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) = [S₀ⁱ(t)]^{exp (β₁age + β₂sex + β₃bm)}

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 β̂ 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 initial parameter value (0, 0, 0) and the second 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 wish to do this in a manner so that the master process is unable to associate the contribution to the likelihood from each site.

The overall likelihood function l(β) for the entire data is the sum of the likelihoods at each site: l(β) = l₁(β) + l₂(β) + l₃(β). How can this likelihood be computed while preventing the master from knowing the individual contributions of each site?

The key to ensuring that each site will not reveal the actual value l_i(β) for any β involves the use of two non-cooperating parties, say, NCP1 and NCP2. Site i sends E(l_i(β) + r_i) to NCP1 and E(l_i(β) − r_i) to NCP2, where E(x) denotes the encrypted value of x and r_i is a random quantity generated anew for each site and each β. NCP1 can compute $\sum_{i=1}^3E(l_i + r_i)$ and NCP2 can compute $\sum_{i=1}^3E(l_i - r_i)$, but individually, neither has a handle on $l = \sum_{i=1}^3 l_i$.

The master process can retrieve $\sum_{i=1}^3E(l_i + r_i)$ and $\sum_{i=1}^3E(l_i - r_i)$ from NCP1 and NCP2 respectively. Each is an encrypted value of the sum of the likelihood contributions from all sites, obfuscated by a random term, and hence is random to the master. However, the master using the associative and homomorphic properties of E(.), can compute:

$$ \sum_{i=1}^3E(l_i + r_i) +\sum_{i=1}^3E(l_i - r_i) = \sum_{i=1}^3E(l_i + r_i + l_i - r_i) = \sum_{i=1}^3E(2l_i) = E(2l) $$

since l = l₁ + l₂ + l₃. The master can now decrypt the result and obtain 2l!

This is pictorially shown below.

The red arrows show the master proposing a value β to each of the sites, which reply back to NCP1 and NCP2. The master then retrieves the values from NCP1 and NCP2 and sums them.

A Modified Topology

The drawback of the above scheme is that channels of communication have to be established from each site to the master process and also to the two non-cooperating parties NCP1 and NCP2. If the number of participating sites in a computation changes, then both the master and NCP1 and NCP2 have to be made aware of the change.

It would be simpler if only NCP1 and NCP2 can talk to both the master and the sites. Such a situation would arise, for example, when the sites are all participating in a disease specific registry. The parties NCP1 and NCP2 would probably be set up once and any new site that has to be onboarded needs only to be known to P1 and P2. This has the added advantage of hiding the number of sites, which could even be 1!

Such a communication topology would mean that the β values have be funneled to the sites through NCP1 and NCP2 and that can be easily accomplished. The picture below shows this configuration and looks more complicated than it actually is.

To summarize, the modified scheme has several characteristics:

The master only communicates with NCP1 and NCP2
NCP1 and NCP2 are the only parties communicating with both the master and sites
NCP1 and NCP2 are the only ones that know how many sites are participating
New sites can be added and only NCP1 and NCP2 need to account for them while the master remains oblivous to the number of sites; so the scheme works even with one site
It appears that there is unnecessary communication of the same information, i.e. β is being sent twice to each site from each of the NCP1 and NCP2. This is easily mitigated by engineering either by using a broker between NCP1 and NCP2, or the sites caching their results for a short period to avoid recomputation.

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, 2²⁵⁶, 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 classes to encapsulate our sites, non-cooperating parties and a master process.

The Site Class

Our site class will compute the partial log likelihood on site data given a parameter β. Note how the private nll 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.)

In the code below, we exploit a feature of coxph: a control parameter can be passed to evaluate the partial likelihood at a given β value. We also use a cache so that we can distribute each piece of the encrypted likelihood E(l_i − r_i) and E(l_i + r_i) to the two non-cooperating parties.

Site <-
    R6::R6Class(
            "Site",
            private = list(
                ## name of the site
                name = NA,
                ## local data
                data = NA,
                ## Control variable for cox regression
                cph.control = NA,
                beta_cache = list(),
                local_nll = function(beta) {
                    ## Check if value is cached
                    beta_hash  <- paste0("b", digest::digest(beta, algo = "xxhash64"))
                    result  <- private$beta_cache[[beta_hash]]
                    if (is.null(result)) {
                        ## We're worker, so compute local negative log likelihood
                        nllValue  <- tryCatch({
                            m <- coxph(formula = Surv(time, event) ~ sex + age + bm,
                                       data = private$data,
                                       init = beta,
                                       control = private$cph.control)
                            -(m$loglik[1])
                        },
                        error = function(e) NA)
                        if (!is.na(nllValue)) {
                            pubkey <- self$pubkey
                            ## Generate random offset for int and frac parts
                            offset <- list(int = random.bigz(nBits = 256),
                                           frac = random.bigz(nBits = 256))
                            ## 2. Add to neg log likelihood
                            result.int <- floor(nllValue)
                            result.frac <- nllValue - result.int
                            ## Approximate fractional part by a rational
                            result.fracnum <- gmp::as.bigz(gmp::numerator(gmp::as.bigq(result.frac) * self$den))
                            result  <- list(
                                int1 = pubkey$encrypt(result.int - offset$int),
                                frac1 = pubkey$encrypt(result.fracnum - offset$frac),
                                int2 = pubkey$encrypt(result.int + offset$int),
                                frac2 = pubkey$encrypt(result.fracnum + offset$frac)
                            )
                            private$beta_cache[[beta_hash]]  <- result
                        } else {
                            result  <- list(int1 = NA, frac1 = NA, int2 = NA, frac2 = NA)
                        }
                    }
                    result
                }
            ),
            public = list(
                count = NA,
                ## Common denominator for approximate real arithmetic
                den = NA,
                ## The master's public key; everyone has this
                pubkey = NA,
                initialize = function(name, data) {
                    private$name <- name
                    private$data <- data
                    private$cph.control <- replace(coxph.control(), "iter.max", 0)
                },
                setPublicKey = function(pubkey) {
                    self$pubkey <- pubkey
                },
                setDenominator = function(den) {
                    self$den = den
                },
                ## neg log lik,
                nll = function(beta, party) {
                    result  <- private$local_nll(beta)
                    if (party == 1) {
                        list(int = result$int1, frac = result$frac1)
                    } else {
                        list(int = result$int2, frac = result$frac2)
                    }
                }
            )
        )

