In theory
Firstly, you can fit a linear model with multiple LHS.
Secondly, explicit data splitting is not the only way (or a recommended way) for group-by regression. See R regression analysis: analyzing data for a certain ethnicity and R: build separate models for each category. So build your model as cbind(x1, x2, x3, x4) ~ day * subject where subject is a factor variable.
Finally, since you have many factor levels and work with a large dataset, lm is infeasible. Consider using speedglm::speedlm with sparse = TRUE, or MatrixModels::glm4 with sparse = TRUE.
In reality
Neither speedlm nor glm4 is under active development. Their functionality is (in my view) primitive.
Neither speedlm nor glm4 supports multiple LHS as lm. So you need do 4 separate models x1 ~ day * subject to x4 ~ day * subject instead.
These two packages have different logic behind sparse = TRUE.
speedlm first constructs a dense design matrix using the standard model.matrix.default, then uses is.sparse to survey whether it is sparse or not. If TRUE, subsequent computations can use sparse methods.glm4 uses model.Matrix to construct a design matrix, and a sparse one can be built directly.
So, it is not surprising that speedlm is as bad as lm in this sparsity issue, and glm4 is the one we really want to use.
glm4 does not have a full, useful set of generic functions for analyzing fitted models. You can extract coefficients, fitted values and residuals by coef, fitted and residuals, but have to compute all statistics (standard error, t-statistic, F-statistic, etc) all by yourself. This is not a big deal for people who know regression theory well, but it is still rather inconvenient.
glm4 still expects you to use the best model formula so that the sparsest matrix can be constructed. The conventional ~ day * subject is really not a good one. I should probably set up a Q & A on this issue later. Basically, if your formula has intercept and factors are contrasted, you loose sparsity. This is the one we should use: ~ 0 + subject + day:subject.
A test with glm4
## use chinsoon12's data in his answer
set.seed(0L)
nSubj <- 200e3
nr <- 1e6
DF <- data.frame(subject = gl(nSubj, 5),
day = 3:7,
y1 = runif(nr),
y2 = rpois(nr, 3),
y3 = rnorm(nr),
y4 = rnorm(nr, 1, 5))
library(MatrixModels)
fit <- glm4(y1 ~ 0 + subject + day:subject, data = DF, sparse = TRUE)
It takes about 6 ~ 7 seconds on my 1.1GHz Sandy Bridge laptop. Let's extract its coefficients:
b <- coef(fit)
head(b)
# subject1 subject2 subject3 subject4 subject5 subject6
# 0.4378952 0.3582956 -0.2597528 0.8141229 1.3337102 -0.2168463
tail(b)
#subject199995:day subject199996:day subject199997:day subject199998:day
# -0.09916175 -0.15653402 -0.05435883 -0.02553316
#subject199999:day subject200000:day
# 0.02322640 -0.09451542
You can do B <- matrix(b, ncol = 2), so that the 1st column is intercept and the 2nd is slope.
My thoughts: We probably need better packages for large regression
Using glm4 here does not give attractive advantage over chinsoon12's data.table solution since it also basically just tells you the regression coefficient. It is also a bit slower than data.table method, because it computes fitted values and residuals.
Analysis of simple regression does not require a proper model fitting routine. I have a few answers on how to do fancy stuff on this kind of regression, like Fast pairwise simple linear regression between all variables in a data frame where details on how to compute all statistics are also given. But as I wrote this answer, I was thinking about something in general regarding large regression problem. We might need better packages otherwise there is no end doing case-to-case coding.
In reply to OP
speedglm::speedlm(x1 ~ 0 + subject + day:subject, data = df, sparse = TRUE) gives Error: cannot allocate vector of size 74.5 Gb
yeah, because it has a bad sparse logic.
MatrixModels::glm4(x1 ~ day * subject, data = df, sparse = TRUE) gives Error in Cholesky(crossprod(from), LDL = FALSE) : internal_chm_factor: Cholesky factorization failed
This is because you have only one datum for some subject. You need at least two data to fit a line. Here is an example (in dense settings):
dat <- data.frame(t = c(1:5, 1:9, 1),
f = rep(gl(3,1,labels = letters[1:3]), c(5, 9, 1)),
y = rnorm(15))
Level "c" of f only has one datum / row.
X <- model.matrix(~ 0 + f + t:f, dat)
XtX <- crossprod(X)
chol(XtX)
#Error in chol.default(XtX) :
# the leading minor of order 6 is not positive definite
Cholesky factorization is unable to resolve rank-deficient model. If we use lm's QR factorization, we will see an NA coefficient.
lm(y ~ 0 + f + t:f, dat)
#Coefficients:
# fa fb fc fa:t fb:t fc:t
# 0.49893 0.52066 -1.90779 -0.09415 -0.03512 NA
We can only estimate an intercept for level "c", not a slope.
Note that if you use the data.table solution, you end up with 0 / 0 when computing slope of this level, and the final result is NaN.
Update: fast solution now available
Check out Fast group-by simple linear regression and general paired simple linear regression.
lmat all, and to implement it in Rcpp with a list of indices. – Phrasing