Helper function to compose a survival distribution (or cumulative hazard)
from the relative risk predictions (linear predictors, lp) of a
proportional hazards model (e.g. a Cox-type model).
Arguments
- times
(
numeric())
Vector of times (train set).- status
(
numeric())
Vector of status indicators (train set). For each observation in the train set, this should be 0 (alive/censored) or 1 (dead).- lp_train
(
numeric())
Vector of linear predictors (train set). These are the relative score predictions (\(lp = \hat{\beta}X_{train}\)) from a proportional hazards model on the train set.- lp_test
(
numeric())
Vector of linear predictors (test set). These are the relative score predictions (\(lp = \hat{\beta}X_{test}\)) from a proportional hazards model on the test set.- eval_times
(
numeric())
Vector of times to compute survival probabilities. IfNULL(default), the unique and sortedtimesfrom the train set will be used, otherwise the unique and sortedeval_times.- type
(
character())
Type of prediction estimates. Default issurvwhich returns the survival probabilities \(S_i(t)\) for each test observation \(i\). Ifcumhaz, the function returns the estimated cumulative hazards \(H_i(t)\).
Value
a matrix (obs x times). Number of columns is equal to eval_times
and number of rows is equal to the number of test observations (i.e. the
length of the lp_test vector). Depending on the type argument, the matrix
can have either survival probabilities (0-1) or cumulative hazard estimates
(0-Inf).
Details
We estimate the survival probability of individual \(i\) (from the test set), at time point \(t\) as follows: $$S_i(t) = e^{-H_i(t)} = e^{-\hat{H}_0(t) \times e^{lp_i}}$$
where:
\(H_i(t)\) is the cumulative hazard function for individual \(i\)
\(\hat{H}_0(t)\) is Breslow's estimator for the cumulative baseline hazard. Estimation requires the training set's
timesandstatusas well the risk predictions (lp_train).\(lp_i\) is the risk prediction (linear predictor) of individual \(i\) on the test set.
Breslow's approach uses a non-parametric maximum likelihood estimation of the cumulative baseline hazard function:
$$\hat{H}_0(t) = \sum_{i=1}^n{\frac{I(T_i \le t)\delta_i} {\sum\nolimits_{j \in R_i}e^{lp_j}}}$$
where:
\(t\) is the vector of time points (unique and sorted, from the train set)
\(n\) is number of events (train set)
\(T\) is the vector of event times (train set)
\(\delta\) is the status indicator (1 = event or 0 = censored)
\(R_i\) is the risk set (number of individuals at risk just before event \(i\))
\(lp_j\) is the risk prediction (linear predictor) of individual \(j\) (who is part of the risk set \(R_i\)) on the train set.
We employ constant interpolation to estimate the cumulative baseline hazards,
extending from the observed unique event times to the specified evaluation
times (eval_times).
Any values falling outside the range of the estimated times are assigned as
follows:
$$\hat{H}_0(eval\_times < min(t)) = 0$$ and
$$\hat{H}_0(eval\_times > max(t)) = \hat{H}_0(max(t))$$
Note that in the rare event of lp predictions being Inf or -Inf, the
resulting cumulative hazard values become NaN, which we substitute with
Inf (and corresponding survival probabilities take the value of \(0\)).
For similar implementations, see gbm::basehaz.gbm(), C060::basesurv() and
xgboost.surv::sgb_bhaz().
References
Breslow N (1972). “Discussion of 'Regression Models and Life-Tables' by D.R. Cox.” Journal of the Royal Statistical Society: Series B, 34(2), 216-217.
Lin, Y. D (2007). “On the Breslow estimator.” Lifetime Data Analysis, 13(4), 471-480. doi:10.1007/s10985-007-9048-y .
