Contents

1 Introduction

RIVER is an R package of a probabilistic modeling framework, called RIVER (RNA-Informed Variant Effect on Regulation), that jointly analyzes personal genome (WGS) and transcriptome data (RNA-seq) to estimate the probability that a variant has regulatory impact in that individual. It is based on a generative model that assumes that genomic annotations, such as the location of a variant with respect to regulatory elements, determine the prior probability that variant is a functional regulatory variant, which is an unobserved variable. The functional regulatory variant status then influences whether nearby genes are likely to display outlier levels of gene expression in that person.

RIVER is a hierarchical Bayesian model that predicts the regulatory effects of rare variants by integrating gene expression with genomic annotations. The RIVER consists of three layers: a set of nodes \(G = G_{1}, ..., G_{P}\) in the topmost layer representing \(P\) observed genomic annotations over all rare variants near a particular gene, a latent binary variable \(FR\) in the middle layer representing the unobserved funcitonal regulatory status of the rare variants, and one binary node \(E\) in the final layer representing expression outlier status of the nearby gene. We model each conditional probability distribution as follows: \[ FR | G \sim Bernoulli(\psi), \psi = logit^{-1}(\beta^T, G) \] \[E | FR \sim Categorical(\theta_{FR}) \] \[ \beta_i \sim N(0, \frac{1}{\lambda})\] \[\theta_{FR} \sim Beta(C,C)\] with parameters \(\beta\) and \(\theta\) and hyper-parameters \(\lambda\) and \(C\).

Because \(FR\) is unobserved, log-likelihood objective of RIVER over instances \(n = 1, ..., N\), \[ \sum_{n=1}^{N} log\sum_{FR_n= 0}^{1} P(E_n, G_n, FR_n | \beta, \theta), \] is non-convex. We therefore optimize model parameters via Expectation-Maximization (EM) as follows:

In the E-step, we compute the posterior probabilities (\(\omega_n^{(i)}\)) of the latent variables \(FR_n\) given current parameters and observed data. For example, at the \(i\)-th iteration, the posterior probability of \(FR_n = 1\) for the \(n\)-th instance is \[ \omega_{1n}^{(i)} = P(FR_n = 1 | G_n, \beta^{(i)}, E_n, \theta^{(i)}) =\frac{P(FR_n = 1 | G_n, \beta^{(i)}) \cdotp P(E_n | FR_n = 1, \theta^{(i)})}{\sum_{FR_n = 0}^1 P(FR_n | G_n, \beta^{(i)}) \cdotp P(E_n | FR_n, \theta^{(i)})}, \] \[\omega_{0n}^{(i)} = 1 - \omega_{1n}^{(i)}.\]

In the M-step, at the \(i\)-th iteration, given the current estimates \(\omega^{(i)}\), the parameters (\(\beta^{(i+1)*}\)) are estimated as \[ \max_{\beta^{(i+1)}} \sum_{n = 1}^N \sum_{FR_n = 0}^1 log(P(FR_n | G_n, \beta^{(i+1)})) \cdotp \omega_{FR, n}^{(i)} - \frac{\lambda}{2}||\beta^{(i+1)}||_2, \] where \(\lambda\) is an L2 penalty hyper-parameter derived from the Gaussian prior on \(\beta\). The parameters \(\theta\) get updated as: \[ \theta_{s,t}^{(i+1)} = \sum_{n = 1}^{N} I(E_n = t) \cdotp \omega_{s,n}^{(i)} + C. \] where \(I\) is an indicator operator, \(t\) is the binary value of expression \(E_n\), \(s\) is the possible binary values of \(FR_n\) and \(C\) is a pseudo count derived from the Beta prior on . The E- and M-steps are applied iteratively until convergence.

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2 Quick Start

The purpose of this section is to provide users a general sense of our package, RIVER, including components and their behaviors and applications. We will briefly go over main functions, observe basic operations and corresponding outcomes. Throughout this section, users may have better ideas about which functions are available, which values to be chosen for necessary parameters, and where to seek help. More detailed descriptions are given in later sections.

First, we load RIVER:

library("RIVER")

RIVER consists of several functions mainly supporting two main functions including evaRIVER and appRIVER, which we are about to show how to use them here. We first load simulated data created beforehand for illustration.

filename <- system.file("extdata", "simulation_RIVER.gz", 
                       package="RIVER")
dataInput <- getData(filename) # import experimental data

getData combines different resources including genomic features, outlier status of gene expression, and N2 pairs having same rare variants into standardized data structures, called ExpressionSet class.

print(dataInput)
## ExpressionSet (storageMode: lockedEnvironment)
## assayData: 18 features, 6122 samples 
##   element names: exprs 
## protocolData: none
## phenoData
##   sampleNames: indiv58:gene1614 indiv5:gene1331 ... indiv18:gene1111
##     (6122 total)
##   varLabels: Outlier N2pair
##   varMetadata: labelDescription
## featureData: none
## experimentData: use 'experimentData(object)'
## Annotation:
Feat <- t(Biobase::exprs(dataInput)) # genomic features (G)
Out <- as.numeric(unlist(dataInput$Outlier))-1 # outlier status (E)

In the simulated data, an input object dataInput consists of 18 genomic features and expression outlier status of 6122 samples and which samples belong to N2 pairs.

