RRreg: Correlation and Regression Analyses for Randomized Response Data

Documented in RRsimu

#' Monte Carlo simulation for one or two RR variables
#'
#' @description Simulate and analyse bivariate data including either one RR
#'   variable (either correlation, logistic, or linear regression model) or two
#'   RR variables (only correlations). Useful for power analysis, parametric
#'   bootstraps or for testing the effects of noncompliance on the stability of
#'   estimates.
#'
#' @param numRep number of replications
#' @param n sample size
#' @param pi true proportion of carriers of sensitive attribute (for 2 RR
#'   variables: \code{vector})
#' @param model either one or two RR model (as \code{vector}), see
#'   \code{\link{RRuni}}
#' @param p randomization probability (for 2 RR variables: a \code{list}). See
#'   \code{\link{RRuni}}  for details.
#' @param cor true Pearson-correlation used for data generation (for
#'   \code{\link{RRcor}}). Can also be used to generate data with two
#'   dichotomous RR variables.
#' @param b.log true regression coefficient in logistic regression (for
#'   \code{\link{RRlog}})
#' @param complyRates vector with two values giving the proportions of
#'   participants who adhere to the instructions in the subset with or without
#'   the sensitive attribute, respectively (for 2 RR variables: a \code{list})
#' @param sysBias probability of responding 'yes' (coded as 1 in the RR
#'   variable) in case of non-compliance for carriers and non-carriers,
#'   respectively. See \code{\link{RRgen}}
#' @param method vector specifying which RR methods to be used in each
#'   replication. For a single RR variable, the methods \code{\link{RRuni}},
#'   \code{\link{RRcor}},\code{\link{RRlog}}, and \code{\link{RRlin}} are
#'   available. For 2 RR variables, only \code{\link{RRcor}} is available.
#' @param alpha significance threshold for testing the logistic regression
#'   parameter \code{beta}
#' @param groupRatio proportion of participants in group 1. Only for two-group
#'   models (e.g., \code{"SLD"}) (for 2 RR variables: \code{vector})
#' @param MLest concerns \code{\link{RRuni}}: whether to use \code{optim} to get
#'   ML instead of moment estimates (only relevant if pi is outside of [0,1])
#' @param getPower whether to compute power for \code{method="RRcor"} (performs
#'   an additional bootstrap assuming independence)
#' @param nCPU either the number of CPU cores or a cluster initialized via
#'   \code{\link[parallel]{makeCluster}}.
#' @return A list containing \item{parEsts}{matrix containing the estimated
#'   parameters} \item{results}{matrix  with mean parameters, standard errors,
#'   and number of samples to which the respective method could not be fitted}
#'   \item{power}{vector with the estimated power of the selected randomized
#'   response procedures}
#'
#' @details  For a single RR variable:
#'
#'   The parameter \code{b.log} is the slope-coefficient for the true, latent
#'   values in a logistic regression model that is used for data generation.
#'
#'   The argument \code{cor} is used for data generation for linear models. The
#'   directly measured covariate is sampled from a normal distribution with
#'   shifted means, depending on the true state on the sensitive attribute
#'   (i.e., the true, underlying values on the RR variable). For dichotomous RR
#'   variables, this corresponds to the assumption of an ordinary t-test, where
#'   the dependent variable is normally distributed within groups with equal
#'   variance. The difference in means is chosen in a way, to obtain the
#'   point-biserial correlation defined by \code{cor}.
#'
#'   For two RR variables:
#'
#'   \code{cor} has to be used. In case of two dichotomous RR variables, the
#'   true group membership of individuals is sampled from a 2x2 cross table.
#'   Within this table, probabilities are chosen in a way, to obtain the
#'   point-tetrachoric correlation defined by \code{cor}
#'
#'   Note, that for the FR model with multiple response categories (e.g., from 0
#'   to 4), the specified \code{cor} is not the exact target of the sampling
#'   procedure. It assumes a normal distribution for each true state, with
#'   constant differences between the groups (i.e., it assumes an interval
#'   scaled variable).
#'   
#' @examples # Not run: Simulate data according to the Warner model
#' # mcsim <-  RRsimu(numRep=100, n=300, pi=.4,
#' #                  model="Warner", p=.2, cor=.3)
#' # print(mcsim)
#' @export
RRsimu <- function(numRep, n, pi, model, p, cor = 0, b.log = 0,
                   complyRates = c(1, 1), sysBias = c(0, 0),
                   method = c("RRuni", "RRcor", "RRlog", "RRlin"),
                   alpha = 0.05, groupRatio = 0.5, MLest = FALSE,
                   getPower = TRUE, nCPU = 1) {
  modelNames <- c(
    "Warner", "UQTknown", "UQTunknown", "Mangat", "Kuk",
    "FR", "Crosswise", "Triangular", "direct", "CDM", "CDMsym", "SLD", "custom"
  )
  model <- pmatch(model, modelNames, duplicates.ok = T)
  model <- modelNames[model]

