# Event History and Survival Analysis

### The eha package

#### Göran Broström

#### 2023-09-29

This vignette is still ongoing work, so if you are looking for something you cannot find, please alert me 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”, Kalbfleisch and Prentice (1980) 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.

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

    ## Call:
    ## survival::coxph(formula = Surv(enter, exit, event) ~ sex + civ +
    ##     imr.birth, data = oldmort)
    ##
    ##   n= 6495, number of events= 1971
    ##
    ##                coef exp(coef) se(coef)     z Pr(>|z|)
    ## sexfemale  -0.24306   0.78423  0.04742 -5.13  3.0e-07 ***
    ## civmarried -0.39549   0.67335  0.08128 -4.87  1.1e-06 ***
    ## civwidow   -0.25980   0.77120  0.07882 -3.30  0.00098 ***
    ## imr.birth   0.00283   1.00284  0.00634  0.45  0.65487
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##            exp(coef) exp(-coef) lower .95 upper .95
    ## sexfemale      0.784      1.275     0.715     0.861
    ## civmarried     0.673      1.485     0.574     0.790
    ## civwidow       0.771      1.297     0.661     0.900
    ## imr.birth      1.003      0.997     0.990     1.015
    ##
    ## Concordance= 0.55  (se = 0.008 )
    ## Likelihood ratio test= 41.4  on 4 df,   p=2e-08
    ## Wald test            = 42.6  on 4 df,   p=1e-08
    ## Score (logrank) test = 42.7  on 4 df,   p=1e-08

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.

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

    ## Covariate             Mean       Coef     Rel.Risk   S.E.    LR p
    ## sex                                                          0.000
    ##             male      0.406     0         1 (reference)
    ##           female      0.594    -0.243     0.784     0.047
    ## civ                                                          0.000
    ##        unmarried      0.080     0         1 (reference)
    ##          married      0.530    -0.395     0.673     0.081
    ##            widow      0.390    -0.260     0.771     0.079
    ## imr.birth            15.162     0.003     1.003     0.006    0.655
    ##
    ## Events                    1971
    ## Total time at risk         37824
    ## Max. log. likelihood      -13558
    ## LR test statistic         41.43
    ## Degrees of freedom        4
    ## Overall p-value           2.18642e-08

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 Hauck and Donner (1977). 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 (Borgan, Goldstein, and Langholz 1995).

### 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.

    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 Broström (2002).

    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 (Broström 2012). 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

Borgan, Ø., L. Goldstein, and B. Langholz. 1995. “Methods for the Analysis of Sampled Cohort Data in the Cox Proportional Hazards Model.” The Annals of Statistics 23: 1749–78.

Broström, G. 2002. “Cox Regression: Ties Without Tears.” Communications in Statistics: Theory and Methods 31: 285–97.

———. 2012. Event History Analysis with R. Boca Raton: Chapman & Hall/CRC.

Hauck, W. W., and A. Donner. 1977. “Wald’s Test as Applied to Hyptheses in Logit Analysis.” Journal of the American Statistical Association 72: 851–53.

Kalbfleisch, J. D., and R. L. Prentice. 1980. The Statistical Analysis of Failure Time Data. First. Hoboken, N.J.: Wiley.