Introduction to TwoTimeScales

One time scale only

For data over one time scale, the function requires as minimal inputs only a vector of exit times (s_out), a vector of events’ indicators (events) and the desired bins’ size (ds). If the data are left truncated, the entry times can be passed to the argument s_in. These can be either extracted from a dataset by using the argument data and passing the vector’s names in each respective argument, as illustrated below, or by directly passing the vectors.

dt1ts <- prepare_data(data = reccolon2ts,
                      s_out = "timesr",
                      events = "status",
                      ds = 30)

> str(dt1ts)

List of 2
 $ bins   :List of 3
  ..$ bins_s: num [1:92] 0 30 60 90 120 150 180 210 240 270 ...
  ..$ mids  : num [1:91] 15 45 75 105 135 165 195 225 255 285 ...
  ..$ ns    : int 91
 $ bindata:List of 2
  ..$ r: num [1:91] 13674 13070 12474 12002 11534 ...
  ..$ y: num [1:91] 14 24 16 11 24 22 20 10 24 26 ...
 - attr(*, "class")= chr "data2ts"

> print(dt1ts)
An object of class 'data2ts'

Data:
List of 2
 $ bins   :List of 3
 $ bindata:List of 2
 - attr(*, "class")= chr "data2ts"
NULL

Range covered by the bins: 
$bins_s
[1]    0 2730


Number of bins: 
$ns
[1] 91


Overview of the binned data:
Total exposure time: 246018
Total number of events: 409

The object dt1ts is a list with two elements. The first element is a list of bins (bins_s) with their midpoints (mids) and the number of bins (ns). The second element of the list is another list with the data binned, a vector of exposures called r and a vector of events’ counts y. By calling the method print(dt1ts), we obtain additional information about the data, such as the range covered by the bins, the total exposure time and the total number of events.

Two time scales

The same function is used to bin the individual data in a two-dimensional grid of rectangles (or squares) over \(u\) and \(s\), or parallelograms in a Lexis diagram over \(t\) and \(s\). The two time scales model is estimated over the half-plane \((u,s)\). Therefore, we will here discuss only preparation of the data over \(u\) and \(s\), and defer the discussion of the preparation over the Lexis diagram to the vignette dedicated to the analysis with two time scales.

dt2ts <- prepare_data(data = reccolon2ts,
                      u = "timer",
                      s_out = "timesr",
                      events = "status",
                      ds = 30)

> str(dt2ts)
List of 2
 $ bins   :List of 6
  ..$ bins_u: num [1:77] 8 38 68 98 128 158 188 218 248 278 ...
  ..$ midu  : num [1:76] 23 53 83 113 143 173 203 233 263 293 ...
  ..$ nu    : int 76
  ..$ bins_s: num [1:92] 0 30 60 90 120 150 180 210 240 270 ...
  ..$ mids  : num [1:91] 15 45 75 105 135 165 195 225 255 285 ...
  ..$ ns    : int 91
 $ bindata:List of 2
  ..$ R: num [1:76, 1:91] 205 243 540 796 474 812 506 746 657 517 ...
  ..$ Y: num [1:76, 1:91] 1 1 0 1 1 2 1 1 1 2 ...
 - attr(*, "class")= chr "data2ts"

> print(dt2ts)
An object of class 'data2ts'

Data:
List of 2
 $ bins   :List of 6
 $ bindata:List of 2
 - attr(*, "class")= chr "data2ts"
NULL

Range covered by the bins: 
$bins_u
[1]    8 2288

$bins_s
[1]    0 2730


Number of bins: 
$nu
[1] 76

$ns
[1] 91


Overview of the binned data:
Total exposure time: 246018
Total number of events: 409

We have now included only one additional input u, that is the vector with the individuals’ time at recurrence of the cancer. The object dt2ts has the same structure as dt1ts, but with additional bins over \(u\) (bins_u, midu and nu). The data are now matrices R and Y of dimension nu by ns.