  checkzero <- c(cor != 0, b.log != 0)
  if (sum(checkzero) > 1) {
    if (cor != 0) {
      if (abs(cor) > 1) {
        warning("True correlation cor must be in the interval [-1, 1].")
      }
      b.log <- 0
    }
    warning(paste("Only the argument 'cor' is used for data generation (linear model)! Currently used: cor =", cor, "; b.log =", b.log))
  }

  nn <- c(
    "pi.true", "pi.RRuni", "pi2.true", "pi2.RRuni", "piSE.RRuni",
    "cor.true", "cor.RRcor",
    "beta.true", "betaSE.true", "beta.RRlog", "betaSE.RRlog",
    "lincoef.true", "lincoefSE.true", "lincoef.RRlin", "lincoefSE.RRlin",
    "beta.deltaG2.RRlog",
    "par2.true", "par2.RRuni", "par2SE.RRuni"
  )
  nvar <- length(nn)

  # check: one or two RR variables
  model <- model[model != "direct"]
  if (length(model) < 1 | length(model) > 2) {
    stop("Definition of either one or two RR variables required (e.g., model='Warner').")
  }
  twoRR <- length(model) == 2
  if (twoRR && b.log != 0) {
    warning("For two RR variables, only the argument cor is used to simulate data.")
  }
  if (cor != 0) {
    if (!twoRR) {
      # if (!inherits(complyRates, "list")) complyRates <- list(complyRates)
      # if only one RR variable included: transform correlation to mean difference
      diffMean <- sqrt(cor^2 / (pi[1] * (1 - pi[1]) * (1 - cor^2))) * sign(cor)
      if (model %in% c("custom", "FR") & length(pi) > 2) {
        ppi <- pi
        if (min(pi) == 0) {
          ppi <- pi + .001
        }
        diffMean <- sign(cor) * mean(sqrt(cor^2 / (ppi * (1 - ppi) * (1 - cor^2)))) * sqrt(pi %*% pi)
      }
    } else {
      # two RR variables: think about data generation !
      if (missing(complyRates)) complyRates <- list(c(1, 1), c(1, 1))
      if (missing(groupRatio)) groupRatio <- c(.5, .5)
      if (length(pi) != 2 || min(pi) <= 0 || max(pi) >= 1) stop("Please provide a vector pi with two values in (0,1)")
    }
  }

  ### CDM and CDMsym not available in RRcor
  if (any(model %in% c("CDM", "CDMsym")) && any(c("RRlin", "RRcor") %in% method)) {
    method <- method[!(method == "RRcor")]
    warning("Models 'CDM' and 'CDMsym' are not available in RRcor because both define a third group of cheaters (see vignette('RRreg') ).")
  }

  ############################################### ONE RR   ###############################################
  getParEsts.oneRR <- function(replic) {
    parEsts <- matrix(NA, nrow = replic, ncol = nvar)
    colnames(parEsts) <- nn
    parEsts <- as.data.frame(parEsts)
    cnt <- 1
    max.tries <- 3 # ; trys <- 0
    #     okcor <- T; oklog <- T; oklin <- T;
    while (cnt <= replic) {
      # generate normal data dependent on true value on RR variable
      # k : loop through true states
      if (cor != 0) {
        # generate continuous and discrete variable (point-biserial correlation)
        dat <- RRgen(
          n = n, pi.true = pi, model = model, p = p, complyRates = complyRates,
          sysBias = sysBias, groupRatio = groupRatio
        )
        n.true <- table(dat$true)
        dat$cov <- rep(0, n)
        for (k in 0:(length(n.true) - 1)) {
          dat[dat$true == k, ]$cov <- rnorm(n.true[k + 1], k * diffMean, 1)
        }
        # generate from logistic regression model
      } else if (b.log != 0) {
        cov <- rnorm(n)
        modelpred <- log(pi / (1 - pi)) + b.log * cov
        probsens <- 1 / (1 + exp(-modelpred))
        true <- rbinom(n, 1, probsens)
        dat <- RRgen(
          n = n, model = model, p = p, complyRates = complyRates,
          sysBias = sysBias, groupRatio = groupRatio, trueState = true
        )
        dat$cov <- cov
      } else { ## use RRlin with b.lin=0 anyways
        #         true <- rbinom(n, 1, pi)
        #         dat <- RRgen(model=model, p=p, complyRates=complyRates,
        #                         sysBias=sysBias, groupRatio=groupRatio, trueState=true)
        #         dat$cov <- b.lin * true + rnorm(n, sd=sig.lin)
        dat <- RRgen(
          n = n, pi.true = pi, model = model, p = p, complyRates = complyRates,
          sysBias = sysBias, groupRatio = groupRatio
        )
        dat$cov <- rnorm(n)
      }