head(Feat)
##                    Feature1   Feature2   Feature3   Feature4   Feature5
## indiv58:gene1614  0.4890338 -1.2143183 -0.5015241 -0.8093881 -0.1916957
## indiv5:gene1331  -4.1665487 -0.6355490 -1.0914989 -2.8069442 -0.3921658
## indiv76:gene447  -0.4469724 -0.9440314  0.7386942  0.6786997  0.6043647
## indiv16:gene126   1.6252083 -2.2686506  1.4156013 -0.5662072 -0.1405224
## indiv45:gene1044  0.3086791  1.0044798  0.8493856  0.3515569 -0.3763434
## indiv6:gene258    0.9046627  1.7618399 -1.6258895  0.3961724 -0.2203089
##                     Feature6   Feature7   Feature8   Feature9   Feature10
## indiv58:gene1614 -0.06614698  0.1993153 -0.7544253 -1.3505036 -0.03477715
## indiv5:gene1331  -0.43831006  1.8175888 -0.8411557  2.1495237 -0.48691686
## indiv76:gene447   1.47627930  0.6521233 -0.6416004  1.0309900 -0.38262446
## indiv16:gene126   2.11676989  1.0670951  0.3404799 -1.5970916 -0.49910751
## indiv45:gene1044  1.45269690 -0.8368466 -1.0016646  0.1908291 -0.26598159
## indiv6:gene258   -0.76821018 -1.3436283 -0.7201516  0.5440035  0.31006097
##                   Feature11  Feature12   Feature13  Feature14  Feature15
## indiv58:gene1614 2.08675352  0.7283183  0.15074710  1.5183579  0.2226134
## indiv5:gene1331  2.57339374  0.4840111  0.58897093 -0.2069596  0.1000502
## indiv76:gene447  0.30621172  0.6690835 -1.39701213  1.4853201  1.6552545
## indiv16:gene126  0.58599041 -0.7052013 -0.07715282 -1.3326831  0.1719152
## indiv45:gene1044 1.45029299 -0.4530850 -0.09610983  1.4731617 -0.9372256
## indiv6:gene258   0.05174989  0.4026417 -0.50911462 -0.3623563 -2.5279770
##                    Feature16  Feature17  Feature18
## indiv58:gene1614  1.88846321 -0.4408236 -0.4558111
## indiv5:gene1331   0.22640429  1.2535793 -1.0163254
## indiv76:gene447   0.71720775  0.2758493  1.1378452
## indiv16:gene126   1.12919669  0.6166061 -1.6069772
## indiv45:gene1044  0.14794114  1.2241711  1.4916670
## indiv6:gene258   -0.01574568 -1.5835940 -1.2366762
head(Out)
## [1] 1 1 1 0 1 1

Feat contains continuous values of genomic features (defined as \(G\) in the objective function) while Out contains binary values representing outlier status of same samples as Feat (defined as \(E\) in the objective function).

For evaluation, we hold out pairs of individuals at genes where only those two individuals shared the same rare variants. Except for the list of instances, we train RIVER with the rest of instances, compute the RIVER score (the posterior probability of having a functional regulatory variant given both WGS and RNA-seq data) from one individual, and assess the accuracy with respect to the second individual’s held-out expression levels. Since there is currently quite few gold standard set of functional rare variants, using this labeled test data allow us to evaluate predictive accuracy of RIVER compared with genomic annotation model, GAM, that uses genomic annotations alone. We can observe a significant improvement by incorporating expression data.

To do so, we simply use evaRIVER:

evaROC <- evaRIVER(dataInput)
##  ::: EM iteration is terminated since it converges within a
##         predefined tolerance (0.001) :::

evaROC is an S4 object of class evaRIVER which contains two AUC values from RIVER and GAM, specificity and sensitivity measures from the two models, and p-value of comparing the two AUC values.

cat('AUC (GAM-genomic annotation model) = ', round(evaROC$GAM_auc,3), '\n')
## AUC (GAM-genomic annotation model) =  0.58
cat('AUC (RIVER) = ', round(evaROC$RIVER_auc,3), '\n')
## AUC (RIVER) =  0.8
cat('P-value ', format.pval(evaROC$pvalue, digits=2, eps=0.001), '***\n')
## P-value  <0.001 ***

We can visualize the ROC curves with AUC values:

par(mar=c(6.1, 6.1, 4.1, 4.1))
plot(NULL, xlim=c(0,1), ylim=c(0,1), 
     xlab="False positive rate", ylab="True positive rate", 
     cex.axis=1.3, cex.lab=1.6)
abline(0, 1, col="gray")
lines(1-evaROC$RIVER_spec, evaROC$RIVER_sens, 
      type="s", col='dodgerblue', lwd=2)
lines(1-evaROC$GAM_spec, evaROC$GAM_sens, 
      type="s", col='mediumpurple', lwd=2)
legend(0.7,0.2,c("RIVER","GAM"), lty=c(1,1), lwd=c(2,2),
       col=c("dodgerblue","mediumpurple"), cex=1.2, 
       pt.cex=1.2, bty="n")
title(main=paste("AUC: RIVER = ", round(evaROC$RIVER_auc,3), 
                 ", GAM = ", round(evaROC$GAM_auc,3), 
                 ", P = ", format.pval(evaROC$pvalue, digits=2, 
                                       eps=0.001),sep=""))

Each ROC curve from either RIVER or GAM is computed by comparing the posterior probability given available data for the 1st individual with the outlier status of the 2nd individual in the list of held-out pairs and vice versa.

To extract posterior probabilities for prioritizing functional rare variants in any downstream analysis such as finding pathogenic rare variants in disease, you simply run appRIVER to obtain the posterior probabilities:

postprobs <- appRIVER(dataInput)
##  ::: EM iteration is terminated since it converges within a
##         predefined tolerance (0.001) :::

postprobs is an S4 object of class appRIVER which contains subject IDs, gene names, \(P(FR = 1 | G)\), \(P(FR = 1 | G, E)\), and fitRIVER, all the relevant information of the fitted RIVER including hyperparamters for further use.

