From 2d2ed29f23fae8132be0f3fd527327d5a95042e8 Mon Sep 17 00:00:00 2001 From: Peter Carbonetto Date: Fri, 19 May 2023 22:13:41 -0500 Subject: [PATCH] A few more improvements to the vignette; rebuilt it for the pkgdown site. --- docs/articles/mr_mash_intro.html | 249 +++++++++++------- .../figure-html/plot-coefs-1.png | Bin 0 -> 38664 bytes .../figure-html/plot-pred-test-1.png | Bin 143192 -> 94347 bytes vignettes/mr_mash_intro.Rmd | 49 +++- 4 files changed, 196 insertions(+), 102 deletions(-) create mode 100644 docs/articles/mr_mash_intro_files/figure-html/plot-coefs-1.png diff --git a/docs/articles/mr_mash_intro.html b/docs/articles/mr_mash_intro.html index dd33009..360c3d1 100644 --- a/docs/articles/mr_mash_intro.html +++ b/docs/articles/mr_mash_intro.html @@ -33,7 +33,7 @@ mr.mash.alpha - 0.2-26 + 0.2-27 @@ -42,6 +42,9 @@
  • Home
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  • @@ -67,7 +70,7 @@

    Introduction to mr.mash

    Peter Carbonetto & Fabio Morgante

    -

    2023-05-16

    +

    2023-05-19

    Source: vignettes/mr_mash_intro.Rmd @@ -76,105 +79,173 @@

    2023-05-16

    -

    The aim of this vignette is to introduce the basic steps of a -mr.mash analysis, fitting the mr.mash model then using -it to make predictions.

    -

    First, we set the seed and load the mr.mash.alpha R -package.

    +

    The aim of this vignette is to introduce the basic steps of a mr.mash +analysis through a toy example. To learn more about mr.mash, please see +the paper.

    +

    First, we set the seed to make the results more easily reproducible, +and we load the “mr.mash.alpha” package.

    -library(mr.mash.alpha)
    -set.seed(123)
    -
    -

    Step 1 – Simulate example data -

    -

    We start by simulating a data set with 800 individuals, 1000 -predictors and 5 responses. We then 5 causal variables (randomly sampled -from the total 1000) are assigned equal effects across responses and -explain 20% of the total per-response variance. This would be roughly -equivalent to one gene in the “equal effects” scenario in the -mr.mash (with the difference being that genotypes are simulated -here).

    +library(mr.mash.alpha) +set.seed(123)
    +

    We illustrate the application of mr.mash to a data set simulated from +a multivariate, multiple linear regression with 5 responses in which the +coefficients are the same for all responses. In the target application +considered in the paper—prediction of multi-tissue gene expression from +genotypes—this would correspond to the situation in which we would like +to predict expression of a single gene in 5 different tissues from +genotype data at multiple SNPs, and the SNPs have the same effects on +gene expression in all 5 tissues. (In multi-tissue gene expression we +would normally like to predict expression of many genes, but to simplify +this vignette here we illustrate the key ideas with a single gene.)

    +

    Although this simulation is not a particularly realistic, this is +meant to illustrate the benefits of mr.mash: by modeling the sharing of +effects across tissues, mr.mash is able to more accurately estimate the +effects in multiple tissues, and therefore is able to obtain better +predictions.

    +

    We start by simulating 150 samples from a multivariate, multiple +linear regression model in which 5 out of the 800 variables (SNPs) +affect the 5 responses (expression levels).

    -n <- 800
    -p <- 1000
    -p_causal <- 5
    -r <- 5
    -pve <- 0.2
    -B_cor <- 1
    -out <- simulate_mr_mash_data(n, p, p_causal, r, pve=pve, B_cor=B_cor,
    -                             B_scale=1, X_cor=0, X_scale=1, V_cor=0)
    - -
    -

    Step 2 – Split the data into training and test sets -

    -

    We then split the data into a training set and a test set.

    +dat <- simulate_mr_mash_data(n = 150,p = 800,p_causal = 5,r = 5,pve = 0.5, + V_cor = 0.25)
    +

    Next we split the samples into a training set (with 100 samples) and +test set (with 50 samples).

