In this vignette we provide details on transforming data into a format suitable to fit piece-wise exponential (additive) models (PAM). Three main cases need to be distinguished
Data without time-dependent covariates
In this simple case the data transformation is relatively straight forward and handled by the as_ped (as piece-wise exponential data) function. This function internally calls survival::survSplit, thus all arguments available for the survSplit function are also available to the as_ped function.
Using defaults
As an example data set we first consider Veterans’ Administration lung cancer study (Kalbfleisch and Prentice 1980) available from the survival package:
data("veteran", package = "survival") veteran <- veteran %>% mutate( id = row_number(), trt = 1L * (trt == 2), prior = 1L * (prior != 0)) %>% select(id, everything()) head(veteran)
## id trt celltype time status karno diagtime age prior
## 1 1 0 squamous 72 1 60 7 69 0
## 2 2 0 squamous 411 1 70 5 64 1
## 3 3 0 squamous 228 1 60 3 38 0
## 4 4 0 squamous 126 1 60 9 63 1
## 5 5 0 squamous 118 1 70 11 65 1
## 6 6 0 squamous 10 1 20 5 49 0Each row contains information on the survival time (time), an event indicator (status) and the treatment information (6-MP vs. placebo) as the only covariate.
To transform the data into piece-wise exponential data (ped) format, we need to
define \(J+1\) interval break points \(t_{\min} = \kappa_0 < \kappa_1 < \cdots < \kappa_J = t_{\max}\)
create pseudo-observations for each interval \(j = 1,\ldots, J\) in which subject \(i, i = 1,\ldots,n\) was under risk.
Using the pammtools package this is easily achieved with the as_ped function as follows:
ped <- as_ped(Surv(time, status) ~ ., data = veteran, id = "id") ped %>% filter(id %in% c(42, 85, 112)) %>% select(-diagtime, -prior)
## id tstart tend interval offset ped_status trt celltype karno age
## 1 42 0 1 (0,1] 0.0000000 0 0 smallcell 50 72
## 2 42 1 2 (1,2] 0.0000000 0 0 smallcell 50 72
## 3 42 2 3 (2,3] 0.0000000 0 0 smallcell 50 72
## 4 42 3 4 (3,4] 0.0000000 0 0 smallcell 50 72
## 5 42 4 7 (4,7] 1.0986123 1 0 smallcell 50 72
## 6 85 0 1 (0,1] 0.0000000 1 1 squamous 50 35
## 7 112 0 1 (0,1] 0.0000000 0 1 adeno 60 62
## 8 112 1 2 (1,2] 0.0000000 0 1 adeno 60 62
## 9 112 2 3 (2,3] 0.0000000 0 1 adeno 60 62
## 10 112 3 4 (3,4] 0.0000000 0 1 adeno 60 62
## 11 112 4 7 (4,7] 1.0986123 0 1 adeno 60 62
## 12 112 7 8 (7,8] 0.0000000 0 1 adeno 60 62
## 13 112 8 10 (8,10] 0.6931472 0 1 adeno 60 62
## 14 112 10 11 (10,11] 0.0000000 0 1 adeno 60 62
## 15 112 11 12 (11,12] 0.0000000 0 1 adeno 60 62
## 16 112 12 13 (12,13] 0.0000000 0 1 adeno 60 62
## 17 112 13 15 (13,15] 0.6931472 0 1 adeno 60 62
## 18 112 15 16 (15,16] 0.0000000 0 1 adeno 60 62
## 19 112 16 18 (16,18] 0.6931472 0 1 adeno 60 62
## 20 112 18 19 (18,19] 0.0000000 0 1 adeno 60 62
## 21 112 19 20 (19,20] 0.0000000 0 1 adeno 60 62
## 22 112 20 21 (20,21] 0.0000000 0 1 adeno 60 62
## 23 112 21 22 (21,22] 0.0000000 0 1 adeno 60 62
## 24 112 22 24 (22,24] 0.6931472 0 1 adeno 60 62
## 25 112 24 25 (24,25] 0.0000000 0 1 adeno 60 62
## 26 112 25 27 (25,27] 0.6931472 0 1 adeno 60 62
## 27 112 27 29 (27,29] 0.6931472 0 1 adeno 60 62
## 28 112 29 30 (29,30] 0.0000000 0 1 adeno 60 62
## 29 112 30 31 (30,31] 0.0000000 0 1 adeno 60 62
## 30 112 31 33 (31,33] 0.6931472 0 1 adeno 60 