Probabilities of rare variants being functional from RIVER and GAM for a few samples are shown below:

example_probs <- data.frame(Subject=postprobs$indiv_name, 
                           Gene=postprobs$gene_name, 
                           RIVERpp=postprobs$RIVER_posterior, 
                           GAMpp=postprobs$GAM_posterior)
head(example_probs)
##   Subject     Gene   RIVERpp     GAMpp
## 1 indiv58 gene1614 0.4303673 0.2319994
## 2  indiv5 gene1331 0.3597397 0.1939262
## 3 indiv76  gene447 0.5249447 0.3061179
## 4 indiv16  gene126 0.1316062 0.2447880
## 5 indiv45 gene1044 0.4488103 0.2370526
## 6  indiv6  gene258 0.3598831 0.1714101

From left to right, it shows subject ID, gene name, posterior probabilities from RIVER, posterior probabilities from GAM.

To observe how much we can obtain additional information on functional rare variants by integrating the outlier status of gene expression into RIVER in the following figure.

plotPosteriors(postprobs, outliers=as.numeric(unlist(dataInput$Outlier))-1)

As shown in this figure, the integration of both genomic features and expression outliers indeed provide higher quantitative power for prioritizing functional rare variants. You can observe a few examples of pathogenic regulatory variants based on posterior probabilities from RIVER in our paper (http://biorxiv.org/content/early/2016/09/09/074443).

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3 Two Main Functions

3.1 Evaluation of RIVER

The function, evaRIVER, is to see how much we can gain additional information by integrating an outlier status of gene expression into integrated models. The prioritization of functional rare variants has difficulty in its evaluation especially due to no gold standard class of the functionality of rare variants. To come up with this limitation, we extract pairs of individuals for genes having same rare variants and hold them out for the evaluation. In other words, we train RIVER with all the instances except for those held-out pairs of individuals, calculate posterior probabilities of functional regulatory variants given genomic features and outlier status for the first individual, and compare the probabilities with the second individual’s outlier status and vice versa. You can simply observe how the entire steps of evaluating models including both RIVER and GAM proceed by using evaRIVER with verbose=TRUE:

filename <- system.file("extdata", "simulation_RIVER.gz", 
                       package="RIVER")
dataInput <- getData(filename) # import experimental data
evaROC <- evaRIVER(dataInput, pseudoc=50, 
                   theta_init=matrix(c(.99, .01, .3, .7), nrow=2), 
                   costs=c(100, 10, 1, .1, .01, 1e-3, 1e-4), 
                   verbose=TRUE)
##  *** best lambda = 0.1 *** 
## 
##  *** RIVER: EM step 1
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.9523
##      M-step: norm(theta difference) = 0.1479, norm(beta difference) = 0.0081 *** 
## 
##  *** RIVER: EM step 2
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.918
##      M-step: norm(theta difference) = 0.0548, norm(beta difference) = 0.0099 *** 
## 
##  *** RIVER: EM step 3
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8853
##      M-step: norm(theta difference) = 0.052, norm(beta difference) = 0.0103 *** 
## 
##  *** RIVER: EM step 4
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8543
##      M-step: norm(theta difference) = 0.0492, norm(beta difference) = 0.0096 *** 
## 
##  *** RIVER: EM step 5
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8248
##      M-step: norm(theta difference) = 0.0464, norm(beta difference) = 0.009 *** 
## 
##  *** RIVER: EM step 6
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7972
##      M-step: norm(theta difference) = 0.0437, norm(beta difference) = 0.0084 *** 
## 
##  *** RIVER: EM step 7
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7711
##      M-step: norm(theta difference) = 0.041, norm(beta difference) = 0.0077 *** 
## 
##  *** RIVER: EM step 8
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7463
##      M-step: norm(theta difference) = 0.0383, norm(beta difference) = 0.0071 *** 
## 
##  *** RIVER: EM step 9
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7235
##      M-step: norm(theta difference) = 0.0358, norm(beta difference) = 0.0066 *** 
## 
##  *** RIVER: EM step 10
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7022
##      M-step: norm(theta difference) = 0.0333, norm(beta difference) = 0.0062 *** 
## 
##  *** RIVER: EM step 11
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6821
##      M-step: norm(theta difference) = 0.031, norm(beta difference) = 0.0059 *** 
## 
##  *** RIVER: EM step 12
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6633
##      M-step: norm(theta difference) = 0.0287, norm(beta difference) = 0.0056 *** 
## 
##  *** RIVER: EM step 13
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6461
##      M-step: norm(theta difference) = 0.0266, norm(beta difference) = 0.0054 *** 
## 
##  *** RIVER: EM step 14
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.63
##      M-step: norm(theta difference) = 0.0246, norm(beta difference) = 0.0053 *** 
## 
##  *** RIVER: EM step 15
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.615
##      M-step: norm(theta difference) = 0.0227, norm(beta difference) = 0.0052 *** 
## 
##  *** RIVER: EM step 16
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6009
##      M-step: norm(theta difference) = 0.0209, norm(beta difference) = 0.005 *** 
## 
##  *** RIVER: EM step 17
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5878
##      M-step: norm(theta difference) = 0.0192, norm(beta difference) = 0.0048 *** 
## 
##  *** RIVER: EM step 18
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5757
##      M-step: norm(theta difference) = 0.0177, norm(beta difference) = 0.0046 *** 
## 
##  *** RIVER: EM step 19
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5645
##      M-step: norm(theta difference) = 0.0163, norm(beta difference) = 0.0045 *** 
## 
##  *** RIVER: EM step 20
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5542
##      M-step: norm(theta difference) = 0.0149, norm(beta difference) = 0.0045 *** 
## 
##  *** RIVER: EM step 21
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5446
##      M-step: norm(theta difference) = 0.0137, norm(beta difference) = 0.0044 *** 
## 
##  *** RIVER: EM step 22
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5355
##      M-step: norm(theta difference) = 0.0126, norm(beta difference) = 0.0043 *** 
## 
##  *** RIVER: EM step 23
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5272
##      M-step: norm(theta difference) = 0.0115, norm(beta difference) = 0.0043 *** 
## 
##  *** RIVER: EM step 24
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5194
##      M-step: norm(theta difference) = 0.0106, norm(beta difference) = 0.0042 *** 
## 
##  *** RIVER: EM step 25
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5121
##      M-step: norm(theta difference) = 0.0097, norm(beta difference) = 0.0041 *** 
## 
##  *** RIVER: EM step 26
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5052
##      M-step: norm(theta difference) = 0.0088, norm(beta difference) = 0.004 *** 
## 
##  *** RIVER: EM step 27
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4988
##      M-step: norm(theta difference) = 0.0081, norm(beta difference) = 0.0038 *** 
## 
##  *** RIVER: EM step 28
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.493
##      M-step: norm(theta difference) = 0.0074, norm(beta difference) = 0.0037 *** 
## 
##  *** RIVER: EM step 29
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4873
##      M-step: norm(theta difference) = 0.0068, norm(beta difference) = 0.0036 *** 
## 
##  *** RIVER: EM step 30
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4821
##      M-step: norm(theta difference) = 0.0062, norm(beta difference) = 0.0034 *** 
## 
##  *** RIVER: EM step 31
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.477
##      M-step: norm(theta difference) = 0.0056, norm(beta difference) = 0.0033 *** 
## 
##  *** RIVER: EM step 32
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4722
##      M-step: norm(theta difference) = 0.0051, norm(beta difference) = 0.0031 *** 
## 
##  *** RIVER: EM step 33
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4678
##      M-step: norm(theta difference) = 0.0047, norm(beta difference) = 0.0029 *** 
## 
##  *** RIVER: EM step 34
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4635
##      M-step: norm(theta difference) = 0.0043, norm(beta difference) = 0.0028 *** 
## 
##  *** RIVER: EM step 35
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4596
##      M-step: norm(theta difference) = 0.0039, norm(beta difference) = 0.0027 *** 
## 
##  *** RIVER: EM step 36
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4558
##      M-step: norm(theta difference) = 0.0035, norm(beta difference) = 0.0025 *** 
## 
##  *** RIVER: EM step 37
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4523
##      M-step: norm(theta difference) = 0.0032, norm(beta difference) = 0.0024 *** 
## 
##  *** RIVER: EM step 38
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4489
##      M-step: norm(theta difference) = 0.0029, norm(beta difference) = 0.0022 *** 
## 
##  *** RIVER: EM step 39
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4457
##      M-step: norm(theta difference) = 0.0026, norm(beta difference) = 0.0021 *** 
## 
##  *** RIVER: EM step 40
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4426
##      M-step: norm(theta difference) = 0.0024, norm(beta difference) = 0.002 *** 
## 
##  *** RIVER: EM step 41
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4397
##      M-step: norm(theta difference) = 0.0021, norm(beta difference) = 0.0019 *** 
## 
##  *** RIVER: EM step 42
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4369
##      M-step: norm(theta difference) = 0.0019, norm(beta difference) = 0.0018 *** 
## 
##  *** RIVER: EM step 43
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4342
##      M-step: norm(theta difference) = 0.0017, norm(beta difference) = 0.0017 *** 
## 
##  *** RIVER: EM step 44
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4316
##      M-step: norm(theta difference) = 0.0015, norm(beta difference) = 0.0016 *** 
## 
##  *** RIVER: EM step 45
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4292
##      M-step: norm(theta difference) = 0.0014, norm(beta difference) = 0.0015 *** 
## 
##  *** RIVER: EM step 46
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4268
##      M-step: norm(theta difference) = 0.0012, norm(beta difference) = 0.0014 *** 
## 
##  *** RIVER: EM step 47
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4245
##      M-step: norm(theta difference) = 0.0011, norm(beta difference) = 0.0013 *** 
## 
##  *** RIVER: EM step 48
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4223
##      M-step: norm(theta difference) = 0.001, norm(beta difference) = 0.0013 *** 
## 
##  *** RIVER: EM step 49
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4202
##      M-step: norm(theta difference) = 8e-04, norm(beta difference) = 0.0012 *** 
## 
##  *** RIVER: EM step 50
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4181
##      M-step: norm(theta difference) = 7e-04, norm(beta difference) = 0.0011 *** 
## 
##  *** RIVER: EM step 51
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.416
##      M-step: norm(theta difference) = 7e-04, norm(beta difference) = 0.0011 *** 
## 
##  *** RIVER: EM step 52
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4141
##      M-step: norm(theta difference) = 8e-04, norm(beta difference) = 0.001 *** 
## 
##  *** RIVER: EM step 53
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4123
##      M-step: norm(theta difference) = 8e-04, norm(beta difference) = 0.001 *** 
## 
##  ::: EM iteration is terminated since it converges within a
##         predefined tolerance (0.001) ::: 
## 
## 
## *** AUC (GAM - genomic annotation model):  0.58 
##     AUC (RIVER):  0.8 
##      P-value:  <0.001 ***