    -Ytrain <- out$Y[-c(1:200),]
    -Xtrain <- out$X[-c(1:200),]
    -Ytest <- out$Y[c(1:200),]
    -Xtest <- out$X[c(1:200),]
    - +ntest <- 50 +Ytrain <- dat$Y[-(1:ntest),] +Xtrain <- dat$X[-(1:ntest),] +Ytest <- dat$Y[1:ntest,] +Xtest <- dat$X[1:ntest,]
    -

    Step 3 – Define the mixture prior +

    Define the mr.mash prior

    -

    To run mr.mash, we need to first specify the covariances in -the mixture-of-normals prior, which are supposed to capture the effect -sharing patterns across responses. In this example, we use a mixture of -“canonical” covariances computed using -compute_canonical_covs(). However, “data-driven” -covariances can also be used – here’s an example -of how to compute these matrices. Regardless of the type of covariance -matrices, these are each multiplied by a grid of scaling factors -computed using autoselect.mixsd(), which are supposed to -capture the magnitude of the effects. The expansion is done using -expand_covs(), which also adds a matrix of all zeros (our -spike) when requested. The grid is derived from the regression -coefficients and their standard errors from univariate simple linear -regression which can be computed using -compute_univariate_sumstats().

    +

    To run mr.mash, we need to first specify the covariances in the +mixture of normals prior. The idea is that the chosen collection of +covariance matrices should include a variety of potential effect sharing +patterns, and in the model fitting stage the prior should then assign +most weight to the sharing patterns that are present in the data, and +little or no weight on patterns that are inconsistent with the data. In +general, we recommend learning “data-driven” +covariance matrices. But here, for simplicity, we instead use +“canonical” covariances which are not adaptive, but nonetheless well +suited for this toy example since the true effects are the same across +responses/tissues.

    -univ_sumstats <- compute_univariate_sumstats(Xtrain, Ytrain,
    -                   standardize=TRUE, standardize.response=FALSE)
    -grid <- autoselect.mixsd(univ_sumstats, mult=sqrt(2))^2
    -S0 <- compute_canonical_covs(ncol(Ytrain), singletons=TRUE,
    -                             hetgrid=c(0, 0.25, 0.5, 0.75, 1))
    -S0 <- expand_covs(S0, grid, zeromat=TRUE)
    +S0 <- compute_canonical_covs(r = 5,singletons = TRUE, + hetgrid = seq(0,1,0.25))
    +

    This gives a mixture of 10 covariance matrices capturing a variety of +“canonical” effect-sharing patterns:

    +
    +names(S0)
    +#  [1] "singleton1"  "singleton2"  "singleton3"  "singleton4"  "singleton5" 
    +#  [6] "independent" "shared0.25"  "shared0.5"   "shared0.75"  "shared1"
    +

    To illustrate the benefits of modeling a variety of effect-sharing +patterns, we also try out mr.mash with a simpler mixture of covariance +matrices in which the effects are effectively independent across +tissues. Although this may seem to be a very poor choice of prior, +particularly for this example, it turns out that several multivariate +regression methods assume, implicitly or explicitly, this prior.

    +
    +S0_ind <- compute_canonical_covs(r = 5,singletons = FALSE,
    +                                 hetgrid = c(0,0.001,0.01))
    +names(S0_ind)
    +# [1] "independent" "shared0.001" "shared0.01"
    +

    Regardless of the covariance matrices are chosen, it is recommended +to also consider a variety of effect scales in the prior. This is +normally achieved in mr.mash by expanding the mixture across a specifed +grid of scaling factors. Here we choose this grid in an adaptive fashion +based on the data:

    +
    +univ_sumstats <- compute_univariate_sumstats(Xtrain,Ytrain,standardize = TRUE)
    +scaling_grid <- autoselect.mixsd(univ_sumstats,mult = sqrt(2))^2
    +S0 <- expand_covs(S0,scaling_grid)
    +S0_ind <- expand_covs(S0_ind,scaling_grid)
    -

    Step 4 – Fit mr.mash to the training data +

    Fit a mr.mash model to the data

    -

    Now we are ready to fit a mr.mash model to the training data using -mr.mash(), to estimate the posterior mean of the regression -coefficients.

    -
    -fit <- mr.mash(Xtrain, Ytrain, S0, update_V=TRUE, verbose=FALSE)
    +

    Having specified the mr.mash prior, we are now ready to fit a mr.mash +model to the training data (this may take a few minutes to run):

    +
    +fit <- mr.mash(Xtrain,Ytrain,S0,update_V = TRUE)
    +

    And for comparison we fit a second mr.mash model using the simpler +and less flexible prior:

    +
    +fit_ind <- mr.mash(Xtrain,Ytrain,S0_ind,update_V = TRUE)
    +

    (Notice that the less complex model also takes less time to fit.)

