Author: Balasubramanian Narasimhan
Date: 2025-04-08

# Distributed Stratified Cox Regression using Homomorphic Computation

#### Balasubramanian Narasimhan

#### 2025-04-08

## Introduction

It is only a short way from the toy MLE example to a more useful example using Cox regression.

But first, we need the `survival` package and the `homomopheR` package.

    if (!require("survival")) {
        stop("this vignette requires the survival package")
    }
    library(homomorpheR)

We generate some simulated data for the purpose of this example. We will have three sites each with patient data (sizes 1000, 500 and 1500) respectively, containing

- `sex` (0, 1) for male/female
- `age` between 40 and 70
- a biomarker `bm`
- a `time` to some event of interest
- an indicator `event` which is 1 if an event was observed and 0 otherwise.

It is common to fit stratified models using sites as strata since the patient characteristics usually differ from site to site. So the baseline hazards (`lambdaT`) are different for each site but they share common coefficients (`beta.1`, `beta.2` and `beta.3` for `age`, `sex` and `bm` respy.) for the model. See (Terry M. Therneau and Patricia M. Grambsch 2000) by Therneau and Grambsch for details. So our model for each site \(i\) is

\[ S(t, age, sex, bm) = [S_0^i(t)]^{\exp(\beta_1 age + \beta_2 sex + \beta_3 bm)} \]

    sampleSize <- c(n1 = 1000, n2 = 500, n3 = 1500)

    set.seed(12345)

    beta.1 <- -.015; beta.2 <- .2; beta.3 <- .001;

    lambdaT <- c(5, 4, 3)
    lambdaC <- 2

    coxData <- lapply(seq_along(sampleSize),
                      function(i) {
                          sex <- sample(c(0, 1), size = sampleSize[i], replace = TRUE)
                          age <- sample(40:70, size = sampleSize[i], replace = TRUE)
                          bm <- rnorm(sampleSize[i])
                          trueTime <- rweibull(sampleSize[i],
                                               shape = 1,
                                               scale = lambdaT[i] * exp(beta.1 * age + beta.2 * sex + beta.3 * bm ))
                          censoringTime <- rweibull(sampleSize[i],
                                                    shape = 1,
                                                    scale = lambdaC)
                          time <- pmin(trueTime, censoringTime)
                          event <- (time == trueTime)
                          data.frame(stratum = i,
                                     sex = sex,
                                     age = age,
                                     bm = bm,
                                     time = time,
                                     event = event)
                      })

So here is a summary of the data for the three sites.

### Site 1

    str(coxData[[1]])

    ## 'data.frame':    1000 obs. of  6 variables:
    ##  $ stratum: int  1 1 1 1 1 1 1 1 1 1 ...
    ##  $ sex    : num  1 0 1 1 1 1 1 0 0 1 ...
    ##  $ age    : int  47 69 70 47 41 51 59 45 43 69 ...
    ##  $ bm     : num  -0.516 -1.375 1.01 0.454 0.275 ...
    ##  $ time   : num  1.37 0.95 2.35 2.48 1.93 ...
    ##  $ event  : logi  FALSE TRUE TRUE TRUE FALSE FALSE ...

### Site 2

    str(coxData[[2]])

    ## 'data.frame':    500 obs. of  6 variables:
    ##  $ stratum: int  2 2 2 2 2 2 2 2 2 2 ...
    ##  $ sex    : num  0 1 0 1 1 1 0 1 1 1 ...
    ##  $ age    : int  54 63 53 70 40 57 48 54 63 47 ...
    ##  $ bm     : num  -0.3243 0.2531 0.0464 0.8149 -0.1921 ...
    ##  $ time   : num  1.10483 0.34804 0.01602 0.68249 0.00157 ...
    ##  $ event  : logi  FALSE FALSE TRUE TRUE FALSE TRUE ...

### Site 3

    str(coxData[[3]])

    ## 'data.frame':    1500 obs. of  6 variables:
    ##  $ stratum: int  3 3 3 3 3 3 3 3 3 3 ...
    ##  $ sex    : num  1 0 0 1 1 1 0 1 0 1 ...
    ##  $ age    : int  55 70 49 60 44 42 58 62 61 68 ...
    ##  $ bm     : num  -0.9554 0.8138 0.0425 -1.2272 0.3244 ...
    ##  $ time   : num  0.0733 1.9869 2.2946 0.1231 1.0602 ...
    ##  $ event  : logi  TRUE FALSE FALSE TRUE FALSE FALSE ...