evaRIVER requires a ExpressionSet class object containing genomic features, outlier status, and a list of N2 pairs as an input and four optional parameters including pseudo count, initial theta, a list of candidate \(\lambda\), and verbose. The input class is obtained by running getData with an original gzipped file. If you would like to know which format you should follow when generating the original compressed file, refer to the section 4 Generation of custumized data for RIVER below. Most of optional parameters are set according to your input data. The pseudoc is a hyperparameter for estimating theta, parameters between an unobserved FR node and observed outlier E node. Lower pseudoc provides higher reliance on observed data. The theta_init is an initial set of theta parameters. From left to right, the elements are \(P(E = 0 | FR = 0)\), \(P(E = 1 | FR = 0)\), \(P(E = 0 | FR = 1)\), and \(P(E = 1 | FR = 1)\), respecitively. The costs are the list of candidate \(\lambda\) for searching the best L2 penaly hyperparameter for both GAM and RIVER. For more information on optional paramters, see Appendix 5.1 for optional parameters and Appendix 5.2 for parameter stabilities across different initializations.

To train RIVER with training data (all instances except for N2 pairs), we first select best lambda value based on 10 cross-validation on training dataset via glmnet. You can see the selected \(\lambda\) parameter at the first line of output. The initial paramters of \(\beta\) in RIVER were set based on the estimated \(\beta\) parameters from GAM. In each EM iteration, the evaRIVER reports both the top 10 % threshold of expected \(P(FR = 1 | G, E)\) and norms of difference between previous and current estimates of parameters. The EM algorithm iteratively find best estimates of both \(\beta\) and \(\theta\) until it converges within the predefined tolerence of the norm (\(0.001\) for both \(\beta\) and \(\theta\)). After the estimates of paramters converge, evaRIVER reports AUC values from both models and its p-value of the difference between them.

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3.2 Application of RIVER

The function, appRIVER, is to train RIVER (and GAM) with all instances and compute posterior probabilities of them for the future analyses (i.e. finding pathogenic rare variants in disease). Same as evaRIVER, this function also requires a ExpressionSet class object as an input and three optional parameters which you can set again based on your data. If you use a certain set of values for the optional parameters, you would use same ones here.

