[R-sig-ME] alternative interaction representations

[Sebastian P. Luque]

On Sun, 22 Aug 2010 14:39:05 -0500,
"Sebastian P. Luque" <spluque-Re5JQEeQqe8AvxtiuMwx3w at public.gmane.org> wrote:

> 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.

Actually I'm having some trouble interpreting the intercept
coefficients, although it may have more to do with piecewise functions
in general, rather than with mixed modelling (so apologies for the
slightly off-topic message).

In a similar case, Fitzmaurice et al.'s say (in Applied Longitudinal
Analysis) that in a model for the mean response Y for a subject i at
time (T) j randomized to 2 groups (G):

B1 + B2*Tij + B3*(Tij-a) + B4*Gi + B5*Tij*Gi + B6*(Tij-a)*Gi

where Yij can be modelled as linear spline with a single knot at 'a'.
The term (Tij-a) is Tij-a when Tij > a and zero otherwise.  The 'B's are
linear coefficients.  Expressing the model in terms of the two lines of
the model for the baseline group:

B1 + B2*Tij                            (Tij <= a)
(B1 - B3) + (B2 + B3)*Tij              (Tij > a)

In the last case (Tij > a), I can't see why the intercept = (B1 - B3).
B3 is a slope, so why is it playing a role (and subtracted from B1) in
the intercept there?

Thanks for any light on this.

-- 
Seb