#

# Aggregated fit

If the data were all aggregated in one place, it would very simple to fit the model. Below, we row-bind the data from the three sites.

    aggModel <- coxph(formula = Surv(time, event) ~ sex +
                                    age + bm + strata(stratum),
                                data = do.call(rbind, coxData))
    aggModel

    ## Call:
    ## coxph(formula = Surv(time, event) ~ sex + age + bm + strata(stratum),
    ##     data = do.call(rbind, coxData))
    ##
    ##          coef exp(coef)  se(coef)      z       p
    ## sex -0.160493  0.851723  0.050627 -3.170 0.00152
    ## age  0.010057  1.010108  0.002835  3.547 0.00039
    ## bm  -0.005989  0.994029  0.025208 -0.238 0.81222
    ##
    ## Likelihood ratio test=22.82  on 3 df, p=4.413e-05
    ## n= 3000, number of events= 1575

Here `age` and `sex` are significant, but `bm` is not. The estimates \(\hat{\beta}\) are `(-0.180, .020, .007)`.

We can also print out the value of the (partial) log-likelihood at the MLE.

    aggModel$loglik

    ## [1] -9534.495 -9523.087

The first is the value at the parameter value `(0, 0, 0)` and the last is the value at the MLE.

## Distributed Computation

Assume now that the data `coxData` is distributed between three sites none of whom want to share actual data among each other or even with a master computation process. They wish to keep their data secret but are willing, together, to provide the sum of their local negative log-likelihoods. They need to do this in a way so that the master process will not be able to associate the contribution to the likelihood from each site.

The overall likelihood function \(l(\lambda)\) for the entire data is therefore the sum of the likelihoods at each site: \(l(\lambda) = l_1(\lambda)+l_2(\lambda)+l_3(\lambda).\) How can this likelihood be computed while maintaining privacy?

Assuming that every site including the master has access to a homomorphic computation library such as `homomorpheR`, the likelihood can be computed in a privacy-preserving manner using the following scheme. We use \(E(x)\) and \(D(x)\) to denote the encrypted and decrypted values of \(x\) respectively.

0. Master generates a public/private key pair. Master distributes the public key to all sites. (The private key is not distributed and kept only by the master.)
1. Master generates a random offset \(r\) to obfuscate the intial likelihood.
2. Master sends \(E(r)\) and a guess \(\lambda_0\) to site 1. Note that \(\lambda\) is not encrypted.
3. Site 1 computes \(l_1 = l(\lambda_0, y_1)\), the local likelihood for local data \(y_1\) using parameter \(\lambda_0\). It then sends on \(\lambda_0\) and \(E(r) + E(l_1)\) to site 2.
4. Site 2 computes \(l_2 = l(\lambda_0, y_2)\), the local likelihood for local data \(y_2\) using parameter \(\lambda_0\). It then sends on \(\lambda_0\) and \(E(r) + E(l_1) + E(l_2)\) to site 3.
5. Site 3 computes \(l_3 = l(\lambda_0, y_3)\), the local likelihood for local data \(y_3\) using parameter \(\lambda_0\). It then sends on \(E(r) + E(l_1) + E(l_2) + E(l_3)\) back to master.
6. Master retrieves \(E(r) + E(l_1) + E(l_2) + E(l_3)\) which, due to the homomorphism, is exactly \(E(r+l_1+l_2+l_3) = E(r+l).\) So the master computes \(D(E(r+l)) - r\) to obtain the value of the overall likelihood at \(\lambda_0\).
7. Master updates \(\lambda_0\) with a new guess \(\lambda_1\) and repeats steps 1-5. This process is iterated to convergence. For added security, even steps 0-5 can be repeated, at additional computational cost.

This is pictorially shown below.

