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Direct effect sizes estimated from latent factor models

Description

This function returns 'direct' effect sizes for the regression of X (of dimension 1) on the matrix Y,as usually computed in genome-wide association studies.

Usage

effect_size(Y, X, lfmm.object)

Arguments

Details

The response variable matrix Y and the explanatory variable are centered.

Value

a vector of length p containing all effect sizes for the regressionof X on the matrix Y

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

Examples

library(lfmm)## Simulation of 1000 genotypes for 100 individuals (y)u <- matrix(rnorm(300, sd = 1), nrow = 100, ncol = 3) v <- matrix(rnorm(3000, sd = 3), nrow = 3, ncol = 1000)w <- u %*% vy <- matrix(rbinom(100000, size = 2,                   prob = 1/(1 + exp(-0.3*(w                  + rnorm(100000, sd = 2))))),                  nrow = 100,                  ncol = 1000)#PCA of genotypes, 3 main axes of variation (K = 2) plot(prcomp(y))  ## Simulation of 1000 phenotypes (x)## Only the last 10 genotypes have significant effect sizes (b)b <- matrix(c(rep(0, 990), rep(6000, 10)))x <- y%*%b + rnorm(100, sd = 100)## Compute effect sizes using lfmm_ridge## Note that centering is important (scale = F).mod.lfmm <- lfmm_ridge(Y = y,                   X = x,                  K = 2)              ## Compute direct effect sizes using lfmm_ridge estimatesb.estimates <- effect_size(y, x, mod.lfmm)## plot the last 30 effect sizes (true values are 0 and 6000)plot(b.estimates[971:1000])abline(0, 0)abline(6000, 0, col = 2)## Prediction of phenotypescandidates <- 991:1000 #set of causal loci x.pred <- scale(y[,candidates], scale = FALSE) %*% matrix(b.estimates[candidates])## Check predictionsplot(x - mean(x), x.pred,      pch = 19, col = "grey",      xlab = "Observed phenotypes (centered)",      ylab = "Predicted from PRS")     abline(0,1)     abline(lm(x.pred ~ scale(x, scale = FALSE)), col = 2)

Genetic and phenotypic data for Arabidopsis thaliana

Description

A dataset containing SNP frequency and simulated phenotypic data for 170 plant accessions.The variables are as follows:

Usage

data(example.data)

Format

A list with 4 arguments: genotype, phenotype, causal.set, chrpos

Details

• genotype: binary (0 or 1) SNP frequency for 170 individuals (26943 SNPs).

• phenotype: simulated phenotypic data for 170 individuals.

• causal.set: set of indices for causal SNPs.

• chrpos: genetic map including chromosome position of each SNP.

Reference: Atwell et al (2010).Genome-wide association study of 107 phenotypes in Arabidopsis thaliana inbred lines.Nature 465, 627–631.

GLM tests with latent factor mixed models

Description

This function returns significance values for the association between each column of theresponse matrix, Y, and the explanatory variables, X, including correction for unobserved confounders(latent factors). The test is based on an LFMM fitted with a ridge or lasso penalty and a generalized linearmodel.

Usage

glm_test(Y, X, lfmm.obj, calibrate = "gif", family = binomial(link = "logit"))

Arguments

Details

The response variable matrix Y and the explanatory variable are centered.

Value

a list with the following attributes:

• B the effect size matrix with dimensions p x d.

• score a p x d matrix which contains z-scores for each explanatory variable (columns of X),

• pvalue a p x d matrix which contains p-values for each explanatory variable,

• calibrated.pvalue a p x d matrix which contains calibrated p-values for each explanatory variable,

• gif a numeric value for the genomic inflation factor.

