount_function` input (see below). Next, choosing `fix_alpha=TRUE` forces a fixed value of $\hat{\alpha}$ (at the `alpha_max` input), as opposed to estimation via the discount function. In the next modeling stage, a Monte Carlo estimation approach is used, requiring several samples from the posterior distributions. Thus, the user can input a sample size greater than or less than the default value of `number_mcmc=10000`. 

An alternate Monte Carlo-based estimation scheme of $\hat{\alpha}$ has been implemented, controlled by the function input `method="mc"`. Here, instead of treating $\hat{\alpha}$ as a fixed quantity, $\hat{\alpha}$ is treated as random. Because $p$ is recomputed at each Monte Carlo draw using a single pair of posterior samples, the resulting sequence of $\hat{\alpha}$ values can exhibit noticeable Monte Carlo variability. First, $p$, is computed as 

$$ \begin{array}{rcl}
Z & = & \displaystyle{\frac{\left|\tilde{\mu}-\mu_0\right|}{\sqrt{v^2 + v^2_0}}} ,\\
\\
p &  =  & 2\left(1-\Phi\left(Z\right)\right),
\end{array}
$$
where $\Phi\left(x\right)$ is the $x$th quantile of a standard normal (i.e., the `pnorm` R function). Here, $v^2$ and $v^2_0$ are the variances of $\tilde{\mu}$ and $\mu_0$, respectively. Next, $p$ is used to construct $\hat{\alpha}$ via the discount function. Since the values $Z$ and $p$ are computed at each iteration of the Monte Carlo estimation scheme, $\hat{\alpha}$ is computed at each iteration of the Monte Carlo estimation scheme, resulting in a distribution of $\hat{\alpha}$ values.


### Discount function
There are currently three discount functions implemented throughout the `bayesDP` package. The discount function is specified using the `discount_function` input with the following choices available:

1. `identity` (default): Identity.

2. `weibull`: Weibull cumulative distribution function (CDF);

3. `scaledweibull`: Scaled Weibull CDF;

First, the identity discount function (default) sets the discount weight $\hat{\alpha}=p$.

Second, the Weibull CDF has two user-specified parameters associated with it, the shape and scale. The default shape is 3 and the default scale is 0.135, each of which are controlled by the function inputs `weibull_shape` and `weibull_scale`, respectively. The form of the Weibull CDF is
$$W(x) = 1 - \exp\left\{- (x/w_{scale})^{w_{shape}}\right\}.$$

The third discount function option is the Scaled Weibull CDF. The Scaled Weibull CDF is the Weibull CDF divided by the value of the Weibull CDF evaluated at 1, i.e., 
$$W^{\ast}(x) = W(x)/W(1).$$
Similar to the Weibull CDF, the Scaled Weibull CDF has two user-specified parameters associated with it, the shape and scale, again controlled by the function inputs `weibull_shape` and `weibull_scale`, respectively.


Using the default shape and scale inputs, each of the discount functions are shown below.
```{r, echo=FALSE}
df2 <- plot(fit_identity, type="discount", print=FALSE)
df2 + ggtitle("Discount function plot", "Identity")
```

```{r, echo=FALSE}
df1 <- plot(fit02, type="discount", print=FALSE)
df1 + ggtitle("Discount function plot", "Weibull distribution with shape=3 and scale=0.135")
```

In each of the above plots, the x-axis is the stochastic comparison between current and historical data, which we've denoted $p$. The y-axis is the discount value $\hat{\alpha}$ that corresponds to a given value of $p$. 