62
## 31 112 33 35 (33,35] 0.6931472 0 1 adeno 60 62
## 32 112 35 36 (35,36] 0.0000000 0 1 adeno 60 62
## 33 112 36 42 (36,42] 1.7917595 0 1 adeno 60 62
## 34 112 42 43 (42,43] 0.0000000 0 1 adeno 60 62
## 35 112 43 44 (43,44] 0.0000000 0 1 adeno 60 62
## 36 112 44 45 (44,45] 0.0000000 0 1 adeno 60 62
## 37 112 45 48 (45,48] 1.0986123 0 1 adeno 60 62
## 38 112 48 49 (48,49] 0.0000000 0 1 adeno 60 62
## 39 112 49 51 (49,51] 0.6931472 1 1 adeno 60 62When no cut points are specified, the default is to use the unique event times. As can be seen from the above output, the function creates an id variable, indicating the subjects \(i = 1,\ldots,n\), with one row per id for each interval the subject “visited”. Thus,
- subject
42was under risk during 5 intervals, - subject
85was under risk during one interval, - and subject
112in 33 intervals.
In addition to the optional id variable the function also creates
-
tstart: the beginning of each interval -
tend: the end of each interval -
interval: a factor variable denoting the interval -
offset: the log of the duration during which the subject was under risk in that interval
Additionally the time-constant covariates (here: trt, celltype, karno, etc.) are repeated \(n_i\) number of times, where \(n_i\) is the number of intervals during which subject \(i\) was in the risk set.
Custom cut points
Per default, the as_ped function uses all unique event times as cut points (which is a sensible default as for example the (cumulative) baseline hazard estimates in Cox-PH functions are also updated at event times).
In some cases, however, one might want to reduce the number of cut points, to reduce the size of the resulting data and/or faster estimation times. To do so the cut argument has to be specified. Following the above example, we can use only 4 cut points \(\kappa_0 = 0, \kappa_1 = 50, \kappa_2 = 200, \kappa_3 = 400\), resulting in \(J = 3\) intervals:
ped2 <- as_ped(Surv(time, status) ~ ., data = veteran, cut = c(0, 50, 200, 400), id = "id") ped2 %>% filter(id %in% c(42, 85, 112)) %>% select(-diagtime, -prior)
## id tstart tend interval offset ped_status trt celltype karno age
## 1 42 0 50 (0,50] 1.945910 1 0 smallcell 50 72
## 2 85 0 50 (0,50] 0.000000 1 1 squamous 50 35
## 3 112 0 50 (0,50] 3.912023 0 1 adeno 60 62
## 4 112 50 200 (50,200] 0.000000 1 1 adeno 60 62Note that now subjects 42 and 85 have only one row in the data set. The fact that subject 42 was in the risk set longer than subject 85 is, however, still accounted for by the offset variable. Note that the offsets for subject 85 and subject 112 in interval (50,200] are only zero because subject 85 had an observed event time of 1, thus log(1-0)=0 and subject 112 had an event at time 51, thus log(51-50)=0:
## id trt celltype time status karno diagtime age prior
## 1 85 1 squamous 1 1 50 7 35 0
## 2 112 1 adeno 51 1 60 5 62 0Data with time-dependent covariates
In case of data with time-dependent covariates (that should not be modeled as cumulative effects), the follow-up is usually split at desired cut-points and additionally at the time-points at which the TDC changes its value.
We assume that the data is provided in two data sets, one that contains time-to-event data and time-constant covariates and one data set that contains information of time-dependent covariates.