postprobs <- appRIVER(dataInput, pseudoc=50, 
                      theta_init=matrix(c(.99, .01, .3, .7), nrow=2), 
                      costs=c(100, 10, 1, .1, .01, 1e-3, 1e-4), 
                      verbose=TRUE)
##  *** best lambda = 0.1 *** 
## 
##  *** RIVER: EM step 1
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.9542
##      M-step: norm(theta difference) = 0.1511, norm(beta difference) = 0.0081 *** 
## 
##  *** RIVER: EM step 2
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.9221
##      M-step: norm(theta difference) = 0.0522, norm(beta difference) = 0.0082 *** 
## 
##  *** RIVER: EM step 3
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8916
##      M-step: norm(theta difference) = 0.0495, norm(beta difference) = 0.0084 *** 
## 
##  *** RIVER: EM step 4
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8625
##      M-step: norm(theta difference) = 0.0469, norm(beta difference) = 0.0078 *** 
## 
##  *** RIVER: EM step 5
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8352
##      M-step: norm(theta difference) = 0.0444, norm(beta difference) = 0.0072 *** 
## 
##  *** RIVER: EM step 6
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.8091
##      M-step: norm(theta difference) = 0.0418, norm(beta difference) = 0.0068 *** 
## 
##  *** RIVER: EM step 7
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7845
##      M-step: norm(theta difference) = 0.0393, norm(beta difference) = 0.0062 *** 
## 
##  *** RIVER: EM step 8
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7612
##      M-step: norm(theta difference) = 0.0369, norm(beta difference) = 0.0058 *** 
## 
##  *** RIVER: EM step 9
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7393
##      M-step: norm(theta difference) = 0.0345, norm(beta difference) = 0.0054 *** 
## 
##  *** RIVER: EM step 10
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.7187
##      M-step: norm(theta difference) = 0.0323, norm(beta difference) = 0.0052 *** 
## 
##  *** RIVER: EM step 11
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6995
##      M-step: norm(theta difference) = 0.0301, norm(beta difference) = 0.0048 *** 
## 
##  *** RIVER: EM step 12
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6815
##      M-step: norm(theta difference) = 0.028, norm(beta difference) = 0.0046 *** 
## 
##  *** RIVER: EM step 13
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6648
##      M-step: norm(theta difference) = 0.026, norm(beta difference) = 0.0044 *** 
## 
##  *** RIVER: EM step 14
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6492
##      M-step: norm(theta difference) = 0.0241, norm(beta difference) = 0.0044 *** 
## 
##  *** RIVER: EM step 15
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6347
##      M-step: norm(theta difference) = 0.0223, norm(beta difference) = 0.0043 *** 
## 
##  *** RIVER: EM step 16
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6212
##      M-step: norm(theta difference) = 0.0206, norm(beta difference) = 0.0042 *** 
## 
##  *** RIVER: EM step 17
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.6084
##      M-step: norm(theta difference) = 0.019, norm(beta difference) = 0.0041 *** 
## 
##  *** RIVER: EM step 18
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5965
##      M-step: norm(theta difference) = 0.0176, norm(beta difference) = 0.004 *** 
## 
##  *** RIVER: EM step 19
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5853
##      M-step: norm(theta difference) = 0.0162, norm(beta difference) = 0.0039 *** 
## 
##  *** RIVER: EM step 20
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5751
##      M-step: norm(theta difference) = 0.0149, norm(beta difference) = 0.0038 *** 
## 
##  *** RIVER: EM step 21
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5654
##      M-step: norm(theta difference) = 0.0137, norm(beta difference) = 0.0038 *** 
## 
##  *** RIVER: EM step 22
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5563
##      M-step: norm(theta difference) = 0.0126, norm(beta difference) = 0.0037 *** 
## 
##  *** RIVER: EM step 23
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5478
##      M-step: norm(theta difference) = 0.0116, norm(beta difference) = 0.0036 *** 
## 
##  *** RIVER: EM step 24
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.54
##      M-step: norm(theta difference) = 0.0107, norm(beta difference) = 0.0035 *** 
## 
##  *** RIVER: EM step 25
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5326
##      M-step: norm(theta difference) = 0.0098, norm(beta difference) = 0.0035 *** 
## 
##  *** RIVER: EM step 26
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5258
##      M-step: norm(theta difference) = 0.009, norm(beta difference) = 0.0034 *** 
## 
##  *** RIVER: EM step 27
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5194
##      M-step: norm(theta difference) = 0.0083, norm(beta difference) = 0.0033 *** 
## 
##  *** RIVER: EM step 28
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5133
##      M-step: norm(theta difference) = 0.0076, norm(beta difference) = 0.0032 *** 
## 
##  *** RIVER: EM step 29
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5077
##      M-step: norm(theta difference) = 0.007, norm(beta difference) = 0.003 *** 
## 
##  *** RIVER: EM step 30
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.5023
##      M-step: norm(theta difference) = 0.0064, norm(beta difference) = 0.0029 *** 
## 
##  *** RIVER: EM step 31
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4974
##      M-step: norm(theta difference) = 0.0058, norm(beta difference) = 0.0028 *** 
## 
##  *** RIVER: EM step 32
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4927
##      M-step: norm(theta difference) = 0.0053, norm(beta difference) = 0.0027 *** 
## 
##  *** RIVER: EM step 33
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4882
##      M-step: norm(theta difference) = 0.0049, norm(beta difference) = 0.0025 *** 
## 
##  *** RIVER: EM step 34
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4841
##      M-step: norm(theta difference) = 0.0045, norm(beta difference) = 0.0024 *** 
## 
##  *** RIVER: EM step 35
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.48
##      M-step: norm(theta difference) = 0.0041, norm(beta difference) = 0.0023 *** 
## 
##  *** RIVER: EM step 36
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4763
##      M-step: norm(theta difference) = 0.0037, norm(beta difference) = 0.0022 *** 
## 
##  *** RIVER: EM step 37
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4728
##      M-step: norm(theta difference) = 0.0034, norm(beta difference) = 0.0021 *** 
## 
##  *** RIVER: EM step 38
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4694
##      M-step: norm(theta difference) = 0.0031, norm(beta difference) = 0.002 *** 
## 
##  *** RIVER: EM step 39
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4662
##      M-step: norm(theta difference) = 0.0028, norm(beta difference) = 0.0018 *** 
## 
##  *** RIVER: EM step 40
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4631
##      M-step: norm(theta difference) = 0.0025, norm(beta difference) = 0.0017 *** 
## 
##  *** RIVER: EM step 41
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4601
##      M-step: norm(theta difference) = 0.0023, norm(beta difference) = 0.0016 *** 
## 
##  *** RIVER: EM step 42
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4573
##      M-step: norm(theta difference) = 0.0021, norm(beta difference) = 0.0016 *** 
## 
##  *** RIVER: EM step 43
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4547
##      M-step: norm(theta difference) = 0.0019, norm(beta difference) = 0.0015 *** 
## 
##  *** RIVER: EM step 44
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4521
##      M-step: norm(theta difference) = 0.0017, norm(beta difference) = 0.0014 *** 
## 
##  *** RIVER: EM step 45
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4497
##      M-step: norm(theta difference) = 0.0015, norm(beta difference) = 0.0013 *** 
## 
##  *** RIVER: EM step 46
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4473
##      M-step: norm(theta difference) = 0.0014, norm(beta difference) = 0.0012 *** 
## 
##  *** RIVER: EM step 47
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.445
##      M-step: norm(theta difference) = 0.0012, norm(beta difference) = 0.0012 *** 
## 
##  *** RIVER: EM step 48
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4427
##      M-step: norm(theta difference) = 0.0011, norm(beta difference) = 0.0011 *** 
## 
##  *** RIVER: EM step 49
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4406
##      M-step: norm(theta difference) = 0.001, norm(beta difference) = 0.001 *** 
## 
##  *** RIVER: EM step 50
##      E-step: Top 10 % Threshold of expected P(FR=1 | G, E): 0.4386
##      M-step: norm(theta difference) = 9e-04, norm(beta difference) = 0.001 *** 
## 
##  ::: EM iteration is terminated since it converges within a
##         predefined tolerance (0.001) :::

Like the reported procedures from evaRIVER, we can recognize which \(\lambda\) is set and variant top 10 % threshold of expected \(P(FR = 1 | G, E)\) and norms of difference during each of EM iteractions.