      if (is.null(dat$group)) {
        group <- rep(1, nrow(dat))
      } else {
        group <- dat$group
      }

      fit.success <- c(RRcor = FALSE, RRlog = FALSE, RRlin = FALSE)
      for (trys in 1:max.tries) {
        suppressWarnings(try(rm(log1), silent = TRUE))
        suppressWarnings(try(rm(linmod), silent = TRUE))

        # analyse with RRcor
        if (!fit.success[1] && "RRcor" %in% method) {
          cor1 <- RRcor(
            x = dat[, c("response", "cov")],
            models = c(model, "d"), p.list = list(p, 1),
            group = group
          )
          if (!is.na(cor1$r["cov", "response"])) {
            parEsts[cnt, "cor.true"] <- cor(dat$true, dat$cov)
            parEsts[cnt, "cor.RRcor"] <- cor1$r["cov", "response"]
            fit.success[1] <- TRUE
          }
        } else {
          fit.success[1] <- TRUE
        }

        # analyse with RRuni
        if (trys == 1) {
          uni1 <- RRuni(response = dat$response, model = model, p = p, group = group, MLest = MLest)
          if ("RRuni" %in% method) {
            parEsts[cnt, "pi.true"] <- table(dat$true)["1"] / n
            parEsts[cnt, "pi.RRuni"] <- uni1$pi[ifelse(model %in% c("custom", "FR"), 2, 1)]
            parEsts[cnt, "piSE.RRuni"] <- uni1$piSE[ifelse(model %in% c("custom", "FR"), 2, 1)]

            if (model %in% c("CDMsym", "CDM")) {
              parEsts[cnt, "pi.true"] <- table(dat[dat$comply == 1, "true"])["1"] / n
              parEsts[cnt, "par2.true"] <- 1 - colMeans(dat)["comply"]
              parEsts[cnt, "par2.RRuni"] <- uni1$gamma
              #         parEsts[cnt,"par2SE.RRuni"] = uni1$gammaSE
            } else if (model == "SLD") {
              parEsts[cnt, "par2.true"] <- colMeans(dat[dat$true == 1, ])["comply"]
              parEsts[cnt, "par2.RRuni"] <- uni1$t
              #         parEsts[cnt,"par2SE.RRuni"] = uni1$tSE
            }
          }
        }

        # analyse with RRlog
        if (!fit.success[2] && "RRlog" %in% method) {
          #         oklog <- F
          parEsts[cnt, "beta.true"] <- NA
          # adjust dependent variable for CDM: having sensitive attribute and commiting in RR design!
          if (model %in% c("CDMsym", "CDM")) {
            dat$true <- dat$true & dat$comply
          }
          if (trys == 1) {
            try(
              {
                glm1 <- glm(true ~ cov, dat, family = binomial(link = "logit"))
                parEsts[cnt, "beta.true"] <- glm1$coefficients["cov"]
                parEsts[cnt, "betaSE.true"] <- summary(glm1)$coef[2, 2]
              },
              silent = T
            )
          }

          try(
            {
              log1 <- RRlog(response ~ cov,
                data = dat, model = model, p = p, group = group,
                LR.test = T, fit.n = 1
              )
            },
            silent = T
          )
          if (exists("log1", envir = environment()) &&
            !is.null(log1$coefficients["cov"]) && !is.na(log1$coefficients["cov"])) {
            if (abs(log1$coefficients["cov"]) > 3) {
              # print(log1)
              try(
                {
                  log1 <- RRlog(response ~ cov,
                    data = dat, model = model, p = p, group = group,
                    LR.test = T, fit.n = 3
                  )
                },
                silent = T
              )
              # print(log1)
            }
            parEsts[cnt, "beta.RRlog"] <- log1$coefficients["cov"]
            suppressWarnings(try(parEsts[cnt, "betaSE.RRlog"] <- sqrt(log1$vcov["cov", "cov"]), silent = TRUE))
            try(parEsts[cnt, "beta.deltaG2.RRlog"] <- -2 * log1$deltaLogLik["cov"], silent = TRUE)
            fit.success[2] <- TRUE
          }
        } else {
          fit.success[2] <- TRUE
        }