For illustration, we use the pbc data from the survival package. Before the actual data transformation we perform the data preprocessing in vignette("timedep", package="survival"):
data("pbc", package = "survival") # loads pbc and pbcseq pbc <- pbc %>% filter(id <= 312) %>% mutate(status = 1L * (status == 2)) %>% select(id:sex, bili, protime)
We want to transform the data such that we could fit a concurrent effect of the time-dependent covariates bili and protime, therefore, the RHS of the formula passed to as_ped needs to contain an additional component, separated by | and all variables for which should be transformed to the concurrent format specified within the concurrent special. The data sets are provided within a list. In concurrent, we also have to specify the name of the variable which stores information on the time points at which the TDCs are updated.
ped_pbc <- as_ped( data = list(pbc, pbcseq), formula = Surv(time, status) ~ . + concurrent(bili, protime, tz_var = "day"), id = "id")
Multiple data sets can be provided within the list and multiple concurrent terms with different tz_var arguments can be specified with different lags. This will increase the data set, as an (additional) split point will be generated for each (lagged) time-point at which the TDC was observed:
pbc_ped <- as_ped(data = list(pbc, pbcseq), formula = Surv(time, status) ~ . + concurrent(bili, tz_var = "day") + concurrent(protime, tz_var = "day", lag = 10), id = "id")
If different tz_var arguments are provided, the union of both will be used as cut points in addition to usual cut points.
Warning: For time-points on the follow up that have no observation of the TDC, the last observation will be carried forward until the next update is available.
Data with time-dependent covariates (with cumulative effects)
Here we demonstrate data transformation of data with TDCs (\(\mathbf{z}=\{z(t_{z,q}),q=1,\ldots,Q\}\), that subsequently will be modeled as cumulative effects defined as
\[ g(\mathbf{z}, t) = \int_{\mathcal{T}(t)} h(t, t_z, z(t_z))\mathrm{d}t_z \] Here
- the three-variate function \(h(t,t_z,z(t_z))\) defines the so-called partial effects of the TDC \(z(t_z)\) observed at exposure time \(t_z\) on the hazard at time \(t\) (Bender et al. 2018). The general partial effect definition given above is only one possibility, other common choices are
-
\(h(t-t_z)z(t_z)\) (This is the WCE model by Sylvestre and Abrahamowicz (2009))
-
\(h(t-t_z, z(t_z))\) (This is the DLNM model by Gasparrini et al. (2017))
-
\(h(t-t_z)z(t_z)\) (This is the WCE model by Sylvestre and Abrahamowicz (2009))
the cumulative effect \(g(\mathbf{z}, t)\) at follow-up time \(t\) is the integral (or sum) of the partial effects over exposure times \(t_z\) contained within \(\mathcal{T}(t)\)
the so called lag-lead window (or window of effectiveness) is denoted by \(\mathcal{T}(t)\). This represents the set of exposure times at which exposures can affect the hazard rate at time \(t\). The most common definition is \(\mathcal{T}(t) = \{t_{z,q}: t \geq t_{z,q}, q=1,\ldots, Q\}\), which means that all exposures that were observed prior to \(t\) or at \(t\) are eligible.
As before we use the function as_ped to transform the data and additionally use the formula special cumulative (for functional covariate) to specify the partial effect structure on the right-hand side of the formula.
Let \(t\) (time) the follow up time, \(\mathbf{z}_1\) (z1) and \(\mathbf{z}_2\) (z2) two TDCs observed at different exposure time grids \(t_z\) (tz1) and \(s_z\) (tz2) with lag-lead windows \(\mathcal{T}_1(t) = \{t_{z,q}: t \geq t_{z,q}\}\) and \(\mathcal{T}_2(t) = \{s_{z,k}: t \geq s_{z,k} + 2\}\) (defined by ll2 <- function(t, tz) { t >= tz + 2}).
