Observations in a linear model [General Sta­tis­tics]

posted by Helmut Homepage – Vienna, Austria, 2022-11-15 11:55 (24 d 11:18 ago) – Posting: # 23367
Views: 610

Dear all,

off-list I was asked whether it makes a difference if we have single observations or replicates. When you look at the standard equations, clearly the answer is no.
If you don’t believe that, an [image]-script for the simulation example at the end.

x = 1, 2, 3, 4, 5
 estimate    a.hat  b.hat
     mean +0.00001 1.9999
x = 1, 1, 3, 5, 5
 estimate    a.hat  b.hat
     mean -0.00003 2.0000
x = 1, 1, 1, 5, 5
 estimate    a.hat  b.hat
     mean +0.00011 1.9999
x = 1, 1, 1, 1, 5
 estimate    a.hat  b.hat
     mean +0.00012 1.9999

Very small differences in the estimated intercept, practically identical slope.


lr <- function(x, y, n) { # standard equations are faster than lm(y ~ x)
  sum.y <- sum(y)
  sum.x <- sum(x)
  b.hat <- (sum(x * y) - sum.x * sum.y / n) / (sum(x^2) - sum.x^2 / n)
  a.hat <- sum.y / n - sum.x / n * b.hat
  return(c(a.hat, b.hat))
}
a     <- 0   # intercept
b     <- 2   # slope
nsims <- 1e5 # number of simulations
x.n   <- 5   # number of x-levels
x.min <- 1   # minimum level
x.max <- 5   # maximum level
x1    <- x2 <- x3 <- x4 <- seq(x.min, x.max, length.out = x.n)
x2[c(c(1:2), c((x.n-1):x.n))] <- c(rep(x.min, 2), rep(x.max, 2))
x3[c(c(1:3), c((x.n-1):x.n))] <- c(rep(x.min, 3), rep(x.max, 2))
x4[1:4] <- rep(x.min, 4)
s1    <- data.frame(sim = rep(1:nsims, each = x.n), x = x1) # singlets
s2    <- data.frame(sim = rep(1:nsims, each = x.n), x = x2) # 2 duplicates
s3    <- data.frame(sim = rep(1:nsims, each = x.n), x = x3) # tri- and duplicate
s4    <- data.frame(sim = rep(1:nsims, each = x.n), x = x4) # quadruplet
t     <- c(paste("x =", paste(signif(x1, 3), collapse = ", "), "\n"),
                 paste("x =", paste(signif(x2, 3), collapse = ", "), "\n"),
                 paste("x =", paste(signif(x3, 3), collapse = ", "), "\n"),
                 paste("x =", paste(signif(x4, 3), collapse = ", "), "\n"))
set.seed(123456)
s1$y  <- rnorm(n = nsims * x.n, mean = a + b * s1$x, sd = 0.5)
set.seed(123456)
s2$y  <- rnorm(n = nsims * x.n, mean = a + b * s2$x, sd = 0.5)
set.seed(123456)
s3$y  <- rnorm(n = nsims * x.n, mean = a + b * s3$x, sd = 0.5)
set.seed(123456)
s4$y  <- rnorm(n = nsims * x.n, mean = a + b * s4$x, sd = 0.5)
c1    <- c2 <- c3 <- c4 <- data.frame(sim = 1:nsims, a.hat = NA_real_, b.hat = NA_real_)
pb    <- txtProgressBar(style = 3)
for (i in 1:nsims) {
  c1[i, 2:3] <- lr(s1$x[s1$sim == i], s1$y[s1$sim == i], x.n)
  c2[i, 2:3] <- lr(s2$x[s2$sim == i], s2$y[s2$sim == i], x.n)
  c3[i, 2:3] <- lr(s3$x[s3$sim == i], s3$y[s3$sim == i], x.n)
  c4[i, 2:3] <- lr(s4$x[s4$sim == i], s4$y[s4$sim == i], x.n)
  setTxtProgressBar(pb, i / nsims)
}
close(pb)
comp  <- data.frame(estimate = rep("mean", 4),
                    a.hat = sprintf("%+.5f", c(mean(c1$a.hat), mean(c2$a.hat),
                                               mean(c3$a.hat), mean(c4$a.hat))),
                    b.hat = sprintf("%.4f", c(mean(c1$b.hat), mean(c2$b.hat),
                                              mean(c3$b.hat), mean(c4$b.hat))))
cat(t[1]); print(comp[1, ], row.names = FALSE)
cat(t[2]); print(comp[2, ], row.names = FALSE)
cat(t[3]); print(comp[3, ], row.names = FALSE)
cat(t[4]); print(comp[4, ], row.names = FALSE)


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