Last updated: 2018-08-20

workflowr checks: (Click a bullet for more information)
  • R Markdown file: up-to-date

    Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

  • Environment: empty

    Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

  • Seed: set.seed(1)

    The command set.seed(1) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

  • Session information: recorded

    Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

  • Repository version: a80bf6f

    Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated.

    Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
    
    Ignored files:
        Ignored:    .DS_Store
        Ignored:    .Rhistory
        Ignored:    .Rproj.user/
        Ignored:    analysis/.DS_Store
        Ignored:    analysis/.Rhistory
        Ignored:    analysis/figure/
        Ignored:    analysis/include/.DS_Store
        Ignored:    data/.DS_Store
        Ignored:    docs/.DS_Store
        Ignored:    output/.DS_Store
    
    Untracked files:
        Untracked:  _workflowr.yml
        Untracked:  analysis/Classify.Rmd
        Untracked:  analysis/EstimateCorMaxEM.Rmd
        Untracked:  analysis/EstimateCorMaxEMGD.Rmd
        Untracked:  analysis/EstimateCorPrior.Rmd
        Untracked:  analysis/EstimateCorSol.Rmd
        Untracked:  analysis/HierarchicalFlashSim.Rmd
        Untracked:  analysis/MashLowSignal.Rmd
        Untracked:  analysis/Mash_GTEx.Rmd
        Untracked:  analysis/MeanAsh.Rmd
        Untracked:  analysis/OutlierDetection.Rmd
        Untracked:  analysis/OutlierDetection2.Rmd
        Untracked:  analysis/OutlierDetection3.Rmd
        Untracked:  analysis/OutlierDetection4.Rmd
        Untracked:  analysis/Test.Rmd
        Untracked:  analysis/mash_missing_row.Rmd
        Untracked:  code/MashClassify.R
        Untracked:  code/MashCorResult.R
        Untracked:  code/MashSource.R
        Untracked:  code/Weight_plot.R
        Untracked:  code/addemV.R
        Untracked:  code/estimate_cor.R
        Untracked:  code/generateDataV.R
        Untracked:  code/johnprocess.R
        Untracked:  code/sim_mean_sig.R
        Untracked:  code/summary.R
        Untracked:  data/Blischak_et_al_2015/
        Untracked:  data/scale_data.rds
        Untracked:  docs/figure/Classify.Rmd/
        Untracked:  docs/figure/OutlierDetection.Rmd/
        Untracked:  docs/figure/OutlierDetection2.Rmd/
        Untracked:  docs/figure/OutlierDetection3.Rmd/
        Untracked:  docs/figure/Test.Rmd/
        Untracked:  docs/figure/mash_missing_whole_row_5.Rmd/
        Untracked:  docs/include/
        Untracked:  output/AddEMV/
        Untracked:  output/CovED_UKBio_strong.rds
        Untracked:  output/CovED_UKBio_strong_Z.rds
        Untracked:  output/Flash_UKBio_strong.rds
        Untracked:  output/MASH.10.em2.result.rds
        Untracked:  output/MASH.10.mle.result.rds
        Untracked:  output/MASH.result.1.rds
        Untracked:  output/MASH.result.10.rds
        Untracked:  output/MASH.result.2.rds
        Untracked:  output/MASH.result.3.rds
        Untracked:  output/MASH.result.4.rds
        Untracked:  output/MASH.result.5.rds
        Untracked:  output/MASH.result.6.rds
        Untracked:  output/MASH.result.7.rds
        Untracked:  output/MASH.result.8.rds
        Untracked:  output/MASH.result.9.rds
        Untracked:  output/Mash_EE_Cov_0_plusR1.rds
        Untracked:  output/Trail 1/
        Untracked:  output/Trail 2/
        Untracked:  output/UKBio_mash_model.rds
    
    Unstaged changes:
        Modified:   analysis/EstimateCorMaxEM2.Rmd
        Modified:   analysis/EstimateCorMaxMash.Rmd
        Modified:   analysis/Mash_UKBio.Rmd
        Modified:   analysis/_site.yml
        Modified:   analysis/chunks.R
        Modified:   analysis/mash_missing_samplesize.Rmd
        Modified:   output/Flash_T2_0.rds
        Modified:   output/Flash_T2_0_mclust.rds
        Modified:   output/Mash_model_0_plusR1.rds
        Modified:   output/PresiAddVarCol.rds
    
