# Haplotype Discrete Survival Models

#### Klaus Holst & Thomas Scheike

#### 2026-05-24

# Haplotype Analysis for discrete TTP

Cycle-specific logistic regression of haplotype effects with known haplotype probabilities. Given observed genotype \(G\) and unobserved haplotypes \(H\), we marginalise over the possible haplotypes using \(P(H|G)\) supplied as input.

\[\begin{align*} S(t|x,G) & = E( S(t|x,H) | G) = \sum_{h \in G} P(h|G) S(t|z,h) \end{align*}\] so survival can be computed by marginalising over possible \(h\) given \(g\).

Survival is based on logistic regression for the discrete hazard function of the form \[\begin{align*} \mbox{logit}(P(T=t \mid T \geq t, x, h)) & = \alpha_t + x(h)^T \beta \end{align*}\] where \(x(h)\) is a regression design of \(x\) and haplotypes \(h=(h_1,h_2)\).

Simple binomial data can also be fitted using this function.

Standard errors are computed assuming that the haplotype probabilities are known.

We are particularly interested in the types haplotypes:

    types <- c("DCGCGCTCACG","DTCCGCTGACG","ITCAGTTGACG","ITCCGCTGAGG")

    ## some haplotypes frequencies for simulations
    data(haplo)
    hapfreqs <- haplo$hapfreqs
    print(hapfreqs)
    #>             index   haplotype     freq
    #> DCGAGCTCACG     1 DCGAGCTCACG 0.010681
    #> DCGCGCTCACG     2 DCGCGCTCACG 0.138387
    #> DTGAGCTCACG     3 DTGAGCTCACG 0.000310
    #> DTGAGCTCACA     4 DTGAGCTCACA 0.006800
    #> DTGAGCTCGCG     5 DTGAGCTCGCG 0.034517
    #> DTGACCTCACG     6 DTGACCTCACG 0.001336
    #> DTGCGCTCACG     7 DTGCGCTCACG 0.009969
    #> DTGCGCTCACA     8 DTGCGCTCACA 0.011833
    #> DTGCGCTCGCG     9 DTGCGCTCGCG 0.302389
    #> DTGCGCCCGCG    10 DTGCGCCCGCG 0.001604
    #> DTGCCCTCACG    11 DTGCCCTCACG 0.003912
    #> DTCAGCTGACG    12 DTCAGCTGACG 0.001855
    #> DTCCGCTGACG    13 DTCCGCTGACG 0.103394
    #> DTCCCCTGACG    14 DTCCCCTGACG 0.000310
    #> ITCAGTTGACG    15 ITCAGTTGACG 0.048124
    #> ITCCGCTGAGG    16 ITCCGCTGAGG 0.291273
    #> ITCCGTTGACG    17 ITCCGTTGACG 0.031089
    #> ITCCGTCGACG    18 ITCCGTCGACG 0.001502
    #> ITCCCCTGAGG    19 ITCCCCTGAGG 0.000653

Among the haplotypes of interest we look up the frequencies and choose a baseline:

    www <-which(hapfreqs$haplotype %in% types)
    hapfreqs$freq[www]
    #> [1] 0.138387 0.103394 0.048124 0.291273

    baseline=hapfreqs$haplotype[9]
    baseline
    #> [1] "DTGCGCTCGCG"

We have cycle-specific data with \(id\) and outcome \(y\):

    haploX  <- haplo$haploX
    dlist(haploX,.~id|id %in% c(1,4,7))
    #> id: 1
    #>   y X1 X2 X3 X4 times lbnr__id Count1
    #> 1 0 0  0  0  0  1     1        0
    #> 2 0 0  0  0  0  2     2        0
    #> 3 0 0  0  0  0  3     3        0
    #> 4 0 0  0  0  0  4     4        0
    #> 5 0 0  0  0  0  5     5        0
    #> 6 0 0  0  0  0  6     6        0
    #> ------------------------------------------------------------
    #> id: 4
    #>    y X1 X2 X3 X4 times lbnr__id Count1
    #> 19 1 0  0  0  0  1     1        0
    #> ------------------------------------------------------------
    #> id: 7
    #>    y X1 X2 X3 X4 times lbnr__id Count1
    #> 37 0 1  0  0  0  1     1        0
    #> 38 0 1  0  0  0  2     2        0
    #> 39 1 1  0  0  0  3     3        0

and a list of possible haplotypes for each id and their probabilities \(p\) (the probabilities within each id sum to 1):

    ghaplos <- haplo$ghaplos
    head(ghaplos)
    #>    id      haplo1      haplo2          p
    #> 1   1 DTGCGCTCGCG DTGAGCTCGCG 1.00000000
    #> 19  2 ITCCGTTGACG DTGAGCTCGCG 0.06867716
    #> 21  2 ITCAGTTGACG DTGCGCTCGCG 0.93132284
    #> 51  3 ITCCGTTGACG DTGAGCTCGCG 0.06867716
    #> 53  3 ITCAGTTGACG DTGCGCTCGCG 0.93132284
    #> 66  4 DTGCGCTCGCG DTGCGCTCGCG 1.00000000

The first id=1 has the haplotype fully observed, but id=2 has two possible haplotypes consistent with the observed genotype for this id, the probabilities are 7% and 93%, respectively.