The Non-cooperating Parties Class

The non-cooperating parties can communicate with the sites. So they have methods for adding sites, passing on public keys from the master etc. The nll method for this class merely calls each site to compute the result and adds them up before sending it on to the master, so that the master has no idea of the individual contributions.

NCParty <-
    R6::R6Class(
            "NCParty",
            private = list(
                ## name of the site
                name = NA,
                ## NC party number
                number = NA,
                ## The master
                master = NA,
                ## The sites
                sites = list()
            ),
            public = list(
                ## The master's public key; everyone has this
                pubkey = NA,
                ## The denoinator for rational arithmetic
                den = NA,
                initialize = function(name, number) {
                    private$name <- name
                    private$number  <- number
                },
                setPublicKey = function(pubkey) {
                    self$pubkey <- pubkey
                    ## Propagate to sites
                    for (site in sites) {
                        site$setPublicKey(pubkey)
                    }
                },
                setDenominator = function(den) {
                    self$den <- den
                    ## Propagate to sites
                    for (site in sites) {
                        site$setDenominator(den)
                    }
                },
                addSite = function(site) {
                    private$sites  <- c(private$sites, list(site))
                },
                ## neg log lik
                nll = function(beta) {
                    pubkey  <- self$pubkey
                    results  <- lapply(sites, function(x) x$nll(beta, private$number))
                    ## Accumulate the integer and fractional parts
                    n  <- length(results)
                    sumInt  <- results[[1L]]$int
                    sumFrac <- results[[1L]]$frac
                    for (i in 2:n) {
                        sumInt  <- pubkey$add(sumInt, results[[i]]$int)
                        sumFrac  <- pubkey$add(sumFrac, results[[i]]$frac)
                    }
                    list(int = sumInt, frac = sumFrac)
                }
            )
        )

The Master Class

The master process

Master  <-
    R6::R6Class(
            "Master",
            private = list(
                ## name of the site
                name = NA,
                ## Private and public keys
                keys = NA,
                ## Non cooperating party 1
                nc_party_1 = NA,
                ## Non cooperating party 2
                nc_party_2 = NA
            ),
            public = list(
                ## Denominator for rational arithmetic
                den  = NA,
                initialize = function(name) {
                    private$name <- name
                    private$keys <- PaillierKeyPair$new(1024) ## Generate new public and private key.
                    self$den <- gmp::as.bigq(2)^256  #Our denominator for rational approximations
                },
                setNCParty1  = function(site) {
                    private$nc_party_1 <- site
                    private$nc_party_1$setPublicKey(private$keys$pubkey)
                    private$nc_party_1$setDenominator(self$den)
                },
                setNCParty2  = function(site) {
                    private$nc_party_2 <- site
                    private$nc_party_2$setPublicKey(private$keys$pubkey)
                    private$nc_party_2$setDenominator(self$den)
                },
                ## neg log lik
                nLL = function(beta) {
                    pubkey  <- private$keys$pubkey
                    privkey  <- private$keys$getPrivateKey()
                    result1  <- private$nc_party_1$nll(beta)
                    result2  <- private$nc_party_2$nll(beta)
                    ## Accumulate the integer and fractional parts
                    sumInt  <- pubkey$add(result1$int, result2$int)
                    sumFrac  <- pubkey$add(result1$frac, result2$frac)
                    intResult <- as.double(privkey$decrypt(sumInt))
                    fracResult <- as.double(gmp::as.bigq(privkey$decrypt(sumFrac)) / self$den)
                    ## Since we 2L, we divide by 2.
                    (intResult + fracResult) / 2.0
                }
            )
        )

Example

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

1. Create sites

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

sites  <- list(site1 = site1, site2 = site2, site3 = site3)

2. Create Non-cooperating parties

ncp1  <- NCParty$new("NCP1", 1)
ncp2  <- NCParty$new("NCP1", 2)

We add sites to the non-cooperating parties.

for (s in sites) {
    ncp1$addSite(s)
    ncp2$addSite(s)
}

3. Create the master process

master  <- Master$new("Master")

We next connect the master to the non-cooperating parties.

master$setNCParty1(ncp1)
master$setNCParty2(ncp2)

At this point the communication graph has been defined between the master and non-cooperating parties and the non-cooperating parties and the sites.

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.

Distributed Stratified Cox Regression using Non-Cooperating Parties