        # analyse with RRlin
        if (!fit.success[3] && "RRlin" %in% method) {
          #         oklin <- F
          parEsts[cnt, "lincoef.true"] <- NA
          # adjust dependent variable for CDM: having sensitive attribute and commiting in RR design!
          if (model %in% c("CDMsym", "CDM")) {
            dat$true <- dat$true & dat$comply
          }
          lm1 <- lm(cov ~ true, dat)
          parEsts[cnt, "lincoef.true"] <- coef(lm1)["true"]
          parEsts[cnt, "lincoefSE.true"] <- sqrt(vcov(lm1)["true", "true"])

          try(linmod <- RRlin(cov ~ response,
            data = dat, models = model, p.list = list(p),
            group = group, fit.n = 1
          ), silent = T)
          if (exists("linmod", envir = environment()) &&
            !is.null(linmod$beta["response"]) && !is.na(linmod$beta["response"])) {
            parEsts[cnt, "lincoef.RRlin"] <- linmod$beta["response"]
            parEsts[cnt, "lincoefSE.RRlin"] <- sqrt(vcov(linmod)["response", "response"])
            parEsts[cnt, "lincoef.deltaG2.RRlin"] <- -2 * logLik(linmod)["response"]
            fit.success[3] <- TRUE
          }
        } else {
          fit.success[3] <- TRUE
        }
        if (all(fit.success)) {
          break
        }
      }
      cnt <- cnt + 1
    }
    parEsts
  }

  ############################################### TWO RR   ###############################################
  getParEsts.twoRR <- function(replic) {
    parEsts <- matrix(NA, nrow = replic, ncol = nvar)
    colnames(parEsts) <- nn
    parEsts <- as.data.frame(parEsts)
    for (i in 1:(replic)) {
      # generate data for two discrete, dichotomous RR variables
      # for a fourfold table, with marginal proportions pi[1] and pi[2]
      # a b
      # c d
      cov <- cor * sqrt(pi[1] * (1 - pi[1]) * pi[2] * (1 - pi[2]))
      a <- pi[1] * pi[2] + cov
      b <- pi[1] * (1 - pi[2]) - cov
      c <- (1 - pi[1]) * pi[2] - cov
      d <- (1 - pi[1]) * (1 - pi[2]) + cov
      if (any(c(a, b, c, d) < 0)) {
        stop("The given correlation coefficient cor=", cor, " (phi coefficient) is not observable with n=", n, " and pi=c(", pi[1], ", ", pi[2], ").")
      }
      dat <- expand.grid(0:1, 0:1)[sample(1:4, size = n, replace = TRUE, prob = c(a, b, c, d)), ]
      colnames(dat) <- c("true1", "true2")
      RR1 <- RRgen(
        model = model[1], pi.true = .3, p = p[[1]], complyRates = complyRates[[1]],
        sysBias = sysBias, groupRatio = groupRatio[1], trueState = dat$true1
      )
      colnames(RR1) <- paste0(colnames(RR1), 1)
      RR2 <- RRgen(
        model = model[2], pi.true = .3, p = p[[2]], complyRates = complyRates[[2]],
        sysBias = sysBias, groupRatio = groupRatio[2], trueState = dat$true2
      )
      colnames(RR2) <- paste0(colnames(RR2), 2)
      dat <- cbind(RR1, RR2)
      group <- dat[, grep("^group", colnames(dat))]

      # analyse with RRcor and RRlog
      if ("RRcor" %in% method) {
        cor1 <- RRcor(
          x = dat[, c("response1", "response2")],
          models = model, p.list = p,
          group = group
        )
        parEsts[i, "cor.true"] <- cor(dat$true1, dat$true2)
        parEsts[i, "cor.RRcor"] <- cor1$r["response1", "response2"]
      }
    }
    parEsts
  }