    
    Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
Expand here to see past versions:
    File Version Author Date Message
    Rmd a80bf6f zouyuxin 2018-08-20 wflow_publish(“analysis/EstimateCor.Rmd”)
    html 6281062 zouyuxin 2018-08-15 Build site.
    Rmd 3e3e128 zouyuxin 2018-08-15 wflow_publish(c(“analysis/EstimateCor.Rmd”, “analysis/EstimateCorMax.Rmd”,
    html 05731eb zouyuxin 2018-08-15 Build site.
    Rmd ccc1607 zouyuxin 2018-08-15 wflow_publish(c(“analysis/EstimateCorIndex.Rmd”, “analysis/EstimateCor.Rmd”))
    html 568fbe6 zouyuxin 2018-08-13 Build site.
    Rmd 3ae3f08 zouyuxin 2018-08-13 wflow_publish(c(“analysis/EstimateCor.Rmd”,
    html 10d4174 zouyuxin 2018-08-13 Build site.
    Rmd b8c1cd8 zouyuxin 2018-08-13 wflow_publish(“analysis/EstimateCor.Rmd”)
    html 3bfa4f5 zouyuxin 2018-08-13 Build site.
    Rmd 49d53fb zouyuxin 2018-08-13 wflow_publish(“analysis/EstimateCor.Rmd”)
    html 6e4f0a1 zouyuxin 2018-08-03 Build site.
    Rmd c330f07 zouyuxin 2018-08-03 wflow_publish(c(“analysis/EstimateCor.Rmd”))
    html 2985466 zouyuxin 2018-07-26 Build site.
    Rmd e2c5ebd zouyuxin 2018-07-26 wflow_publish(“analysis/EstimateCor.Rmd”)


library(mashr)
Loading required package: ashr
library(knitr)
library(kableExtra)
source('../code/generateDataV.R')
source('../code/summary.R')

We illustrate the problem about estimating the correlation matrix in mashr.

In my simple simulation, the current approach underestimates the null correlation. We want to find better positive definite estimator. We could try to estimate the pairwise correlation, ie. mle of \(\sum_{l,k} \pi_{lk} N_{2}(0, V + w_{l}U_{k})\) for any pair of conditions.

Problem

Simple simulation in \(R^2\) to illustrate the problem: \[ \hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix}\right)) \]

\[ \beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}\right)) \]

\(\Rightarrow\) \[ \hat{\beta} \sim \frac{1}{4}N_{2}(0, \left( \begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 2 & 0.5 \\ 0.5 & 1 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 1 & 0.5 \\ 0.5 & 2 \end{matrix} \right)) + \frac{1}{4}N_{2}(0, \left( \begin{matrix} 2 & 1.5 \\ 1.5 & 2 \end{matrix} \right)) \]

n = 4000

set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)

Let’s check the result of mash under different correlation matrix:

  1. Identity \[ V.I = I_{2} \]
m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)
m.I = mash(m.data, U.c, verbose= FALSE)
  1. The current approach: truncated empirical correlation \(V.trun\)
Vhat = estimate_null_correlation(m.data, apply_lower_bound = FALSE)
Vhat
          [,1]      [,2]
[1,] 1.0000000 0.3439205
[2,] 0.3439205 1.0000000

It underestimates the correlation.

# Use underestimate cor
m.data.V = mash_set_data(data$Bhat, data$Shat, V=Vhat)
m.V = mash(m.data.V, U.c, verbose = FALSE)
  1. Overestimate correlation \[ V.o = \left( \begin{matrix} 1 & 0.65 \\ 0.65 & 1\end{matrix} \right) \]
# If we overestimate cor
V.o = matrix(c(1,0.65,0.65,1),2,2)
m.data.Vo = mash_set_data(data$Bhat, data$Shat, V=V.o)
m.Vo = mash(m.data.Vo, U.c, verbose=FALSE)
  1. mash.1by1

We run ash for each condition, and estimate correlation matrix based on the non-significant genes. The estimated cor is closer to the truth.

m.1by1 = mash_1by1(m.data)
strong = get_significant_results(m.1by1)
V.mash = cor(data$Bhat[-strong,])
V.mash
          [,1]      [,2]
[1,] 1.0000000 0.4597745
[2,] 0.4597745 1.0000000
m.data.1by1 = mash_set_data(data$Bhat, data$Shat, V=V.mash)
m.V1by1 = mash(m.data.1by1, U.c, verbose = FALSE)
  1. True correlation
# With correct cor
m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)

