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/femaleagebetween 40 and 70- a biomarker
bm - a
timeto some event of interest - an indicator
eventwhich 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= 1575Here 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 -9523.087The 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.
- 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.)
- Master generates a random offset \(r\) to obfuscate the intial likelihood.
- Master sends \(E(r)\) and a guess \(\lambda_0\) to site 1. Note that \(\lambda\) is not encrypted.
- 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.
- 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.
- 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.
- 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 \(\lambda_0\).
- 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$new(1024) ## Generate new public and private key.
den <- gmp::as.bigq(2)^256 #Our denominator for rational approximations2. Create sites
site1 <- Site$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$nLL(c(age, sex, bm))
fit <- mle(nll, start = list(age = 0, sex = 0, bm = 0))5. Compare the results
The summary will show the results.
summary(fit)## Maximum likelihood estimation
##
## Call:
## mle(minuslogl = nll, start = list(age = 0, sex = 0, bm = 0))
##
## Coefficients:
## Estimate Std. Error
## age -0.160493329 0.050626611
## sex 0.010057265 0.002835374
## bm -0.005988214 0.025208370
##
## -2 log L: 19046.17Note how the estimated coefficients and standard errors closely match the full model summary below.
summary(aggModel)## Call:
## coxph(formula = Surv(time, event) ~ sex + age + bm + strata(stratum),
## data = do.call(rbind, coxData))
##
## n= 3000, number of events= 1575
##
## coef exp(coef) se(coef) z Pr(>|z|)
## sex -0.160493 0.851723 0.050627 -3.170 0.00152 **
## age 0.010057 1.010108 0.002835 3.547 0.00039 ***
## bm -0.005989 0.994029 0.025208 -0.238 0.81222
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## exp(coef) exp(-coef) lower .95 upper .95
## sex 0.8517 1.174 0.7713 0.9406
## age 1.0101 0.990 1.0045 1.0157
## bm 0.9940 1.006 0.9461 1.0444
##
## Concordance= 0.536 (se = 0.009 )
## Likelihood ratio test= 22.82 on 3 df, p=4e-05
## Wald test = 22.81 on 3 df, p=4e-05
## Score (logrank) test = 22.85 on 3 df, p=4e-05And the log likelihood of the distributed homomorphic fit also matches that of the model on aggregated data:
cat(sprintf("logLik(MLE fit): %f, logLik(Agg. fit): %f.\n", logLik(fit), aggModel$loglik[2]))## logLik(MLE fit): -9523.087001, logLik(Agg. fit): -9523.087001.