An advanced input for the plot function is `print`. The default value is `print = TRUE`, which simply returns the graphics. Alternately, users can specify `print = FALSE`, which returns a `ggplot2` object. Below is an example using the discount function plot:
```{r}
p1 <- plot(fit02, type="discount", print=FALSE)
p1 + ggtitle("Discount Function Plot :-)")
```

## Estimation of the posterior distribution of the current data, conditional on the historical data
With $\hat{\alpha}$ in hand, we can now estimate the posterior distribution of the mean of the current data. Using the notation of the previous section, the posterior distribution is 
$$\mu \sim \mathcal{N}ormal\left( \frac{\sigma^2_0N\bar{y} + \tilde{\sigma}^2N_0\bar{y}_0\hat{\alpha}}{N\sigma^2_0 + \tilde{\sigma}^2N_0\hat{\alpha}},\,\frac{\tilde{\sigma}^2\sigma^2_0}{N\sigma^2_0 + \tilde{\sigma}^2N_0\hat{\alpha}}      \right).$$
At this model stage, we have in hand `number_mcmc` simulations from the augmented mean distribution. If there are no control data, i.e., a one-arm trial, then the modeling stops and we generate summaries of the posterior distribution of $\mu$. Otherwise, if there are control data, we proceed to a third step and compute a comparison between treatment and control data.


## Estimation of the posterior treatment effect: treatment versus control
This step of the model is carried out on-the-fly using the `summary` or `print` methods. Let $\mu_T$ and $\mu_C$ denote posterior mean estimates of the treatment and control arms, respectively. Currently, the implemented comparison between treatment and control is the difference, i.e., summary statistics related to the posterior difference: $\mu_T - \mu_C$. In a future release, we may consider implementing additional comparison types. 


## Inputting Data
The data inputs for `bdpnormal` are `mu_t`, `sigma_t`, `N_t`, `mu0_t`, `sigma0_t`, `N0_t`, `mu_c`, `sigma_c`, `N_c`, `mu0_c`, `sigma0_c`, and `N0_c`. The data must be input as (`mu`, `sigma`, `N`) triplets. For example, `mu_t`, the sample mean of the current treatment group, must be accompanied by `sigma_t` and `N_t`, the sample standard deviation and sample size, respectively. Historical data inputs are not necessary, but using this function would not be necessary either. 

__At the minimum, `mu_t`, `sigma_t`, and `N_t` must be input__. In the case that only `mu_t`, `sigma_t`, and `N_t` are input, the analysis is analogous to a one-sample t-test. Each of the following input combinations are allowed:

- (`mu_t`, `sigma_t`, `N_t`) - one-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu0_t`, `sigma0_t`, `N0_t`) - one-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu_c`, `sigma_c`, `N_c`) - two-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu0_c`, `sigma0_c`, `N0_c`) - two-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu0_t`, `sigma0_t`, `N0_t`) + (`mu_c`, `sigma_c`, `N_c`) - two-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu0_t`, `sigma0_t`, `N0_t`) + (`mu0_c`, `sigma0_c`, `N0_c`) - two-arm trial
- (`mu_t`, `sigma_t`, `N_t`) + (`mu0_t`, `sigma0_t`, `N0_t`) + (`mu_c`, `sigma_c`, `N_c`) + (`mu0_c`, `sigma0_c`, `N0_c`) - two-arm trial



# Examples
## One-arm trial
Suppose we have historical data with a mean of `mu0_t=50`, standard deviation of `sigma0_t=10`, and sample size of `N0_t=50` patients. Also suppose that we have current data with a mean of `mu_t=45`, standard deviation of `sigma_t=10`, and sample size of `N_t=50`. To illustrate the approach, let's first give full weight to the historical data. This is accomplished by setting `alpha_max=1` and `fix_alpha=TRUE` as follows:
```{r}
set.seed(42)
fit1 <- bdpnormal(mu_t      = 45,
                  sigma_t   = 10,
                  N_t       = 50, 
                  mu0_t     = 50,
                  sigma0_t  = 10,
                  N0_t      = 50,
                  alpha_max = 1,
                  fix_alpha = TRUE,
                  method    = "fixed")
summary(fit1)
```

Based on the `summary` output of `fit1`, we can see that the value of `alpha` was held fixed at 1. Note that the `print` and `summary` methods result in the same output. The resulting treatment group mean is approximately 47.5, which is the average of 45 and 50, as expected.