The results are summarized in table:

null.ind = which(apply(data$B,1,sum) == 0)
V.trun = c(get_loglik(m.V), length(get_significant_results(m.V)), sum(get_significant_results(m.V) %in% null.ind))
V.I = c(get_loglik(m.I), length(get_significant_results(m.I)), sum(get_significant_results(m.I) %in% null.ind))
V.over = c(get_loglik(m.Vo), length(get_significant_results(m.Vo)), sum(get_significant_results(m.Vo) %in% null.ind))
V.1by1 = c(get_loglik(m.V1by1), length(get_significant_results(m.V1by1)), sum(get_significant_results(m.V1by1) %in% null.ind))
V.correct = c(get_loglik(m.correct), length(get_significant_results(m.correct)), sum(get_significant_results(m.correct) %in% null.ind))
temp = cbind(V.I, V.trun, V.1by1, V.correct, V.over)
colnames(temp) = c('Identity','truncate', 'm.1by1', 'true', 'overestimate')
row.names(temp) = c('log likelihood', '# significance', '# False positive')
temp %>% kable() %>% kable_styling()
Identity truncate m.1by1 true overestimate
log likelihood -12390.14 -12307.65 -12304.13 -12302.62 -12301.81
# significance 166.00 30.00 25.00 25.00 70.00
# False positive 14.00 1.00 0.00 0.00 4.00

The estimated pi is

par(mfrow=c(2,3))
barplot(get_estimated_pi(m.I), las=2, cex.names = 0.7, main='Identity', ylim=c(0,0.8))
barplot(get_estimated_pi(m.V), las=2, cex.names = 0.7, main='Truncate', ylim=c(0,0.8))
barplot(get_estimated_pi(m.V1by1), las=2, cex.names = 0.7, main='m.1by1', ylim=c(0,0.8))
barplot(get_estimated_pi(m.correct), las=2, cex.names = 0.7, main='True', ylim=c(0,0.8))
barplot(get_estimated_pi(m.Vo), las=2, cex.names = 0.7, main='OverEst', ylim=c(0,0.8))

Expand here to see past versions of unnamed-chunk-10-1.png:
Version Author Date
10d4174 zouyuxin 2018-08-13
3bfa4f5 zouyuxin 2018-08-13
2985466 zouyuxin 2018-07-26

The ROC curve:

m.I.seq = ROC.table(data$B, m.I)
m.V.seq = ROC.table(data$B, m.V)
m.Vo.seq = ROC.table(data$B, m.Vo)
m.V1by1.seq = ROC.table(data$B, m.V1by1)
m.correct.seq = ROC.table(data$B, m.correct)

Expand here to see past versions of unnamed-chunk-12-1.png:
Version Author Date
10d4174 zouyuxin 2018-08-13
3bfa4f5 zouyuxin 2018-08-13
2985466 zouyuxin 2018-07-26

Comparing accuracy

rrmse = rbind(RRMSE(data$B, data$Bhat, list(m.I = m.I, m.V = m.V, m.1by1 = m.V1by1, m.true = m.correct, m.over = m.Vo)))
colnames(rrmse) = c('Identity','V.trun','V.1by1','V.true','V.over')
row.names(rrmse) = 'RRMSE'
rrmse %>% kable() %>% kable_styling()
Identity V.trun V.1by1 V.true V.over
RRMSE 0.6522463 0.5925754 0.5811472 0.5817699 0.6052702
barplot(rrmse, ylim=c(0,(1+max(rrmse))/2), las=2, cex.names = 0.7, main='RRMSE')

Expand here to see past versions of unnamed-chunk-14-1.png:
Version Author Date
568fbe6 zouyuxin 2018-08-13
3bfa4f5 zouyuxin 2018-08-13

Solution: MLE

K=1

Suppose a simple extreme case \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right)| \left(\begin{matrix} x \\ y \end{matrix} \right) \sim N_{2}(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); \left(\begin{matrix} x \\ y \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix}\right)) \] \[ \left(\begin{matrix} x \\ y \end{matrix} \right) \sim \delta_{0} \] \(\Rightarrow\) \[ \left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right) \sim N_{2}(\left(\begin{matrix} \hat{x} \\ \hat{y} \end{matrix} \right); \left(\begin{matrix} 0 \\ 0 \end{matrix} \right), \left( \begin{matrix} 1 & \rho \\ \rho & 1 \end{matrix}\right)) \]

