Dear R-helpers,
I'd like to understand how to test the statistical significance of a
random effect in gamm. I am using gamm because I want to test a model
with an AR(1) error structure, and it is my understanding neither gam
nor gamm4 will do the latter.
The data set includes nine short interrupted time series (single case
designs in education, sometimes called N-of-1 trials in medicine) from
one study. They report a proportion as outcome (y), computed from a
behavior either observed or not out of 10 trials per time point. Hence I
use binomial (I believe quasi-binomial is not available in gamm). Each
of the nine series has an average of 30 observations give or take (total
264 observations), some under treatment (z) and some not. xc is centered
session number, int is the z*xc interaction. Based on prior work, xc is
also smoothed
Consider, for example, two models, both with AR(1) but one allowing a
random effect on xc:
g1 <- gamm(y ~ s(xc) +z+ int,family=binomial, weights=trial,
correlation=corAR1())
g2 <- gamm(y ~ s(xc) +z+ int,family=binomial, weights=trial, random =
list(xc=~1),correlation=corAR1())
I include the output for g1 and g2 below, but the question is how to
test the significance of the random effect on xc. I considered a test
comparing the Log-Likelihoods, but have no idea what the degrees of
freedom would be given that s(xc) is smoothed. I also tried:
anova(g1$gam, g2$gam)
that did not seem to return anything useful for this question.
A related question is how to test the significance of adding a second
random effect to a model that already has a random effect, such as:
g3 <- gamm(y ~ xc +z+ s(int),family=binomial, weights=trial, random =
list(Case=~1, z=~1),correlation=corAR1())
g4 <- gamm(y ~ xc +z+ s(int),family=binomial, weights=trial, random =
list(Case=~1, z=~1, int=~1),correlation=corAR1())
Any help would be appreciated.
Thanks.
Will Shadish
********************************************
g1
$lme
Linear mixed-effects model fit by maximum likelihood
Data: data
Log-likelihood: -437.696
Fixed: fixed
X(Intercept) Xz Xint Xs(xc)Fx1
0.6738466 -2.5688317 0.0137415 -0.1801294
Random effects:
Formula: ~Xr - 1 | g
Structure: pdIdnot
Xr1 Xr2 Xr3 Xr4
Xr5 Xr6 Xr7 Xr8 Residual
StdDev: 0.0004377781 0.0004377781 0.0004377781 0.0004377781 0.0004377781
0.0004377781 0.0004377781 0.0004377781 1.693177
Correlation Structure: AR(1)
Formula: ~1 | g
Parameter estimate(s):
Phi
0.3110725
Variance function:
Structure: fixed weights
Formula: ~invwt
Number of Observations: 264
Number of Groups: 1
$gam
Family: binomial
Link function: logit
Formula:
y ~ s(xc) + z + int
Estimated degrees of freedom:
1 total = 4
attr(,"class")
[1] "gamm" "list"
****************************
> g2
$lme
Linear mixed-effects model fit by maximum likelihood
Data: data
Log-likelihood: -443.9495
Fixed: fixed
X(Intercept) Xz Xint Xs(xc)Fx1
0.720018143 -2.562155820 0.003457463 -0.045821030
Random effects:
Formula: ~Xr - 1 | g
Structure: pdIdnot
Xr1 Xr2 Xr3 Xr4
Xr5 Xr6 Xr7 Xr8
StdDev: 7.056078e-06 7.056078e-06 7.056078e-06 7.056078e-06 7.056078e-06
7.056078e-06 7.056078e-06 7.056078e-06
Formula: ~1 | xc %in% g
(Intercept) Residual
StdDev: 6.277279e-05 1.683007
Correlation Structure: AR(1)
Formula: ~1 | g/xc
Parameter estimate(s):
Phi
0.1809409
Variance function:
Structure: fixed weights
Formula: ~invwt
Number of Observations: 264
Number of Groups:
g xc %in% g
1 34
$gam
Family: binomial
Link function: logit
Formula:
y ~ s(xc) + z + int
Estimated degrees of freedom:
1 total = 4
attr(,"class")
[1] "gamm" "list"
--
William R. Shadish
Distinguished Professor
Founding Faculty
Mailing Address:
William R. Shadish
University of California
School of Social Sciences, Humanities and Arts
5200 North Lake Rd
Merced CA 95343
Physical/Delivery Address:
University of California Merced
ATTN: William Shadish
School of Social Sciences, Humanities and Arts
Facilities Services Building A
5200 North Lake Rd.
Merced, CA 95343
209-228-4372 voice
209-228-4007 fax (communal fax: be sure to include cover sheet)
wshad...@ucmerced.edu
http://faculty.ucmerced.edu/wshadish/index.htm
http://psychology.ucmerced.edu
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