---
title: "Event History and Survival Analysis"
subtitle: "The eha package"
author: "Göran Broström"
date: "`r Sys.Date()`"
slug: eha
output: 
    knitr:::html_vignette:
        toc: yes
bibliography: mybib.bib
vignette: >
  %\VignetteIndexEntry{Event History and Survival Analysis}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  \usepackage[utf8]{inputenc}
---

```{r setup, include = FALSE}
library(eha)
options(digits = 4)
```

This vignette is still ongoing work, so if you are looking for something you cannot
find, please [alert me](https://github.com/goranbrostrom/eha/issues/) and I will
do something about it.

# Background

## Early eighties

In the year 1979, I got my first real job after finishing my PhD studies in 
mathematical statistics at Umeå University the same year. My new job was as a 
statistical consultant at the Demographic Data Base (DDB), Umeå University, and
I soon got 
involved in a research project concerning infant mortality in the 19th century
northern Sweden. Individual data were collected from church books registered at
the DDB, and we searched for relevant statistical methods for the analysis of the 
data in relation to our research questions.

Almost parallel in time, the now classical book *"The Statistical Analysis of 
Failure Time Data"*, @kp80 was published, and I realized that I had found the 
answer to our prayers. There was only one small problem, lack of suitable
software. However, there was in the book an appendix with Fortran code for 
*Cox regression*, and since I had taken a course in Fortran77, I decided to 
transfer the appendix to to punch cards(!) and feed them to the mainframe at
the university, of course with our infant mortality data. Bad luck: It turned out 
that by accident two pages in the appendix had been switched, without introducing 
any syntactic error! It however introduced an infinite loop in the code, so the 
prize for the first run was high (this error was corrected in later printings of 
the book).

This was the starting point of my development of software for survival analysis.
The big challenge was to find ways to illustrate results graphically. It led to
translating the Fortran code to *Turbo Pascal* (Borland, MS Dos) in the mid and late 
eighties. 

I soon regretted that choice, and the reason was *portability*: That Pascal code 
didn't work well on Unix work stations and other environments outside MS Dos.
And on top of that, another possibility for getting 
graphics into the picture showed soon up: **R**. So the Pascal code was 
quickly translated into *C*, and it was an easy process to call the C and Fortran
functions 
from R and on top of that write simple routines for presenting results 
graphically and as tables. And so in 2003, the *eha* package was introduced on 
*CRAN*.

## Extensions to Cox regression

During the eighties and nineties, focus was on *Cox regression* and extensions
thereof, but after the transform to an R package, the *survival* package has 
been allowed to take over most of the stuff that was (and still is) found in the 
**eha::coxreg** function.

Regarding *Cox regression*, the **eha** package can be seen as a complement to 
the recommended package **survival**: 
In fact, **eha** *imports* some functions from **survival**, and for *standard Cox regression*, 
`eha::coxreg()` simply calls `survival::agreg.fit()` or `survival::coxph.fit()`,
functions *exported* by **survival**. The simple reason for this is that the
underlying code in these **survival** functions is very fast and efficient.
However, `eha::coxreg()` has some unique features: *Sampling of risk sets*, 
*The "weird" bootstrap*, and *discrete time modeling* via maximum likelihood, 
which motivates the continued support of it.  

I have put effort in producing nice and relevant printouts of regression results,
both on screen and to $\LaTeX$ documents (HTML output may come next). By
*relevant* output I basically mean *avoiding misleading p-values*, show all 
*factor levels*, and use the *likelihood ratio test* instead of the *Wald test* 
where possible. 

To summarize, the extensions are (in descending order of importance, by my own 
judgement).

### Relevant printed output

Consider the following standard way of performing a Cox regression and reporting 
the result.

```{r coxph}
fit <- survival::coxph(Surv(enter, exit, event) ~ sex + civ + imr.birth, 
                       data = oldmort)
summary(fit)
```

The presentation of the covariates and corresponding coefficient estimates follows
common **R** standard, but it is a little bit confusing regarding covariates that 
are represented as *factors*. In this example we have *civ* (civil status), with
three levels, `unmarried`, `married`, and `widow`, but only two are shown, the 
*reference category*, `unmarried`, is hidden. Moreover, the variable *name* is 
joined with the variable *labels*, in a somewhat hard-to-read fashion. I prefer
the following result presentation.

```{r coxreg}
fit <- coxreg(Surv(enter, exit, event) ~ sex + civ + imr.birth, data = oldmort)
summary(fit)
```

There are a few things to notice here. First, the layout: Variables are clearly 
separated from their labels, and *all* categories for factors are shown, even
the reference categories. Second, *p*-values are given for *variables*, not for 
levels, and third, the *p*-values are *likelihood ratio* (LR) based, and not Wald
tests. The importance of this was explained by @hd77. They considered logistic 
regression, but their conclusions in fact hold for most nonlinear models. The 
very short version is that a large Wald *p*-value can mean one of two things: 
(i) Your finding is statistically non-significant (what you expect), or (ii)
your finding is strongly significant (surprise!). Experience shows that condition 
(ii) is quite rare, but why take chances? (The truth is that it is more time 
consuming and slightly more complicated to program, so standard statistical
software, including **R**, avoids it.)

