3b: Generalized linear mixed models

Author

Douglas Bates and Claudia Solis-Lemus

Published

2022-07-30

Load the packages to be used

Code

using AlgebraOfGraphics
using CairoMakie
using DataFrameMacros
using DataFrames
using MixedModels
using MixedModelsMakie
using ProgressMeter

CairoMakie.activate!(; type = "svg");
ProgressMeter.ijulia_behavior(:clear);

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.

Matrix notation for the sleepstudy model

sleepstudy = DataFrame(MixedModels.dataset(:sleepstudy))

180 rows × 3 columns

	subj	days	reaction
	String	Int8	Float64
1	S308	0	249.56
2	S308	1	258.705
3	S308	2	250.801
4	S308	3	321.44
5	S308	4	356.852
6	S308	5	414.69
7	S308	6	382.204
8	S308	7	290.149
9	S308	8	430.585
10	S308	9	466.353
11	S309	0	222.734
12	S309	1	205.266
13	S309	2	202.978
14	S309	3	204.707
15	S309	4	207.716
16	S309	5	215.962
17	S309	6	213.63
18	S309	7	217.727
19	S309	8	224.296
20	S309	9	237.314
21	S310	0	199.054
22	S310	1	194.332
23	S310	2	234.32
24	S310	3	232.842
25	S310	4	229.307
26	S310	5	220.458
27	S310	6	235.421
28	S310	7	255.751
29	S310	8	261.012
30	S310	9	247.515
⋮	⋮	⋮	⋮

contrasts = Dict(:subj => Grouping())
m1 = let
  form = @formula(reaction ~ 1 + days + (1 + days | subj))
  fit(MixedModel, form, sleepstudy; contrasts)
end
println(m1)

Minimizing 58    Time: 0:00:00 ( 5.19 ms/it)

Linear mixed model fit by maximum likelihood
 reaction ~ 1 + days + (1 + days | subj)

   logLik   -2 logLik     AIC       AICc        BIC    
  -875.9697  1751.9393  1763.9393  1764.4249  1783.0971

Variance components:


            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.

m1.y

180-element view(::Matrix{Float64}, :, 3) with eltype Float64:
 249.56
 258.7047
 250.8006
 321.4398
 356.8519
 414.6901
 382.2038
 290.1486
 430.5853
 466.3535
 222.7339
 205.2658
 202.9778
   ⋮
 350.7807
 369.4692
 269.4117
 273.474
 297.5968
 310.6316
 287.1726
 329.6076
 334.4818
 343.2199
 369.1417
 364.1236

m1.β

2-element Vector{Float64}:
 251.40510484848375
  10.467285959595792

m1.X

180×2 Matrix{Float64}:
 1.0  0.0
 1.0  1.0
 1.0  2.0
 1.0  3.0
 1.0  4.0
 1.0  5.0
 1.0  6.0
 1.0  7.0
 1.0  8.0
 1.0  9.0
 1.0  0.0
 1.0  1.0
 1.0  2.0
 ⋮    
 1.0  8.0
 1.0  9.0
 1.0  0.0
 1.0  1.0
 1.0  2.0
 1.0  3.0
 1.0  4.0
 1.0  5.0
 1.0  6.0
 1.0  7.0
 1.0  8.0
 1.0  9.0

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.

m1.b

1-element Vector{Matrix{Float64}}:
 [2.8158191325913045 -40.04844170604461 … 0.7232620377478233 12.118907799956531; 9.07551171690184 -8.644079452057246 … -0.9710526342424729 1.31069806890679]

only(m1.b)   # only one grouping factor

2×18 Matrix{Float64}:
 2.81582  -40.0484   -38.4331  22.8321   …  -24.7101   0.723262  12.1189
 9.07551   -8.64408   -5.5134  -4.65872       4.6597  -0.971053   1.3107

There is a model matrix, \(\mathbf{Z}\), 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.

In this case there is only one grouping factor.

