A GLMM (Generalized Linear Mixed Model) is used instead of a LMM (Linear Mixed Model) when the response is binary or, perhaps, a count with a low expected count.
The specification of the model includes the distribution family for the response and, possibly, the link function, g, relating the mean response, μ, to the value of the linear predictor, η.
To explain the model it helps to consider the linear mixed model in some detail first.
Column Variance Std.Dev. Corr.
subj (Intercept) 565.51067 23.78047
days 32.68212 5.71683 +0.08
Residual 654.94145 25.59182
Number of obs: 180; levels of grouping factors: 18
Fixed-effects parameters:
──────────────────────────────────────────────────
Coef. Std. Error z Pr(>|z|)
──────────────────────────────────────────────────
(Intercept)
251.405 6.63226 37.91 <1e-99
days 10.4673 1.50224 6.97 <1e-11
──────────────────────────────────────────────────
The response vector, y, has 180 elements. The fixed-effects coefficient vector, β, has 2 elements and the fixed-effects model matrix, X, is of size 180 × 2.
The second column of X is just the days vector and the first column is all 1’s.
There are 36 random effects, 2 for each of the 18 levels of subj. The “estimates” (technically, the conditional means or conditional modes) are returned as a vector of matrices, one matrix for each grouping factor. In this case there is only one grouping factor for the random effects so there is one one matrix which contains 18 intercept random effects and 18 slope random effects.
There is a model matrix, Z, for the random effects. In general it has one chunk of columns for the first grouping factor, a chunk of columns for the second grouping factor, etc.
The defining property of a linear model or linear mixed model is that the fitted values are linear combinations of the fixed-effects parameters and the random effects. We can write the fitted values as
In symbols we would write the linear predictor expression as \[
\boldsymbol{\eta} = \mathbf{X}\boldsymbol{\beta} +\mathbf{Z b}
\] where \(\boldsymbol{\eta}\) has 180 elements, \(\boldsymbol{\beta}\) has 2 elements, \(\bf b\) has 36 elements, \(\bf X\) is of size 180 × 2 and \(\bf Z\) is of size 180 × 36.
For a linear model or linear mixed model the linear predictor is the mean response, \(\boldsymbol\mu\). That is, we can write the probability model in terms of a 180-dimensional random variable, \(\mathcal Y\), for the response and a 36-dimensional random variable, \(\mathcal B\), for the random effects as \[
\begin{aligned}
(\mathcal{Y} | \mathcal{B}=\bf{b}) &\sim\mathcal{N}(\bf{ X\boldsymbol\beta + Z b},\sigma^2\bf{I})\\\\
\mathcal{B}&\sim\mathcal{N}(\bf{0},\boldsymbol{\Sigma}_{\boldsymbol\theta}) .
\end{aligned}
\] where \(\boldsymbol{\Sigma}_\boldsymbol{\theta}\) is a 36 × 36 symmetric covariance matrix that has a special form - it consists of 18 diagonal blocks, each of size 2 × 2 and all the same.
Recall that this symmetric matrix can be constructed from the parameters \(\boldsymbol\theta\), which generate the lower triangular matrix \(\boldsymbol\lambda\), and the estimate \(\widehat{\sigma^2}\).
Compare the diagonal elements to the Variance column of
VarCorr(m1)
Column
Variance
Std.Dev
Corr.
subj
(Intercept)
565.51067
23.78047
days
32.68212
5.71683
+0.08
Residual
654.94145
25.59182
Linear predictors in LMMs and GLMMs
Writing the model for \(\mathcal Y\) as \[
(\mathcal{Y} | \mathcal{B}=\bf{b})\sim\mathcal{N}(\bf{ X\boldsymbol\beta + Z b},\sigma^2\bf{I})
\] may seem like over-mathematization (or “overkill”, if you prefer) relative to expressions like \[
y_i = \beta_1 x_{i,1} + \beta_2 x_{i,2}+ b_1 z_{i,1} +\dots+b_{36} z_{i,36}+\epsilon_i
\] but this more abstract form is necessary for generalizations.
The way that I read the first form is
The conditional distribution of the response vector, \(\mathcal Y\), given that the random effects vector, \(\mathcal B =\bf b\), is a multivariate normal (or Gaussian) distribution whose mean, \(\boldsymbol\mu\), is the linear predictor, \(\boldsymbol\eta=\bf{X\boldsymbol\beta+Zb}\), and whose covariance matrix is \(\sigma^2\bf I\). That is, conditional on \(\bf b\), the elements of \(\mathcal Y\) are independent normal random variables with constant variance, \(\sigma^2\), and means of the form \(\boldsymbol\mu = \boldsymbol\eta = \bf{X\boldsymbol\beta+Zb}\).
So the only things that differ in the distributions of the \(y_i\)’s are the means and they are determined by this linear predictor, \(\boldsymbol\eta = \bf{X\boldsymbol\beta+Zb}\).