## Implementation

The above implementation assumes that the encryption and decryption can happen with real numbers which is not the actual situation. Instead, we use rational approximations using a large denominator, \(2^{256}\), say. In the future, of course, we need to build an actual library is built with rigorous algorithms guaranteeing precision and overflow/undeflow detection. For now, this is just an ad hoc implementation.

Also, since we are only using homomorphic additive properties, a partial homomorphic scheme such as the Paillier Encryption system will be sufficient for our computations.

We define a class to encapsulate our sites that will compute the Poisson likelihood on site data given a parameter \(\lambda\). Note how the `addNLLAndForward` method takes care to split the result into an integer and fractional part while performing the arithmetic operations. (The latter is approximated by a rational number.)

We define a class to encapsulate our sites that will compute the partial log likelihood on site data given a parameter \(\beta\).

In the code below, we exploit, for expository purposes, a feature of `coxph`: a control parameter can be passed to evaluate the partial likelihood at a given \(\beta\) value.

    Site <- R6::R6Class("Site",
                        private = list(
                            ## name of the site
                            name = NA,
                            ## only master has this, NA for workers
                            privkey = NA,
                            ## local data
                            data = NA,
                            ## The next site in the communication: NA for master
                            nextSite = NA,
                            ## is this the master site?
                            iAmMaster = FALSE,
                            ## intermediate result variable
                            intermediateResult = NA,
                            ## Control variable for cox regression
                            cph.control = NA
                        ),
                        public = list(
                            count = NA,
                            ## Common denominator for approximate real arithmetic
                            den = NA,
                            ## The public key; everyone has this
                            pubkey = NA,
                            initialize = function(name, data, den) {
                                private$name <- name
                                private$data <- data
                                self$den <- den
                                private$cph.control <- replace(coxph.control(), "iter.max", 0)
                            },
                            setPublicKey = function(pubkey) {
                                self$pubkey <- pubkey
                            },
                            setPrivateKey = function(privkey) {
                                private$privkey <- privkey
                            },
                            ## Make me master
                            makeMeMaster = function() {
                                private$iAmMaster <- TRUE
                            },
                            ## add neg log lik and forward to next site
                            addNLLAndForward = function(beta, enc.offset) {
                                if (private$iAmMaster) {
                                    ## We are master, so don't forward
                                    ## Just store intermediate result and return
                                    private$intermediateResult <- enc.offset
                                } else {
                                    ## We are workers, so add and forward
                                    ## add negative log likelihood and forward result to next site
                                    ## Note that offset is encrypted
                                    nllValue <- self$nLL(beta)
                                    result.int <- floor(nllValue)
                                    result.frac <- nllValue - result.int
                                    result.fracnum <- gmp::as.bigq(gmp::numerator(gmp::as.bigq(result.frac) * self$den))
                                    pubkey <- self$pubkey
                                    enc.result.int <- pubkey$encrypt(result.int)
                                    enc.result.fracnum <- pubkey$encrypt(result.fracnum)
                                    result <- list(int = pubkey$add(enc.result.int, enc.offset$int),
                                                   frac = pubkey$add(enc.result.fracnum, enc.offset$frac))
                                    private$nextSite$addNLLAndForward(beta, enc.offset = result)
                                }
                                ## Return a TRUE result for now.
                                TRUE
                            },
                            ## Set the next site in the communication graph
                            setNextSite = function(nextSite) {
                                private$nextSite <- nextSite
                            },
                            ## The negative log likelihood
                                nLL = function(beta) {
                                if (private$iAmMaster) {
                                    ## We're master, so need to get result from sites
                                    ## 1. Generate a random offset and encrypt it
                                    pubkey <- self$pubkey
                                    offset <- list(int = random.bigz(nBits = 256),
                                                   frac = random.bigz(nBits = 256))
                                    enc.offset <- list(int = pubkey$encrypt(offset$int),
                                                       frac = pubkey$encrypt(offset$frac))
                                    ## 2. Send off to next site
                                    throwaway <- private$nextSite$addNLLAndForward(beta, enc.offset)
                                    ## 3. When the call returns, the result will be in
                                    ##    the field intermediateResult, so decrypt that.
                                    sum <- private$intermediateResult
                                    privkey <- private$privkey
                                    intResult <- as.double(privkey$decrypt(sum$int) - offset$int)
                                    fracResult <- as.double(gmp::as.bigq(privkey$decrypt(sum$frac) - offset$frac) / den)
                                    intResult + fracResult
                                } else {
                                    ## We're worker, so compute local negative log likelihood
                                    tryCatch({
                                        m <- coxph(formula = Surv(time, event) ~ sex + age + bm,
                                                             data = private$data,
                                                             init = beta,
                                                             control = private$cph.control)
                                        -(m$loglik[1])
                                    },
                                    error = function(e) NA)
                                }
                            })
                        )

We are now ready to use our sites in the computation.

