# Survival Analysis in mlr3proba

#### Raphael Sonabend

This vignette is an introduction to performing survival analysis in mlr3proba.

## A very quick introduction to survival analysis

Survival analysis is a sub-field of supervised machine learning in which the aim is to predict the survival distribution of a given individual. Arguably the main feature of survival analysis is that unlike classification and regression, learners are trained on two features: 1. the time until the event takes place, 2. the event type: either censoring or death. At a particular time-point, an individual is either: alive, dead, or censored. Censoring occurs if it is unknown if an individual is alive or dead. For example, say we are interested in patients in hospital and every day it is recorded if they are alive or dead, then after a patient leaves it is unknown if they are alive or dead, hence they are censored.

In the case that there is no censoring, but a predicted probability distribution is still the goal, then probabilistic regression learners are advised instead.

## Survival Tasks

Unlike `TaskClassif` and `TaskRegr` which have a single ‘target’ argument, `TaskSurv` mimics the `survival::Surv` object and has three-four target arguments (dependent on censoring type)

    library(mlr3proba); library(mlr3); library(survival)

    # type = "right" is default

    TaskSurv$new(id = "right_censored", backend = survival::rats,
                 time = "time", event = "status", type = "right")
    #> <TaskSurv:right_censored> (300 x 5)
    #> * Target: time, status
    #> * Properties: -
    #> * Features (3):
    #>   - int (1): litter
    #>   - dbl (1): rx
    #>   - chr (1): sex

    task = TaskSurv$new(id = "interval_censored", backend = survival::bladder2[,-c(1, 7)],
                        time = "start", time2 = "stop", type = "interval2")
    task
    #> <TaskSurv:interval_censored> (178 x 6)
    #> * Target: start, stop
    #> * Properties: -
    #> * Features (4):
    #>   - dbl (2): enum, rx
    #>   - int (2): number, size
    task$truth()[1:10]
    #>  [1] [0,  1] [0,  4] [0,  7] [0, 10] [0,  6] [6, 10] [0, 14] [0, 18] [0,  5]
    #> [10] [5, 18]

## Train and Predict

    # create task and learner

    veteran = mlr3misc::load_dataset("veteran", package = "survival")
    task_veteran = TaskSurv$new(id = "veteran", backend = veteran, time = "time", event = "status")
    learner = lrn("surv.coxph")

    # train/test split

    train_set = sample(task_veteran$nrow, 0.8 * task_veteran$nrow)
    test_set = setdiff(seq_len(task_veteran$nrow), train_set)

    # fit Cox PH and inspect model

    learner$train(task_veteran, row_ids = train_set)
    learner$model
    #> Call:
    #> survival::coxph(formula = task$formula(), data = task$data(),
    #>     x = TRUE)
    #>
    #>                        coef exp(coef)  se(coef)      z        p
    #> age               -0.007685  0.992345  0.011751 -0.654 0.513130
    #> celltypesmallcell  0.976473  2.655076  0.317421  3.076 0.002096
    #> celltypeadeno      1.173923  3.234657  0.354138  3.315 0.000917
    #> celltypelarge      0.460501  1.584868  0.334837  1.375 0.169039
    #> diagtime           0.001186  1.001187  0.009848  0.120 0.904155
    #> karno             -0.029826  0.970614  0.006497 -4.590 4.42e-06
    #> prior              0.010400  1.010454  0.026531  0.392 0.695071
    #> trt                0.225305  1.252705  0.247522  0.910 0.362695
    #>
    #> Likelihood ratio test=47.36  on 8 df, p=1.307e-07
    #> n= 109, number of events= 100

    # make predictions for new data

    prediction = learner$predict(task_veteran, row_ids = test_set)
    prediction
    #> <PredictionSurv> for 28 observations:
    #>     row_id time status      crank                distr         lp
    #>          3  228   TRUE -0.6936862 <VectorDistribution> -0.6936862
    #>          4  126   TRUE -0.7746938 <VectorDistribution> -0.7746938
    #>          8  110   TRUE -1.4899216 <VectorDistribution> -1.4899216
    #> ---
    #>        119    7   TRUE  1.1495552 <VectorDistribution>  1.1495552
    #>        128   19   TRUE  0.9844033 <VectorDistribution>  0.9844033
    #>        137   49   TRUE  0.8945899 <VectorDistribution>  0.8945899

## Evaluate - crank, lp, and distr

Every `PredictionSurv` object can predict one or more of:

- `lp` - Linear predictor calculated as the fitted coefficients multiplied by the test data.
- `distr` - Predicted survival distribution, either discrete or continuous. Implemented in distr6.
- `crank` - Continuous risk ranking.