    +

    For prediction, the key output is the posterior mean estimtes of the +regression coefficients, stored in the “mu1” output. Let’s compare the +estimates to the ground truth:

    +
    +par(mfrow = c(1,2))
    +plot(dat$B,fit_ind$mu1,pch = 20,xlab = "true",ylab = "estimated",
    +     main = sprintf("cor = %0.3f",
    +                    cor(as.vector(dat$B),as.vector(fit_ind$mu1))))
    +abline(a = 0,b = 1,col = "royalblue",lty = "dotted")
    +plot(dat$B,fit$mu1,pch = 20,xlab = "true",ylab = "estimated",
    +     main = sprintf("cor = %0.3f",
    +                    cor(as.vector(dat$B),as.vector(fit$mu1))))
    +abline(a = 0,b = 1,col = "royalblue",lty = "dotted")
    +

    +

    As expected, the coefficients on the left-hand side obtained using an +“independent effects” prior are not as accurate as the the coefficients +estimated using the more flexible prior (right-hand side).

    +

    While perhaps not of primary interest, for diagnostic purposes it is +often helpfl to examine the estimated mixture weights in the prior as +well as the estimated residual covariance matrix.

    +

    Inspecting the top prior mixture weights from the better model, it is +helpful to see that the “null” and “shared1” components are among the +top components by weight. (The top component is the null component +because most of the SNPs have no effect on gene expression.)

    +
    +head(sort(fit$w0,decreasing = TRUE),n = 10)
    +#             null singleton2_grid1 singleton1_grid1 singleton5_grid1 
    +#       0.04741471       0.04243749       0.04202594       0.04141742 
    +# singleton4_grid1 singleton3_grid1 singleton2_grid2    shared1_grid1 
    +#       0.04038929       0.04022118       0.03798986       0.03788223 
    +# singleton1_grid2 singleton5_grid2 
    +#       0.03724834       0.03618774
    +

    Also, reassuringly, the estimated residual variance-covariance matrix +is close to the matrix used to simulate the data:

    +
    +dat$V
    +#           [,1]      [,2]      [,3]      [,4]      [,5]
    +# [1,] 3.5145375 0.8786344 0.8786344 0.8786344 0.8786344
    +# [2,] 0.8786344 3.5145373 0.8786343 0.8786343 0.8786343
    +# [3,] 0.8786344 0.8786343 3.5145373 0.8786343 0.8786343
    +# [4,] 0.8786344 0.8786343 0.8786343 3.5145373 0.8786343
    +# [5,] 0.8786344 0.8786343 0.8786343 0.8786343 3.5145373
    +fit$V
    +#           [,1]      [,2]      [,3]      [,4]      [,5]
    +# [1,] 3.2407233 0.8502217 0.8841572 1.3563118 1.0189386
    +# [2,] 0.8502217 3.5174997 1.2133736 0.6212035 0.1021557
    +# [3,] 0.8841572 1.2133736 2.5068761 0.6631672 1.0174394
    +# [4,] 1.3563118 0.6212035 0.6631672 2.7869990 0.7854292
    +# [5,] 1.0189386 0.1021557 1.0174394 0.7854292 2.9687632
    -

    Step 5 – Predict responses in the test +

    Use the fitted mr.mash model to make predictions

    -

    We then use the fitted model from step 4 to predict the response -values in the test set. In this plot, we compare the true and predicted -values

    -
    -Ytest_est <- predict(fit,Xtest)
    -plot(Ytest_est,Ytest,pch = 20,col = "darkblue",xlab = "true",
    -     ylab = "predicted")
    -abline(a = 0,b = 1,col = "magenta",lty = "dotted")
    -

    -

    However, we also want to assess prediction accuracy more formally. -Here, we do that in terms of \(R^2\) -which is easy to interpret – its maximum value would be the proportion -of variance explained, 0.2 in this case).

    -
    -r2 <- vector("numeric", r)
    -for(i in 1:r){
    -  fit_acc  <- lm(Ytest[, i] ~ Ytest_est[, i])
    -  r2[i] <- summary(fit_acc)$r.squared
    -}
    -
    -r2
    -# [1] 0.1426443 0.1937896 0.1248826 0.1756781 0.2181555
    -

    We can see that the predictions are pretty accurate.

    +

    We can use the fitted mr.mash model to predict gene expression from a +genotype sample, including a sample not included in the training set. +This is implemented by the “predict” method. Let’s compare the +predictions from the two mr.mash models:

    +
    +par(mfrow = c(1,2))
    +Ypred <- predict(fit,Xtest)
    +Ypred_ind <- predict(fit_ind,Xtest)
    +plot(Ytest,Ypred_ind,pch = 20,col = "darkblue",xlab = "true",
    +     ylab = "predicted",
    +     main = sprintf("cor = %0.3f",cor(as.vector(Ytest),as.vector(Ypred_ind))))
    +abline(a = 0,b = 1,col = "magenta",lty = "dotted")
    +plot(Ytest,Ypred,pch = 20,col = "darkblue",xlab = "true",
    +     ylab = "predicted",
    +     main = sprintf("cor = %0.3f",cor(as.vector(Ytest),as.vector(Ypred))))
    +abline(a = 0,b = 1,col = "magenta",lty = "dotted")
    +

    +

    Indeed, mr.mash with the more flexible prior (right-hand plot) +produces more accurate predictions than mr.mash with the “independent +effects” prior.