### 1. Generate public and private key pair

We also choose a denominator for all our rational approximations.

    keys <- PaillierKeyPair$new(1024) ## Generate new public and private key.
    den <- gmp::as.bigq(2)^256  #Our denominator for rational approximations

### 2. Create sites

    site1 <- Site$new(name = "Site 1", data = coxData[[1]], den = den)
    site2 <- Site$new(name = "Site 2", data = coxData[[2]], den = den)
    site3 <- Site$new(name = "Site 3", data = coxData[[3]], den = den)

The master process is also a site but has no data. So has to be thus designated.

    ## Master has no data!
    master <- Site$new(name = "Master", data = c(), den = den)
    master$makeMeMaster()

### 2. Distribute public key to sites

    site1$setPublicKey(keys$pubkey)
    site2$setPublicKey(keys$pubkey)
    site3$setPublicKey(keys$pubkey)
    master$setPublicKey(keys$pubkey)

Only master has private key for decryption.

    master$setPrivateKey(keys$getPrivateKey())

### 3. Define the communication graph

Master will always send to the first site, and then the others have to forward results in turn with the last site returning to the master.

    master$setNextSite(site1)
    site1$setNextSite(site2)
    site2$setNextSite(site3)
    site3$setNextSite(master)

### 4. Perform the likelihood estimation

    library(stats4)
    nll <- function(age, sex, bm) master$nLL(c(age, sex, bm))
    fit <- mle(nll, start = list(age = 0, sex = 0, bm = 0))

### 5. Compare the results

The summary will show the results.

    summary(fit)

    ## Maximum likelihood estimation
    ##
    ## Call:
    ## mle(minuslogl = nll, start = list(age = 0, sex = 0, bm = 0))
    ##
    ## Coefficients:
    ##         Estimate  Std. Error
    ## age -0.160493329 0.050626611
    ## sex  0.010057265 0.002835374
    ## bm  -0.005988214 0.025208370
    ##
    ## -2 log L: 19046.17

Note how the estimated coefficients and standard errors closely match the full model summary below.

    summary(aggModel)

    ## Call:
    ## coxph(formula = Surv(time, event) ~ sex + age + bm + strata(stratum),
    ##     data = do.call(rbind, coxData))
    ##
    ##   n= 3000, number of events= 1575
    ##
    ##          coef exp(coef)  se(coef)      z Pr(>|z|)
    ## sex -0.160493  0.851723  0.050627 -3.170  0.00152 **
    ## age  0.010057  1.010108  0.002835  3.547  0.00039 ***
    ## bm  -0.005989  0.994029  0.025208 -0.238  0.81222
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##     exp(coef) exp(-coef) lower .95 upper .95
    ## sex    0.8517      1.174    0.7713    0.9406
    ## age    1.0101      0.990    1.0045    1.0157
    ## bm     0.9940      1.006    0.9461    1.0444
    ##
    ## Concordance= 0.536  (se = 0.009 )
    ## Likelihood ratio test= 22.82  on 3 df,   p=4e-05
    ## Wald test            = 22.81  on 3 df,   p=4e-05
    ## Score (logrank) test = 22.85  on 3 df,   p=4e-05

And the log likelihood of the distributed homomorphic fit also matches that of the model on aggregated data:

    cat(sprintf("logLik(MLE fit): %f, logLik(Agg. fit): %f.\n", logLik(fit), aggModel$loglik[2]))

    ## logLik(MLE fit): -9523.087001, logLik(Agg. fit): -9523.087001.

## References

Terry M. Therneau, and Patricia M. Grambsch. 2000. Modeling Survival Data: Extending the Cox Model. New York: Springer-Verlag.