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

See Also

lfmm_test

Examples

library(lfmm)## An EWAS example with Y = methylation data ## and X = "exposure" ## Simulate the datadat <- lfmm_sampler(n = 100,                     p = 500,                    K = 3,                    outlier.prop = 0.01,                    cs = 0.1,                    sigma = 0.2,                    B.sd = 5,                    B.mean = 0,                    U.sd = 1.0,                    V.sd = 1.0)                    Y <- pnorm(dat$Y)X <- dat$X## Fit an LFMM with 2 latent factorsmod.lfmm <- lfmm_ridge(Y = Y,                       X = X,                        K = 3)                       ## Perform association testing using the fitted model:pv <- glm_test(Y = pnorm(Y),                 X = X,                lfmm.obj = mod.lfmm,                 family = binomial(link = "probit"),                calibrate = "gif")                ## Manhattan plot with true associations showncausal <- dat$outlierpvalues <- pv$calibrated.pvalueplot(-log10(pvalues),      pch = 19,      cex = .3,     xlab = "Probe",     col = "grey")     points(causal,       -log10(pvalues)[causal],        col = "blue")

R package : Fast and Accurate statistical methods for adjusting confounding factorsin association studies.

Description

Implements statistical methods for adjusting confounding factorsin association studies.

References

Caye, K., B. Jumentier, J. Lepeule, and O. François, 2019 LFMM 2: fast and accurate inference of gene-environment associations in genome-widestudies. Mol. Biol. Evol. 36: 852–860.https://doi.org/10.1093/molbev/msz008

B. Jumentier, Caye, K., J. Lepeule, and O. François, 2019 Sparse latent factor regression models for genome-wide and epigenome-wide association studies (in prep)

LFMM least-squares estimates with lasso penalty (Sparse LFMM)

Description

This function computes regularized least squares estimatesfor latent factor mixed models using a lasso penalty.

Usage

lfmm_lasso(  Y,  X,  K,  nozero.prop = 0.01,  mu.num = 100,  mu.min.ratio = 0.01,  mu = NULL,  it.max = 100,  relative.err.epsilon = 1e-06)

Arguments

Details

The algorithm minimizes the following penalized least-squares criterion

The response variable matrix Y and the explanatory variable are centered.

Value

an object of classlfmm with the following attributes:

• U the latent variable score matrix with dimensions n x K,

• V the latent variable axes matrix with dimensions p x K,

• B the effect size matrix with dimensions p x d.

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

References

B. Jumentier, Caye, K., J. Lepeule, and O. François, 2019 Sparse latent factor regression models for genome-wide and epigenome-wide association studies (in prep)

Examples

library(lfmm)## An EWAS example with Y = methylation data ## and X = exposure## Simulate the data dat <- lfmm_sampler(n = 100,                     p = 1000,                    K = 3,                    outlier.prop = 0.02,                    cs = 0.1,                    sigma = 0.2,                    B.sd = 5,                    B.mean = 0,                    U.sd = 1.0,                    V.sd = 1.0)Y <- scale(dat$Y)X <- scale(dat$X)## Fit an LFMM with 2 latent factorsmod.lfmm <- lfmm_lasso(Y = Y,                       X = X,                        K = 3,                       nozero.prop = 0.02)                       ## Manhattan plot of sparse effect sizeseffect <- mod.lfmm$Bcausal <- dat$outlierplot(effect,      pch = 19,      cex = .3,     xlab = "Probe",     col = "grey")     points(causal,        effect[causal],        col = "blue")

LFMM least-squares estimates with ridge penalty

Description

This function computes regularized least squares estimatesfor latent factor mixed models using a ridge penalty.

Usage

lfmm_ridge(  Y,  X,  K,  lambda = 1e-05,  algorithm = "analytical",  it.max = 100,  relative.err.min = 1e-06)

Arguments

Details

The algorithm minimizes the following penalized least-squares criterion

L(U, V, B) = \frac{1}{2} ||Y - U V^{T} - X B^T||_{F}^2 + \frac{\lambda}{2} ||B||^{2}_{2} ,

where Y is a response data matrix, X contains all explanatory variables,U denotes the score matrix, V is the loading matrix, B is the (direct) effectsize matrix, and lambda is a regularization parameter.

The response variable matrix Y and the explanatory variable are centered.

Value

an object of classlfmm with the following attributes:

• U the latent variable score matrix with dimensions n x K,

• V the latent variable axis matrix with dimensions p x K,

• B the effect size matrix with dimensions p x d.