The table below gives a selection of possible partial effects and the usage of cumulative to create matrices needed to estimate different types of cumulative effects of \(\mathbf{z}_1\) and \(\mathbf{z}_2\):
| partial effect |
as_ped specification |
|---|---|
| \(\int_{\mathcal{T}_1}h(t-t_z, z_1(t_z))\) | cumulative(latency(tz1), z1, tz_var= "tz1") |
| \(\int_{\mathcal{T}_1}h(t, t-t_z, z_1(t_z))\) | cumulative(time, latency(tz1), z1, tz_var="tz1") |
| \(\int_{\mathcal{T}_1}h(t, t_z, z_1(t_z))\) | cumulative(time, tz1, z1, tz_var="tz1") |
| \(\int_{\mathcal{T}_1}h(t, t_z, z_1(t_z)) + \int_{\mathcal{T}_2}h(t-s_z, z_2(s_z))\) | cumulative(time, tz1, z1, tz_var="tz1") + cumulative(latency(tz2), z2, tz_var="tz2", ll_fun=ll2) |
| … | … |
Note that
the variable representing follow-up time \(t\) in
cumulative(heretime) must match the time variable specified on the left-hand side of theformulathe variable representing exposure time \(t_z\) (here
tz1andtz2) must be wrapped withinlatency()to tellas_pedto calculate \(t-t_z\)by default, \(\mathcal{T}(t)\) is defined as
function(t, tz) {t >= tz}, thus for \(\mathcal{T}_1\) it is not necessary to explicitly specify the lag-lead window. In case you want to use a custom lag-lead window, provide the respective function to thell_funargument incumulative(seell2in the examples above)cumulativemakes no difference between partial effects such as \(h(t-t_z, z(t_z))\) and \(h(t-t_z)z(t_z)\) as the necessary data transformation is the same in both cases. Later, however, when fitting the model, a distinction must be mademore than one \(z\) variable can be provided to
cumulative, which can be convenient if multiple covariates should have the same time components and the same lag-lead windowmultiple
cumulativeterms can be specified, having different exposure timest_z,s_zand/or different lag-lead windows for different covariates \(\mathbf{z}_1\), \(\mathbf{z}_2\)To tell
cumulativewhich of the variables specified is the exposure time \(t_z\), thetz_varargument must be specified within eachcumulativeterm. The follow-up time component \(t\) will be automatically recognized via the left-hand side of theformula.
One time-dependent covariate
For illustration we use the ICU patients data sets patient and daily provided in the pammtools package, with follow-up time survhosp (\(t\)), exposure time Study_Day (\(t_z\)) and TDCs caloriesPercentage \(\mathbf{z}_1\) and proteinGproKG (\(\mathbf{z}_2\)):
head(patient)
## Year CombinedicuID CombinedID Survdays PatientDied survhosp Gender Age
## 1 2007 1114 1110 30.1 0 30.1 Male 68
## 2 2007 1114 1111 30.1 0 30.1 Female 57
## 3 2007 1114 1116 9.8 1 9.8 Female 68
## 4 2007 598 1316 30.1 0 30.1 Male 47
## 5 2007 365 1410 30.1 0 30.1 Male 69
## 6 2007 365 1414 30.1 0 5.4 Male 47
## AdmCatID ApacheIIScore BMI DiagID2
## 1 Surgical Elective 20 31.60321 Cardio-Vascular
## 2 Medical 22 24.34176 Respiratory
## 3 Surgical Elective 25 17.99308 Cardio-Vascular
## 4 Medical 16 33.74653 Orthopedic/Trauma
## 5 Surgical Elective 20 38.75433 Respiratory
## 6 Medical 21 45.00622 Respiratoryhead(daily)
## # A tibble: 6 x 4
## CombinedID Study_Day caloriesPercentage proteinGproKG
## <int> <int> <dbl> <dbl>
## 1 1110 1 0 0
## 2 1110 2 0 0
## 3 1110 3 4.05 0
## 4 1110 4 35.1 0.259
## 5 1110 5 77.2 0.647
## 6 1110 6 17.3 0Below we illustrate the usage of as_ped and cumulative to obtain covariate matrices for the latency matrix of follow up time and exposure time and a matrix of the TDC (caloriesPercentage). The follow up will be split in 30 intervals with interval breaks points 0:30. By default, the exposure time provided to tz_var will be used as a suffix for the names of the new matrix columns to indicate the exposure they refer to, however, you can override the suffix by specifying the suffix argument to cumulative.
ped <- as_ped( data = list(patient, daily), formula = Surv(survhosp, PatientDied) ~ . + cumulative(latency(Study_Day), caloriesPercentage, tz_var = "Study_Day"), cut = 0:30, id = "CombinedID") str(ped, 1)
## Classes 'fped', 'ped' and 'data.frame': 37258 obs. of 18 variables:
## $ CombinedID : int 1110 1110 1110 1110 1110 1110 1110 1110 1110 1110 ...
## $ tstart : num 0 1 2 3 4 5 6 7 8 9 ...
## $ tend : int 1 2 3 4 5 6 7 8 9 10 ...
## $ interval : Factor w/ 30 levels "(0,1]","(1,2]",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ offset : num 0 0 0 0 0 0 0 0 0 0 ...
