# Discrete Interval Censored Survival Models

#### Klaus Holst & Thomas Scheike

#### 2025-08-30

# Discrete Inteval Censored survival times

We consider the cumulative odds model for the probability of dying before time t: \[\begin{align*} \mbox{logit}(P(T \leq t | x)) & = \log(G(t)) + x^T \beta \\ P(T \leq t | x) & = \frac{G(t) exp( x^T \beta)}{1 + G(t) exp( x^T \beta) } \\ P(T >t | x) & = \frac{1}{1 + G(t) exp( x^T \beta) } \end{align*}\]

Input are intervals given by \(]t_l,t_r]\) where t_r can be infinity for right-censored intervals. When the data is discrete, in contrast to grouping of continuous data, \(]0,1]\) then the intervals \(]j,j+1]\) will be equvilant to an observation at j+1 (see below example).

Likelihood is maximized: \[\begin{align*} \prod_i P(T_i >t_{il} | x) - P(T_i> t_{ir}| x). \end{align*}\]

This model is also called the cumulative odds model \[\begin{align*} P(T \leq t | x) & = \frac{ G(t) exp( x^T \beta) }{1 + G(t) exp( x^T \beta) }. \end{align*}\] and \(\beta\) says something about the OR of probability of being before \(t\).

The baseline is parametrized as \[\begin{align*} G(t) & = \sum_{j \leq t} \exp( \alpha_j ) \end{align*}\]

An important consequence of the model is that for all cut-points \(t\) we have the same OR parameters for the OR of being early or later than \(t\).

# Discrete TTP

First we look at some time to pregnancy data (simulated discrete survival data) that is right-censored, and set it up to fit the cumulative odds model by constructing the intervals appropriately:

    library(mets)

    data(ttpd)
    dtable(ttpd,~entry+time2)
    #>
    #>       time2   1   2   3   4   5   6 Inf
    #> entry
    #> 0           316   0   0   0   0   0   0
    #> 1             0 133   0   0   0   0   0
    #> 2             0   0 150   0   0   0   0
    #> 3             0   0   0  23   0   0   0
    #> 4             0   0   0   0  90   0   0
    #> 5             0   0   0   0   0  68   0
    #> 6             0   0   0   0   0   0 220
    out <- interval.logitsurv.discrete(Interval(entry,time2)~X1+X2+X3+X4,ttpd)
    summary(out)
    #> $baseline
    #>       Estimate Std.Err   2.5%   97.5%   P-value
    #> time1  -2.0064  0.1523 -2.305 -1.7079 1.273e-39
    #> time2  -2.1749  0.1599 -2.488 -1.8614 4.118e-42
    #> time3  -1.4581  0.1544 -1.761 -1.1554 3.636e-21
    #> time4  -2.9260  0.2453 -3.407 -2.4453 8.379e-33
    #> time5  -1.2051  0.1706 -1.539 -0.8706 1.633e-12
    #> time6  -0.9102  0.1860 -1.275 -0.5457 9.843e-07
    #>
    #> $logor
    #>    Estimate Std.Err    2.5%  97.5%   P-value
    #> X1   0.9913  0.1179 0.76024 1.2223 4.100e-17
    #> X2   0.6962  0.1162 0.46847 0.9238 2.064e-09
    #> X3   0.3466  0.1159 0.11941 0.5738 2.788e-03
    #> X4   0.3223  0.1151 0.09668 0.5478 5.111e-03
    #>
    #> $or
    #>    Estimate     2.5%    97.5%
    #> X1 2.694610 2.138791 3.394874
    #> X2 2.006032 1.597554 2.518953
    #> X3 1.414239 1.126834 1.774950
    #> X4 1.380231 1.101503 1.729490

