Using doubly-robust gradient boosting to generate estimates of the prediction function that maximizes the C-index
Usage
boost_c_index(
time,
event,
X,
newX,
S_hat,
G_hat,
V,
approx_times,
tuning,
produce_fit = TRUE,
subsample_n,
boosting_params
)Arguments
- time
n x 1numeric vector of observed follow-up times. If there is censoring, these are the minimum of the event and censoring times.- event
n x 1numeric vector of status indicators of whether an event was observed.- X
n x pdata.frame of observed covariate values- newX
m x pdata.frame of new observed covariate values at which to obtainmpredictions for the estimated algorithm. Must have the same names and structure asX.- S_hat
n x J2matrix of conditional event time survival function estimates- G_hat
n x J2matrix of conditional censoring time survival function estimates- V
Number of cross-validation folds for selection of tuning parameters
- approx_times
Numeric vector of length J2 giving times at which to approximate C-index integral. Note that the last time in
approx_timesis taken to be the restriction time (i.e., the maximum follow-up) for comparison of pairs of individuals. Essentially, this time should be chosen such that the conditional survival function is identified at this time for all covariate valuesXpresent in the data. Choosing the restriction time such that roughly 10% of individuals remain at-risk at that time has been shown to work reasonably well in simulations.- tuning
Logical, whether or not to use cross-validation to select tuning parameters
- produce_fit
Logical, whether to produce a fitted prediction function using the selected optimal parameters.
- subsample_n
Number of samples to use for boosting procedure. Using a subsample of the full sample can greatly reduce runtime
- boosting_params
Named list of parameter values for the boosting procedure. Elements of this list include
mstop(number of boosting iterations),nu(learning rate),sigma(smoothness parameter for sigmoid approximation, with smaller meaning less smoothing), andlearner(base learner, can take values"glm","gam", or"tree")
Examples
# This is a small simulation example
set.seed(123)
n <- 250
X <- data.frame(X1 = rnorm(n), X2 = rbinom(n, size = 1, prob = 0.5))
T <- rexp(n, rate = exp(-2 + X[,1] - X[,2] + .5 * X[,1] * X[,2]))
C <- rexp(n, exp(-2 -.5 * X[,1] - .25 * X[,2] + .5 * X[,1] * X[,2]))
C[C > 15] <- 15
time <- pmin(T, C)
event <- as.numeric(T <= C)
# Note that this a very small Super Learner library, for computational purposes.
SL.library <- c("SL.mean", "SL.glm")
# Note that we do not use times beyond the 90th percentile of observed follow-up times
approx_times <- c(0, unique(quantile(time, probs = seq(0, 0.9, by = 0.01))))
# estimate conditional survival functions at approx_times
fit <- stackG(time = time,
event = event,
X = X,
newX = X,
newtimes = approx_times,
direction = "prospective",
bin_size = 0.1,
time_basis = "continuous",
surv_form = "PI",
learner = "SuperLearner",
time_grid_approx = approx_times,
SL_control = list(SL.library = SL.library,
V = 3))
# use boosting to estimate optimal (according to C-index) prediction function
boosted_preds <- boost_c_index(time = time,
event = event,
X = X,
newX = X,
approx_times = approx_times,
S_hat = fit$S_T_preds,
G_hat = fit$S_C_preds,
V = 3,
tuning = TRUE,
produce_fit = TRUE,
subsample_n = 200,
boosting_params = list(mstop = c(100, 200),
nu = 0.1,
sigma = 0.1,
learner = "glm"))
boosted_preds
#> 1 2 3 4 5
#> -0.0686025275 -0.0322597503 0.1645717386 0.0008247817 0.0072922855
#> 6 7 8 9 10
#> 0.1817756968 0.0437814373 -0.1461282309 -0.0825078385 -0.0559695540
#> 11 12 13 14 15
#> 0.1277527111 0.0326571208 0.0371636969 0.0052451712 -0.0680925911
#> 16 17 18 19 20
#> 0.1896811642 0.0478453232 -0.2233205796 0.0702370686 -0.0589546118
#> 21 22 23 24 25
#> -0.1244261433 -0.0309170984 -0.1198247614 -0.0871333302 -0.0757064766
#> 26 27 28 29 30
#> -0.1925204988 0.0852486165 0.0099424053 -0.1321627232 0.1310242531
#> 31 32 33 34 35