Generalized Linear Mixed Models
Consider first a GLMM for a vector, \(\bf y\), of binary (i.e. yes/no) responses. The probability model for the conditional distribution \(\mathcal Y|\mathcal B=\bf b\) consists of independent Bernoulli distributions where the mean, \(\mu_i\), for the i’th response is again determined by the i’th element of a linear predictor, \(\boldsymbol\eta = \mathbf{X}\boldsymbol\beta+\mathbf{Z b}\).
However, in this case we will run into trouble if we try to make \(\boldsymbol\mu=\boldsymbol\eta\) because \(\mu_i\) is the probability of “success” for the i’th response and must be between 0 and 1. We can’t guarantee that the i’th component of \(\boldsymbol\eta\) will be between 0 and 1. To get around this problem we apply a transformation to take \(\eta_i\) to \(\mu_i\). For historical reasons this transformation is called the inverse link, written \(g^{-1}\), and the opposite transformation - from the probability scale to an unbounded scale - is called the link, g.
Each probability distribution in the exponential family (which is most of the important ones), has a canonical link which comes from the form of the distribution itself. The details aren’t as important as recognizing that the distribution itself determines a preferred link function.
For the Bernoulli distribution, the canonical link is the logit or log-odds function, \[
\eta = g(\mu) = \log\left(\frac{\mu}{1-\mu}\right),
\] (it’s called log-odds because it is the logarithm of the odds ratio, \(p/(1-p)\)) and the canonical inverse link is the logistic\[
\mu=g^{-1}(\eta)=\frac{1}{1+\exp(-\eta)}.
\] This is why fitting a binary response is sometimes called logistic regression.
For later use we define a Julia logistic function. See this presentation for more information than you could possible want to know on how Julia converts code like this to run on the processor.
increment(x) = x +one(x)logistic(η) =inv(increment(exp(-η)))
logistic (generic function with 1 method)
To reiterate, the probability model for a Generalized Linear Mixed Model (GLMM) is \[
\begin{aligned}
(\mathcal{Y} | \mathcal{B}=\bf{b}) &\sim\mathcal{D}(\bf{g^{-1}(X\boldsymbol\beta + Z b)},\phi)\\\\
\mathcal{B}&\sim\mathcal{N}(\bf{0},\Sigma_{\boldsymbol\theta}) .
\end{aligned}
\] where \(\mathcal{D}\) is the distribution family (such as Bernoulli or Poisson), \(g^{-1}\) is the inverse link and \(\phi\) is a scale parameter for \(\mathcal{D}\) if it has one. The important cases of the Bernoulli and Poisson distributions don’t have a scale parameter - once you know the mean you know everything you need to know about the distribution. (For those following the presentation, this poem by John Keats is the one with the couplet “Beauty is truth, truth beauty - that is all ye know on earth and all ye need to know.”)
An example of a Bernoulli GLMM
The contra dataset in the MixedModels package is from a survey on the use of artificial contraception by women in Bangladesh.
contra =DataFrame(MixedModels.dataset(:contra))
1,934 rows × 5 columns
dist
urban
livch
age
use
String
String
String
Float64
String
1
D01
Y
3+
18.44
N
2
D01
Y
0
-5.56
N
3
D01
Y
2
1.44
N
4
D01
Y
3+
8.44
N
5
D01
Y
0
-13.56
N
6
D01
Y
0
-11.56
N
7
D01
Y
3+
18.44
N
8
D01
Y
3+
-3.56
N
9
D01
Y
1
-5.56
N
10
D01
Y
3+
1.44
N
11
D01
Y
0
-11.56
Y
12
D01
Y
0
-2.56
N
13
D01
Y
1
-4.56
N
14
D01
Y
3+
5.44
N
15
D01
Y
3+
-0.56
N
16
D01
Y
3+
4.44
Y
17
D01
Y
0
-5.56
N
18
D01
Y
3+
-0.56
Y
19
D01
Y
1
-6.56
Y
20
D01
Y
2
-3.56
N
21
D01
Y
0
-4.56
N
22
D01
Y
0
-9.56
N
23
D01
Y
3+
2.44
N
24
D01
Y
2
2.44
Y
25
D01
Y
1
-4.56
Y
26
D01
Y
3+
14.44
N
27
D01
Y
0
-6.56
Y
28
D01
Y
1
-3.56
Y
29
D01
Y
1
-5.56
Y
30
D01
Y
1
-1.56
Y
⋮
⋮
⋮
⋮
⋮
⋮
combine(groupby(contra, :dist), nrow)
60 rows × 2 columns
dist
nrow
String
Int64
1
D01
117
2
D02
20
3
D03
2
4
D04
30
5
D05
39
6
D06
65
7
D07
18
8
D08
37
9
D09
23
10
D10
13
11
D11
21
12
D12
29
13
D13
24
14
D14
118
15
D15
22
16
D16
20
17
D17
24
18
D18
47
19
D19
26
20
D20
15
21
D21
18
22
D22
20
23
D23
15
24
D24
14
25
D25
67
26
D26
13
27
D27
44
28
D28
49
29
D29
32
30
D30
61
⋮
⋮
⋮
The information recorded included woman’s age, the number of live children she has, whether she lives in an urban or rural setting, and the political district in which she lives.