  ###############################################

  # multi core processing
  if (is.numeric(nCPU) && nCPU == 1) {
    cl.tmp <- NULL
    if (!twoRR) {
      parEsts <- getParEsts.oneRR(numRep)
    } else {
      parEsts <- getParEsts.twoRR(numRep)
    }
  } else if (is.numeric(nCPU) && nCPU > 1) {
    cl.tmp <- makeCluster(nCPU)
  } else if (any(
    inherits(nCPU, "SOCKcluster"),
    inherits(nCPU, "cluster")
  )) {
    cl.tmp <- nCPU
  } else {
    stop("'nCPU' must be integer or a cluster (see ?parallel::makeCluster)")
  }
  if (!is.null(cl.tmp)) {
    registerDoParallel(cl.tmp)
    if (!twoRR) {
      parEsts <- foreach(k = seq_along(cl.tmp), .combine = rbind, .packages = "RRreg") %dopar% {
        getParEsts.oneRR(ceiling(numRep / length(cl.tmp)))
      }
    } else {
      parEsts <- foreach(k = seq_along(cl.tmp), .combine = rbind, .packages = "RRreg") %dopar% {
        getParEsts.twoRR(ceiling(numRep / length(cl.tmp)))
      }
    }
    if (is.numeric(nCPU)) stopCluster(cl.tmp)
  }

  # significance testing of beta coefficients
  # parEsts[,"beta.perctSignif.RRlog"] <- parEsts[,"beta.prob.RRlog"]  <alpha
  parEsts <- as.matrix(parEsts)
  if ("RRlog" %in% method) {
    cutoff <- max(parEsts[, "beta.true"], na.rm = T)
    parEsts[parEsts[, "beta.RRlog"] > max(10, 5 * cutoff), "beta.RRlog"] <- NA
  }

  # summarize simulation results
  NAs <- colSums(is.na(parEsts))
  results <- data.frame(
    Mean = colMeans(parEsts, na.rm = T),
    #                         SE=bs.se(parEsts,)
    SD = apply(parEsts, 2, sd, na.rm = T)
  )
  # use online estimates between quantiles(.01, .99)
  for (i in 1:nrow(results)) {
    if (!is.na(results[i, 1])) {
      sel <- parEsts[, i] >= quantile(parEsts[, i], probs = .01, na.rm = TRUE) &
        parEsts[, i] <= quantile(parEsts[, i], probs = .99, na.rm = TRUE)
      results[i, 1] <- mean(parEsts[sel, i], na.rm = T)
      results[i, 2] <- sd(parEsts[sel, i], na.rm = T)
    }
  }
  ####################################### COMPUTE POWER ###########################################
  power <- rep(-1, 4)
  names(power) <- c("RRuni(z-test)", "RRcor(par_bootstrap)", "RRlog(LR-test)", "RRlin(Wald-test)")
  if (!twoRR && "RRuni" %in% method) {
    z_val <- parEsts[, "pi.RRuni"] / parEsts[, "piSE.RRuni"]
    power[1] <- sum(pnorm(z_val, lower.tail = F) < alpha, na.rm = T) / (nrow(parEsts) - NAs["pi.RRuni"])
  }
  if ("RRcor" %in% method && getPower) {
    simH0 <- RRsimu(
      numRep = numRep, n = n, pi = pi, model = model, cor = 0,
      p = p, MLest = TRUE, complyRates = complyRates,
      sysBias = sysBias, groupRatio = groupRatio,
      method = "RRcor", getPower = F, nCPU = nCPU
    )
    crit <- quantile(ecdf(abs(simH0$parEsts[, "cor.RRcor"])), 1 - alpha)
    power[2] <- sum(abs(parEsts[, "cor.RRcor"]) > crit, na.rm = T) /
      (nrow(parEsts) - NAs["cor.RRcor"])
  }
  if ("RRlog" %in% method) {
    prob <- pchisq(parEsts[, "beta.deltaG2.RRlog"], 1, lower.tail = F)
    power[3] <- sum(prob < alpha, na.rm = T) / (nrow(parEsts) - NAs["beta.deltaG2.RRlog"])
  }
  if (!twoRR && "RRlin" %in% method) {
    z_val <- parEsts[, "lincoef.RRlin"] / parEsts[, "lincoefSE.RRlin"]
    power[4] <- sum(pnorm(abs(z_val), lower.tail = F) < alpha / 2, na.rm = T) / (nrow(parEsts) - NAs["lincoef.RRlin"])
  }
  power <- power[power != -1]


  results <- cbind(results, NAs)
  results <- results[!is.na(results[, 1]), ]
  parEsts <- parEsts[, NAs != numRep]
  # filtering for
  # sim$parEsts <- sim$parEsts[sim$parEsts[,"beta.deltaG2.RRlog"]<200,]

  sim <- list(
    parEsts = parEsts, results = results, power = power,
    model = model, p = p, n = n, numRep = numRep, alpha = alpha,
    complyRates = complyRates, sysBias = sysBias, groupRatio = groupRatio,
    method = method
  )
  class(sim) <- "RRsimu"
  sim
}


