parfrailty fits shared frailty gamma-Weibull models. It is
specifically designed to work with the function standardize_parfrailty, which
performs regression standardization in shared frailty gamma-Weibull models.
Arguments
- formula
an object of class "
formula", in the same format as accepted by the coxph function.- data
a data frame containing the variables in the model.
- clusterid
a string containing the name of a cluster identification variable.
- init
an optional vector of initial values for the model parameters.
Value
An object of class "parfrailty" which is a list containing:
- est
the Maximum Likelihood (ML) estimates \(\{\log(\hat{\alpha}),\log(\hat{\eta}), \log(\hat{\phi}),\hat{\beta}\}\).
- vcov
the variance-covariance vector of the ML estimates.
- score
a matrix containing the cluster-specific contributions to the ML score equations.
Details
parfrailty fits the shared frailty gamma-Weibull model
$$\lambda(t_{ij}|C_{ij})=\lambda(t_{ij};\alpha,\eta)U_i\exp\{h(C_{ij};\beta)\},$$
where \(t_{ij}\) and \(C_{ij}\) are the survival time and covariate
vector for subject \(j\) in cluster \(i\), respectively.
\(\lambda(t;\alpha,\eta)\) is the Weibull baseline hazard function
$$\eta t^{\eta-1}\alpha^{-\eta},$$ where \(\eta\) is the shape
parameter and \(\alpha\) is the scale parameter. \(U_i\) is the
unobserved frailty term for cluster \(i\), which is assumed to have a
gamma distribution with scale = 1/shape = \(\phi\). \(h(X;\beta)\) is
the regression function as specified by the formula argument,
parameterized by a vector \(\beta\). The ML estimates
\(\{\log(\hat{\alpha}),\log(\hat{\eta}),\log(\hat{\phi}),\hat{\beta}\}\) are
obtained by maximizing the marginal (over \(U\)) likelihood.
Note
If left truncation is present, it is assumed that it is strong left truncation. This means that even if the truncation time may be subject-specific, the whole cluster is unobserved if at least one subject in the cluster dies before his/her truncation time. If all subjects in the cluster survive beyond their subject-specific truncation times, then the whole cluster is observed (Van den Berg and Drepper, 2016).
References
Dahlqwist E., Pawitan Y., Sjölander A. (2019). Regression standardization and attributable fraction estimation with between-within frailty models for clustered survival data. Statistical Methods in Medical Research 28(2), 462-485.
Van den Berg G.J., Drepper B. (2016). Inference for shared frailty survival models with left-truncated data. Econometric Reviews, 35(6), 1075-1098.
Examples
require(survival)
#> Loading required package: survival
# simulate data
set.seed(5)
n <- 200
m <- 3
alpha <- 1.5
eta <- 1
phi <- 0.5
beta <- 1
id <- rep(1:n, each = m)
U <- rep(rgamma(n, shape = 1 / phi, scale = phi), each = m)
X <- rnorm(n * m)
# reparameterize scale as in rweibull function
weibull.scale <- alpha / (U * exp(beta * X))^(1 / eta)
T <- rweibull(n * m, shape = eta, scale = weibull.scale)
# right censoring
C <- runif(n * m, 0, 10)
D <- as.numeric(T < C)
T <- pmin(T, C)
# strong left-truncation
L <- runif(n * m, 0, 2)
incl <- T > L
incl <- ave(x = incl, id, FUN = sum) == m
dd <- data.frame(L, T, D, X, id)
dd <- dd[incl, ]
fit <- parfrailty(formula = Surv(L, T, D) ~ X, data = dd, clusterid = "id")
print(fit)
#> $formula
#> Surv(L, T, D) ~ X
#> <environment: 0x63e9453a3990>
#>
#> $data
#> L T D X id
#> 4 1.78722376 3.03496785 1 0.634371955 2
