---
title: "psc-vignette"
author: "Richard Jackson"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{psc-vignette}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height= 5
)
```

# Introduction

The psc.R package implements the methods for applying Personalised Synthetic 
Controls, which allows for patients receiving some experimental treatment to be 
compared against a model which predicts their reponse to some control.  This is 
a form of causal inference which differes from other approaches in that

\item Data are only required on a single treatment - all counterfactual evidence 
is supplied by a parametric model
\item Causal inference, in theory at least, is estimated at a patient level - as 
opposed to estimating average effects over a population

The causal estimand obtained is the Average Treatment Effect of the Treated (ATT)
which differs from the Average Treatment Effect (ATE) obtained in other settings 
and addresses the question of whether treatments are effective in the population 
of patients who are treated.  This estimand then targets efficacy over effectivness.


In its basic form, this method creates a likelihood to compare a cohort of data 
to a parametric model.  See (X) for disucssion on it's use as a causal inference 
tool.  To use this package, two basic peices of information are required, a 
dataset and a model against which they can be compared.

In this vignette, we will detail how the psc.r package is constructed and give 
some examples for it's application in practice.


# Methodology

The `pscfit` function compares a dataset ('DC') against a parametric model. This 
is done by selecting a likelihood which is identified by the type of CFM that is 
supplied. At present, two types of model are supported, a flexible parmaeteric 
survival model of type 'flexsurvreg' and a geleneralised linear model of type 
'glm'.

Where the CFM is of type 'flexsurvreg' the likeihood supplied is of the form:


$$L(D∣\Lambda,\Gamma_i)=\prod_{i=1}^{n} f(t_i∣\Lambda,\Gamma_i)^{c_i} 
S(t_i∣\Gamma,\Lambda_i)^{(1−c_i)}$$

Where $\Gamma$ defines the cumulative baseline hazard function, $\Lambda$ is the 
linear predictor and $t$ and $c$ are the event time and indicator variables.

Where the CFM is of the type 'glm' the likelihood supplied is of the form:

$$L(x∣\Gamma_i) = \prod_{i=1}^{n} b (x∣ \Gamma_i )\exp\{\Gamma_i t(x)−
c(\Gamma_i)\}$$


Where $b(.)$, $t(.)$ and $c(.)$ represent the functions of the exponential 
family. In both cases, $\Gamma$ is defiend as:

$$ \Gamma_i = \gamma x_i+\beta $$

Where $\gamma$ are the model coefficients supplied by the CFM and $\beta$ is the 
parameter set to measure the difference between the CFM and the DC.

Estimation is performed using a Bayesian MCMC procedure. Prior distributions for 
$\Gamma$ (& $\Lambda$) are derived directly from the model coefficients (mean 
and variance covariance matrix) or the CFM. A bespoke MCMC routine is performed 
to estimate $\beta$. Please see '?mcmc' for more detials.

For the standard example where the DC contains information from only a single 
treatment, trt need not be specified. Where comparisons between the CFM and 
multiple treatments are require, a covariate of treamtne allocations must be 
specified sperately (using the 'trt' option).


# Package Structure

The main function for using applying Personal Synthetic Controls is the pscfit() 
function which has two inputs, a Counter-Factual Model (CFM) and a data cohort 
(DC).  Further arguments include

* nsim which sets the number of MCMC iterations (defaults to 5000)
* 'id' if the user wishes to restrict estimation to a sub-set (or individual) 
within the DC
* 'trt' to be used as an initial identifier if mulitple treatment comparisons 
are to be made (please see the Mulitple Treatment Comparison below)


## psc object

The output of the "pscfit()" function is an object of class 'psc'.  This class 
contains the following attributes

* A definition of the calss of the model supplied
* A 'cleaned' dataset including extracted components of the CFM and the cleaned DC included in the procedure
* An object defingin the class of model (and therefore the procedure applied - see above)
* A matrix containing the draws of the posterior distributions

## Postestimation functions

basic post estimation functions have been developed to work with the psc object, 
namely "print()", "coef()", "summary()" and "plot()".  For the first three of 
these these provided basic summaries of the efficacy parameter obtained from the 
posterior distribution.


# Motivating Example


The psc.r package includes as example a dataset which is derived from patients 
with pancreatic ductal adenocarcinoma (PDAC) who have all received some 
experimental treatment, in this case GemCap.  The dataset is named 'e4_data' and is loaded into the enviroment using the "data()" function

```{r}
#install.packages("psc")
library(psc)
library(ggpubr)
e4_data <- psc::e4_data
```

Included is a list of prognostic covariates:

* nodes: patient lymph node status; negative (n=1) or positive (n=2) lymph nodes
* grade: tumour grade; 1,2 or 3
* lca199: log transformed ca19.9
* t: T-stage; (1,2 or 3)


Also included are the following structures

* time - survival time
* cen - censoring indictor


We give esamples of how the 'pscfit()' function can be used to comapre data 
against models with survival outcomes (with a 'flexsurvreg' model). Examples on 
how to perform analyses using GLM model objects are available from the github 
repo https://github.com/richJJackson/psc

## Survival Example

For an example with a survival outcome a model must be supplied which is 
contructed ont he basis of flexible parametric splines.  This is contructed 
using the "flexsurvreg" function within the "flexsurv" package.  An example is 
included within the 'psc.r' package names 'surv.mod' and is loaded using the 
'data()" function:

```{r}
gemCFM <- psc::gemCFM
```


The 'gemCFM' is an object of calss pscCFM which means it contains all of the structures required for analysis but has stripped the model object of any patient level data.  Included instead are a summary table:

```{r}
gemCFM$datasumm
```

...and a set of visualisations which we arrange using the ggarrange


```{r}
ggarrange(plotlist=gemCFM$datavis)
```


In this example you can see that this is a model constructed with 3 internal 
knots and hence 5 parameters to describe the baseline cumulative hazard 
function.  There are also prognostic covariates which match with the prognostic 
covariates in the data cohort.

Comparing the dataset to the model is then performed using

```{r,include=F}
surv.psc <- pscfit(gemCFM,e4_data)
```

and we can view the attributes of the psc object that is created

```{r}
attributes(surv.psc)
```

For example to view the matrix contianing the draws of the posterior 
distribution we use

```{r}
surv.post <- surv.psc$posterior
head(surv.post)
```

Inspection will show that there is a column for each parameter in the original model as well as 'beta' and 'DIC' vcolumns which give teh posterior estiamtes for $\beta$ and the Deviance Informaiton Criterion respectively.

We can inspect the poterior distribution using the autocorrelation function, trace and stardard summary statistics:

#### Autocorrelation
```{r}
acf(surv.post$beta)
```

#### Trace
```{r}
plot(surv.post$beta,typ="s")
```

#### Summary


Standard 'summary()' function wil summarise the model fit

```{r}
summary(surv.psc)
```

To visualise the original model and the fit of the data, the plot function has 
been developed

```{r}
plot(surv.psc)
```







