alternative interaction representations

Hi,

## With the CO2 data, suppose we want to build a LME model of 'uptake' with
## 'conc' (continuous) and want to know whether there is a change in slope
## at conc=300, with random slopes for plants

CO2new <- subset(CO2, Type == "Quebec" & Treatment == "nonchilled")
CO2new <- within(CO2new, {
? ?## The more intuitive way to set up the interaction is to first define
? ?## a factor breaking up the 'conc' predictor
? ?stage1 <- cut(conc, breaks=c(floor(min(conc)), 300,
? ? ? ? ? ? ? ? ? ? ? ? ?ceiling(max(conc))),
? ? ? ? ? ? ? ? ?labels=c("pre", "post"), include.lowest=TRUE)
? ?## Alternative, direct representation of interaction
? ?stage2 <- ifelse(conc > 300, conc - 300, 0)
? ?## We center conc at 300 for interpreting intercept here
? ?conc <- conc - 300
})
str(CO2new)
xyplot(uptake ~ conc, data=CO2new, groups=Plant, type="b")

## Consider a model with fixed effects for intercept, conc, and varying
## slopes. ?Using the more intuitive representation:

(fm1 <- lmer(uptake ~ conc + conc:stage1 + (conc:stage1 | Plant), data=CO2new))

## And using the direct representation of the interaction

(fm2 <- lmer(uptake ~ conc + stage2 + (conc + stage2 | Plant), data=CO2new))

## In this simple case, it doesn't seem to matter which representation is
## used. ?For other models where an interaction with another factor, say
## Type, is needed in the model to indicate 3-way interactions with conc
## then the latter seems to allow for a simpler model (which may impact
## lmer performance) because the interaction would then be modelled as a
## 2-way interaction.
##
## Is this a fair comparison of using direct representations of
## interactions compared to the more natural factor-based representations?
## Overall, is it preferable to use one rather than the other?

Cheers,

--
Seb

# This representation fits two linear slopes, one below and one after
conc = 300, splicing them at 0:
CO2new$conc1 <- ifelse(CO2new$conc < 300, CO2new$conc - 300, 0)
CO2new$conc2 <- ifelse(CO2new$conc > 300, CO2new$conc - 300, 0)

# Basic LMM
print(LMM <- lmer(uptake ~ conc1 + conc2 + (1 | Plant), data=CO2new), cor=FALSE)
#  ... the linear uptake is significant below 300, no longer
significant after 300
# ... the intercept estimates the upake at conc=300

# To test whether there is significant between-plant variance in
slopes below and above conc:
# Varying-slopes LMM
print(LMM.conc.1 <-   lmer(uptake ~ conc1 + conc2 + (1 | Plant) +
(0+conc1 | Plant), data=CO2new), cor=FALSE)
print(LMM.conc.2 <-   lmer(uptake ~ conc1 + conc2 + (1 | Plant) +
(0+conc2 | Plant), data=CO2new), cor=FALSE)
#print(LMM.conc.1.2 <- lmer(uptake ~ conc1 + conc2 + (1 | Plant) +
(0+conc1 | Plant) + (0+conc2 | Plant), data=CO2new), cor=FALSE)
#print(LMM.conc.12 <-  lmer(uptake ~ conc1 + conc2 + (1 + conc1 +
conc2 | Plant), data=CO2new), cor=FALSE)

anova(LMM, LMM.conc.1)
anova(LMM, LMM.conc.2)
#  Apparently there is not enough information in the data to test the
between-slope variance.

On Sun, 22 Aug 2010 08:47:47 +0200,
Reinhold Kliegl <reinhold.kliegl-Re5JQEeQqe8AvxtiuMwx3w-XMD5yJDbdMReXY1tMh2IBg at public.gmane.org> wrote:

# This representation fits two linear slopes, one below and one after

conc = 300, splicing them at 0: CO2new$conc1 <- ifelse(CO2new$conc <
300, CO2new$conc - 300, 0) CO2new$conc2 <- ifelse(CO2new$conc > 300,
CO2new$conc - 300, 0)

# Basic LMM print(LMM <- lmer(uptake ~ conc1 + conc2 + (1 | Plant),
data=CO2new), cor=FALSE) # ... the linear uptake is significant below
300, no longer significant after 300 # ... the intercept estimates
the upake at conc=300

# To test whether there is significant between-plant variance in
slopes below and above conc: # Varying-slopes LMM print(LMM.conc.1 <-
lmer(uptake ~ conc1 + conc2 + (1 | Plant) + (0+conc1 | Plant),
data=CO2new), cor=FALSE) print(LMM.conc.2 <- lmer(uptake ~ conc1 +
conc2 + (1 | Plant) + (0+conc2 | Plant), data=CO2new), cor=FALSE)
#print(LMM.conc.1.2 <- lmer(uptake ~ conc1 + conc2 + (1 | Plant) +
(0+conc1 | Plant) + (0+conc2 | Plant), data=CO2new), cor=FALSE)
#print(LMM.conc.12 <- lmer(uptake ~ conc1 + conc2 + (1 + conc1 +
conc2 | Plant), data=CO2new), cor=FALSE)

anova(LMM, LMM.conc.1) anova(LMM, LMM.conc.2) # Apparently there is
not enough information in the data to test the between-slope
variance.

Thanks Reinhold, it seems as if these piecewise linear splines with
one or two knots are easier to fit and interpret than using a factor.