User:Timothee Flutre/Notebook/Postdoc/2011/06/28
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Linear regression by ordinary least squares
In matrix notation: y = Xθ + ε with ε˜N_{N}(0,σ^{2}I_{N}) and θ^{T} = (μ,β)
Here is the ordinary-least-square (OLS) estimator of θ:
Let's now define 4 summary statistics, very easy to compute:
This allows to obtain the estimate of the effect size only by having the summary statistics available:
The same works for the estimate of the standard deviation of the errors:
We can also benefit from this for the standard error of the parameters:
V(y) = V(μ + βg + ε) = V(μ) + V(βg) + V(ε) = β^{2}V(g) + σ^{2} The most intuitive way to simulate data is therefore to fix the proportion of variance in y explained by the genotype, for instance PVE = 60%, as well as the standard deviation of the errors, typically σ = 1. From this, we can calculate the corresponding effect size β of the genotype:
Therefore: Note that g is the random variable corresponding to the genotype encoded in allele dose, such that it is equal to 0, 1 or 2 copies of the minor allele. For our simulation, we will fix the minor allele frequency f (eg. f = 0.3) and we will assume Hardy-Weinberg equilibrium. Then g is distributed according to a binomial distribution with 2 trials for which the probability of success is f. As a consequence, its variance is V(g) = 2f(1 − f). Here is some R code implementing all this: set.seed(1859) N <- 100 # sample size mu <- 4 pve <- 0.6 sigma <- 1 maf <- 0.3 beta <- sigma * sqrt(pve / ((1 - pve) * 2 * maf * (1 - maf))) # 1.88 g <- sample(x=0:2, size=N, replace=TRUE, prob=c(maf^2, 2*maf*(1-maf), (1-maf)^2)) y <- mu + beta * g + rnorm(n=N, mean=0, sd=sigma) ols <- lm(y ~ g) summary(ols) # muhat=3.5, betahat=2.1, R2=0.64 sqrt(mean(ols$residuals^2)) # sigmahat = 0.98 plot(x=0, type="n", xlim=range(g), ylim=range(y), xlab="genotypes", ylab="phenotypes", main="Simple linear regression") for(i in unique(g)) points(x=jitter(g[g == i]), y=y[g == i], col=i+1, pch=19) abline(a=coefficients(ols)[1], b=coefficients(ols)[2])
As above, we want , and . To efficiently get them, we start with the singular value decomposition of X: X = UDV^{T} This allows us to get the Moore-Penrose pseudoinverse matrix of X: X^{ + } = (X^{T}X)^{ − 1}X^{T} X^{ + } = VD^{ − 1}U^{T} From this, we get the OLS estimate of the effect sizes:
Then it's straightforward to get the residuals:
With them we can calculate the estimate of the error variance:
And finally the standard errors of the estimates of the effect sizes:
We can check this with some R code: ## simulate the data set.seed(1859) N <- 100 mu <- 5 Xg <- sample(x=0:2, size=N, replace=TRUE, prob=c(0.5, 0.3, 0.2)) # genotypes beta.g <- 0.5 Xc <- sample(x=0:1, size=N, replace=TRUE, prob=c(0.7, 0.3)) # gender beta.c <- 0.3 pve <- 0.8 betas.gc.bar <- mean(beta.g * Xg + beta.c * Xc) # 0.405 sigma <- sqrt((1/N) * sum((beta.g * Xg + beta.c * Xc - betas.gc.bar)^2) * (1-pve) / pve) # 0.2 y <- mu + beta.g * Xg + beta.c * Xc + rnorm(n=N, mean=0, sd=sigma) ## perform the OLS analysis with the SVD of X X <- cbind(rep(1,N), Xg, Xc) Xp <- svd(x=X) B.hat <- Xp$v %*% diag(1/Xp$d) %*% t(Xp$u) %*% y E.hat <- y - X %*% B.hat sigma.hat <- as.numeric(sqrt((1/(N-3)) * t(E.hat) %*% E.hat)) # 0.211 var.theta.hat <- sigma.hat^2 * Xp$v %*% diag((1/Xp$d)^2) %*% t(Xp$v) sqrt(diag(var.theta.hat)) # 0.0304 0.0290 0.0463 ## check all this ols <- lm(y ~ Xg + Xc) summary(ols) # muhat=4.99+-0.03, beta.g.hat=0.52+--.-29, beta.c.hat=0.24+-0.046, R2=0.789 Such an analysis can also be done easily in a custom C/C++ program thanks to the GSL (here). |