Including covariates

If we want to estimate a proportional hazard model with covariates a different structure of the data is required (see Carollo et al. (2024)). The data are augmented by one dimension. So, for the model with one time scale only, we need to create a matrix of exposure times R with dimension n (number of individuals in the data) by ns (number of bins). The same goes for the matrix of event counts. For the model with two time scales, each observation in the sample will be represented by its own matrix of exposure times and matrix of event indicators. Therefore, the data for the whole sample can be arranged in three-dimensional arrays, with dimensions nu by ns by n. The function prepare_data() does exactly so by specifying the optional input individual = TRUE. It is possible, and recommended, to provide the covariates that one wishes to use in the PH model in the extra argument covs. The function will take care of the preparation of the regression matrix Z. However, if desired, this can also be prepared externally (in which case a warning will be returned).

dt2ts_cov <- prepare_data(data = reccolon2ts,
                          u = "timer",
                          s_in = "entrys",
                          s_out = "timesr",
                          events = "status",
                          ds = 30,
                          individual = TRUE, 
                          covs = c("rx", "node4", "sex"))

> str(dt2ts_cov)
List of 2
 $ bins   :List of 6
  ..$ bins_u: num [1:77] 8 38 68 98 128 158 188 218 248 278 ...
  ..$ midu  : num [1:76] 23 53 83 113 143 173 203 233 263 293 ...
  ..$ nu    : int 76
  ..$ bins_s: num [1:92] 0 30 60 90 120 150 180 210 240 270 ...
  ..$ mids  : num [1:91] 15 45 75 105 135 165 195 225 255 285 ...
  ..$ ns    : int 91
 $ bindata:List of 3
  ..$ R: num [1:76, 1:91, 1:461] 0 0 0 0 0 0 0 0 0 0 ...
  ..$ Y: num [1:76, 1:91, 1:461] 0 0 0 0 0 0 0 0 0 0 ...
  ..$ Z: num [1:461, 1:6] 0 0 0 0 0 1 0 1 0 1 ...
  .. ..- attr(*, "dimnames")=List of 2
  .. .. ..$ : NULL
  .. .. ..$ : chr [1:6] "rx_Lev" "rx_Lev+5FU" "sex_male" "adhere" ...
 - attr(*, "class")= chr "data2ts"

> print(dt2ts_cov)
An object of class 'data2ts'

Data:
List of 2
 $ bins   :List of 6
 $ bindata:List of 3
 - attr(*, "class")= chr "data2ts"
NULL

Range covered by the bins: 
$bins_u
[1]    8 2288

$bins_s
[1]    0 2730


Number of bins: 
$nu
[1] 76

$ns
[1] 91


Overview of the binned data:
Total exposure time: 246018
Total number of events: 409
Covariates:
[1] "rx_Lev"     "rx_Lev+5FU" "sex_male"   "adhere"    
[5] "obstruct"   "node4" 

The data are now the three-dimensional arrays, R and Y and a regression matrix Z, where the covariates are transformed in dummy variables (where needed). The print method returns also the names of the dummy variables, so we can see that from the variable rx we obtained the two dummies rx_Lev and rx_Lev+5FU and from the variable sex we obtained now the dummy variable sex_male (Obs and female are reference categories, as well as not having more than four nodes).

One time scale

To estimate the model we use the function fit1ts(). As input to this function we pass the object returned from prepare_data. Without specifying additional parameters, the function estimates a \(P\)-splines model, by using numerical optimization of the AIC to find the optimal smoothing parameter. The \(B\)-splines basis is built using default options (see help(fit1ts) for details). However, for illustration, we now show how to specify the \(B\)-splines basis to more closely match the data structure.

m1ts <- fit1ts(data1ts = dt1ts,
               Bbases_spec = list(bdeg = 3,
                                  nseg_s = 20,
                                  min_s = 0,
                                  max_s = 2730))

> str(m1ts)
List of 3
 $ optimal_model :List of 9
  ..$ alpha   : num [1:23, 1] -6.77 -6.58 -6.39 -6.25 -6.19 ...
  ..$ SE_alpha: num [1:23] 0.2276 0.123 0.0763 0.0744 0.0797 ...
  ..$ eta     : num [1:91, 1] -6.56 -6.52 -6.48 -6.44 -6.4 ...
  ..$ H       : num [1:23, 1:23] 0.07508 0.04686 0.02027 0.00324 -0.00366 ...
  ..$ deviance: num 79.4
  ..$ ed      : num 4.25
  ..$ aic     : num 87.9
  ..$ bic     : num 98.6
  ..$ Bbases  :List of 1
  .. ..$ Bs: num [1:91, 1:23] 1.18e-01 5.02e-02 1.52e-02 2.05e-03 2.21e-07 ...
  .. .. ..- attr(*, "x")= num [1:91] 15 45 75 105 135 165 195 225 255 285 ...
  .. .. ..- attr(*, "xl")= num 0
  .. .. ..- attr(*, "xr")= num 2730
  .. .. ..- attr(*, "nseg")= num 20
  .. .. ..- attr(*, "bdeg")= num 3
  .. .. ..- attr(*, "type")= chr "bbase"
 $ optimal_logrho: num 2.02
 $ P_optim       : num [1:23, 1:23] 105 -211 105 0 0 ...
 - attr(*, "class")= chr "haz1ts"