## $ ped_status : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Year : Factor w/ 4 levels "2007","2008",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ CombinedicuID : Factor w/ 456 levels "21","24","25",..: 355 355 355 355 355 355 355 355 355 355 ...
## $ Survdays : num 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 ...
## $ Gender : Factor w/ 2 levels "Female","Male": 2 2 2 2 2 2 2 2 2 2 ...
## $ Age : int 68 68 68 68 68 68 68 68 68 68 ...
## $ AdmCatID : Factor w/ 3 levels "Medical","Surgical Elective",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ ApacheIIScore : int 20 20 20 20 20 20 20 20 20 20 ...
## $ BMI : num 31.6 31.6 31.6 31.6 31.6 ...
## $ DiagID2 : Factor w/ 9 levels "Gastrointestinal",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ Study_Day_latency : num [1:37258, 1:12] 0 0 1 2 3 4 5 6 7 8 ...
## $ caloriesPercentage: num [1:37258, 1:12] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "dimnames")=List of 2
## $ LL : num [1:37258, 1:12] 0 1 1 1 1 1 1 1 1 1 ...
## - attr(*, "breaks")= int [1:31] 0 1 2 3 4 5 6 7 8 9 ...
## - attr(*, "id_var")= chr "CombinedID"
## - attr(*, "intvars")= chr [1:6] "CombinedID" "tstart" "tend" "interval" ...
## - attr(*, "trafo_args")=List of 3
## - attr(*, "time_var")= chr "survhosp"
## - attr(*, "status_var")= chr "PatientDied"
## - attr(*, "func")=List of 1
## - attr(*, "ll_funs")=List of 1
## - attr(*, "tz")=List of 1
## - attr(*, "tz_vars")=List of 1
## - attr(*, "ll_weights")=List of 1
## - attr(*, "func_mat_names")=List of 1As you can see, the data is transformed to the PED format as before and two additional matrix columns are added, Study_Day_latency and caloriesPercentage.
Two time-dependent covariates observed on the same exposure time grid
Using the same data, we show a slightly more complex example with two functional covariate terms with partial effects \(h(t, t-t_z, z_1(t_z))\) and \(h(t-t_z, z_2(t_z))\). As in the case of ICU patient data, both TDCs were observed on the same exposure time grid, we want to use the same lag-lead window and the latency term \(t-t_z\) occurs in both partial effects, we only need to specify one cumulative term.
ped <- as_ped( data = list(patient, daily), formula = Surv(survhosp, PatientDied) ~ . + cumulative(survhosp, latency(Study_Day), caloriesPercentage, proteinGproKG, tz_var = "Study_Day"), cut = 0:30, id = "CombinedID") str(ped, 1)
## Classes 'fped', 'ped' and 'data.frame': 37258 obs. of 20 variables:
## $ CombinedID : int 1110 1110 1110 1110 1110 1110 1110 1110 1110 1110 ...
## $ tstart : num 0 1 2 3 4 5 6 7 8 9 ...
## $ tend : int 1 2 3 4 5 6 7 8 9 10 ...
## $ interval : Factor w/ 30 levels "(0,1]","(1,2]",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ offset : num 0 0 0 0 0 0 0 0 0 0 ...
## $ ped_status : num 0 0 0 0 0 0 0 0 0 0 ...
## $ Year : Factor w/ 4 levels "2007","2008",..: 1 1 1 1 1 1 1 1 1 1 ...
## $ CombinedicuID : Factor w/ 456 levels "21","24","25",..: 355 355 355 355 355 355 355 355 355 355 ...
## $ Survdays : num 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 30.1 ...
## $ Gender : Factor w/ 2 levels "Female","Male": 2 2 2 2 2 2 2 2 2 2 ...
## $ Age : int 68 68 68 68 68 68 68 68 68 68 ...
## $ AdmCatID : Factor w/ 3 levels "Medical","Surgical Elective",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ ApacheIIScore : int 20 20 20 20 20 20 20 20 20 20 ...