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

References

Caye, K., B. Jumentier, J. Lepeule, and O. François, 2019 LFMM 2: fast and accurate inference of gene-environment associations in genome-widestudies. Mol. Biol. Evol. 36: 852–860.https://doi.org/10.1093/molbev/msz008

Examples

library(lfmm)## a GWAS example with Y = SNPs and X = phenotypedata(example.data)Y <- example.data$genotype[, 1:10000]X <- example.data$phenotype## Fit an LFMM with K = 6 factorsmod.lfmm <- lfmm_ridge(Y = Y,                        X = X,                        K = 6)## Perform association testing using the fitted model:pv <- lfmm_test(Y = Y,                 X = X,                 lfmm = mod.lfmm,                 calibrate = "gif")## Manhattan plot with causal loci shownpvalues <- pv$calibrated.pvalueplot(-log10(pvalues), pch = 19,      cex = .2, col = "grey", xlab = "SNP")points(example.data$causal.set[1:5],       -log10(pvalues)[example.data$causal.set[1:5]],        type = "h", col = "blue")## An EWAS example with Y = methylation data and X = exposureY <- skin.exposure$beta.valueX <- as.numeric(skin.exposure$exposure)## Fit an LFMM with 2 latent factorsmod.lfmm <- lfmm_ridge(Y = Y,                       X = X,                        K = 2)                       ## Perform association testing using the fitted model:pv <- lfmm_test(Y = Y,                 X = X,                lfmm = mod.lfmm,                 calibrate = "gif")                ## Manhattan plot with true associations shownpvalues <- pv$calibrated.pvalueplot(-log10(pvalues),      pch = 19,      cex = .3,     xlab = "Probe",     col = "grey")     causal.set <- seq(11, 1496, by = 80)points(causal.set,       -log10(pvalues)[causal.set],        col = "blue")

LFMM generative data sampler

Description

Simulate data from the latent factor model.

Usage

lfmm_sampler(  n,  p,  K,  outlier.prop,  cs,  sigma = 0.2,  B.sd = 1,  B.mean = 0,  U.sd = 1,  V.sd = 1)

Arguments

Details

lfmm_sampler() sample a response matrix Y and a primary variable X such that

     Y = U t(V) + X t(B) + Epsilon.

U,V, B and Epsilon are simulated according to normal multivariate distributions.Moreover U and X are such thatcor(U[,i], X) = cs[i].

Value

A list with simulated data.

Author(s)

kevin caye, olivier francois

Examples

dat <- lfmm_sampler(n = 100,                     p = 1000,                     K = 3,                    outlier.prop = 0.1,                    cs = c(0.8),                    sigma = 0.2,                    B.sd = 1.0,                     B.mean = 0.0,                    U.sd = 1.0,                     V.sd = 1.0)

Statistical tests with latent factor mixed models (linear models)

Description

This function returns significance values for the association between each column of theresponse matrix, Y, and the explanatory variables, X, including correction for unobserved confounders(latent factors). The test is based on an LFMM fitted with a ridge or lasso penalty (linear model).

Usage

lfmm_test(Y, X, lfmm, calibrate = "gif")

Arguments

Details

The response variable matrix Y and the explanatory variable are centered.

Value

a list with the following attributes:

• B the effect size matrix with dimensions p x d.

• score a p x d matrix which contains z-scores for each explanatory variable (columns of X),

• pvalue a p x d matrix which contains p-values for each explanatory variable,

• calibrated.pvalue a p x d matrix which contains calibrated p-values for each explanatory variable,

• gif a numeric value for the genomic inflation factor.