`lp` and `crank` can be used with measures of discrimination such as the concordance index. Whilst `lp` is a specific mathematical prediction, `crank` is any continuous ranking that identifies who is more or less likely to experience the event. So far the only implemented learner that only returns a continuous ranking is `surv.svm`. If a `PredictionSurv` returns an `lp` then the `crank` is identical to this. Otherwise `crank` is calculated as the expectation of the predicted survival distribution. Note that for linear proportional hazards models, the ranking (but not necessarily the `crank` score itself) given by `lp` and the expectation of `distr`, is identical.

    # In the previous example, Cox model predicts `lp` so `crank` is identical

    all(prediction$lp == prediction$crank)
    #> [1] TRUE
    prediction$lp[1:10]
    #>  [1] -0.69368619 -0.77469379 -1.48992162 -1.02294630  0.10603704  0.35349280
    #>  [7]  0.19769846 -0.24070498  1.61950353  0.02121054

    # These are evaluated with measures of discrimination and calibration.
    # As all PredictionSurv objects will return crank, Harrell's C is the default measure.

    prediction$score()
    #> surv.harrellC
    #>     0.7526596

    # distr is evaluated with probabilistic scoring rules.

    measure = lapply(c("surv.graf", "surv.grafSE"), msr)
    prediction$score(measure)
    #>   surv.graf surv.grafSE
    #>   0.1120402   0.0129813

    # Often measures can be integrated over mutliple time-points, or return
    # predictions for single time-points

    measure = msr("surv.graf", times = 60)
    prediction$score(measure)
    #> surv.graf
    #> 0.1457346

### Probability distributions with distr6

Predicted distributions are implemented in distr6, which contains functionality for plotting and further analysis of probability distributions. See here for full tutorials. Briefly we will go over the most important parts for mlr3proba.

    task = tgen("simsurv")$generate(20)
    learner = lrn("surv.coxph")

    # In general it is not advised to train/predict on same data

    prediction = learner$train(task)$predict(task)

    # The predicted `distr` is a VectorDistribution consisting of 20 separate distributions

    prediction$distr
    #> WeightDisc1 WeightDisc2 ... WeightDisc19 WeightDisc20

    # These can be extracted and queried either invidually...

    prediction$distr[1]$survival(60:70)
    #>  [1] 0 0 0 0 0 0 0 0 0 0 0
    prediction$distr[1]$mean()
    #> [1] 1.694734

    # ...or together

    prediction$distr$cdf(60)[,1:10]
    #>    WeightDisc1 WeightDisc2 WeightDisc3 WeightDisc4 WeightDisc5 WeightDisc6
    #> 1:           1           1           1           1           1           1
    #>    WeightDisc7 WeightDisc8 WeightDisc9 WeightDisc10
    #> 1:           1           1           1            1
    prediction$distr$mean()[,1:10]
    #>    WeightDisc1 WeightDisc2 WeightDisc3 WeightDisc4 WeightDisc5 WeightDisc6
    #> 1:    1.694734    1.259515    1.490812    1.630679    1.756059    1.724937
    #>    WeightDisc7 WeightDisc8 WeightDisc9 WeightDisc10
    #> 1:    1.680944    1.753342    1.662975     1.696142

    # As well as plotted

    plot(prediction$distr[1], "survival", main = "First 2 Survival Curves")
    lines(prediction$distr[2], "survival", col = 2)

## Composition

Finally we take a look at the `PipeOp`s implemented in mlr3proba, which are used for composition of predict types. For example, if a learner only returns a linear predictor, then `PipeOpDistrCompositor` can be used to estimate a survival distribution. Or, if a learner returns a `distr` then `PipeOpCrankCompositor` can be used to estimate `crank` from `distr`. See mlr3pipelines for full tutorials and details on `PipeOp`s.

### PipeOpDistrCompositor

    library(mlr3pipelines)

    # PipeOpDistrCompositor - Train one model with a baseline distribution,
    # (Kaplan-Meier or Nelson-Aalen), and another with a predicted linear predictor.
    task = tgen("simsurv")$generate(30)
    leaner_lp = lrn("surv.gbm", bag.fraction = 1, n.trees = 50L)
    leaner_distr = lrn("surv.kaplan")
    prediction_lp = leaner_lp$train(task)$predict(task)
    prediction_distr = leaner_distr$train(task)$predict(task)
    prediction_lp$distr
    #> NULL

    # Doesn't need training. Base = baseline distribution. ph = Proportional hazards.

    pod = po("distrcompose", param_vals = list(form = "ph", overwrite = FALSE))
    prediction = pod$predict(list(base = prediction_distr, pred = prediction_lp))$output