Here, we specify a cubic (bdeg = 3) \(B\)-splines basis that covers the range of \(s\) from 0 to 2730 days (min_s and max_s respectively), and is built on 20 segments (nseg_s = 20). The object returned by fit1ts is of class "haz1ts". The first element is a list with the results of the optimal model. The second element is the optimal smoothing parameter (on the \(\log_{10}\)-scale) and the last element is the penalty matrix, which incorporates the optimal value of \(\varrho\), P_optim.

The vignette Smooth hazards with one time scale discusses alternative options for estimation, in particular concerning the search for the optimal smoothing parameter.

Two time scales

Estimation over the \((u,s)\)-plane is performed in the same way. Here, we illustrate how to use the function fit2ts(), and again we will change the default specifications of the marginal \(B\)-spline bases, but otherwise use default values for the remaining inputs.

m2ts <- fit2ts(data2ts = dt2ts,
               Bbases_spec = list(bdeg = 3,
                                  nseg_s = 20,
                                  min_s = 0,
                                  max_s = 2730,
                                  nseg_u = 16,
                                  min_u = 0,
                                  max_u = 2300))

> summary(m2ts)
Number of events =  409 
Model specifications:
  nu =  76 
  ns =  91 
  cu =  19 
  cs =  23 

Optimal smoothing: 
  log10(rho_u) =  2.101313 
  log10(rho_s) =  0.4089259 
  rho_u =  126.2736 
  rho_s =  2.564046 

Model with no covariates

Model diagnostics: 
  AIC =  1249.329 
  BIC =  1313.998 
  ED =  11.02155

Whereas the estimates from fit1ts are vectors, the ones from fit2ts are organized in matrices: The matrix optimal_model$Alpha, containing the estimated \(B\)-splines’ coefficients \(\alpha_{lm}\), for \(l \in 1,\dots,c_u\) (here \(c_u = 19\)) and \(m \in 1,\dots, c_s\) (\(c_s = 23\)). And the matrix Eta of dimension \(n_u\) by \(n_s\) (90 by 91). Additionally, we have a list with the marginal \(B\)-splines used for estimation (optimal_model$Bbases), a vector of \(\log_{10}\varrho_u\) and \(\log_{10}\varrho_s\) values (optimal_logrho) and finally the penalty matrix (P_optim) that incorporates the optimal penalty parameters.

The summary method is implemented for objects of class haz2ts, and it returns the model specifications, the optimal smoothing parameters, the effective dimension (ED) and the model fit as measured by the AIC and BIC of the model.

One time scale

Once the model is estimated, we can represent the results with the function plot(). The minimal input for plot() is the fitted model. Here, we show how to evaluate the \(B\)-splines basis on a finer grid of points that are only 10 days apart (as opposite to 30, option that was used for the estimating \(B\)-splines). Moreover, we pass as inputs some graphical parameters to obtain a nicer plot. The result is a plot of the one-dimensional hazard curve over \(s\) (time since recurrence), with accompany 95% confidence bands.

plot(x = m1ts,
     plot_grid = c("smin" = 0, "smax" = 2730, "ds" = 10),
     plot_options= list(
       col = "darkblue",
       main = "Hazard",
       ylab = "hazard",
       xlab = "time since recurrence",
       cex_main = 1))

#### Two time scales

There are several options to represent the smooth hazard with two time scales. Here, we will present only the most simple, yet very informative, way, which is also the default plot returned by the function plot(), applied to an object of class 'haz2ts'. This is a surface plot of the hazard over the \((u,s)\)-plane with white contour lines. We provide a vignette that only focus on visualization of the hazard with two time scales Visualize hazards with two time scales.

plot(x = m2ts,
     plot_grid = list(c("umin" = 0, "umax" = 2300, "du" = 10),
                      c("smin" = 0, "smax" = 2730, "ds" = 10)),
     plot_options= list(
       main = "Bi-dimensional hazard",
       ylab = "time since recurrence",
       xlab = "time since randomization",
       cex_main = 1))