With the baseline given above we can specify a regression design that gives an effect if a haplotype of the specified type is present (`sm=0`), or an additive effect of haplotypes (`sm=1`):

    designftypes <- function(x,sm=0) {
    hap1=x[1]
    hap2=x[2]
    if (sm==0) y <- 1*( (hap1==types) | (hap2==types))
    if (sm==1) y <- 1*(hap1==types) + 1*(hap2==types)
    return(y)
    }

To fit the model we first construct a time design matrix (named `X`) and supply the haplotype distributions for each id:

    haploX$time <- haploX$times
    Xdes <- model.matrix(~factor(time),haploX)
    colnames(Xdes) <- paste("X",1:ncol(Xdes),sep="")
    X <- dkeep(haploX,~id+y+time)
    X <- cbind(X,Xdes)
    Haplos <- dkeep(ghaplos,~id+"haplo*"+p)
    desnames=paste("X",1:6,sep="")   # six X's related to 6 cycles
    head(X)
    #>   id y time X1 X2 X3 X4 X5 X6
    #> 1  1 0    1  1  0  0  0  0  0
    #> 2  1 0    2  1  1  0  0  0  0
    #> 3  1 0    3  1  0  1  0  0  0
    #> 4  1 0    4  1  0  0  1  0  0
    #> 5  1 0    5  1  0  0  0  1  0
    #> 6  1 0    6  1  0  0  0  0  1

We can now fit the model with the design given by the design function:

    out <- haplo_surv_discrete(X=X,y="y",time.name="time",
          Haplos=Haplos,desnames=desnames,designfunc=designftypes)
    names(out$coef) <- c(desnames,types)
    out$coef
    #>          X1          X2          X3          X4          X5          X6
    #> -1.82153345 -0.61608261 -0.17143057 -1.27152045 -0.28635976 -0.19349091
    #> DCGCGCTCACG DTCCGCTGACG ITCAGTTGACG ITCCGCTGAGG
    #>  0.79753613  0.65747412  0.06119231  0.31666905
    summary(out)
    #>             Estimate Std.Err     2.5%   97.5%   P-value
    #> X1          -1.82153  0.1619 -2.13892 -1.5041 2.355e-29
    #> X2          -0.61608  0.1895 -0.98748 -0.2447 1.149e-03
    #> X3          -0.17143  0.1799 -0.52398  0.1811 3.406e-01
    #> X4          -1.27152  0.2631 -1.78719 -0.7559 1.346e-06
    #> X5          -0.28636  0.2030 -0.68425  0.1115 1.584e-01
    #> X6          -0.19349  0.2134 -0.61184  0.2249 3.647e-01
    #> DCGCGCTCACG  0.79754  0.1494  0.50465  1.0904 9.445e-08
    #> DTCCGCTGACG  0.65747  0.1621  0.33971  0.9752 5.007e-05
    #> ITCAGTTGACG  0.06119  0.2145 -0.35931  0.4817 7.755e-01
    #> ITCCGCTGAGG  0.31667  0.1361  0.04989  0.5834 1.999e-02

Haplotypes `"DCGCGCTCACG"` and `"DTCCGCTGACG"` give an increased hazard of pregnancy.

The data were generated with these true coefficients:

    tcoef=c(-1.93110204,-0.47531630,-0.04118204,-1.57872602,-0.22176426,-0.13836416,
    0.88830288,0.60756224,0.39802821,0.32706859)

    cbind(out$coef,tcoef)
    #>                               tcoef
    #> X1          -1.82153345 -1.93110204
    #> X2          -0.61608261 -0.47531630
    #> X3          -0.17143057 -0.04118204
    #> X4          -1.27152045 -1.57872602
    #> X5          -0.28635976 -0.22176426
    #> X6          -0.19349091 -0.13836416
    #> DCGCGCTCACG  0.79753613  0.88830288
    #> DTCCGCTGACG  0.65747412  0.60756224
    #> ITCAGTTGACG  0.06119231  0.39802821
    #> ITCCGCTGAGG  0.31666905  0.32706859

The fitted design matrix can be found in the output:

    head(out$X,10)
    #>    X1 X2 X3 X4 X5 X6 haplo1 haplo2 haplo3 haplo4
    #> 1   1  0  0  0  0  0      0      0      0      0
    #> 2   1  1  0  0  0  0      0      0      0      0
    #> 3   1  0  1  0  0  0      0      0      0      0
    #> 4   1  0  0  1  0  0      0      0      0      0
    #> 5   1  0  0  0  1  0      0      0      0      0
    #> 6   1  0  0  0  0  1      0      0      0      0
    #> 8   1  0  0  0  0  0      0      0      1      0
    #> 10  1  1  0  0  0  0      0      0      1      0
    #> 12  1  0  1  0  0  0      0      0      1      0
    #> 14  1  0  0  1  0  0      0      0      1      0

# SessionInfo

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /home/kkzh/.asdf/installs/r/4.6.0/lib/R/lib/libRblas.so
    #> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Europe/Copenhagen
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats     graphics  grDevices utils     datasets  methods   base
    #>
    #> other attached packages:
    #> [1] timereg_2.0.7  survival_3.8-6 mets_1.3.10
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] cli_3.6.6              knitr_1.51             rlang_1.2.0
    #>  [4] xfun_0.57              otel_0.2.0             future.apply_1.20.2
    #>  [7] listenv_0.10.1         lava_1.9.1             stats4_4.6.0
    #> [10] grid_4.6.0             evaluate_1.0.5         yaml_2.3.12
    #> [13] mvtnorm_1.3-7          numDeriv_2016.8-1.1    compiler_4.6.0
    #> [16] codetools_0.2-20       Rcpp_1.1.1-1.1         ucminf_1.2.3
    #> [19] future_1.70.0          lattice_0.22-9         digest_0.6.39
    #> [22] parallelly_1.47.0      parallel_4.6.0         splines_4.6.0
    #> [25] Matrix_1.7-5           tools_4.6.0            RcppArmadillo_15.2.6-1
    #> [28] globals_0.19.1