The age was centered. Unfortunately, the version of the data to which I had access did not record what the centering value was.
A data plot, drawn using lattice graphics in R, shows that the probability of contraception use is not linear in age - it is low for younger women, higher for women in the middle of the range (assumed to be women in late 20’s to early 30’s) and low again for older women (late 30’s to early 40’s in this survey).
If we fit a model with only the age term in the fixed effects, that term will not be significant. This doesn’t mean that there is no “age effect”, it only means that there is no significant linear effect for age.
contrasts =Dict(:dist =>Grouping(),:urban =>HelmertCoding(),:livch =>DummyCoding(), # default, but no harm in being explicit)nAGQ =9dist =Bernoulli()gm1 =let form =@formula( use ~1+ age +abs2(age) + urban + livch + (1| dist) )fit(MixedModel, form, contra, dist; nAGQ, contrasts)end
Minimizing 218 Time: 0:00:00 ( 2.29 ms/it)
Est.
SE
z
p
σ_dist
(Intercept)
-0.6871
0.1686
-4.08
<1e-04
0.4786
age
0.0035
0.0092
0.38
0.7022
abs2(age)
-0.0046
0.0007
-6.29
<1e-09
urban: Y
0.3484
0.0600
5.81
<1e-08
livch: 1
0.8151
0.1622
5.02
<1e-06
livch: 2
0.9165
0.1851
4.95
<1e-06
livch: 3+
0.9154
0.1858
4.93
<1e-06
Notice that the linear term for age is not significant but the quadratic term for age is highly significant. We usually retain the lower order term, even if it is not significant, if the higher order term is significant.
Notice also that the parameter estimates for the treatment contrasts for livch are similar. Thus the distinction of 1, 2, or 3+ childen is not as important as the contrast between having any children and not having any. Those women who already have children are more likely to use artificial contraception.
Furthermore, the women without children have a different probability vs age profile than the women with children. To allow for this we define a binary children factor and incorporate an age&children interaction.
VarCorr(gm1)
Column
Variance
Std.Dev
dist
(Intercept)
0.229094
0.478637
Notice that there is no “residual” variance being estimated. This is because the Bernoulli distribution doesn’t have a scale parameter.
Convert livch to a binary factor
@transform!(contra, :children =:livch ≠"0")# add the associated contrast specifiercontrasts[:children] =EffectsCoding()
EffectsCoding(nothing, nothing)
gm2 =let form =@formula( use ~1+ age * children +abs2(age) + children + urban + (1| dist) )fit(MixedModel, form, contra, dist; nAGQ, contrasts)end
Because these models are not nested, we cannot do a likelihood ratio test. Nevertheless we see that the deviance is much lower in the model with age & children even though the 3 levels of livch have been collapsed into a single level of children. There is a substantial decrease in the deviance even though there are fewer parameters in model gm2 than in gm1. This decrease is because the flexibility of the model - its ability to model the behavior of the response - is being put to better use in gm2 than in gm1.
At present the calculation of the geomdof as sum(influence(m)) is not correctly defined in our code for a GLMM so we need to do some more work before we can examine those values.
Using urban&dist as a grouping factor
It turns out that there can be more difference between urban and rural settings within the same political district than there is between districts. To model this difference we build a model with urban&dist as a grouping factor.
gm3 =let form =@formula( use ~1+ age * children +abs2(age) + children + urban + (1| urban & dist) )fit(MixedModel, form, contra, dist; nAGQ, contrasts)end
Notice that the parameter count in gm3 is the same as that of gm2 - the thing that has changed is the number of levels of the grouping factor- resulting in a much lower deviance for gm3. This reinforces the idea that a simple count of the number of parameters to be estimated does not always reflect the complexity of the model.
gm2
Est.
SE
z
p
σ_dist
(Intercept)
-0.3614
0.1275
-2.84
0.0046
0.4757
age
-0.0131
0.0110
-1.19
0.2352
children: true
0.6054
0.1035
5.85
<1e-08
abs2(age)
-0.0058
0.0008
-6.89
<1e-11
urban: Y
0.3567
0.0602
5.93
<1e-08
age & children: true
0.0342
0.0127
2.69
0.0072
gm3
Est.
SE
z
p
σ_urban & dist
(Intercept)
-0.3415
0.1269
-2.69
0.0071
0.5761
age
-0.0129
0.0112
-1.16
0.2471
children: true
0.6065
0.1045
5.80
<1e-08
abs2(age)
-0.0056
0.0008
-6.66
<1e-10
urban: Y
0.3936
0.0859
4.58
<1e-05
age & children: true
0.0332
0.0128
2.59
0.0096
The coefficient for age may be regarded as insignificant but we retain it for two reasons: we have a term of age² (written abs2(age)) in the model and we have a significant interaction age & children in the model.
Predictions for some subgroups
For a “typical” district (random effect near zero) the predictions on the linear predictor scale for a woman whose age is near the centering value (i.e. centered age of zero) are: