Premise
Small update to a similar thread from last week, on the comparison of slope and intercept terms fit to a multi-level model. I finally figured out (thanks R-Help mailing list!) how to efficiently use contrasts in R. The C() function can be called within a model formula, to reset the base level of an un-ordered factor. The UCLA Stats Library has an extensive description of this topic here. This approach can be used to sequentially test for differences between slope and intercept terms from a multi-level model, by re-setting the base level of a factor. See example data and figure below.
Note that the multcomp package has a much more robust approach to this type of operation. Details below.
Example Multi-Level Data
# need these
library(lattice)
# replicate an important experimental dataset
set.seed(10101010)
x <- rnorm(100)
y1 <- x[1:25] * 2 + rnorm(25, mean=1)
y2 <- x[26:50] * 2.6 + rnorm(25, mean=1.5)
y3 <- x[51:75] * 2.9 + rnorm(25, mean=5)
y4 <- x[76:100] * 3.5 + rnorm(25, mean=5.5)
d <- data.frame(x=x, y=c(y1,y2,y3,y4), f=factor(rep(letters[1:4], each=25)))
# plot
xyplot(y ~ x, groups=f, data=d,
auto.key=list(columns=4, title='Beard Type', lines=TRUE, points=FALSE, cex=0.75),
type=c('p','r'), ylab='Number of Pirates', xlab='Distance from Land')
Example Multi-Level Model II
Default Contrasts (contr.treatment for regular factors, contr.poly for ordered factors)
# standard comparison to base level of f
summary(lm(y ~ x * f, data=d))
# output:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.0747 0.1889 5.689 1.51e-07 ***
x 1.9654 0.1799 10.927 < 2e-16 ***
fb 0.3673 0.2724 1.348 0.1808
fc 4.1310 0.2714 15.221 < 2e-16 ***
fd 4.4309 0.2731 16.223 < 2e-16 ***
x:fb 0.5951 0.2559 2.326 0.0222 *
x:fc 1.0914 0.2449 4.456 2.35e-05 ***
x:fd 1.3813 0.2613 5.286 8.38e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Setting the "base level" in the Model Formula This allows us to compare all slope and intercept terms to the slope and intercept from level 4 of our factor ('d' in our example).
# compare to level 4 of f
summary(lm(y ~ x * C(f, base=4), data=d))
# output:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.5055 0.1972 27.911 < 2e-16 ***
x 3.3467 0.1896 17.653 < 2e-16 ***
C(f, base = 4)1 -4.4309 0.2731 -16.223 < 2e-16 ***
C(f, base = 4)2 -4.0635 0.2783 -14.603 < 2e-16 ***
C(f, base = 4)3 -0.2999 0.2773 -1.081 0.28230
x:C(f, base = 4)1 -1.3813 0.2613 -5.286 8.38e-07 ***
x:C(f, base = 4)2 -0.7862 0.2628 -2.992 0.00356 **
x:C(f, base = 4)3 -0.2899 0.2521 -1.150 0.25327
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Testing with Multcomp Package using data from above example
# need these
library(multcomp)
library(sandwich)
# open this vignette, lots of good information
vignette("generalsiminf", package = "multcomp")
# fit two models
l.1 <- lm(y ~ x + f, data=d)
l.2 <- lm(y ~ x * f, data=d)
# note that: tests are AGAINST the null hypothesis
summary(glht(l.1))
# see the plotting methods:
plot(glht(l.1))
plot(glht(l.2))
# pair-wise comparisons
summary(glht(l.1, linfct=mcp(f='Tukey')))
# pair-wise comparisons
# may not be appropriate for model with interaction
summary(glht(l.2, linfct=mcp(f='Tukey')))
# when variance is not homogenous between groups:
summary(glht(l.1, linfct=mcp(f='Tukey'), vcov=sandwich))
Links:
Comparison of Slope and Intercept Terms for Multi-Level Model
R: advanced statistical package
Computing Statistics from Poorly Formatted Data (plyr and reshape packages for R)
