# BayesSUR with random effects

#### Zhi Zhao

The BayesSUR model has been extended to include mandatory variables by assigning Gaussian priors as random effects rather than spike-and-slab priors, named as SSUR-MRF with random effects in Zhao et al. 2023. The R code for the simulated data and real data analyses in Zhao et al. 2023 can be found at the GitHub repository BayesSUR-RE.

Here, we show some small examples to run the BayesSUR mdoel with random effects. To get started, load the package with

    library("BayesSUR")

## Simulate data

We design a network as the following figure (a) to construct a complex structure between \(20\) response variables and \(300\) predictors. It assumes that the responses are divided into six groups, and the first \(120\) predictors are divided into nine groups.

Simulation scenarios: True relationships between response variables and predictors. (a) Network structure between \(\mathbf Y\) and \(\mathbf X\). (b) Spare latent indicator variable \(\Gamma\) for the associations between \(\mathbf Y\) and \(\mathbf X\) in the SUR model. Black blocks indicate nonzero coefficients and white blocks indicate zero coefficients. (c) Additional structure in the residual covariance matrix between response variables not explained by \(\mathbf X\mathbf B\). Black blocks indicate correlated residuals of the corresponding response variables and white blocks indicate uncorrelated residuals of the corresponding response variables.

Load the simulation function `sim.ssur()` as follows.

    library("gRbase")
    sim.ssur <- function(n, s, p, t0 = 0, seed = 123, mv = TRUE,
                         t.df = Inf, random.intercept = 0, intercept = TRUE) {
      # set seed to fix coefficients
      set.seed(7193)
      sd_b <- 1
      mu_b <- 1
      b <- matrix(rnorm((p + ifelse(t0 == 0, 1, 0)) * s, mu_b, sd_b), p + ifelse(t0 == 0, 1, 0), s)

      # design groups and pathways of Gamma matrix
      gamma <- matrix(FALSE, p + ifelse(t0 == 0, 1, 0), s)
      if (t0 == 0) gamma[1, ] <- TRUE
      gamma[2:6 - ifelse(t0 == 0, 0, 1), 1:5] <- TRUE
      gamma[11:21 - ifelse(t0 == 0, 0, 1), 6:12] <- TRUE
      gamma[31:51 - ifelse(t0 == 0, 0, 1), 1:5] <- TRUE
      gamma[31:51 - ifelse(t0 == 0, 0, 1), 13:15] <- TRUE
      gamma[52:61 - ifelse(t0 == 0, 0, 1), 1:12] <- TRUE
      gamma[71:91 - ifelse(t0 == 0, 0, 1), 6:15] <- TRUE
      gamma[111:121 - ifelse(t0 == 0, 0, 1), 1:15] <- TRUE
      gamma[122 - ifelse(t0 == 0, 0, 1), 16:18] <- TRUE
      gamma[123 - ifelse(t0 == 0, 0, 1), 19] <- TRUE
      gamma[124 - ifelse(t0 == 0, 0, 1), 20] <- TRUE

      G_kron <- matrix(0, s * p, s * p)
      G_m <- bdiag(matrix(1, ncol = 5, nrow = 5),
                   matrix(1, ncol = 7, nrow = 7),
                   matrix(1, ncol = 8, nrow = 8))
      G_p <- bdiag(matrix(1, ncol = 5, nrow = 5), diag(3),
                   matrix(1, ncol = 11, nrow = 11), diag(9),
                   matrix(1, ncol = 21, nrow = 21),
                   matrix(1, ncol = 10, nrow = 10), diag(9),
                   matrix(1, ncol = 21, nrow = 21), diag(19),
                   matrix(1, ncol = 11, nrow = 11), diag(181))
      G_kron <- kronecker(G_m, G_p)