If you would like to observe estimated parameters associated with genomic features (\(\beta\)) and outliers (\(\theta\)), you can simply use print for the corresponding parameters of interest.

print(postprobs$fitRIVER$beta)
##      Feature1      Feature2      Feature3      Feature4      Feature5 
##  0.0468452545  0.0006976417  0.0723175214  0.0763155944  0.0427705093 
##      Feature6      Feature7      Feature8      Feature9     Feature10 
##  0.0269997566  0.0734000766 -0.0206468997  0.0248426648 -0.0012436251 
##     Feature11     Feature12     Feature13     Feature14     Feature15 
##  0.0060632810  0.0358184099 -0.0044748430  0.0453044233  0.0481758514 
##     Feature16     Feature17     Feature18 
## -0.0005280174  0.0079277389 -0.0084118400
print(postprobs$fitRIVER$theta)
##           [,1]      [,2]
## [1,] 0.8394424 0.4784675
## [2,] 0.1605576 0.5215325

These parameters can be used for computing test posterior probabilities of new instances given their \(G\) and \(E\) for further analyses.

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4 Generation of custumized data for RIVER

An original compressed file, generated from all necessary processed data including genomic features from various genomic annotations, Z-scores from gene expression, and a list of N2 pairs based on WGS, contains all the information.

filename <- system.file("extdata", "simulation_RIVER.gz", 
                       package = "RIVER")
system(paste('zcat ', filename, " | head  -2", sep=""), 
       ignore.stderr=TRUE)

From right to left column in each row, the data includes subject ID, gene name, values of genomic features of interest (18 features here), Z-scores of corresponding gene expression, and either integer values or NA representing the existence/absence in N2 pairs sharing same rare variants. If one subject has a unique set of rare variants compared to other subjects near a particular gene, NA is assigned in N2pair column. Otherwise, two subjects sharing same rare variants in any gene have same integers as unique identifiers of each of N2 pairs.

If you would like to train RIVER with your own data, you need to generate your own compressed file having same file format as explained above. Then, you simply put an entire path of your compressed data file into getData which generates a ExpressionSet class object (YourInputToRIVER) with all necessary information for running RIVER with your own data.

YourInputToRIVER <- getData(filename) # import experimental data

For our paper, genomic features were generated from various genomic annotations including conservation scores like Gerp, chromatin states from chromHMM or segway, and other summary scores such as CADD and DANN. The intances were selected based on two criteria: (1) any genes having at least one individual outlier in term of z-scores of gene expression and (2) any individuals having at least one rare variant within specific regions near each gene. In each instance, the feature values within regions of interest were aggreated into one summary statistics by applying relevant mathematical operations like max. In more details of a list of genomic annotations used for constructing features and how to generate the features and outlier status, please refer to our publication pre-print.

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5 Basics

5.1 Installation of RIVER

R is an open-source statistical environment which can be easily modified to enhance its functionality via packages. RIVER is a R package available via the Bioconductor repository for packages. R can be installed on any operating system from CRAN after which you can install RIVER by using the following commands in your R session:

## try http:// if https:// URLs are not supported
source("https://bioconductor.org/biocLite.R")
biocLite("RIVER")

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5.2 Session Info

Here is the output of sessionInfo() on the system on which this document was compiled:

## Session info
library('devtools')
options(width=120)
session_info()
## Session info ----------------------------------------------------------------------------------------------------------
##  setting  value                                      
##  version  R version 3.5.0 Patched (2018-05-03 r74699)
##  system   x86_64, linux-gnu                          
##  ui       X11                                        
##  language (EN)                                       
##  collate  C                                          
##  tz       America/New_York                           
##  date     2018-06-13
## Packages --------------------------------------------------------------------------------------------------------------
##  package      * version date       source        
##  backports      1.1.2   2017-12-13 CRAN (R 3.5.0)
##  base         * 3.5.0   2018-05-09 local         
##  Biobase        2.41.0  2018-06-13 Bioconductor  
##  BiocGenerics   0.27.0  2018-06-13 Bioconductor  
##  BiocStyle    * 2.9.3   2018-06-13 Bioconductor  
##  bookdown       0.7     2018-02-18 CRAN (R 3.5.0)
##  codetools      0.2-15  2016-10-05 CRAN (R 3.5.0)
##  colorspace     1.3-2   2016-12-14 CRAN (R 3.5.0)
##  compiler       3.5.0   2018-05-09 local         
##  datasets     * 3.5.0   2018-05-09 local         
##  devtools     * 1.13.5  2018-02-18 CRAN (R 3.5.0)
##  digest         0.6.15  2018-01-28 CRAN (R 3.5.0)
##  evaluate       0.10.1  2017-06-24 CRAN (R 3.5.0)
##  foreach        1.4.4   2017-12-12 CRAN (R 3.5.0)
##  ggplot2        2.2.1   2016-12-30 CRAN (R 3.5.0)
##  glmnet         2.0-16  2018-04-02 CRAN (R 3.5.0)
##  graphics     * 3.5.0   2018-05-09 local         
##  grDevices    * 3.5.0   2018-05-09 local         
##  grid           3.5.0   2018-05-09 local         
##  gtable         0.2.0   2016-02-26 CRAN (R 3.5.0)
##  htmltools      0.3.6   2017-04-28 CRAN (R 3.5.0)
##  iterators      1.0.9   2017-12-12 CRAN (R 3.5.0)
##  knitr          1.20    2018-02-20 CRAN (R 3.5.0)
##  labeling       0.3     2014-08-23 CRAN (R 3.5.0)
##  lattice        0.20-35 2017-03-25 CRAN (R 3.5.0)
##  lazyeval       0.2.1   2017-10-29 CRAN (R 3.5.0)
##  magrittr       1.5     2014-11-22 CRAN (R 3.5.0)
##  Matrix         1.2-14  2018-04-13 CRAN (R 3.5.0)
##  memoise        1.1.0   2017-04-21 CRAN (R 3.5.0)
##  methods      * 3.5.0   2018-05-09 local         
##  munsell        0.5.0   2018-06-12 CRAN (R 3.5.0)
##  parallel       3.5.0   2018-05-09 local         
##  pillar         1.2.3   2018-05-25 CRAN (R 3.5.0)
##  plyr           1.8.4   2016-06-08 CRAN (R 3.5.0)
##  pROC           1.12.1  2018-05-06 CRAN (R 3.5.0)
##  Rcpp           0.12.17 2018-05-18 CRAN (R 3.5.0)
##  RIVER        * 1.5.0   2018-06-13 Bioconductor  
##  rlang          0.2.1   2018-05-30 CRAN (R 3.5.0)
##  rmarkdown      1.10    2018-06-11 CRAN (R 3.5.0)
##  rprojroot      1.3-2   2018-01-03 CRAN (R 3.5.0)
##  scales         0.5.0   2017-08-24 CRAN (R 3.5.0)
##  stats        * 3.5.0   2018-05-09 local         
##  stringi        1.2.3   2018-06-12 CRAN (R 3.5.0)
##  stringr        1.3.1   2018-05-10 CRAN (R 3.5.0)
##  tibble         1.4.2   2018-01-22 CRAN (R 3.5.0)
##  tools          3.5.0   2018-05-09 local         
##  utils        * 3.5.0   2018-05-09 local         
##  withr          2.1.2   2018-03-15 CRAN (R 3.5.0)
##  xfun           0.1     2018-01-22 CRAN (R 3.5.0)
##  yaml           2.1.19  2018-05-01 CRAN (R 3.5.0)