    dfactor(ttpd) <- entry.f~entry
    out <- cumoddsreg(entry.f~X1+X2+X3+X4,ttpd)
    summary(out)
    #> $baseline
    #>       Estimate Std.Err   2.5%   97.5%   P-value
    #> time1  -2.0064  0.1523 -2.305 -1.7079 1.273e-39
    #> time2  -2.1749  0.1599 -2.488 -1.8614 4.118e-42
    #> time3  -1.4581  0.1544 -1.761 -1.1554 3.636e-21
    #> time4  -2.9260  0.2453 -3.407 -2.4453 8.379e-33
    #> time5  -1.2051  0.1706 -1.539 -0.8706 1.633e-12
    #> time6  -0.9102  0.1860 -1.275 -0.5457 9.843e-07
    #>
    #> $logor
    #>    Estimate Std.Err    2.5%  97.5%   P-value
    #> X1   0.9913  0.1179 0.76024 1.2223 4.100e-17
    #> X2   0.6962  0.1162 0.46847 0.9238 2.064e-09
    #> X3   0.3466  0.1159 0.11941 0.5738 2.788e-03
    #> X4   0.3223  0.1151 0.09668 0.5478 5.111e-03
    #>
    #> $or
    #>    Estimate     2.5%    97.5%
    #> X1 2.694610 2.138791 3.394874
    #> X2 2.006032 1.597554 2.518953
    #> X3 1.414239 1.126834 1.774950
    #> X4 1.380231 1.101503 1.729490

We note that the probability of dying is increased considerably for all covariates.

Now using this discrete survival model we simulate some data from this model

    set.seed(1000) # to control output in simulatins for p-values below.
    n <- 200
    Z <- matrix(rbinom(n*4,1,0.5),n,4)
    outsim <- simlogitSurvd(out$coef,Z)
    outsim <- transform(outsim,left=time,right=time+1)
    outsim <- dtransform(outsim,right=Inf,status==0)
    outss <- interval.logitsurv.discrete(Interval(left,right)~+X1+X2+X3+X4,outsim)
    summary(outss)
    #> $baseline
    #>       Estimate Std.Err    2.5%   97.5%   P-value
    #> time1  -2.0154  0.3698 -2.7402 -1.2906 5.036e-08
    #> time2  -1.5474  0.3473 -2.2281 -0.8666 8.385e-06
    #> time3  -0.8119  0.3411 -1.4804 -0.1434 1.729e-02
    #> time4  -2.0085  0.5102 -3.0084 -1.0086 8.248e-05
    #> time5  -0.2185  0.3858 -0.9746  0.5376 5.711e-01
    #> time6   0.2637  0.4618 -0.6415  1.1689 5.681e-01
    #>
    #> $logor
    #>    Estimate Std.Err    2.5%  97.5%   P-value
    #> X1  1.27893  0.2804  0.7293 1.8286 5.106e-06
    #> X2  0.39293  0.2635 -0.1235 0.9094 1.359e-01
    #> X3 -0.09008  0.2524 -0.5847 0.4045 7.211e-01
    #> X4  0.20766  0.2627 -0.3072 0.7225 4.292e-01
    #>
    #> $or
    #>    Estimate      2.5%    97.5%
    #> X1 3.592796 2.0735647 6.225116
    #> X2 1.481310 0.8838237 2.482711
    #> X3 0.913858 0.5572845 1.498582
    #> X4 1.230798 0.7355301 2.059553

    pred <- predictlogitSurvd(out,se=TRUE)
    plotSurvd(pred,se=TRUE)

Finally, we look at some data and compare with the icenReg package that can also fit the proportional odds model for continous or discrete data. We make the data fully interval censored/discrete by letting also exact obsevations be only observed to be in an interval.