Regarding *p*-values for *levels*, it is a strongly discouraged practice. *Only* if 
a test shows a small enough *p*-value for the *variable*, dissemination of the 
significance of contrasts is allowed, and then only if treated as a *mass 
significance* problem .

### Sampling of risk sets

The idea is that you can regard what happens in a risk set as a very unbalanced 
two-sample problem: The two groups are (i) those who dies at the given time 
point, often only one individual, and (ii) those who survive (many persons).
The questions is which covariate values that tend to create a death, and we simply 
can do without so many survivors, so we take a random sample of them. It will 
save computer time and storage, and in some cases the prize for collecting the
information about all survivors is high [@bgl95].

### The weird bootstrap

The weird bootstrap is aimed at getting estimates of the uncertainty of the
regression parameter estimates in a Cox regression. It works by regarding each
risk set and the number of failures therein as given and by simulation determining
who will be failures. It is assumed that what happens in one riskset is independent 
of what has happened in other risk sets (this is the "weird" part in the procedure). 
This is repeated many times, resulting in a collection of parameter estimates,
the bootstrap samples. 




```{r boot, eval = FALSE}
library(eha)
fit <- coxreg(Surv(enter, exit, event) ~ sex, data = oldmort, boot = 100, 
              control = list(trace = TRUE))
```

### Discrete time methods

These methods are supposed to be used with data that are heavily tied, so that 
a discrete time model may be reasonable.

The method *ml* performs a maximum likelihood estimation, and the "mppl" method 
is a compromise between the ML method and Efron's metod of handling ties. These 
methods are described in detail in @gb02.

```{r mppl, cache = FALSE, eval = FALSE}
library(eha)
fit <- coxreg(Surv(enter, exit, event) ~ sex, data = oldmort, method = "ml")
plot(c(60, fit$hazards[[1]][, 1]), c(0, cumsum(fit$hazards[[1]][, 2])), type = "l")
```

The importance of discrete time methods in the framework of Cox regression, as 
a means of treating tied data, has diminished in the light of *Efron*'s 
approximation, today the default method in the *survival* and *eha* packages.


## Later development

The development before 2010 is summarized in the book *Event History Analysis 
with R* [@gb12]. Later focus has 
been on parametric survival models, who are more suitable to handle huge amounts 
of data through methods based on the theory of sufficient statistics, 
in particular *piecewise constant hazards models*. In demographic applications 
(in particular *mortality* studies), the *Gompertz* survival distribution is 
important, because adult mortality universally shows a pattern of increasing 
exponentially with increasing age.

# Parametric survival models


There is a special vignette describing the theory and implementation of the 
parametric failure time models. It is *not* very useful as a *user's manual*. It also 
has the flaw that it only considers the theory in the case of time fixed covariates, 
although it also works for time-varying ones. This is fairy trivial for the PH models,
but some care is needed to be taken with the AFT models.

## Accelerated Failure Time (AFT) models

The parametric accelerated failure time (AFT) models are present via `eha::aftreg()`,
which is corresponding to `survival::survreg()`. An important difference is that 
`eha::aftreg()` allows for *left truncated data*.

## Proportional Hazards (PH) models

Parametric proportional hazards (PH) modeling is available through the functions
`eha::phreg()` and `eha::weibreg()`, the latter still in the package for 
historical reasons. It will eventually be removed, since the Weibull distribution 
is also available in `eha::phreg()`.

## PH models with tabular data 

Cox regression is not very suitable in the analysis of huge data sets with a lot
of events (e.g., deaths). For instance, consider analyzing the mortality of 
the Swedish population aged 60--110 during the years 1968-2019, where we can 
count to more than four million deaths. This is elaborated in a separate 
vignette.


## The Gompertz distribution

The *Gompertz* distribution has long been part of the possible distributions in 
the `phreg` and `aftreg` functions, but it will be placed in a function of its 
own, `gompreg`, see the separate vignette on this topic.

# Utilities

The primary applications in mind for **eha** were *demography* and *epidemiology*.
There are some functions in **eha** that makes certain common tasks in that context 
easy to perform, for instance *rectangular cuts* in the *Lexis diagram*, creating 
*period* and *cohort* statistics, etc.

## Lexis cuts

## Tabulation

# References
