Interpretation of ordered and non-ordered factors, vs. numerical predictors in model summary
I have fitted a model where:
Y ~ A + A^2 + B + mixed.effect(C)
Y is continuous A is continuous B actually refers to a DAY and currently looks like this:
Levels: 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 11 < 12
I can easily change the data type, but I'm not sure whether it is more appropriate to treat B as numeric, a factor, or as an ordered factor. AND when treated as numeric or ordered factor, I'm not quite sure how to interpret the output.
When treated as an ordered factor, summary(my.model) outputs something like this:
Linear mixed model fit by REML ['lmerMod']
Formula: Y ~ A + I(A^2) + B + (1 | mixed.effect.C)
Fixed effects:
Estimate Std. Error t value
(Intercept) 19.04821 0.40926 46.54
A -151.01643 7.19035 -21.00
I(A^2) 457.19856 31.77830 14.39
B.L -3.00811 0.29688 -10.13
B.Q -0.12105 0.24561 -0.49
B.C 0.35457 0.24650 1.44
B^4 0.09743 0.24111 0.40
B^5 -0.08119 0.22810 -0.36
B^6 0.19640 0.22377 0.88
B^7 0.02043 0.21016 0.10
B^8 -0.48931 0.20232 -2.42
B^9 -0.43027 0.17798 -2.42
B^10 -0.13234 0.15379 -0.86
What are L, Q, and C? I need to know the effect of each additional day (B) on the response (Y). How do I get this information from the output?
When I treat B as.numeric, I get something like this as output:
Fixed effects:
Estimate Std. Error t value
(Intercept) 20.79679 0.39906 52.11
A -152.29941 7.17939 -21.21
I(A^2) 461.89157 31.79899 14.53
B -0.27321 0.02391 -11.42
To get the effect of each additional day (B) on the response (Y), am I supposed to multiply the coefficient of B times B (the day number)? Not sure what to do with this output...
Answer
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.