We consider the interval censored survival times for time from onset of diabetes to to diabetic nephronpathy, then modify it to observe only that the event times are in certain intervals.

    test <- 0
    if (test==1) {

    require(icenReg)
    data(IR_diabetes)
    IRdia <- IR_diabetes
    ## removing fully observed data in continuous version, here making it a discrete observation
    IRdia <- dtransform(IRdia,left=left-1,left==right)
    dtable(IRdia,~left+right,level=1)

    ints <- with(IRdia,dInterval(left,right,cuts=c(0,5,10,20,30,40,Inf),show=TRUE) )
    }

We note that the gender effect is equivalent for the two approaches.

    if (test==1) {
    ints$Ileft <- ints$left
    ints$Iright <- ints$right
    IRdia <- cbind(IRdia,data.frame(Ileft=ints$Ileft,Iright=ints$Iright))
    dtable(IRdia,~Ileft+Iright)
    #
    #       Iright   1   2   3   4   5 Inf
    # Ileft
    # 0             10   1  34  25   4   0
    # 1              0  55  19  17   1   1
    # 2              0   0 393  16   4   0
    # 3              0   0   0 127   1   0
    # 4              0   0   0   0  21   0
    # 5              0   0   0   0   0   2

    outss <- interval.logitsurv.discrete(Interval(Ileft,Iright)~+gender,IRdia)
    #            Estimate Std.Err    2.5%    97.5%   P-value
    # time1        -3.934  0.3316 -4.5842 -3.28418 1.846e-32
    # time2        -2.042  0.1693 -2.3742 -1.71038 1.710e-33
    # time3         1.443  0.1481  1.1530  1.73340 1.911e-22
    # time4         3.545  0.2629  3.0295  4.06008 1.976e-41
    # time5         6.067  0.7757  4.5470  7.58784 5.217e-15
    # gendermale   -0.385  0.1691 -0.7165 -0.05351 2.283e-02
    summary(outss)
    outss$ploglik
    # [1] -646.1946

    fit <- ic_sp(cbind(Ileft, Iright) ~ gender, data = IRdia, model = "po")
    #
    # Model:  Proportional Odds
    # Dependency structure assumed: Independence
    # Baseline:  semi-parametric
    # Call: ic_sp(formula = cbind(Ileft, Iright) ~ gender, data = IRdia,
    #     model = "po")
    #
    #            Estimate Exp(Est)
    # gendermale    0.385     1.47
    #
    # final llk =  -646.1946
    # Iterations =  6
    # Bootstrap Samples =  0
    # WARNING: only  0  bootstrap samples used for standard errors.
    # Suggest using more bootstrap samples for inference
    summary(fit)

    ## sometimes NR-algorithm needs modifications of stepsize to run
    ## outss <- interval.logitsurv.discrete(Interval(Ileft,Iright)~+gender,IRdia,control=list(trace=TRUE,stepsize=1.0))
    }

Also agrees with the cumulative link regression of the ordinal package, although the baseline is parametrized differently. In additon the clm is describing the probability of surviving rather than the probabibility of dying.

    data(ttpd)
    dtable(ttpd,~entry+time2)
    #>
    #>       time2   1   2   3   4   5   6 Inf
    #> entry
    #> 0           316   0   0   0   0   0   0
    #> 1             0 133   0   0   0   0   0
    #> 2             0   0 150   0   0   0   0
    #> 3             0   0   0  23   0   0   0
    #> 4             0   0   0   0  90   0   0
    #> 5             0   0   0   0   0  68   0
    #> 6             0   0   0   0   0   0 220
    ttpd <- dfactor(ttpd,fentry~entry)
    out <- cumoddsreg(fentry~X1+X2+X3+X4,ttpd)
    summary(out)
    #> $baseline
    #>       Estimate Std.Err   2.5%   97.5%   P-value
    #> time1  -2.0064  0.1523 -2.305 -1.7079 1.273e-39
    #> time2  -2.1749  0.1599 -2.488 -1.8614 4.118e-42
    #> time3  -1.4581  0.1544 -1.761 -1.1554 3.636e-21
    #> time4  -2.9260  0.2453 -3.407 -2.4453 8.379e-33
    #> time5  -1.2051  0.1706 -1.539 -0.8706 1.633e-12
    #> time6  -0.9102  0.1860 -1.275 -0.5457 9.843e-07
    #>
    #> $logor
    #>    Estimate Std.Err    2.5%  97.5%   P-value
    #> X1   0.9913  0.1179 0.76024 1.2223 4.100e-17
    #> X2   0.6962  0.1162 0.46847 0.9238 2.064e-09
    #> X3   0.3466  0.1159 0.11941 0.5738 2.788e-03
    #> X4   0.3223  0.1151 0.09668 0.5478 5.111e-03
    #>
    #> $or
    #>    Estimate     2.5%    97.5%
    #> X1 2.694610 2.138791 3.394874
    #> X2 2.006032 1.597554 2.518953
    #> X3 1.414239 1.126834 1.774950
    #> X4 1.380231 1.101503 1.729490