      combn11 <- combn(rep((1:5 - 1) * p, each = length(1:5)) +
                         rep(1:5, times = length(1:5)), 2)
      combn12 <- combn(rep((1:5 - 1) * p, each = length(30:60)) +
                         rep(30:60, times = length(1:5)), 2)
      combn13 <- combn(rep((1:5 - 1) * p, each = length(110:120)) +
                         rep(110:120, times = length(1:5)), 2)
      combn21 <- combn(rep((6:12 - 1) * p, each = length(10:20)) +
                         rep(10:20, times = length(6:12)), 2)
      combn22 <- combn(rep((6:12 - 1) * p, each = length(51:60)) +
                         rep(51:60, times = length(6:12)), 2)
      combn23 <- combn(rep((6:12 - 1) * p, each = length(70:90)) +
                         rep(70:90, times = length(6:12)), 2)
      combn24 <- combn(rep((6:12 - 1) * p, each = length(110:120)) +
                         rep(110:120, times = length(6:12)), 2)
      combn31 <- combn(rep((13:15 - 1) * p, each = length(30:50)) +
                         rep(30:50, times = length(13:15)), 2)
      combn32 <- combn(rep((13:15 - 1) * p, each = length(70:90)) +
                         rep(70:90, times = length(13:15)), 2)
      combn33 <- combn(rep((13:15 - 1) * p, each = length(110:120)) +
                         rep(110:120, times = length(13:15)), 2)
      combn4 <- combn(rep((16:18 - 1) * p, each = length(121)) +
                        rep(121, times = length(16:18)), 2)
      combn5 <- matrix(rep((19 - 1) * p, each = length(122)) +
                         rep(122, times = length(19)), nrow = 1, ncol = 2)
      combn6 <- matrix(rep((20 - 1) * p, each = length(123)) +
                         rep(123, times = length(20)), nrow = 1, ncol = 2)

      combnAll <- rbind(t(combn11), t(combn12), t(combn13),
                        t(combn21), t(combn22), t(combn23), t(combn24),
                        t(combn31), t(combn32), t(combn33),
                        t(combn4), combn5, combn6)

      set.seed(seed + 7284)
      sd_x <- 1
      x <- matrix(rnorm(n * p, 0, sd_x), n, p)

      if (t0 == 0 & intercept) x <- cbind(rep(1, n), x)
      if (!intercept) {
        gamma <- gamma[-1, ]
        b <- b[-1, ]
      }
      xb <- matrix(NA, n, s)
      if (mv) {
        for (i in 1:s) {
          if (sum(gamma[, i]) >= 1) {
            if (sum(gamma[, i]) == 1) {
              xb[, i] <- x[, gamma[, i]] * b[gamma[, i], i]
            } else {
              xb[, i] <- x[, gamma[, i]] %*% b[gamma[, i], i]
            }
          } else {
            xb[, i] <- sapply(1:s, function(i) rep(1, n) * b[1, i])
          }
        }
      } else {
        if (sum(gamma) >= 1) {
          xb <- x[, gamma] %*% b[gamma, ]
        } else {
          xb <- sapply(1:s, function(i) rep(1, n) * b[1, i])
        }
      }

      corr_param <- 0.9
      M <- matrix(corr_param, s, s)
      diag(M) <- rep(1, s)

      ## wanna make it decomposable
      Prime <- list(c(1:(s * .4), (s * .8):s),
                    c((s * .4):(s * .6)),
                    c((s * .65):(s * .75)),
                    c((s * .8):s))
      G <- matrix(0, s, s)
      for (i in 1:length(Prime)) {
        G[Prime[[i]], Prime[[i]]] <- 1
      }

      # check
      dimnames(G) <- list(1:s, 1:s)
      length(gRbase::mcsMAT(G - diag(s))) > 0

      var <- solve(BDgraph::rgwish(n = 1, adj = G, b = 3, D = M))

      # change seeds to add randomness on error
      set.seed(seed + 8493)
      sd_err <- 0.5
      if (is.infinite(t.df)) {
        err <- matrix(rnorm(n * s, 0, sd_err), n, s) %*% chol(as.matrix(var))
      } else {
        err <- matrix(rt(n * s, t.df), n, s) %*% chol(as.matrix(var))
      }

      if (t0 == 0) {
        b.re <- NA
        z <- NA
        y <- xb + err
        if (random.intercept != 0) {
          y <- y + matrix(rnorm(n * s, 0, sqrt(random.intercept)), n, s)
        }

        z <- sample(1:4, n, replace = T, prob = rep(1 / 4, 4))