## $ BMI : num 31.6 31.6 31.6 31.6 31.6 ...
## $ DiagID2 : Factor w/ 9 levels "Gastrointestinal",..: 2 2 2 2 2 2 2 2 2 2 ...
## $ survhosp_mat : int [1:37258, 1:12] 0 1 2 3 4 5 6 7 8 9 ...
## $ Study_Day_latency : num [1:37258, 1:12] 0 0 1 2 3 4 5 6 7 8 ...
## $ caloriesPercentage: num [1:37258, 1:12] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "dimnames")=List of 2
## $ proteinGproKG : num [1:37258, 1:12] 0 0 0 0 0 0 0 0 0 0 ...
## ..- attr(*, "dimnames")=List of 2
## $ LL : num [1:37258, 1:12] 0 1 1 1 1 1 1 1 1 1 ...
## - attr(*, "breaks")= int [1:31] 0 1 2 3 4 5 6 7 8 9 ...
## - attr(*, "id_var")= chr "CombinedID"
## - attr(*, "intvars")= chr [1:6] "CombinedID" "tstart" "tend" "interval" ...
## - attr(*, "trafo_args")=List of 3
## - attr(*, "time_var")= chr "survhosp"
## - attr(*, "status_var")= chr "PatientDied"
## - attr(*, "func")=List of 1
## - attr(*, "ll_funs")=List of 1
## - attr(*, "tz")=List of 1
## - attr(*, "tz_vars")=List of 1
## - attr(*, "ll_weights")=List of 1
## - attr(*, "func_mat_names")=List of 1Multiple TDCs observed at different exposure time grids
To illustrate data transformation when TDCs \(\mathbf{z}_1\) and \(\mathbf{z}_2\) were observed on different exposure time grids (\(t_z\) and \(s_z\)) and are assumed to have different lag_lead windows \(\mathcal{T}_1(t)\) and \(\mathcal{T}_2(t)\), we use simulated data simdf_elra contained in pammtools (see example in sim_pexp for data generation).
simdf_elra## # A tibble: 250 x 9
## id time status x1 x2 tz1 z.tz1 tz2 z.tz2
## * <int> <dbl> <int> <dbl> <dbl> <list> <list> <list> <list>
## 1 1 3.22 1 1.59 4.61 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 2 2 10 0 -0.530 0.178 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 3 3 0.808 1 -2.43 3.25 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 4 4 3.04 1 0.696 1.26 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 5 5 3.36 1 2.54 3.71 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 6 6 4.07 1 -0.0231 0.607 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 7 7 0.534 1 1.68 1.74 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 8 8 1.57 1 -1.89 2.51 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 9 9 2.39 1 0.126 3.38 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## 10 10 6.61 1 1.79 1.06 <int [10]> <dbl [10]> <int [11]> <dbl [11]>
## # … with 240 more rows## Warning: `cols` is now required when using unnest().
## Please use `cols = c(tz1)`## # A tibble: 10 x 2
## id tz1
## <int> <int>
## 1 1 1
## 2 1 2
## 3 1 3
## 4 1 4
## 5 1 5
## 6 1 6
## 7 1 7
## 8 1 8
## 9 1 9
## 10 1 10## Warning: `cols` is now required when using unnest().
## Please use `cols = c(tz2)`## # A tibble: 11 x 2
## id tz2
## <int> <int>
## 1 1 -5
## 2 1 -4
## 3 1 -3
## 4 1 -2
## 5 1 -1
## 6 1 0
## 7 1 1
## 8 1 2
## 9 1 3
## 10 1 4
## 11 1 5Note that tz1 has a maximum length of 10, while tz2 has a maximum length of 11 and while tz1 was observed during the follow up, tz2 was partially observed before the beginning of the follow up.