Examples
task = tsk("rats")
part = partition(task, ratio = 0.8)
learner = lrn("surv.coxph")
learner$train(task, part$train)
p_train = learner$predict(task, part$train)
p_test = learner$predict(task, part$test)
surv = breslow(times = task$times(part$train), status = task$status(part$train),
lp_train = p_train$lp, lp_test = p_test$lp)
head(surv)
#> 23 32 39 40 45 49 51 53
#> [1,] 1 1 0.9942569 0.9884994 0.9827474 0.9827474 0.9827474 0.9827474
#> [2,] 1 1 0.9972328 0.9944503 0.9916620 0.9916620 0.9916620 0.9916620
#> [3,] 1 1 0.9971625 0.9943096 0.9914509 0.9914509 0.9914509 0.9914509
#> [4,] 1 1 0.9971625 0.9943096 0.9914509 0.9914509 0.9914509 0.9914509
#> [5,] 1 1 0.9969409 0.9938657 0.9907852 0.9907852 0.9907852 0.9907852
#> [6,] 1 1 0.9996131 0.9992232 0.9988315 0.9988315 0.9988315 0.9988315
#> 54 55 61 62 63 64 65
#> [1,] 0.9768072 0.9708613 0.9708613 0.9708613 0.9708613 0.9647664 0.9647664
#> [2,] 0.9887737 0.9858733 0.9858733 0.9858733 0.9858733 0.9828908 0.9828908
#> [3,] 0.9884899 0.9855167 0.9855167 0.9855167 0.9855167 0.9824596 0.9824596
#> [4,] 0.9884899 0.9855167 0.9855167 0.9855167 0.9855167 0.9824596 0.9824596
#> [5,] 0.9875950 0.9843925 0.9843925 0.9843925 0.9843925 0.9811003 0.9811003
#> [6,] 0.9984248 0.9980153 0.9980153 0.9980153 0.9980153 0.9975932 0.9975932
#> 66 67 68 69 70 71 72
#> [1,] 0.9586665 0.9525531 0.9464029 0.9464029 0.9402276 0.9340419 0.9278944
#> [2,] 0.9798960 0.9768846 0.9738450 0.9738450 0.9707826 0.9677046 0.9646351
#> [3,] 0.9793901 0.9763039 0.9731889 0.9731889 0.9700509 0.9668972 0.9637524
#> [4,] 0.9793901 0.9763039 0.9731889 0.9731889 0.9700509 0.9668972 0.9637524
#> [5,] 0.9777957 0.9744738 0.9711219 0.9711219 0.9677460 0.9643540 0.9609726
#> [6,] 0.9971682 0.9967397 0.9963061 0.9963061 0.9958680 0.9954265 0.9949851
#> 73 74 75 76 77 78 79
#> [1,] 0.9154402 0.9154402 0.9089700 0.9089700 0.9022524 0.8953582 0.8884508
#> [2,] 0.9583841 0.9583841 0.9551191 0.9551191 0.9517166 0.9482109 0.9446843
#> [3,] 0.9573488 0.9573488 0.9540046 0.9540046 0.9505198 0.9469296 0.9433185
#> [4,] 0.9573488 0.9573488 0.9540046 0.9540046 0.9505198 0.9469296 0.9433185
#> [5,] 0.9540897 0.9540897 0.9504966 0.9504966 0.9467535 0.9428984 0.9390219
#> [6,] 0.9940822 0.9940822 0.9936086 0.9936086 0.9931136 0.9926020 0.9920857
#> 80 81 82 83 84 85 86
#> [1,] 0.8742574 0.8669508 0.8669508 0.8669508 0.8516968 0.8516968 0.8440066
#> [2,] 0.9373931 0.9336158 0.9336158 0.9336158 0.9256761 0.9256761 0.9216453
#> [3,] 0.9358534 0.9319866 0.9319866 0.9319866 0.9238601 0.9238601 0.9197352
#> [4,] 0.9358534 0.9319866 0.9319866 0.9319866 0.9238601 0.9238601 0.9197352
#> [5,] 0.9310119 0.9268648 0.9268648 0.9268648 0.9181537 0.9181537 0.9137343
#> [6,] 0.9910130 0.9904544 0.9904544 0.9904544 0.9892739 0.9892739 0.9886713
#> 87 88 89 90 91 92 93
#> [1,] 0.8440066 0.8360258 0.8280218 0.8280218 0.8280218 0.8191580 0.8191580
#> [2,] 0.9216453 0.9174421 0.9132057 0.9132057 0.9132057 0.9084894 0.9084894
#> [3,] 0.9197352 0.9154343 0.9110999 0.9110999 0.9110999 0.9062751 0.9062751
#> [4,] 0.9197352 0.9154343 0.9110999 0.9110999 0.9110999 0.9062751 0.9062751
#> [5,] 0.9137343 0.9091280 0.9044876 0.9044876 0.9044876 0.8993242 0.8993242
#> [6,] 0.9886713 0.9880405 0.9874021 0.9874021 0.9874021 0.9866885 0.9866885
#> 94 95 96 97 98 99 100
#> [1,] 0.8098601 0.8098601 0.7901883 0.7901883 0.7901883 0.7901883 0.7901883
#> [2,] 0.9035135 0.9035135 0.8928872 0.8928872 0.8928872 0.8928872 0.8928872
#> [3,] 0.9011855 0.9011855 0.8903188 0.8903188 0.8903188 0.8903188 0.8903188
#> [4,] 0.9011855 0.9011855 0.8903188 0.8903188 0.8903188 0.8903188 0.8903188
#> [5,] 0.8938797 0.8938797 0.8822633 0.8822633 0.8822633 0.8822633 0.8822633
#> [6,] 0.9859321 0.9859321 0.9843047 0.9843047 0.9843047 0.9843047 0.9843047
#> 101 102 103 104
#> [1,] 0.7798095 0.7591196 0.7270081 0.7153434
#> [2,] 0.8872254 0.8758210 0.8577967 0.8511473
#> [3,] 0.8845302 0.8728733 0.8544577 0.8476664
#> [4,] 0.8845302 0.8728733 0.8544577 0.8476664
#> [5,] 0.8760798 0.8636375 0.8440075 0.8367767
#> [6,] 0.9834308 0.9816558 0.9788096 0.9777466