        return(list(y = y, x = x, b = b, gamma = gamma, z = model.matrix(~ factor(z) + 0)[, ],
                    b.re = b.re, Gy = G, mrfG = combnAll))
      } else {
        # add random effects
        z <- t(rmultinom(n, size = 1, prob = c(.1, .2, .3, .4)))
        z <- sample(1:t0, n, replace = T, prob = rep(1 / t0, t0))
        set.seed(1683)
        b.re <- rnorm(t0, 0, 2)
        y <- matrix(b.re[z], nrow = n, ncol = s) + xb + err

        return(list(
          y = y, x = x, b = b, gamma = gamma, z = model.matrix(~ factor(z) + 0)[, ],
          b.re = b.re, Gy = G, mrfG = combnAll
        ))
      }
    }

To simulate data with sample size \(n=250\), responsible variables \(s=20\) and covariates \(p=300\), we can specify the corresponding parameters in the function `sim.ssur()` as follows.

    library("Matrix")
    n <- 250
    s <- 20
    p <- 300
    sim1 <- sim.ssur(n, s, p, seed = 1)

To simulate data from \(4\) individual groups with group indicator variables following the defaul multinomial distribution \(multinomial(0.1,0.2,0.3,0.4)\), we can simply add the argument `t0 = 4` in the function `sim.ssur()` as follows.

    t0 <- 4
    sim2 <- sim.ssur(n, s, p, t0, seed = 1) # learning data
    sim2.val <- sim.ssur(n, s, p, t0, seed=101) # validation data

## Run BayesSUR model with random effects

According to the guideline of prior specification in Zhao et al. 2023, we first set the following parameters `hyperpar` and then running the BayesSUR model with random effects via `betaPrior = "reGroup"` (default `betaPrior = "independent"` with spike-and-slab priors for all coefficients). For illustration, we run a short MCMC with `nIter = 300` and `burnin = 100`. Note that here the graph used for the Markov random field prior is the true graph from the returned object of the simulation `sim2$mrfG`.

    hyperpar <- list(mrf_d = -2, mrf_e = 1.6, a_w0 = 100, b_w0 = 500, a_w = 15, b_w = 60)
    set.seed(1038)
    fit2 <- BayesSUR(
      data = cbind(sim2$y, sim2$z, sim2$x),
      Y = 1:s,
      X_0 = s + 1:t0,
      X = s + t0 + 1:p,
      outFilePath = "sim2_mrf_re",
      hyperpar = hyperpar,
      gammaInit = "0",
      betaPrior = "reGroup",
      nIter = 300, burnin = 100,
      covariancePrior = "HIW",
      standardize = F,
      standardize.response = F,
      gammaPrior = "MRF",
      mrfG = sim2$mrfG,
      output_CPO = T
    )

    ## BayesSUR -- Bayesian Seemingly Unrelated Regression Modelling
    ## Reading input files ... ... successfull!
    ## Clearing and initialising output files
    ## Initialising the (SUR) MCMC Chain ...  ...  DONE!
    ## Drafting the output files with the start of the chain ... DONE!
    ##
    ## Starting 2 (parallel) chain(s) for 300 iterations:
    ## Temperature ladder updated, new temperature ratio : 1.1
    ##  MCMC ends.   --- Saving results and exiting
    ## Saved to :   sim2_mrf_re/data_SSUR_****_out.txt
    ## Final w0  : 4.9971
    ## Final w   : 2.29497
    ## Final tau : 0.293487    w/ proposal variance: 1.2229
    ## Final eta : 0.0471505
    ##   -- Average Omega : 0
    ## Final temperature ratio : 1.1
    ##
    ## DONE, exiting!