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zU9Qm2!;z&Cp&!LsQn)spSKlkPY89!Rw(%cn6=n#vW+6}$|M~nA4Iq=LbjY~k7>mGC QB-q5@qN#4Rj(fu203af=tN;K2 diff --git a/vignettes/mr_mash_intro.Rmd b/vignettes/mr_mash_intro.Rmd index 63fd66c..4e6da3a 100644 --- a/vignettes/mr_mash_intro.Rmd +++ b/vignettes/mr_mash_intro.Rmd @@ -19,7 +19,7 @@ mr.mash analysis through a toy example. To learn more about mr.mash, please see the [paper][mr-mash-biorxiv]. ```{r knitr-opts, include=FALSE} -knitr::opts_chunk$set(comment = "#",collapse = TRUE,results = "hold", +knitr::opts_chunk$set(comment = "#",collapse = TRUE, fig.align = "center",dpi = 120) ``` @@ -128,14 +128,14 @@ Fit a mr.mash model to the data Having specified the mr.mash prior, we are now ready to fit a mr.mash model to the training data (this may take a few minutes to run): -```{r fit-mr-mash-1} +```{r fit-mr-mash-1, results="hide"} fit <- mr.mash(Xtrain,Ytrain,S0,update_V = TRUE) ``` And for comparison we fit a second mr.mash model using the simpler and less flexible prior: -```{r fit-mr-mash-2} +```{r fit-mr-mash-2, results="hide"} fit_ind <- mr.mash(Xtrain,Ytrain,S0_ind,update_V = TRUE) ``` @@ -165,25 +165,48 @@ While perhaps not of primary interest, for diagnostic purposes it is often helpfl to examine the estimated mixture weights in the prior as well as the estimated residual covariance matrix. +Inspecting the top prior mixture weights from the better model, it is +helpful to see that the "null" and "shared1" components are among the +top components by weight. (The top component is the null component +because most of the SNPs have no effect on gene expression.) + ```{r prior-mixture-weights} +head(sort(fit$w0,decreasing = TRUE),n = 10) +``` + +Also, reassuringly, the estimated residual variance-covariance matrix +is close to the matrix used to simulate the data: +```{r resid-var} +dat$V +fit$V ``` -## Step 5 -- Predict responses in the test -We then use the fitted model from step 4 to predict the response values in the -test set. In this plot, we compare the true and predicted values +Use the fitted mr.mash model to make predictions +------------------------------------------------ -```{r plot-pred-test, fig.height=5, fig.width=5} +We can use the fitted mr.mash model to predict gene expression from +a genotype sample, including a sample not included in the training +set. This is implemented by the "predict" method. Let's compare the +predictions from the two mr.mash models: + +```{r plot-pred-test, fig.height=3.5, fig.width=6} par(mfrow = c(1,2)) -Ytest_est <- predict(fit_ind,Xtest) -plot(Ytest_est,Ytest,pch = 20,col = "darkblue",xlab = "true", - ylab = "predicted",main = cor(as.vector(Ytest_est),as.vector(Ytest))) +Ypred <- predict(fit,Xtest) +Ypred_ind <- predict(fit_ind,Xtest) +plot(Ytest,Ypred_ind,pch = 20,col = "darkblue",xlab = "true", + ylab = "predicted", + main = sprintf("cor = %0.3f",cor(as.vector(Ytest),as.vector(Ypred_ind)))) abline(a = 0,b = 1,col = "magenta",lty = "dotted") -Ytest_est <- predict(fit,Xtest) -plot(Ytest_est,Ytest,pch = 20,col = "darkblue",xlab = "true", - ylab = "predicted",main = cor(as.vector(Ytest_est),as.vector(Ytest))) +plot(Ytest,Ypred,pch = 20,col = "darkblue",xlab = "true", + ylab = "predicted", + main = sprintf("cor = %0.3f",cor(as.vector(Ytest),as.vector(Ypred)))) abline(a = 0,b = 1,col = "magenta",lty = "dotted") ``` +Indeed, mr.mash with the more flexible prior (right-hand plot) +produces more accurate predictions than mr.mash with the "independent +effects" prior. + [mr-mash-biorxiv]: https://doi.org/10.1101/2022.11.22.517471 [mashr-dd-vignette]: https://stephenslab.github.io/mashr/articles/intro_mash_dd.html