---
title: "Discrete Interval Censored Survival Models"
author: Klaus Holst & Thomas Scheike
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    fig_caption: yes
    fig_width: 7.15
    fig_height: 5.5    
vignette: >
  %\VignetteIndexEntry{Discrete Interval Censored Survival Models}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(mets)
```

Discrete Inteval Censored survival times 
========================================

\begin{align*}
   \mbox{logit}(P(T >t | x)) & = \log(G(t)) + 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 truly discrete,
in contrast to grouping of continuous data, $]0,1]$ will be an
observation at 1, and $]j,j+1]$ will be an observation at j+1.

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: 

```{r}
library(mets)

data(ttpd) 
dtable(ttpd,~entry+time2)
out <- interval.logitsurv.discrete(Interval(entry,time2)~X1+X2+X3+X4,ttpd)
summary(out)
```

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

```{r}
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)

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. 


```{r}
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. 


```{r}
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.

```{r}
if (test==1) {

###
### data(ttpd) 
### dtable(ttpd,~entry+time2)
### out <- interval.logitsurv.discrete(Interval(entry,time2)~X1+X2+X3+X4,ttpd)
### summary(out)

#       Estimate Std.Err     2.5%   97.5%   P-value
# time1  -2.0064  0.1461 -2.29277 -1.7201 6.466e-43
# time2  -2.1749  0.1543 -2.47725 -1.8725 3.869e-45
# time3  -1.4581  0.1496 -1.75132 -1.1648 1.936e-22
# time4  -2.9260  0.2436 -3.40344 -2.4486 3.078e-33
# time5  -1.2051  0.1655 -1.52935 -0.8808 3.267e-13
# time6  -0.9102  0.1790 -1.26103 -0.5594 3.671e-07
# X1      0.9913  0.1171  0.76175  1.2208 2.557e-17
# X2      0.6962  0.1156  0.46953  0.9228 1.739e-09
# X3      0.3466  0.1150  0.12110  0.5721 2.590e-03
# X4      0.3223  0.1147  0.09749  0.5470 4.952e-03
 out$ploglik
# [1] -1676.456

### library(ordinal)
### ttpd <- dfactor(ttpd,fentry~entry)
### 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
============


```{r}
sessionInfo()
```


