# Benjamini-Hochberg Procedure with the Simulator

Suppose we wish to test \(n\) hypotheses, \(H_1,\ldots, H_n\), and we have a p-value \(\hat p_i\) for each hypothesis \(H_i\). That is,

\[ \mathbb{P}_{H_i}(\hat p_i\le \alpha) = \alpha. \]

Benjamini and Hochberg (1995) design a procedure (for when these p-values are independent) that controls what they call the false discovery rate (FDR), which is the expected proportion of the rejected tests that should not have been rejected:

\[ \mathbb{E}_{\{H_i:i\in S\}}\left[\frac{\sum_{i\in S}1\{H_i\text{ rejected}\}}{\max[1,\sum_{i=1}^n1\{H_i\text{ rejected}\}]}\right]. \]

Given a desired FDR \(q\), the BH procedure finds a data-adaptive threshold level \(\hat p(q)\) and rejects all \(H_i\) for which \(\hat p_i\le \hat p(q)\). The threshold level is given by comparing the sorted p-values \(\hat p_{(1)}\le \cdots \le \hat p_{(n)}\) to a line of slope \(q/n\) and identifying the largest p-value that is below this line. That is, \(\hat p(q)=\hat p_{(\hat i)}\) where \[ \hat i = \max\{i: \hat p_i \le q i / n\}. \]

In this simulation, we verify that the BH procedure works, and we investigate the effect that correlation has on the FDR control.

# Main simulation

In the Models section below, we show the code for `make_correlated_pvalues`, a function that generates a model object given parameters \(n\) (the number of hypotheses), \(\pi_0\) (the fraction of hypotheses that are null), and \(\rho\) (the correlation between any pair of test statistics). In the simulation below, we fix \(n=20\) and vary \(\pi_0\) and \(\rho\).

    library(simulator)

    name_of_simulation <- "fdr"
    sim <- new_simulation(name = name_of_simulation,
                          label = "False Discovery Rate") %>%
      generate_model(make_correlated_pvalues, seed = 123,
                     n = 20,
                     pi0 = list(0.8, 1),
                     rho = list(-0.01, 0, 0.1, 0.9),
                     vary_along = c("pi0", "rho")) %>%
      simulate_from_model(nsim = 25, index = 1:4) %>%
      run_method(bh_methods, parallel = list(socket_names = 2)) %>%
      evaluate(list(fdp, nd))

The variable `bh_methods` is defined in the Methods section below and corresponds to the BH procedure for four different values of \(q\).

We begin by looking at the results when \(\rho=0\) (i.e., independent tests).

    sim %>%
      subset_simulation(rho == 0) %>%
      tabulate_eval(metric_name = "fdp", output_type = "html",
                    format_args = list(digits = 1, nsmall = 2))

A comparison of Mean false discovery proportion (averaged over 100 replicates).
| BH (q = 0.05) | BH (q = 0.1) | BH (q = 0.2)
--- | --- | --- | ---
pi0 = 0.8, rho = 0 | 0.04 (0.01) | 0.08 (0.02) | 0.17 (0.02)
pi0 = 1, rho = 0 | 0.04 (0.02) | 0.08 (0.03) | 0.25 (0.04)

It appears that the BH procedure does control the FDR at each stated \(q\) value. However, we also see that when \(\pi_0\) is less than 1, it tends to be more conservative. Indeed, Benjamini and Hochberg show that the FDR of BH does not exceed \(\pi_0q\).

# Adding to a simulation

We might like to increase the number of simulations.

Suppose we return to this simulation several days later and wish to double the number of random draws used. In the above code, we had 100 draws, which were simulated in 4 chunks of 25 draws each. The simulator allows us to add to a simulation without requiring us to re-run parts of the simulation that have already been run.

If we had closed the R session without saving the image[^]: for the sake of reproducibility, I like to always start with a fresh workspace, so I can be sure that my functions aren’t calling a global variable that I have forgotten about), we can open a new one in the directory that has the `files` directory in it. We start by loading `sim`, which is the Simulation object (containing all the necessary pointers to saved files). Loading `sim` is fast because it only loads the file locations, not the files themselves.

    sim <- load_simulation("fdr")

We do so by simply adding 4 more chunks, with `index=5:8`. Each distinct value of `index` corresponds to a separate random number generator stream. This is convenient because it means that we do not have to rely on the state of the RNG after completing one chunk to start the next one.

    sim <- sim %>%
      simulate_from_model(nsim = 25, index = 5:8) %>%
      run_method(bh_methods, parallel = list(socket_names = 2)) %>%
      evaluate(list(fdp, nd))

We can look again at the table.

    sim %>%
      subset_simulation(rho == 0) %>%
      tabulate_eval(metric_name = "fdp", output_type = "html",
                    format_args = list(digits = 1, nsmall = 2))

A comparison of Mean false discovery proportion (averaged over 200 replicates).
| BH (q = 0.05) | BH (q = 0.1) | BH (q = 0.2)
--- | --- | --- | ---
pi0 = 0.8, rho = 0 | 0.03 (0.009) | 0.07 (0.010) | 0.15 (0.014)
pi0 = 1, rho = 0 | 0.03 (0.013) | 0.07 (0.018) | 0.19 (0.028)

## Some plots

The FDR is the average of the false discovery proportion (FDP). We can look at these raw values (200 for each method-model pair).

    sim %>%
      subset_simulation(rho == 0) %>%
    plot_eval(metric_name = "fdp")

When \(\pi_0=1\), we see that the FDP is either 0 or 1. This is because if we make any number of discoveries, then they will all be false (but if we do not make any discoveries, we have FDP=0).