#> 5 0.87593944 3.43397538 0 -0.231192902 2
#> 6 1.36879064 1.62048311 0 -1.368194204 2
#> 13 1.93077359 3.88018888 1 -0.869256130 5
#> 14 1.77243226 7.36242205 1 -1.332288341 5
#> 15 1.56056491 2.34032225 0 0.070562870 5
#> 19 1.22215738 9.84817895 0 -2.334691775 7
#> 20 1.21702364 5.95522302 0 -1.730891047 7
#> 21 1.49315787 2.32677008 1 0.825009586 7
#> 34 0.05522163 0.07623505 0 1.806258708 12
#> 35 0.46919068 5.65551975 0 -1.912519898 12
#> 36 0.60892023 0.64452417 0 0.199282075 12
#> 64 0.63330686 1.58382458 1 0.744600427 22
#> 65 1.19372559 8.92000775 0 -0.690190218 22
#> 66 1.57970257 2.76014646 1 -0.791411761 22
#> 67 1.81702192 3.16249173 1 -0.262081191 23
#> 68 0.48221653 7.21706324 0 -0.407917108 23
#> 69 0.18189881 0.28269222 1 0.201311056 23
#> 94 1.87380979 2.72313373 1 -0.454146942 32
#> 95 1.11637152 3.55521551 1 0.194874583 32
#> 96 0.24535298 3.02099790 1 0.031498073 32
#> 109 0.90027766 4.55355698 0 -1.515531972 37
#> 110 0.76864173 1.05942131 1 0.266538188 37
#> 111 0.05512638 2.86769961 1 0.291532260 37
#> 118 1.42239958 1.89974406 1 -0.394103521 40
#> 119 0.47835233 8.89930612 0 -1.510240230 40
#> 120 1.12377144 1.83634274 1 0.005989837 40
#> 127 0.69530126 8.08615804 0 -0.017375770 43
#> 128 1.82338834 1.94208484 0 0.541082912 43
#> 129 0.08598422 5.68351861 0 -0.540041112 43
#> 130 0.51413870 1.38826341 0 -0.080849396 44
#> 131 1.67780447 2.02832169 1 -0.029633958 44
#> 132 0.17971137 3.31347489 0 -0.145250484 44
#> 136 0.51457090 8.44363529 0 -1.154205453 46
#> 137 1.93562494 9.85561167 0 0.422597786 46
#> 138 1.77824194 7.14841753 0 -0.386989726 46
#> 139 1.87523751 5.23743428 0 -1.240525985 47
#> 140 1.81141152 3.71216593 0 0.456962145 47
#> 141 1.03691251 6.76273142 0 -1.410793353 47
#> 166 0.56609081 1.35006822 0 0.757933789 56
#> 167 0.33184205 0.37859500 1 0.359560263 56
#> 168 0.61275686 2.24092552 1 -0.829235675 56
#> 169 0.06145601 0.94865387 1 -0.445752122 57
#> 170 0.53187267 7.37740149 0 -0.670912047 57
#> 171 1.11085148 7.33086430 0 -0.800916902 57
#> 175 0.33909702 0.41699230 1 1.289449757 59
#> 176 1.97795961 5.06034775 1 -0.143912736 59
#> 177 0.17176757 0.67997250 0 -2.288008838 59
#> 250 0.10938821 3.22209051 0 -0.680978405 84
#> 251 0.50699914 6.26629895 0 -2.794959603 84
#> 252 1.25613561 3.98106755 0 -1.084586126 84
#> 259 1.52728552 6.91797197 0 -1.017287856 87
#> 260 1.25885725 1.91230933 1 1.014988922 87
#> 261 1.50238017 7.78931048 0 -0.651507685 87
#> 262 0.58556676 4.63435814 1 0.038215849 88
#> 263 0.96045975 2.80293191 1 0.184078667 88
#> 264 0.15020993 1.11106961 1 0.451722586 88
#> 274 0.18629962 0.90293922 0 -0.335845291 92
#> 275 1.92073994 2.92865786 1 -0.010529493 92
#> 276 0.32608739 1.05008773 1 -0.139135003 92
#> 286 0.73120809 2.17042107 1 0.126737533 96
#> 287 0.62003899 1.88088515 1 -0.485950257 96
#> 288 0.39135693 8.31694088 0 -1.003831691 96
#> 289 0.73954127 8.36045880 0 1.784203198 97
#> 290 0.70089152 1.28215405 1 -0.867144559 97
#> 291 0.54075855 8.57589681 0 -0.970574785 97
#> 298 0.27611811 0.82661051 1 -0.873557167 100
#> 299 0.15071073 0.49972060 1 0.111954035 100
#> 300 0.27641701 0.67741073 1 1.514231380 100
#> 316 0.92415468 5.11869150 0 0.082773252 106
#> 317 1.18738138 4.50767761 0 -1.585977289 106
#> 318 0.11445318 0.93428980 0 -1.529555425 106
#> 358 1.90755391 2.36375300 0 -0.628017639 120
#> 359 1.85712558 5.53994832 1 0.253333621 120
