Interpretation of ordered and non-ordered factors, vs. numerical predictors in model summary

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This is not really a mixed-model specific question, but rather a general question about model parameterization in R.

Let’s try a simple example.

set.seed(101)
d <- data.frame(x=sample(1:4,size=30,replace=TRUE))
d$y <- rnorm(30,1+2*d$x,sd=0.01)

x as numeric

This just does a linear regression: the x parameter denotes the change in y per unit of change in x; the intercept specifies the expected value of y at x=0.

coef(lm(y~x,d))
## (Intercept)           x 
##   0.9973078   2.0001922

x as (unordered/regular) factor

coef(lm(y~factor(x),d))
## (Intercept)  factor(x)2  factor(x)3  factor(x)4 
##    3.001627    1.991260    3.995619    5.999098

The intercept specifies the expected value of y in the baseline level of the factor (x=1); the other parameters specify the difference between the expected value of y when x takes on other values.

x as ordered factor

coef(lm(y~ordered(x),d))
##  (Intercept) ordered(x).L ordered(x).Q ordered(x).C 
##  5.998121421  4.472505514  0.006109021 -0.003125958

Now the intercept specifies the value of y at the mean factor level (halfway between 2 and 3); the L (linear) parameter gives a measure of the linear trend (not quite sure I can explain the particular value …), Q and C specify quadratic and cubic terms (which are close to zero in this case because the pattern is linear); if there were more levels the higher-order contrasts would be numbered 5, 6, …

successive-differences contrasts

coef(lm(y~factor(x),d,contrasts=list(`factor(x)`=MASS::contr.sdif)))
##  (Intercept) factor(x)2-1 factor(x)3-2 factor(x)4-3 
##     5.998121     1.991260     2.004359     2.003478

This contrast specifies the parameters as the differences between successive levels, which are all a constant value of (approximately) 2.

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