\[ f(\hat{x},\hat{y}) = \prod_{i=1}^{n} \frac{1}{2\pi\sqrt{1-\rho^2}} \exp \{-\frac{1}{2(1-\rho^2)}\left[ \hat{x}_{i}^2 + \hat{y}_{i}^2 - 2\rho \hat{x}_{i}\hat{y}_{i}\right] \} \] The MLE of \(\rho\): \[ \begin{align*} l(\rho) &= -\frac{n}{2}\log(1-\rho^2) - \frac{1}{2(1-\rho^2)}\left( \sum_{i=1}^{n} x_{i}^2 + y_{i}^2 - 2\rho x_{i}y_{i} \right) \\ l(\rho)' &= \frac{n\rho}{1-\rho^2} - \frac{\rho}{(1-\rho^2)^2} \sum_{i=1}^{n} (x_{i}^2 + y_{i}^2) + \frac{\rho^2 + 1}{(1-\rho^2)^2} \sum_{i=1}^{n} x_{i}y_{i} = 0 \\ &= \rho^{3} - \rho^{2}\frac{1}{n}\sum_{i=1}^{n} x_{i}y_{i} - \left( 1- \frac{1}{n} \sum_{i=1}^{n} x_{i}^{2} + y_{i}^{2} \right) \rho - \frac{1}{n}\sum_{i=1}^{n} x_{i}y_{i} = 0 \\ l(\rho)'' &= \frac{n(\rho^2+1)}{(1-\rho^2)^2} - \frac{1}{2}\left( \frac{8\rho^2}{(1-\rho^2)^{3}} + \frac{2}{(1-\rho^2)^2} \right)\sum_{i=1}^{n}(x_{i}^2 + y_{i}^2) + \{ \left( \frac{8\rho^2}{(1-\rho^2)^{3}} + \frac{2}{(1-\rho^2)^2} \right)\rho + \frac{4\rho}{(1-\rho^2)^2} \}\sum_{i=1}^{n}x_{i}y_{i} \end{align*} \]

The log likelihood is not a concave function in general. The score function has either 1 or 3 real solutions.

Kendall and Stuart (1979) noted that at least one of the roots is real and lies in the interval [−1, 1]. However, it is possible that all three roots are real and in the admissible interval, in which case the likelihood can be evaluated at each root to determine the true maximum likelihood estimate.

I simulate the data with \(\rho=0.6\) and plot the loglikelihood function:

Expand here to see past versions of unnamed-chunk-15-1.png:
Version Author Date
568fbe6 zouyuxin 2018-08-13

\(l(\rho)'\) has one real solution

polyroot(c(- sum(data$Bhat[,1]*data$Bhat[,2]),  - (n - sum(data$Bhat[,1]^2 + data$Bhat[,2]^2)), - sum(data$Bhat[,1]*data$Bhat[,2]), n))
[1] 0.6193031+0.000000i 0.0058209+1.009339i 0.0058209-1.009339i

In general

The general derivation is in estimate correlation mle

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] kableExtra_0.9.0 knitr_1.20       mashr_0.2-11     ashr_2.2-10     

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      highr_0.7         pillar_1.3.0     
 [4] compiler_3.5.1    git2r_0.23.0      plyr_1.8.4       
 [7] workflowr_1.1.1   R.methodsS3_1.7.1 R.utils_2.6.0    
[10] iterators_1.0.10  tools_3.5.1       digest_0.6.15    
[13] viridisLite_0.3.0 tibble_1.4.2      evaluate_0.11    
[16] lattice_0.20-35   pkgconfig_2.0.1   rlang_0.2.1      
[19] Matrix_1.2-14     foreach_1.4.4     rstudioapi_0.7   
[22] yaml_2.2.0        parallel_3.5.1    mvtnorm_1.0-8    
[25] xml2_1.2.0        httr_1.3.1        stringr_1.3.1    
[28] REBayes_1.3       hms_0.4.2         rprojroot_1.3-2  
[31] grid_3.5.1        R6_2.2.2          rmarkdown_1.10   
[34] rmeta_3.0         readr_1.1.1       magrittr_1.5     
[37] whisker_0.3-2     scales_0.5.0      backports_1.1.2  
[40] codetools_0.2-15  htmltools_0.3.6   MASS_7.3-50      
[43] rvest_0.3.2       assertthat_0.2.0  colorspace_1.3-2 
[46] stringi_1.2.4     Rmosek_8.0.69     munsell_0.5.0    
[49] doParallel_1.0.11 pscl_1.5.2        truncnorm_1.0-8  
[52] SQUAREM_2017.10-1 crayon_1.3.4      R.oo_1.22.0      

This reproducible R Markdown analysis was created with workflowr 1.1.1