Check some summarized information of the results:

    summary(fit2)

    ## Call:
    ##   BayesSUR(data = cbind(sim2$y, sim2$z, sim2$x), ...)
    ##
    ## CPOs:
    ##         Min.      1st Qu.       Median      3rd Qu.         Max.
    ## 0.0001896996 0.0242732651 0.0348570615 0.0465600279 0.1312571329
    ##
    ## Number of selected predictors (mPIP > 0.5): 19 of 20x300
    ##
    ## Top 10 predictors on average mPIP across all responses:
    ##     X.74     X.69     X.77     X.82    X.114    X.116    X.122    X.157    X.265
    ## 0.081840 0.049500 0.049500 0.049500 0.049500 0.049500 0.049500 0.049500 0.049500
    ##     X.16
    ## 0.047015
    ##
    ## Top 10 responses on average mPIP across all predictors:
    ##         X.8        X.10         X.5        X.19        X.11         X.9         X.6
    ## 0.012703000 0.010049000 0.007943333 0.006600000 0.005522333 0.004029667 0.003001667
    ##        X.12         X.2         X.4
    ## 0.002852333 0.002769667 0.002404667
    ##
    ## Expected log pointwise predictive density (elpd) estimates:
    ##   elpd.LOO = -16803.49,  elpd.WAIC = -16799.94
    ##
    ## MCMC specification:
    ##   iterations = 300,  burn-in = 100,  chains = 2
    ##   gamma local move sampler: bandit
    ##   gamma initialisation: 0
    ##
    ## Model specification:
    ##   covariance prior: HIW
    ##   gamma prior: MRF
    ##
    ## Hyper-parameters:
    ##   a_w   b_w    nu a_tau b_tau a_eta b_eta mrf_d mrf_e  a_w0  b_w0
    ##  15.0  60.0  22.0   0.1  10.0   0.1   1.0  -2.0   1.6 100.0 500.0

Show the estimates of regression coefficients, variable selection indicators and residual graph

    # show estimates
    plot(fit2, estimator=c("beta","gamma","Gy"), type="heatmap", name.predictors = "auto")

Simulation result: Estimates of regression coefficients, variable selection indicators and residual graph.

Compute the model performance with respect to variable selection

    # compute accuracy, sensitivity, specificity of variable selection
    gamma <- getEstimator(fit2)
    (accuracy <- sum(data.matrix(gamma > 0.5) == sim2$gamma) / prod(dim(gamma)))

    ## [1] 0.8725

    (sensitivity <- sum((data.matrix(gamma > 0.5) == 1) & (sim2$gamma == 1)) / sum(sim2$gamma == 1))

    ## [1] 0.01558442

    (specificity <- sum((data.matrix(gamma > 0.5) == 0) & (sim2$gamma == 0)) / sum(sim2$gamma == 0))

    ## [1] 0.9986616

Compute the model performance with respect to response prediction

    # compute RMSE and RMSPE for prediction performance
    beta <- getEstimator(fit2, estimator = "beta", Pmax = .5, beta.type = "conditional")
    (RMSE <- sqrt(sum((sim2$y - cbind(sim2$z, sim2$x) %*% beta)^2) / prod(dim(sim2$y))))

    ## [1] 8.723454

    (RMSPE <- sqrt(sum((sim2.val$y - cbind(sim2.val$z, sim2.val$x) %*% beta)^2) / prod(dim(sim2.val$y))))

    ## [1] 8.859939

Compute the model performance with respect to coefficient bias

    # compute bias of beta estimates
    b <- sim2$b
    b[sim2$gamma == 0] <- 0
    (beta.l2 <- sqrt(sum((beta[-c(1:4), ] - b)^2) / prod(dim(b))))

    ## [1] 0.5039502

Compute the model performance with respect to covariance selection

    g.re <- getEstimator(fit2, estimator = "Gy")
    (g.accuracy <- sum((g.re > 0.5) == sim2$Gy) / prod(dim(g.re)))

    ## [1] 0.585

    (g.sensitivity <- sum(((g.re > 0.5) == sim2$Gy)[sim2$Gy == 1]) / sum(sim2$Gy == 1))

    ## [1] 0.2475248

    (g.specificity <- sum(((g.re > 0.5) == sim2$Gy)[sim2$Gy == 0]) / sum(sim2$Gy == 0))

    ## [1] 0.9292929

## Referrence

> Zhi Zhao, Marco Banterle, Alex Lewin, Manuela Zucknick (2023). Multivariate Bayesian structured variable selection for pharmacogenomic studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), qlad102. DOI: 10.1093/jrsssc/qlad102.