> > If one compares the random effect estimates, in fact, one sees that
> > they are in the correct proportion, with the expected signs. They are
> > just approximately eight orders of magnitude too small. Is this a bug?
>
> BLUPs are essentially shrinkage estimates, where shrinkage is
> determined with magnitude of variance. Lower variance more
> shrinkage towards the mean - zero in this case. So this is not a bug.
>
> Gregor
I doubled each data set by duplicating each Subject. There are now
46 subjects in each group instead of 23. So now A and B differ in 2
observations out of 322. The lmer resuls are sensible.
I know BLUPs are not like regular parameters, but doubling (or cutting
in half) the data should not, in my opinion, cause this behavior. There
is a lot of room for the original A variance estimate to be lower than B,
maybe it should be 0.05, 0.01, 0.05, or whatever, but not < .0000000005.
"DOUBLED" (Laplace)
A
Groups Name Variance Std.Dev.
Subject (Intercept) 1.759 1.326
Item (Intercept) 0.178 0.422
number of obs: 322, groups: Subject, 46; Item, 7
B
Groups Name Variance Std.Dev.
Subject (Intercept) 1.892 1.376
Item (Intercept) 0.319 0.564
number of obs: 322, groups: Subject, 46; Item, 7
"ORIGINAL" (Laplace)
A
Groups Name Variance Std.Dev.
Subject (Intercept) 1.63 1.28
Item (Intercept) 5.00e-10 2.24e-05
number of obs: 161, groups: Subject, 23; Item, 7
B
Groups Name Variance Std.Dev.
Subject (Intercept) 1.712 1.308
Item (Intercept) 0.109 0.330
number of obs: 161, groups: Subject, 23; Item, 7
By the way, using the PQL method instead of Laplace "fails"
even more badly with the original data (and gives very
different estimates with the doubled data).
"DOUBLED" (PQL)
A
Groups Name Variance Std.Dev.
Subject (Intercept) 2.997 1.731
Item (Intercept) 0.509 0.713
number of obs: 322, groups: Subject, 46; Item, 7
B
Subject (Intercept) 3.317 1.821
Item (Intercept) 0.725 0.852
number of obs: 322, groups: Subject, 46; Item, 7
"ORIGINAL" (PQL)
A
1: Estimated variance for factor Item
is effectively zero
in: LMEopt(x = mer, value = cv)
2: Estimated variance for factors Subject, Item
is effectively zero
in: LMEopt(x = mer, value = cv)
B
Estimated variance for factors Subject, Item
is effectively zero
in: LMEopt(x = mer, value = cv)
I understand that the between-Item variance is low, and probably
it is no greater than what you would expect to occur by chance,
but isn't that what hypothesis testing is for (anova, etc.)?
Is my best way around the algorithm returning zero to do
what I have done above, with my real data? That is, duplicate
(or triplicate) Subjects to increase my data size, and thereby
get a reasonably comparable (if wrong) estimate of the Item variance?
Zero is not a reasonable estimate in any of these data sets.
Thanks,
Daniel
> A.fit # "DOUBLED" DATA A
Generalized linear mixed model fit using Laplace
Formula: Response ~ (1 | Subject) + (1 | Item)
Data: A
Family: binomial(logit link)
AIC BIC logLik deviance
232 243 -113 226
Random effects:
Groups Name Variance Std.Dev.
Subject (Intercept) 1.759 1.326
Item (Intercept) 0.178 0.422
number of obs: 322, groups: Subject, 46; Item, 7
Estimated scale (compare to 1 ) 0.81576
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.601 0.327 -7.95 1.9e-15 ***
---
> B.fit # "DOUBLED" DATA B
Generalized linear mixed model fit using Laplace
Formula: Response ~ (1 | Subject) + (1 | Item)
Data: B
Family: binomial(logit link)
AIC BIC logLik deviance
237 249 -116 231
Random effects:
Groups Name Variance Std.Dev.
Subject (Intercept) 1.892 1.376
Item (Intercept) 0.319 0.564
number of obs: 322, groups: Subject, 46; Item, 7
Estimated scale (compare to 1 ) 0.8052
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.61 0.36 -7.25 4.1e-13 ***
> ranef(A.fit)$Item # "DOUBLED" DATA A
(Intercept)
1 -0.084969
2 -0.084969
3 -0.084969
4 0.325780
5 -0.308044
6 0.515590 # 10 1's, 36 0's
7 0.325780> ranef(B.fit)$Item # "DOUBLED" DATA B
(Intercept)
1 -0.11962
2 -0.11962
3 -0.11962
4 0.42389
5 -0.43555
6 0.88322 # 12 1's, 34 0's
7 0.42389
On Dec 31, 2006, at 1:19 AM, Daniel Ezra Johnson wrote:
I am fitting models to the responses to a questionnaire that has seven
yes/no questions (Item). For each combination of Subject and Item, the
variable Response is coded as 0 or 1.
I want to include random effects for both Subject and Item. While I
understand that the datasets are fairly small, and there are a lot of
invariant subjects, I do not understand something that is happening here,
and in comparing other subsets of the data.
In the data below, which has been adjusted to show this phenomenon
clearly, the Subject random effect variance is comparable for A (1.63) and
B (1.712), but the Item random effect variance comes out as 0.109 for B
and essentially zero for A (5.00e-10).
Note that the only difference between data set A and data set B occurs on
row 19, where a single instance of Response is changed.