Now, let's relax the constraint on fixing `alpha` at 1 and let the function estimate `alpha`. We'll also take this opportunity to describe the output of the plot method.
```{r}
set.seed(42)
fit1a <- bdpnormal(mu_t      = 45, sigma_t  = 10, N_t  = 50, 
                   mu0_t     = 50, sigma0_t = 10, N0_t = 50,
                   fix_alpha = FALSE,
                   method    = "fixed")
summary(fit1a)
```

When `alpha` is not constrained to one, it is estimated based on a comparison between the current and historical data. We see that the stochastic comparison, `p_hat`, between current historical and control is 0.0134. Here, `p_hat` is the posterior probability of the comparison between the current and historical data. With the present example, the low value of `p_hat` implies that the current and historical sample means are different. The result is that the weight given to the historical data is shrunk towards zero. Thus, the estimate of `alpha` from the discount function is 0.0134 and the augmented posterior estimate of the mean is approximately the mean of the current data.

Many of the the values presented in the `summary` method are accessible from the fit object. For instance, `alpha` is found in `fit1a$posterior_treatment$alpha_discount` and `p_hat` is located at `fit1a$posterior_treatment$p_hat`. The augmented mean and CI are computed at run-time. The results can be replicated as:
```{r}
mean_augmented <- round(median(fit1a$posterior_treatment$posterior_mu),4)
mean_augmented

CI95_augmented <- round(quantile(fit1a$posterior_treatment$posterior_mu, prob=c(0.025, 0.975)),4)
CI95_augmented
```

Finally, we'll explore the `plot` method. 
```{r}
plot(fit1a, type="posteriors")
plot(fit1a, type="density")
plot(fit1a, type="discount")
```

The top plot displays three density curves. The blue curve is the density of the historical mean, the green curve is the density of the current mean, and the red curve is the density of the current mean augmented by historical data. Since little weight was given to the historical data, the current and posterior means essentially overlap.

The middle plot simply re-displays the posterior mean.

The bottom plot displays the discount function (solid curve) as well as `alpha` (horizontal dashed line) and `p_hat` (vertical dashed line). In the present example, the discount function is the identity. 


## Two-arm trial
On to two-arm trials. In this package, we define a two-arm trial as an analysis where a current and/or historical control arm is present. Suppose we have the same treatment data as in the one-arm example, but now we introduce control data: `mu_c = 40`, `sigma_c = 10`, `N_c = 50`, `mu0_c = 40`, `sigma0_c = 10`, and `N0_c = 50`.  

Before proceeding, it is worth pointing out that the discount function is applied separately to the treatment and control data. Now, let's carry out the two-arm analysis using default inputs:

```{r}
set.seed(42)
fit2 <- bdpnormal(mu_t      = 45, sigma_t  = 10, N_t  = 50, 
                  mu0_t     = 50, sigma0_t = 10, N0_t = 50,
                  mu_c      = 40, sigma_c  = 10, N_c  = 50, 
                  mu0_c     = 40, sigma0_c = 10, N0_c = 50,
                  fix_alpha = FALSE,
                  method    = "fixed")
summary(fit2)
```

The `summary` method of a two-arm analysis is slightly different than a one-arm analysis. First, we see `p_hat` and `alpha` reported for the control data. In the present analysis, the current and historical control data have sample means that are very close, thus the historical control data is given nearly full weight. Again, little weight is given to the historical treatment data.

The CI is computed at run time and is the interval estimate of the difference between the posterior means of the treatment and control groups. With a 95% CI of `(1.7412, 8.5362)`, we would conclude that the treatment and control arms are not significantly different since zero is outside of the CI.

The `plot` method of a two-arm analysis is slightly different than a one-arm analysis as well:
```{r}
plot(fit2, type="posteriors")
plot(fit2, type="density")
plot(fit2, type="discount")
```
Each of the three plots are analogous to the one-arm analysis, but each plot now presents additional data related to the control arm.


