# Calculate stderr, t-value, p-value, predict value for linear regression

I use the data from `MatrixModels:::lm.fit.sparse` example. I built a custom function `summary_sparse` to perform a summary for this model. All matrix operations are performed with `Matrix` package. Results are compared with dense type model.

Note `lm.fit.sparse` have to be evaluated with `method = "chol"` to get proper results.

Functions:

``````summary_sparse <- function(l, X) {

XXinv <- Matrix::chol2inv(Matrix::chol(Matrix::crossprod(X)))
se <- sqrt(Matrix::diag(XXinv*sum(l\$residuals**2)/(nrow(X)-ncol(X))))
ts <- l\$coef/se
pvals <- 2*c(1 - pnorm(abs(ts)))

list(coef = l\$coef, se = se, t = ts, p = pvals)

}

predict_sparse <- function(X, coef) {
X %*% coef
}
``````

Application:

``````dd <- expand.grid(a = as.factor(1:3),
b = as.factor(1:4),
c = as.factor(1:2),
d= as.factor(1:8))
n <- nrow(dd <- dd[rep(seq_len(nrow(dd)), each = 10), ])

set.seed(17)

dM <- cbind(dd, x = round(rnorm(n), 1))
## randomly drop some
n <- nrow(dM <- dM[- sample(n, 50),])
dM <- within(dM, { A <- c(2,5,10)[a]
B <- c(-10,-1, 3:4)[b]
C <- c(-8,8)[c]
D <- c(10*(-5:-2), 20*c(0, 3:5))[d]
Y <- A + B + A*B + C + D + A*D + C*x + rnorm(n)/10
wts <- sample(1:10, n, replace=TRUE)
rm(A,B,C,D)
})

X <- Matrix::sparse.model.matrix( ~ (a+b+c+d)^2 + c*x, data = dM)

Xd <- as(X,"matrix")

fmDense <- lm(dM[,"Y"]~Xd-1)

ss <- summary(fmDense)

r1 <- MatrixModels:::lm.fit.sparse(X, y = dM[,"Y"], method = "chol")

f <- summary_sparse(r1, X)

all.equal(do.call(cbind, f), ss\$coefficients, check.attributes = F)
#TRUE

all.equal(predict_sparse(X, r1\$coef)@x, predict(fmDense), check.attributes = F, check.names=F)
#TRUE
``````