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

See Also

glm_test

Examples

library(lfmm)## a GWAS example with Y = SNPs and X = phenotypedata(example.data)Y <- example.data$genotype[, 1:10000]X <- example.data$phenotype## Fit an LFMM with K = 6 factorsmod.lfmm <- lfmm_ridge(Y = Y,                        X = X,                        K = 6)## Perform association testing using the fitted model:pv <- lfmm_test(Y = Y,                 X = X,                 lfmm = mod.lfmm,                 calibrate = "gif")## Manhattan plot with causal loci shownpvalues <- pv$calibrated.pvalueplot(-log10(pvalues), pch = 19,      cex = .2, col = "grey", xlab = "SNP")points(example.data$causal.set[1:5],       -log10(pvalues)[example.data$causal.set[1:5]],        type = "h", col = "blue")## An EWAS example with Y = methylation data and X = exposuredata("skin.exposure")Y <- scale(skin.exposure$beta.value)X <- scale(as.numeric(skin.exposure$exposure))## Fit an LFMM with 2 latent factorsmod.lfmm <- lfmm_ridge(Y = Y,                       X = X,                        K = 2)                       ## Perform association testing using the fitted model:pv <- lfmm_test(Y = Y,                 X = X,                lfmm = mod.lfmm,                 calibrate = "gif")                ## Manhattan plot with true associations shownpvalues <- pv$calibrated.pvalueplot(-log10(pvalues),      pch = 19,      cex = .3,     xlab = "Probe",     col = "grey")     causal.set <- seq(11, 1496, by = 80)points(causal.set,       -log10(pvalues)[causal.set],        col = "blue")

Predict polygenic scores from latent factor models

Description

This function computes polygenic risk scores from the estimates of latent factor models.It uses the indirect' effect sizes for the regression of X (a single phenotype) on the matrix Y,for predicting phenotypic values for new genotype data.

Usage

predict_lfmm(Y, X, lfmm.object, fdr.level = 0.1, newdata = NULL)

Arguments

Details

The response variable matrix Y and the explanatory variable are centered.

Value

a list with the following attributes:

• prediction: a vector of length n containing the predicted values for X. If newdata= NULL, the fitted values are returned.

• candidates: a vector of candidate columns of Y on which the predictions are built.

Author(s)

Kevin Caye, Basile Jumentier, Olivier Francois

Examples

library(lfmm)## Simulation of 1000 genotypes for 100 individuals (y)u <- matrix(rnorm(300, sd = 1), nrow = 100, ncol = 3) v <- matrix(rnorm(3000, sd = 3), nrow = 3, ncol = 1000)w <- u %*% vy <- matrix(rbinom(100000, size = 2,                   prob = 1/(1 + exp(-0.3 * (w                   + rnorm(100000, sd = 2))))),                  nrow = 100,                  ncol = 1000)#PCA of genotypes, 2 main axes of variation (K = 2) plot(prcomp(y))  ## Simulation of 1000 phenotypes (x)## Only the last 10 genotypes have significant effect sizes (b)b <- matrix(c(rep(0, 990), rep(6000, 10)))x <- y%*%b + rnorm(100, sd = 100)## Compute effect sizes using lfmm_ridgemod <- lfmm_ridge(Y = y,                   X = x,                  K = 2)              x.pred <- predict_lfmm(Y = y,                        X = x,                       fdr.level = 0.25,                        mod)                    x.pred$candidates##Compare simulated and predicted/fitted phenotypesplot(x - mean(x), x.pred$pred,      pch = 19, col = "grey",      xlab = "Observed phenotypes (centered)",      ylab = "Predicted from PRS")abline(0,1)abline(lm(x.pred$pred ~ scale(x, scale = FALSE)), col = 2)

Simulated (and real) methylation levels for sun exposed patient patients

Description

A data set containing normalized beta values, and sun exposure and simulatedphenotypic data for 78 tissue samples.

Usage

data("skin.exposure")

Format

A list with 6 arguments: beta.value, phenotype, causal.set, chrpos

Details

The variables are:

• beta.value: 1496 filtered normalized beta values (methyation probabilities)for 78 tissue samples.

• exposure: Sun exposure levels for 78 tissue samples.

• phenotype: Simulated binary phenotypic data for 78 tissue samples.

• age: age of patients.

• gender: sex of patients.

• tissue: category for tissue samples.

Reference: Vandiver et al (2015).Age and sun exposure-related widespread genomic blocks of hypomethylation in nonmalignant skin.Genome Biol. 16.