If we wanted to estimate the following cumulative effects
- \(g(\mathbf{z}_1, t) = \int_{\mathcal{T}_1(t)} h(t, t-t_z, z_1(s_z))\) and
- \(g(\mathbf{z}_2, t) = \int_{\mathcal{T}_2(t)} h(t-t_z, z_2(s_z))\)
with \(\mathcal{T}_1(t) = \{t_z: t \geq t_z + 2\}\) and \(\mathcal{T}_2(t) = \{s_z: t \geq s_z\}\) the data transformation function must reflect the different exposure times as well as different lag-lead window specifications as illustrated below:
ped_sim <- as_ped( data = simdf_elra, formula = Surv(time, status) ~ . + cumulative(time, latency(tz1), z.tz1, tz_var = "tz1") + cumulative(latency(tz2), z.tz2, tz_var = "tz2"), cut = 0:10, id = "id") str(ped_sim, 1)
## Classes 'fped', 'ped' and 'data.frame': 1004 obs. of 15 variables:
## $ id : int 1 1 1 1 2 2 2 2 2 2 ...
## $ tstart : num 0 1 2 3 0 1 2 3 4 5 ...
## $ tend : int 1 2 3 4 1 2 3 4 5 6 ...
## $ interval : Factor w/ 10 levels "(0,1]","(1,2]",..: 1 2 3 4 1 2 3 4 5 6 ...
## $ offset : num 0 0 0 -1.53 0 ...
## $ ped_status : num 0 0 0 1 0 0 0 0 0 0 ...
## $ x1 : num 1.59 1.59 1.59 1.59 -0.53 ...
## $ x2 : num 4.612 4.612 4.612 4.612 0.178 ...
## $ time_tz1_mat: int [1:1004, 1:10] 0 1 2 3 0 1 2 3 4 5 ...
## $ tz1_latency : num [1:1004, 1:10] 0 0 1 2 0 0 1 2 3 4 ...
## $ z.tz1_tz1 : num [1:1004, 1:10] -2.014 -2.014 -2.014 -2.014 -0.978 ...
## ..- attr(*, "dimnames")=List of 2
## $ LL_tz1 : num [1:1004, 1:10] 0 1 1 1 0 1 1 1 1 1 ...
## $ tz2_latency : num [1:1004, 1:11] 5 6 7 8 5 6 7 8 9 10 ...
## $ z.tz2_tz2 : num [1:1004, 1:11] -0.689 -0.689 -0.689 -0.689 0.693 ...
## ..- attr(*, "dimnames")=List of 2
## $ LL_tz2 : num [1:1004, 1:11] 1 1 1 1 1 1 1 1 1 1 ...
## - attr(*, "breaks")= int [1:11] 0 1 2 3 4 5 6 7 8 9 ...
## - attr(*, "id_var")= chr "id"
## - attr(*, "intvars")= chr [1:6] "id" "tstart" "tend" "interval" ...
## - attr(*, "trafo_args")=List of 3
## - attr(*, "time_var")= chr "time"
## - attr(*, "status_var")= chr "status"
## - attr(*, "func")=List of 2
## - attr(*, "ll_funs")=List of 2
## - attr(*, "tz")=List of 2
## - attr(*, "tz_vars")=List of 2
## - attr(*, "ll_weights")=List of 2
## - attr(*, "func_mat_names")=List of 2Note how the matrix covariates associated with tz1 have 10 columns while matrix covariates associated with tz2 have 11 columns. Note also that the lag-lead matrices differ
gg_laglead(ped_sim)
References
Bender, Andreas, Fabian Scheipl, Wolfgang Hartl, Andrew G. Day, and Helmut Küchenhoff. 2018. “Penalized Estimation of Complex, Non-Linear Exposure-Lag-Response Associations.” Biostatistics. https://doi.org/10.1093/biostatistics/kxy003.
Gasparrini, Antonio, Fabian Scheipl, Ben Armstrong, and Michael G. Kenward. 2017. “A Penalized Framework for Distributed Lag Non-Linear Models.” Biometrics 73: 938–48. https://doi.org/10.1111/biom.12645.
Kalbfleisch, J., and R. Prentice. 1980. The Statistical Analysis of Failure Time Data. New York: Wiley.
Sylvestre, Marie-Pierre, and Michal Abrahamowicz. 2009. “Flexible Modeling of the Cumulative Effects of Time-Dependent Exposures on the Hazard.” Statistics in Medicine 28 (27): 3437–53. https://doi.org/10.1002/sim.3701.