    # Now we have a predicted distr!

    prediction$distr
    #> WeightDisc1 WeightDisc2 ... WeightDisc29 WeightDisc30

    # This can all be simplified by using the distrcompose wrapper

    gbm.distr = distrcompositor(learner = lrn("surv.gbm", bag.fraction = 1, n.trees = 50L),
                                 estimator = "kaplan",
                                 form = "aft",
                                 overwrite = FALSE)
    gbm.distr$train(task)$predict(task)
    #> <PredictionSurv> for 30 observations:
    #>     row_id     time status       crank                distr          lp
    #>          1 5.000000  FALSE -0.19367047 <VectorDistribution> -0.19367047
    #>          2 3.568860   TRUE -0.66748168 <VectorDistribution> -0.66748168
    #>          3 1.128017   TRUE -0.05205574 <VectorDistribution> -0.05205574
    #> ---
    #>         28 1.635774   TRUE -0.17106619 <VectorDistribution> -0.17106619
    #>         29 5.000000  FALSE -0.47381121 <VectorDistribution> -0.47381121
    #>         30 3.155294   TRUE -0.66748168 <VectorDistribution> -0.66748168

### PipeOpCrankCompositor

Note that a `PredictionSurv` will always return `crank`, but this may either be the same as the `lp` or the expectation of `distr`. This compositor allows you to change the estimation method.

    # PipeOpCrankCompositor - Only one model required.
    leaner = lrn("surv.coxph")
    prediction = leaner$train(task)$predict(task)

    # Doesn't need training - Note: no `overwrite` option as `crank` is always
    # present so the compositor if used will always overwrite.

    poc = po("crankcompose", param_vals = list(method = "mean"))
    composed_prediction = poc$predict(list(prediction))$output

    # Note that whilst the actual values of `lp` and `crank` are different,
    # the rankings are the same, so discrimination measures are unchanged.

    prediction$crank[1:10]
    #>  [1]  0.19248733 -0.43871848  0.39422978  0.24340349 -0.20020759  0.15094767
    #>  [7]  0.60484819 -0.56638731  0.01629422  0.42356422
    composed_prediction$crank[1:10]
    #>  [1] 1.6122990 1.0944552 1.7651577 1.6525095 1.2873051 1.5788921 1.8987710
    #>  [8] 0.9969524 1.4680414 1.7856538
    all(order(prediction$crank) == order(composed_prediction$crank))
    #> [1] TRUE
    cbind(Original = prediction$score(), Composed = composed_prediction$score())
    #>                Original  Composed
    #> surv.harrellC 0.5854342 0.5854342

    # Again a wrapper can be used to simplify this
    crankcompositor(lrn("surv.coxph"), method = "mean")$train(task)$predict(task)
    #> <PredictionSurv> for 30 observations:
    #>     row_id     time status    crank                distr          lp
    #>          1 5.000000  FALSE 1.612299 <VectorDistribution>  0.19248733
    #>          2 3.568860   TRUE 1.094455 <VectorDistribution> -0.43871848
    #>          3 1.128017   TRUE 1.765158 <VectorDistribution>  0.39422978
    #> ---
    #>         28 1.635774   TRUE 1.232586 <VectorDistribution> -0.26655353
    #>         29 5.000000  FALSE 1.436710 <VectorDistribution> -0.02125514
    #>         30 3.155294   TRUE 1.073284 <VectorDistribution> -0.46592203

## All Together Now

Putting all of this together we can perform a (overly simplified) benchmark experiment to find the best learner for making predictions on a simulated dataset.

    library(mlr3pipelines); library(mlr3); library(mlr3tuning); library(paradox)
    set.seed(42)

    task = tgen("simsurv")$generate(50)

    composed_lrn_gbm = distrcompositor(lrn("surv.gbm", bag.fraction = 1, n.trees = 50L),
                                       "kaplan", "ph")

    lrns = lapply(paste0("surv.", c("kaplan", "coxph")), lrn)

    design = benchmark_grid(tasks = task, learners = c(lrns, list(composed_lrn_gbm)),
                            resamplings = rsmp("cv", folds = 2))

    bm = benchmark(design)
    bm$aggregate(lapply(c("surv.harrellC","surv.graf","surv.grafSE"), msr))[,c(4, 7:9)]
    #>                           learner_id surv.harrellC surv.graf surv.grafSE
    #> 1:                       surv.kaplan     0.5000000 0.2107141  0.02003976
    #> 2:                        surv.coxph     0.5003190 0.2259178  0.03210873
    #> 3: surv.kaplan.surv.gbm.distrcompose     0.4811204 0.2774313  0.04296744