# Print results of RRsimu

#' @export
print.RRsimu <- function(x, ...) {
  if (length(x$model) == 1) {
    cat(paste0(
      x$model, " ; n= ", x$n,
      "; randomization probability: ",
      gsub(", ", ", ", toString(round(x$p, 3))), "\n"
    ))
  }
  if (length(x$model) == 2) {
    cat(paste0(
      "\n", x$model[1],
      " ; n= ", x$n,
      "; randomization probability: ",
      gsub(", ", ", ", toString(round(x$p[[1]], 3)))
    ))
    cat(paste0(
      "\n", x$model[2],
      " ; n= ", x$n,
      "; randomization probability: ",
      gsub(", ", ", ", toString(round(x$p[[2]], 3))), "\n"
    ))
  }
  cat(paste0("\nBootstrapped mean and SD of parameters (", x$numRep, " replications):\n"))
  print(round(x$results[, 1:2], 5))
  cat(paste0("\nPower on alpha=", x$alpha, " level:\n"))
  print(round(x$power, 5))
  sel <- x$results[, 3] > x$numRep * .05
  if (any(sel)) {
    warning(
      "Fitting routines did fail in \n  ",
      paste(x$results[, 3] * 100 / x$numRep, collapse = ","),
      "% of bootstrap samples for:\n      ",
      paste(rownames(x$results)[sel], collapse = "; "), ", respectively.",
      "\n  This might be due to extreme prevalence rates (e.g., pi=.01).\n"
    )
  }

  # if("RRlog" %in% x$method){
  #     warning("Fitting routines did fail in more than 5% of bootstrap samples for:\n      ",
  #             paste(rownames(x$results)[sel], collapse="; "),
  #             "\n  This might be due to extreme prevalence rates (e.g., pi=.01).")
  # }
}

# Plot Results of Monte Carlo Simulation
#
# Generic method to plot histograms of the parameter estimates of a Monte Carlo simulation for RR models. Mean parameter estimates are shown in red.
# @param x an \code{\link{RRsimu}} object
# @param ... ignored
#
#' @export
plot.RRsimu <- function(x, ...) {
  parEsts <- x$parEsts
  results <- x$results
  nvar <- ncol(parEsts)
  nn <- colnames(parEsts)

  # how many plots?
  root <- sqrt(nvar)
  mfrow <- c(floor(root), 1 + floor(nvar / floor(root)))
  plot.new()
  par(mfrow = mfrow)

  for (i in 1:nvar) {
    if (!all(is.na(parEsts[, i]))) {
      hist(
        x = parEsts[, i], freq = FALSE, nclass = max(25, round(x$numRep / 25)),
        main = colnames(parEsts)[i], col = "grey",
        xlab = paste0("mean=", round(results[i, 1], 4), ",sd=", round(results[i, 2], 4))
      )
      abline(v = results[i, 1], col = "red")
      if (nn[i] %in% c("pi.RRuni", "pi.RRlog")) {
        dd <- function(x) dnorm(x, mean = results[i, 1], sd = results["piSE.RRuni", 1])
        curve(dd, col = "blue", n = 500, add = T)
      } else if (nn[i] %in% c("par2.RRuni")) {
        dd <- function(x) dnorm(x, mean = results[i, 1], sd = results["par2SE.RRuni", 1])
        curve(dd, col = "blue", n = 500, add = T)
      } else if (nn[i] == "beta.RRlog") {
        dd <- function(x) dnorm(x, mean = results[i, 1], sd = results["betaSE.RRlog", 1])
        curve(dd, col = "blue", n = 500, add = T)
      }
      # asymptotic distribution of beta in RRlog. should be chisq for diffMean=0
      else if (nn[i] == "beta.deltaG2.RRlog") {
        wr <- function(x) dchisq(x, 1)
        curve(wr, col = "blue", add = TRUE, n = 500)
      }
      #     else if (nn[i]=="beta.prob.RRlog"){
      #       curve(dunif, col = "blue", add = TRUE,n=500)
      #     }
    }
  }
  par(mfrow = c(1, 1))
}