Item avg. Response in A avg. Response in B
1 9% 9%
2 9% 9%
3 9% 9%
4 17% 17%
5 4% 4%
6 22% <-> 26%
7 17% 17%
Why does the Item random effect sometimes "crash" to zero? Surely
there is
some more reasonable estimate of the Item effect than zero. The items
still have clearly different Response behavior.
If one compares the random effect estimates, in fact, one sees that they
are in the correct proportion, with the expected signs. They are just
approximately eight orders of magnitude too small. Is this a bug?
More broadly, is it hopeless to analyze this data in this manner, or else,
what should I try doing differently? It would be very useful to be able to
have reliable estimates of random effect sizes, even when they are rather
small.
I've included replicable code below, sorry that I did not know how to make
it more compact!
a1 <- c(0,0,0,0,0,0,0)
a2 <- c(0,0,0,0,0,0,0)
a3 <- c(0,0,0,0,0,0,0)
a4 <- c(0,0,0,0,0,0,0)
a5 <- c(0,0,0,0,0,0,0)
a6 <- c(0,0,0,0,0,0,0)
a7 <- c(0,0,0,0,0,0,0)
a8 <- c(0,0,0,0,0,0,0)
a9 <- c(0,0,0,0,0,0,0)
a10 <- c(0,0,0,0,0,0,0)
a11 <- c(0,0,0,0,0,0,0)
a12 <- c(0,0,0,0,0,0,0)
a13 <- c(0,0,0,0,0,0,1)
a14 <- c(0,0,0,0,0,0,1)
a15 <- c(0,0,0,0,0,1,0)
a16 <- c(0,0,0,0,1,0,0)
a17 <- c(0,0,0,1,0,0,0)
a18 <- c(0,0,1,0,0,0,0)
a19 <- c(0,1,0,0,0,0,0) # differs
a20 <- c(0,1,0,0,0,1,0)
a21 <- c(0,0,0,1,0,1,1)
a22 <- c(1,0,0,1,0,1,1)
a23 <- c(1,0,1,1,0,1,0)
aa <-
rbind(a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a20,a21,a22,a23)
b1 <- c(0,0,0,0,0,0,0)
b2 <- c(0,0,0,0,0,0,0)
b3 <- c(0,0,0,0,0,0,0)
b4 <- c(0,0,0,0,0,0,0)
b5 <- c(0,0,0,0,0,0,0)
b6 <- c(0,0,0,0,0,0,0)
b7 <- c(0,0,0,0,0,0,0)
b8 <- c(0,0,0,0,0,0,0)
b9 <- c(0,0,0,0,0,0,0)
b10 <- c(0,0,0,0,0,0,0)
b11 <- c(0,0,0,0,0,0,0)
b12 <- c(0,0,0,0,0,0,0)
b13 <- c(0,0,0,0,0,0,1)
b14 <- c(0,0,0,0,0,0,1)
b15 <- c(0,0,0,0,0,1,0)
b16 <- c(0,0,0,0,1,0,0)
b17 <- c(0,0,0,1,0,0,0)
b18 <- c(0,0,1,0,0,0,0)
b19 <- c(0,1,0,0,0,1,0) # differs
b20 <- c(0,1,0,0,0,1,0)
b21 <- c(0,0,0,1,0,1,1)
b22 <- c(1,0,0,1,0,1,1)
b23 <- c(1,0,1,1,0,1,0)
bb <-
rbind(b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13,b14,b15,b16,b17,b18,b19,b20,b21,b22,b23)
a <- array(0, c(161,3), list(NULL,
c("Subject","Item","Response")))
for (s in c(1:23))
for (i in c(1:7))
a[7*(s-1)+i,] <- c(s,i,aa[s,i])
A <- data.frame(a)
b <- array(0, c(161,3),
list(NULL,c("Subject","Item","Response")))
for (s in c(1:23))
for (i in c(1:7))
b[7*(s-1)+i,] <- c(s,i,bb[s,i])
B <- data.frame(b)
A.fit <- lmer(Response~(1|Subject)+(1|Item),A,binomial)
B.fit <- lmer(Response~(1|Subject)+(1|Item),B,binomial)
A.fit
B.fit
ranef(A.fit)$Item
ranef(B.fit)$Item
Generalized linear mixed model fit using Laplace
Formula: Response ~ (1 | Subject) + (1 | Item)
Data: A
Family: binomial(logit link)
AIC BIC logLik deviance
120 129 -56.8 114
Random effects:
Groups Name Variance Std.Dev.
Subject (Intercept) 1.63e+00 1.28e+00
Item (Intercept) 5.00e-10 2.24e-05
number of obs: 161, groups: Subject, 23; Item, 7
Estimated scale (compare to 1 ) 0.83326
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.517 0.395 -6.38 1.8e-10 ***
Data: B
Family: binomial(logit link)
AIC BIC logLik deviance
123 133 -58.6 117
Random effects:
Groups Name Variance Std.Dev.
Subject (Intercept) 1.712 1.308
Item (Intercept) 0.109 0.330
number of obs: 161, groups: Subject, 23; Item, 7
Estimated scale (compare to 1 ) 0.81445
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.498 0.415 -6.02 1.8e-09 ***
---> ranef(A.fit)$Item
(Intercept)
1 -2.8011e-10
2 -2.8011e-10
3 -2.8011e-10
4 7.1989e-10
5 -7.8011e-10
6 1.2199e-09 # 5 1's, 18 0's
7 7.1989e-10> ranef(B.fit)$Item
(Intercept)
1 -0.056937
2 -0.056937
3 -0.056937
4 0.120293
5 -0.146925
6 0.293893 # 6 1's, 17 0's
7 0.120293
