# mashr with common baseline at mean

#### Yuxin Zou

#### 2022-01-24

    library(MASS)
    library(kableExtra)

# Introduction

In the previous vignette mash common baseline, we estimate the change in some quantity computed in multiple conditions over a common control condition. However, there might be no common control condition in a study. In this case, we define the reference condition as the mean over different conditions. Deviation in any condition is defined as a difference in the quantity over the mean. We want to estimate the change in some quantity computed in multiple conditions over their mean.

For example, we measure the gene expression under multiple conditions. We want to estimate the change in expression in multiple conditions over their mean.

As in the mash common baseline vignette, we include the additional burden of comparing all conditions to the same reference condition. To deal with these additional correlations, mashr allows the user to specify the reference condition using `mash_update_data` with `ref = 'mean'`, after setting up the data in `mash_set_data`.

# Illustration

    generate_data = function(n, p, V, Utrue, err_sd=0.01, pi=NULL){
      if (is.null(pi)) {
        pi = rep(1, length(Utrue)) # default to uniform distribution
      }
      assertthat::are_equal(length(pi), length(Utrue))

      for (j in 1:length(Utrue)) {
        assertthat::are_equal(dim(Utrue[j]), c(p, p))
      }

      pi <- pi / sum(pi) # normalize pi to sum to one
      which_U <- sample(1:length(pi), n, replace=TRUE, prob=pi)

      Beta = matrix(0, nrow=n, ncol=p)
      for(i in 1:n){
        Beta[i,] = mvrnorm(1, rep(0, p), Utrue[[which_U[i]]])
      }
      Shat = matrix(err_sd, nrow=n, ncol=p, byrow = TRUE)
      E = mvrnorm(n, rep(0, p), Shat[1,]^2 * V)
      Bhat = Beta + E
      return(list(B = Beta, Bhat=Bhat, Shat = Shat, whichU = which_U))
    }

Here we simulate data for illustration. This simulation routine creates a dataset with 5 conditions and 2000 samples. Half of the samples have equal expression among conditions. In the rest samples, half have higher and equal expression in the first 2 conditions, half have higher expression in the last condition.

    set.seed(1)
    n = 2000
    R = 5
    V = diag(R)
    U0 = matrix(0, R, R)
    U1 = matrix(1, R, R)
    U2 = U0; U2[1:2,1:2] = 4
    U3 = U0; U3[5,5] = 4
    simdata = generate_data(n, R, V, list(U0=U0, U1=U1, U2=U2, U3 = U3), err_sd = 1)

1. Read in the data, and set the reference condition as mean

    library(mashr)
    data = mash_set_data(simdata$Bhat, simdata$Shat)

    data.L = mash_update_data(data, ref = 'mean')

The updated mash data object (data.L) includes the induced correlation internally.

2. We proceed the analysis using the simple canonical covariances as in the initial introductory vignette, and the data driven covariances as in the Introduction to mash: data-driven covariances.

- Canonical covariances

    U.c = cov_canonical(data.L)

- Data-driven covariances

    m.1by1 = mash_1by1(data.L)
    strong = get_significant_results(m.1by1)
    U.pca = cov_pca(data.L,2,subset=strong)
    U.ed = cov_ed(data.L, U.pca, subset=strong)

3. Fit mash model

    m = mash(data.L, c(U.c,U.ed), algorithm.version = 'R')

    #  - Computing 2000 x 181 likelihood matrix.
    #  - Likelihood calculations took 0.11 seconds.
    #  - Fitting model with 181 mixture components.
    #  - Model fitting took 0.96 seconds.
    #  - Computing posterior matrices.
    #  - Computation allocated took 0.02 seconds.

The log likelihood is

    print(get_loglik(m),digits=10)

    # [1] -10893.2688

Use `get_significant_results` to find the indices of effects that are ‘significant’:

    length(get_significant_results(m))

    # [1] 139

The number of false positive is 1.