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:
with and V(ε) = σ^{2} This way, by also fixing β, it is easy to calculate the corresponding σ:
Here is some R code implementing this: set.seed(1859) N <- 100 mu <- 5 g <- sample(x=0:2, size=N, replace=TRUE, prob=c(0.5, 0.3, 0.2)) # MAF=0.2 beta <- 0.5 pve <- 0.8 beta.g.bar <- mean(beta * g) sigma <- sqrt((1/N) * sum((beta * g - beta.g.bar)^2) * (1-pve) / pve) # 0.18 y <- mu + beta * g + rnorm(n=N, mean=0, sd=sigma) plot(x=0, type="n", xlim=range(g), ylim=range(y), xlab="genotypes (allele dose)", 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) ols <- lm(y ~ g) summary(ols) # muhat=5.01, betahat=0.46, R2=0.779 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). |