#> 0.0399906780 -0.0394000504 0.0915575968 0.0896879443 0.0834654710
#> 36 37 38 39 40
#> 0.0688379632 0.0540144065 -0.0137454306 -0.0405984093 -0.0487965777
#> 41 42 43 44 45
#> -0.0833720298 -0.0298104545 -0.1461651039 0.2317174186 0.1259790461
#> 46 47 48 49 50
#> -0.1305091502 -0.0512627767 -0.0582794605 0.0788864576 -0.0161063881
#> 51 52 53 54 55
#> 0.0209394187 -0.0100742774 -0.0116503174 0.1436543313 -0.0317749018
#> 56 57 58 59 60
#> 0.1599243143 -0.1773428752 0.0573919046 0.0066944373 0.0168268254
#> 61 62 63 64 65
#> 0.0348385420 -0.0622040291 -0.0435961517 -0.1190073398 -0.1248626905
#> 66 67 68 69 70
#> 0.0264640498 0.0423833464 -0.0011012053 0.0945440156 0.2186379862
#> 71 72 73 74 75
#> -0.0609615364 -0.2610116188 0.1037283539 -0.0849667839 -0.0826350075
#> 76 77 78 79 80
#> 0.1059105661 -0.0382669069 -0.1412491036 0.0130155890 -0.0222155182
#> 81 82 83 84 85
#> -0.0062990372 0.0354592134 -0.0477170746 0.0639676182 -0.0311934555
#> 86 87 88 89 90
#> 0.0295727689 0.1137521602 0.0409498411 -0.0427955941 0.1194702770
#> 91 92 93 94 95
#> 0.1023821707 0.0534069634 0.0193344323 -0.0760219121 0.1427796092
#> 96 97 98 99 100
#> -0.0729799630 0.2337394469 0.1617002043 -0.0328674329 -0.1198705837
#> 101 102 103 104 105
#> -0.0850994584 0.0213316979 -0.0340768321 -0.0451734586 -0.1116400670
#> 106 107 108 109 110
#> -0.0118876820 -0.0932964790 -0.1904572810 -0.0487696774 0.0941841224
#> 111 112 113 114 115
#> -0.0702388216 0.0599611731 -0.1849492534 -0.0130467668 0.0502172143
#> 116 117 118 119 120
#> 0.0262026973 0.0046943025 -0.0774302914 -0.1004264233 -0.1196183824
#> 121 122 123 124 125
#> 0.0060114085 -0.1111841069 -0.0609094126 -0.0351111506 0.1959472604
#> 126 127 128 129 130
#> -0.0786674594 0.0189663633 0.0016447772 -0.1127665637 -0.0147793158
#> 131 132 133 134 135
#> 0.1520109692 0.0427458161 -0.0023964045 -0.0534206889 -0.2328525042
#> 136 137 138 139 140
#> 0.1175480046 -0.1676478123 0.0744833117 0.2031258037 -0.1658051462
#> 141 142 143 144 145
#> 0.0702842093 -0.0357829178 -0.1799166310 -0.1735924788 -0.1831506407
#> 146 147 148 149 150
#> -0.0653490304 -0.1677705525 0.0687583584 0.2241421657 -0.1485455113
#> 151 152 153 154 155
#> 0.0797418027 0.0776846110 0.0296190493 -0.1178851664 -0.0200766677
#> 156 157 158 159 160
#> -0.0377852307 0.0550125874 -0.0479127880 0.1005633196 -0.0481484840
#> 161 162 163 164 165
#> 0.1088967967 -0.1223744430 -0.1455884238 0.3496790479 -0.0528002017
#> 166 167 168 169 170
#> 0.0258807741 0.0631086261 -0.0601637579 0.0499371699 0.0336639744
#> 171 172 173 174 175
#> -0.0306316353 0.0002509347 -0.0106816990 0.2272607475 -0.0885026416
#> 176 177 178 179 180
#> -0.1275259763 -0.0027754061 0.0272289917 0.0410975003 -0.0573673077
#> 181 182 183 184 185
#> -0.1239312745 0.1320552643 -0.0454053790 -0.1021658378 -0.0329311635
#> 186 187 188 189 190
#> -0.0286285778 0.1151914959 0.0023903907 0.0760354280 -0.0618704796
#> 191 192 193 194 195
#> 0.0166621917 -0.0426585323 0.0034737741 -0.1054502942 -0.1511610458
#> 196 197 198 199 200
#> 0.2128205456 0.0591628490 -0.1446109282 -0.0741799888 -0.1373718999
#> 201 202 203 204 205
#> 0.2350023028 0.1374718097 -0.0361072393 0.0528344871 -0.0525231852
#> 206 207 208 209 210
#> -0.0593348198 -0.0937034113 -0.0723591373 0.1747164312 -0.0128779980
#> 211 212 213 214 215
#> 0.0061873072 0.0198797084 0.1286763136 -0.0637158859 -0.1161390465
#> 216 217 218 219 220
#> 0.1774440213 -0.0554745552 -0.0864923756 -0.1429606684 -0.1482908184
#> 221 222 223 224 225
#> -0.0700876969 0.0610638403 0.1151835571 0.0709228267 -0.0469465622
#> 226 227 228 229 230
#> -0.0003589733 -0.0844601719 -0.0858489400 0.0904050125 -0.1186791412
#> 231 232 233 234 235
#> 0.2082081429 -0.0168711561 0.0166724814 -0.0881936336 -0.0701333826
#> 236 237 238 239 240
#> -0.1518448395 -0.0270605915 0.0391674522 0.0287500048 -0.0929258992
#> 241 242 243 244 245
#> -0.0937055165 -0.0621903045 0.1576786048 -0.1320710332 -0.0266343571
#> 246 247 248 249 250
#> 0.2023840077 -0.0180435601 -0.1565568363 -0.0800779954 0.0464819939