#> 360 1.67931147 2.57420883 0 -0.025188786 120
#> 394 0.71279878 1.32882429 1 0.460090646 132
#> 395 0.42105843 0.85011831 1 -0.218147958 132
#> 396 1.33021640 2.55882491 1 1.323011066 132
#> 403 1.20056311 5.05213969 0 -0.127865246 135
#> 404 1.10080783 8.38274394 0 -0.890580307 135
#> 405 0.26320789 1.16477906 1 0.896985420 135
#> 406 0.13875032 0.33482656 1 1.350929367 136
#> 407 1.88179018 6.89945541 0 -1.032052945 136
#> 408 0.57307729 1.69205188 1 0.513654664 136
#> 424 0.38881144 3.35369691 0 -0.966578842 142
#> 425 0.27245622 0.63939907 1 0.375867015 142
#> 426 1.36902177 6.59952559 0 -2.020810170 142
#> 427 0.19699762 1.46133286 1 0.244792596 143
#> 428 0.14960346 1.54345792 1 0.024357435 143
#> 429 0.43622702 6.56075123 0 -0.937410840 143
#> 445 0.57758966 4.79877953 0 -1.654511931 149
#> 446 1.86685356 8.30339929 0 -1.984728500 149
#> 447 0.47436808 6.92532206 1 -0.514325863 149
#> 448 1.77473551 3.51203199 0 -1.963281412 150
#> 449 1.85746704 4.18200700 0 0.303169082 150
#> 450 0.64277312 0.86599516 0 0.613763724 150
#> 469 0.16195196 0.44779638 1 0.354026642 157
#> 470 1.21458744 3.67393086 1 -0.764599960 157
#> 471 1.12302965 1.65112343 1 -0.509099170 157
#> 529 0.48063096 0.80881503 1 1.249536805 177
#> 530 1.28349557 3.27313504 0 0.496885618 177
#> 531 1.33671206 2.47686403 1 -0.309076753 177
#> 571 0.38581986 6.00508764 0 -0.913394120 191
#> 572 0.82877988 0.93946968 0 1.225612645 191
#> 573 1.85757261 4.85840462 0 -1.153224135 191
#> 589 0.70452328 1.60237702 0 -1.441137171 197
#> 590 0.22542309 0.72176056 1 1.638578200 197
#> 591 1.73939423 3.62888657 0 -1.923365406 197
#> 598 0.62471978 1.17165533 0 -0.453984276 200
#> 599 1.52603893 2.83751867 0 -0.097612023 200
#> 600 0.21336639 2.14748748 1 -0.337073721 200
#>
#> $clusterid
#> [1] "id"
#>
#> $ncluster
#> [1] 37
#>
#> $n
#> [1] 111
#>
#> $X
#> X
#> 4 0.634371955
#> 5 -0.231192902
#> 6 -1.368194204
#> 13 -0.869256130
#> 14 -1.332288341
#> 15 0.070562870
#> 19 -2.334691775
#> 20 -1.730891047
#> 21 0.825009586
#> 34 1.806258708
#> 35 -1.912519898
#> 36 0.199282075
#> 64 0.744600427
#> 65 -0.690190218
#> 66 -0.791411761
#> 67 -0.262081191
#> 68 -0.407917108
#> 69 0.201311056
#> 94 -0.454146942
#> 95 0.194874583
#> 96 0.031498073
#> 109 -1.515531972
#> 110 0.266538188
#> 111 0.291532260
#> 118 -0.394103521
#> 119 -1.510240230
#> 120 0.005989837
#> 127 -0.017375770
#> 128 0.541082912
#> 129 -0.540041112
#> 130 -0.080849396
#> 131 -0.029633958
#> 132 -0.145250484
#> 136 -1.154205453
#> 137 0.422597786
#> 138 -0.386989726
#> 139 -1.240525985
#> 140 0.456962145
#> 141 -1.410793353
#> 166 0.757933789
#> 167 0.359560263
#> 168 -0.829235675
#> 169 -0.445752122
#> 170 -0.670912047
#> 171 -0.800916902
#> 175 1.289449757
#> 176 -0.143912736
#> 177 -2.288008838
#> 250 -0.680978405
#> 251 -2.794959603
#> 252 -1.084586126
#> 259 -1.017287856
#> 260 1.014988922
#> 261 -0.651507685
#> 262 0.038215849
#> 263 0.184078667
#> 264 0.451722586
#> 274 -0.335845291
#> 275 -0.010529493
#> 276 -0.139135003
#> 286 0.126737533
#> 287 -0.485950257
#> 288 -1.003831691
#> 289 1.784203198
#> 290 -0.867144559
#> 291 -0.970574785
#> 298 -0.873557167
#> 299 0.111954035
#> 300 1.514231380
#> 316 0.082773252
#> 317 -1.585977289
#> 318 -1.529555425
#> 358 -0.628017639
#> 359 0.253333621
#> 360 -0.025188786
#> 394 0.460090646
#> 395 -0.218147958
#> 396 1.323011066
#> 403 -0.127865246
#> 404 -0.890580307
