Read in the data:
x <- read.table("data/infiltration_data.tsv", header=TRUE)
range(rowSums(x[, c("Sand", "Clay", "Silt")]))
## [1] 100 100
Fit the model:
the_lm <- lm( BIR ~ Sand + Silt + Clay, data=x)
summary(the_lm)
##
## Call:
## lm(formula = BIR ~ Sand + Silt + Clay, data = x)
##
## Residuals:
## Min 1Q Median 3Q Max
## -7.39 -1.52 0.63 1.99 4.27
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -62.112 14.617 -4.25 0.00033 ***
## Sand 0.831 0.160 5.21 3.2e-05 ***
## Silt 0.473 0.234 2.02 0.05577 .
## Clay NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.9 on 22 degrees of freedom
## Multiple R-squared: 0.627, Adjusted R-squared: 0.593
## F-statistic: 18.5 on 2 and 22 DF, p-value: 1.96e-05
There is a small difference between our predictions and theirs - due to rounding, it will turn out.
preds <- predict(the_lm)
ours <- -62.1123 + 0.8315 * x$Sand + 0.4733 * x$Silt
theirs <- 14195.35 - 141.75 * x$Sand - 142.10 * x$Silt - 142.56 * x$Clay
layout(1:2)
plot(preds, ours)
abline(0, 1)
plot(ours, theirs)
abline(0, 1)
Ok, so Sand + Silt + Clay is always equal to 100, so for any number a, it’s always true that a * Sand + a * Silt + a * Clay - a * 100 = 0. So, we can add this to our formula and get exactly the same thing. Our model has no coefficient for Clay, so to make their model look like ours we can do this with a equal to their coefficient for Clay:
clay_coef <- 142.56
# this is the same as
theirs_noclay <- 14195.35 - clay_coef * 100 + (clay_coef - 141.75) * x$Sand + (clay_coef - 142.10) * x$Silt + (clay_coef - 142.56) * x$Clay
plot(theirs, theirs_noclay)
abline(0, 1)
So, what are these numbers?
The new coefficients that are equivalent to theirs are nearly the same as ours - the difference is because of rounding error!
coefs <- as.data.frame(coef(the_lm))
names(coefs)[[1]] <- "lm"
coefs$theirs <- c(14195.35, - 141.75, - 142.10, - 142.56)
coefs$theirs_noclay <- c(
14195.35 - clay_coef * 100,
(clay_coef - 141.75),
(clay_coef - 142.10),
(clay_coef - 142.56)
)
coefs
## lm theirs theirs_noclay
## (Intercept) -62.11 14195 -60.65
## Sand 0.83 -142 0.81
## Silt 0.47 -142 0.46
## Clay NA -143 0.00