    out$ploglik
    #> [1] -1676.456

    if (test==1) {
    ### library(ordinal)
    ### out1 <- clm(fentry~X1+X2+X3+X4,data=ttpd)
    ### summary(out1)

    # formula: fentry ~ X1 + X2 + X3 + X4
    # data:    ttpd
    #
    #  link  threshold nobs logLik   AIC     niter max.grad cond.H
    #  logit flexible  1000 -1676.46 3372.91 6(2)  1.17e-12 5.3e+02
    #
    # Coefficients:
    #    Estimate Std. Error z value Pr(>|z|)
    # X1  -0.9913     0.1171  -8.465  < 2e-16 ***
    # X2  -0.6962     0.1156  -6.021 1.74e-09 ***
    # X3  -0.3466     0.1150  -3.013  0.00259 **
    # X4  -0.3223     0.1147  -2.810  0.00495 **
    # ---
    # Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
    #
    # Threshold coefficients:
    #     Estimate Std. Error z value
    # 0|1  -2.0064     0.1461 -13.733
    # 1|2  -1.3940     0.1396  -9.984
    # 2|3  -0.7324     0.1347  -5.435
    # 3|4  -0.6266     0.1343  -4.667
    # 4|5  -0.1814     0.1333  -1.361
    # 5|6   0.2123     0.1342   1.582
    }

# SessionInfo

    sessionInfo()
    #> R version 4.5.1 (2025-06-13)
    #> Platform: aarch64-apple-darwin24.5.0
    #> Running under: macOS Sequoia 15.6.1
    #>
    #> Matrix products: default
    #> BLAS:   /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRblas.dylib
    #> LAPACK: /Users/kkzh/.asdf/installs/R/4.5.1/lib/R/lib/libRlapack.dylib;  LAPACK version 3.12.1
    #>
    #> locale:
    #> [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
    #>
    #> time zone: Europe/Copenhagen
    #> tzcode source: internal
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] timereg_2.0.7  survival_3.8-3 mets_1.3.7
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] cli_3.6.5           knitr_1.50          rlang_1.1.6
    #>  [4] xfun_0.53           jsonlite_2.0.0      future.apply_1.20.0
    #>  [7] listenv_0.9.1       lava_1.8.1          htmltools_0.5.8.1
    #> [10] sass_0.4.10         rmarkdown_2.29      grid_4.5.1
    #> [13] evaluate_1.0.5      jquerylib_0.1.4     fastmap_1.2.0
    #> [16] numDeriv_2016.8-1.1 yaml_2.3.10         mvtnorm_1.3-3
    #> [19] lifecycle_1.0.4     compiler_4.5.1      codetools_0.2-20
    #> [22] ucminf_1.2.2        Rcpp_1.1.0          future_1.67.0
    #> [25] lattice_0.22-7      digest_0.6.37       R6_2.6.1
    #> [28] parallelly_1.45.1   parallel_4.5.1      splines_4.5.1
    #> [31] Matrix_1.7-4        bslib_0.9.0         tools_4.5.1
    #> [34] globals_0.18.0      cachem_1.1.0