#> 405 0.896985420
#> 406 1.350929367
#> 407 -1.032052945
#> 408 0.513654664
#> 424 -0.966578842
#> 425 0.375867015
#> 426 -2.020810170
#> 427 0.244792596
#> 428 0.024357435
#> 429 -0.937410840
#> 445 -1.654511931
#> 446 -1.984728500
#> 447 -0.514325863
#> 448 -1.963281412
#> 449 0.303169082
#> 450 0.613763724
#> 469 0.354026642
#> 470 -0.764599960
#> 471 -0.509099170
#> 529 1.249536805
#> 530 0.496885618
#> 531 -0.309076753
#> 571 -0.913394120
#> 572 1.225612645
#> 573 -1.153224135
#> 589 -1.441137171
#> 590 1.638578200
#> 591 -1.923365406
#> 598 -0.453984276
#> 599 -0.097612023
#> 600 -0.337073721
#>
#> $fit
#> $fit$par
#> [1] 0.3746027 -0.2732453 -1.0648873 1.2779566
#>
#> $fit$value
#> [1] -111.5306
#>
#> $fit$counts
#> function gradient
#> 45 16
#>
#> $fit$convergence
#> [1] 0
#>
#> $fit$message
#> NULL
#>
#>
#> $est
#> log(α) log(η) log(ϕ) X
#> 0.3746027 -0.2732453 -1.0648873 1.2779566
#>
#> $score
#> log(α) log(η) log(ϕ) X
#> [1,] 0.013953486 0.20289224 -0.11965534 0.37373130
#> [2,] -0.497884032 2.52001507 -0.46772296 -1.42384130
#> [3,] -0.069007646 0.27713823 -0.23097814 0.66557460
#> [4,] 0.137859989 -0.14641165 0.11013129 0.08105783
#> [5,] -0.093520015 0.33405218 -0.30480286 -0.05516350
#> [6,] -0.146292463 -1.03889812 -0.30025772 0.56123686
#> [7,] -0.290499183 2.16086109 -0.47596378 -0.38636190
#> [8,] 0.011099791 0.79398549 -0.21691275 0.29200148
#> [9,] -0.535543940 1.18487038 -0.42506246 0.44181409
#> [10,] 0.472335268 -1.88851477 1.03243696 0.46707458
#> [11,] 0.095094051 0.20316653 -0.06270845 0.11113571
#> [12,] 0.608980969 -2.91764013 1.55679132 -0.03715771
#> [13,] 0.347705224 -1.22974595 0.54900599 0.16346613
#> [14,] -0.367475159 0.77234663 -0.45571856 -1.15674320
#> [15,] 0.425117806 -1.67705538 0.05311604 0.62175200
#> [16,] -0.398640488 1.06649391 -0.57673102 0.65703311
#> [17,] 0.511627932 -0.79298325 0.18290053 0.62054930
#> [18,] 0.053590571 -0.51220966 -0.03847510 1.12121813
#> [19,] -0.079267003 1.46932616 -0.33441943 0.13383099
#> [20,] -0.486755926 1.62765783 -0.44031649 0.01347658
#> [21,] -0.137988012 0.44503587 -0.27985220 0.45230331
#> [22,] 0.497364950 -4.91200778 2.74404636 -3.73290563
#> [23,] -0.495455064 1.69041058 -0.59856851 -2.12530681
#> [24,] 0.644289075 -1.65400943 0.56131700 0.08090285
#> [25,] 0.062392090 0.16838772 -0.04211771 0.04566587
#> [26,] -0.282676261 1.14737959 -0.90297310 -1.35697090
#> [27,] 0.387083643 -1.36082538 0.33269226 0.72348819
#> [28,] -0.139397789 0.18403258 -0.34557360 0.97437992
#> [29,] -0.077711350 -0.26019634 -0.16042625 0.90063440
#> [30,] 0.111303122 0.43070182 -0.18646515 0.65027345
#> [31,] 0.297062945 -0.01523354 -0.05276062 0.60089034
#> [32,] 0.258589968 -0.99229482 0.45701684 -0.06569007
#> [33,] -1.050040656 2.42648364 -0.34545708 -0.40334180
#> [34,] -0.155333955 0.63706623 -0.49576788 -0.24783674
#> [35,] 0.180210992 -0.77083636 0.34050852 0.53482735
#> [36,] 0.191151851 0.01906961 0.08256861 -0.21955355
#> [37,] -0.003373936 0.40771967 -0.14288170 -0.07758479
#>
#> $vcov
#> log(α) log(η) log(ϕ) X
#> log(α) 0.38182395 0.09552250 -0.02123094 0.01369009
#> log(η) 0.09552250 0.08284338 0.11211763 0.04021393
#> log(ϕ) -0.02123094 0.11211763 0.32410342 0.07514008
#> X 0.01369009 0.04021393 0.07514008 0.07210706
#>
#> $call
#> parfrailty(formula = Surv(L, T, D) ~ X, data = dd, clusterid = "id")
#>
#> attr(,"class")
#> [1] "parfrailty"