We now investigate the effect of \(\rho\), the correlation between the test statistics. We now fix \(\pi_0=0.8\) and look at how the plots vary with \(\rho\).

    sim %>%
      subset_simulation(pi0 == 0.8) %>%
    plot_eval(metric_name = "fdp", varying = "rho")

Since \(\rho\) is numeric, it might be more informative to look at the FDR as a function of \(\rho\).

    sim %>%
      subset_simulation(pi0 == 0.8) %>%
    plot_eval_by(metric_name = "fdp", varying = "rho")

We see that the procedure becomes more conservative as the dependence increases, but still does control FDR (which was shown for positive dependence in Benjamini and Yekutieli, 2001). Looking at \(\pi_0=1\), we would like to check whether for negative \(\rho\) the procedure is anti-conservative.

    sim %>%
      subset_simulation(pi0 == 1) %>%
    plot_eval_by(metric_name = "fdp", varying = "rho")

# Creating a simulation based on a preexisting one

To investigate this particular question in greater depth, we might create a new simulation object based on the previous one. We’d like to increase the number of random draws for this particular setting, but don’t care about doing so for the others.

    sim2 <- subset_simulation(sim, pi0 == 1 & rho == -0.01) %>%
      rename("fdr-negdep") %>%
      relabel("BH Procedure under negative dependence") %>%
      simulate_from_model(nsim = 25, index = 9:20) %>%
      run_method(bh_methods, parallel = list(socket_names = 2)) %>%
      evaluate(list(fdp, nd))

We remake the table (on the basis of 500 random draws) to check for anti-conservativeness.

    tabulate_eval(sim2, metric_name = "fdp", output_type = "html",
                    format_args = list(digits = 1, nsmall = 2))

A comparison of Mean false discovery proportion (averaged over 500 replicates).
| BH (q = 0.05) | BH (q = 0.1) | BH (q = 0.2)
--- | --- | --- | ---
pi0 = 1, rho = -0.01 | 0.04 (0.009) | 0.08 (0.012) | 0.21 (0.018)

Observe that at this point, `sim` and `sim2` are two separate simulation objects that refer to some common simulation results. For example, their `Model` and `Draws` objects are the same.

    m <- model(sim, pi0 == 1 & rho == -0.01)
    m2 <- model(sim2)
    all.equal(m, m2)

    ## [1] TRUE

    d <- draws(sim, pi0 == 1 & rho == -0.01)
    d2 <- draws(sim2, index = 1:8)
    all.equal(d, d2)

    ## [1] TRUE

When `model` and `draws` (and likewise `output` and `evals`) are called on a simulation object, they load the appropriate files referenced by the `Simulation` object. The models `m` and `m2` are identical (and likewise for `d` and `d2`) because both `Simulation` objects refer to the same saved files. Thus, having multiple simulation objects does not lead to copies of the (potentially large) results files being made. Instead, only the references themselves are duplicated.

# Components

## Models

    library(mvtnorm)
    make_correlated_pvalues <- function(n, pi0, rho) {
      # Gaussian copula model...
      #
      # n pvalues, the first n*pi0 of which are null, coming from a multivariate
      # normal with all correlations rho.
      sigma <- matrix(rho, n, n)
      diag(sigma) <- 1
      n0 <- round(n * pi0)
      delta <- 2 # size of signal
      mu <- rep(c(0, delta), c(n0, n - n0)) # n0 are null
      new_model(name = "correlated-pvalues",
                label = sprintf("pi0 = %s, rho = %s", pi0, rho),
                params = list(n = n, rho = rho, sigma = sigma,
                              pi0 = pi0, mu = mu, delta = delta,
                              nonnull = which(mu != 0)),
                simulate = function(n, mu, sigma, nsim) {
                  # this function must return a list of length nsim
                  x <- rmvnorm(nsim, mean = mu, sigma = sigma)
                  pvals <- 1 - pnorm(x)
                  return(split(pvals, row(pvals))) # make each row its own list element
                })
    }

## Methods

    make_bh <- function(q) {
      # q is the desired level of control for the FDR
      new_method(name = paste0("bh", q),
                 label = sprintf("BH (q = %s)", q),
                 settings = list(q = q),
                 method = function(model, draw, q) {
                   p <- sort(draw)
                   cutline <- seq(model$n) * q / model$n
                   threshold <- max(p[p <= cutline], 0)
                   list(rejected = which(draw <= threshold))
                 })
    }

    qvalues <- c(0.05, 0.1, 0.2)
    bh_methods <- sapply(qvalues, make_bh)

## Metrics

    fdp <- new_metric(name = "fdp",
                      label = "false discovery proportion",
                      metric = function(model, out) {
                        fp <- setdiff(out$rejected, model$nonnull)
                        nd <- max(length(out$rejected), 1)
                        return(length(fp) / nd)
                        })

    nd <- new_metric(name = "nd",
                     label = "number of discoveries",
                     metric = function(model, out) length(out$rejected))

# Conclusion

To cite the `simulator`, please use

    citation("simulator")

To cite package ‘simulator’ in publications use:

Bien J (2016). “The simulator: An Engine to Streamline Simulations.” arXiv:1607.00021. https://arxiv.org/abs/1607.00021.

A BibTeX entry for LaTeX users is

@Article{, title = {The {simulator}: An Engine to Streamline Simulations}, author = {Jacob Bien}, year = {2016}, url = {https://arxiv.org/abs/1607.00021}, journal = {arXiv:1607.00021}, }