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5.3 Asking for Help

As package developers, we try to explain clearly how to use our packages and in which order to use the functions. But R and Bioconductor have a steep learning curve so it is critical to learn where to ask for help. The blog post quoted above mentions some but we would like to highlight the Bioconductor support site as the main resource for getting help. Other alternatives are available such as creating GitHub issues and tweeting. However, please note that if you want to receive help you should adhere to the posting guidlines. It is particularly critical that you provide a small reproducible example and your session information so package developers can track down the source of the error.

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5.4 References

Xin Li\(^{*}\), Yungil Kim\(^{*}\), Emily K. Tsang\(^{*}\), Joe R. Davis\(^{*}\), Farhan N. Damani, Colby Chiang, Zachary Zappala, Benjamin J. Strober, Alexandra J. Scott, Andrea Ganna, Jason Merker, GTEx Consortium, Ira M. Hall, Alexis Battle\(^{\#}\), Stephen B. Montgomery\(^{\#}\) (2016).
The impact of rare variation on gene expression across tissues
(in arXiv, submitted, *: equal contribution, #: corresponding authors)

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6 Appendices

6.1 Optional parameters

Functions within RIVER have a set of optional parameters which control some aspects of the computation of RIVER scores. The factory default settings are expected to serve in many cases, but users might need to make changes based on the input data.

There are four parameters that users can change:

pseudoc - Pseudo count (hyperparameter) in a beta distribution for \(\theta\); factory default = 50

theta_init - Initial values of \(\theta\); factory default = (P(E = 0 | FR = 0), P(E = 1 | FR = 0), P(E = 0 | FR = 1), P(E = 1 | FR = 1)) = (0.99, 0.01, 0.3, 0.7)

costs - List of candidate \(\lambda\) values for finding a best \(\lambda\) (hyperparameter). A best \(\lambda\) value among the candidate list is selected from L2-regularized logistic regression (GAM) via 10 cross-validation; factory default = (100, 10, 1, .1, .01, 1e-3, 1e-4)

verbose - If you set this parameter as TRUE, you observe how parameters including \(\theta\) and \(\beta\) converge until their updates at each EM iteration are within predefined tolerance levels (one norm of the difference between current and previous parameters < 1e-3); factory default = FALSE

Note that initial values of \(\beta\) are generated from L2-regularized logistic regression (GAM) with pre-selected \(\lambda\) from 10 cross-validation.

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6.2 Stability of Estimated Parameters with Different Parameter Initializations

In this section, we reports how several different initialization parameters for either \(\beta\) or \(\theta\) affect the estimated parameters. We initialized a noisy \(\beta\) by adding K% Gaussian noise compared to the mean of \(\beta\) with fixed \(\theta\) (for K = 10, 20, 50 100, 200, 400, 800). For \(\theta\), we fixed P(E = 1 | FR = 0) and P(E = 0 | FR = 0) as 0.01 and 0.99, respectively, and initialized (P(E = 1 | FR = 1), P(E = 0 | FR = 1)) as (0.1, 0.9), (0.4, 0.6), and (0.45, 0.55) instead of (0.3, 0.7) with \(\beta\) fixed. For each parameter initialization, we computed Spearman rank correlations between parameters from RIVER using the original initialization and the alternative initializations. We also investigated how many instances within top 10% of posterior probabilities from RIVER under the original settings were replicated in the top 10% of posterior probabilities under the alternative initializations. We also tried five different values of pseudoc as 10, 20, 30, 75, and 100 with default settings of \(\beta\) and \(\theta\) and computed both Spearman rank correlations and accuracy as explained above.

Parameter Initialization Spearman ρ Accuracy
10% noise > .999 0.880
25% noise > .999 0.862
50% noise > .999 0.849
\(\beta\) 100% noise > .999 0.848
200% noise > .999 0.843
400% noise > .999 0.846
800% noise > .999 0.846
[0.1, 0.9] > .999 0.841
\(\theta\) [0.4, 0.6] > .999 1.000
[0.45, 0.55] > .999 1.000
10 .988 0.934
20 .996 0.955
pseudoc 30 .999 0.972
75 .999 0.979
100 .998 0.967

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