Interpretation of ordered and non-ordered factors,

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...

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.