Int.(only(m1.reterms))

180×36 Matrix{Int64}:
 1  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0
 1  1  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  2  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  3  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  4  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  5  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0
 1  6  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  7  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  8  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 1  9  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 0  0  1  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0
 0  0  1  1  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 0  0  1  2  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0
 ⋮              ⋮              ⋮        ⋱     ⋮              ⋮              ⋮
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  1  8  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  1  9  0  0
 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  1  0
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  1
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  2
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  3
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  4
 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  1  5
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  6
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  7
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  8
 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  1  9

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

m1.X * m1.β + only(m1.reterms) * vec(only(m1.b))

180-element Vector{Float64}:
 254.22092398107506
 273.7637216575727
 293.3065193340703
 312.849317010568
 332.3921146870656
 351.93491236356317
 371.47771004006086
 391.0205077165585
 410.5633053930561
 430.1061030695537
 211.35666314243912
 213.1798696499777
 215.0030761575162
   ⋮
 328.09823348905815
 337.5944668144114
 263.52401264844025
 275.3019966769429
 287.07998070544545
 298.85796473394805
 310.63594876245065
 322.4139327909532
 334.1919168194558
 345.9699008479584
 357.74788487646094
 369.5258689049635

fitted(m1)   # just to check that these are indeed the same as calculated above

180-element Vector{Float64}:
 254.22092398107506
 273.76372165757266
 293.3065193340703
 312.8493170105679
 332.3921146870656
 351.93491236356317
 371.47771004006086
 391.0205077165585
 410.5633053930561
 430.1061030695537
 211.35666314243912
 213.1798696499777
 215.0030761575162
   ⋮
 328.0982334890581
 337.5944668144114
 263.52401264844025
 275.3019966769429
 287.07998070544545
 298.857964733948
 310.63594876245065
 322.4139327909532
 334.19191681945574
 345.9699008479584
 357.74788487646094
 369.5258689049635

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.

This symmetric matrix can be constructed from the parameters \(\boldsymbol\theta\), which generate the lower triangular matrix \(\boldsymbol\lambda\), and the estimate \(\widehat{\sigma^2}\).

m1.θ

3-element Vector{Float64}:
 0.9292213186823386
 0.018168380890783722
 0.22264487369155986

λ = only(m1.λ)  # with multiple grouping factors there will be multiple λ's

2×2 LinearAlgebra.LowerTriangular{Float64, Matrix{Float64}}:
 0.929221    ⋅ 
 0.0181684  0.222645

Σ = varest(m1) * (λ * λ')

2×2 Matrix{Float64}:
 565.511  11.057
  11.057  32.6821

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, Figure 1, 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.

Code

draw(
  data(
    @transform(
      contra,
      :numuse = Int(:use == "Y"),
      :urb = ifelse(:urban == "Y", "Urban", "Rural")
    )
  ) *
  mapping(
    :age => "Centered age (yr)",
    :numuse => "Frequency of contraception use";
    col = :urb,
    color = :livch,
  ) *
  smooth();
  figure = (; resolution = (800, 450)),
)

Figure 1: Smoothed relative frequency of contraception use versus centered age for women in the 1989 Bangladesh Fertility Survey

contrasts = Dict(
  :dist => Grouping(),
  :urban => HelmertCoding(),
  :children => HelmertCoding(),
)
nAGQ = 9
dist = Bernoulli()
gm1 = let
  form = @formula(use ~ 1 + age + abs2(age) + urban + livch + (1 | dist))
  fit(MixedModel, form, contra, dist; nAGQ, contrasts)
end

Minimizing 217   Time: 0:00:00 ( 1.61 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	0.8152	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)	0.229090	0.478634

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")

1,934 rows × 6 columns

	dist	urban	livch	age	use	children
	String	String	String	Float64	String	Bool
1	D01	Y	3+	18.44	N	1
2	D01	Y	0	-5.56	N	0
3	D01	Y	2	1.44	N	1
4	D01	Y	3+	8.44	N	1
5	D01	Y	0	-13.56	N	0
6	D01	Y	0	-11.56	N	0
7	D01	Y	3+	18.44	N	1
8	D01	Y	3+	-3.56	N	1
9	D01	Y	1	-5.56	N	1
10	D01	Y	3+	1.44	N	1
11	D01	Y	0	-11.56	Y	0
12	D01	Y	0	-2.56	N	0
13	D01	Y	1	-4.56	N	1
14	D01	Y	3+	5.44	N	1
15	D01	Y	3+	-0.56	N	1
16	D01	Y	3+	4.44	Y	1
17	D01	Y	0	-5.56	N	0
18	D01	Y	3+	-0.56	Y	1
19	D01	Y	1	-6.56	Y	1
20	D01	Y	2	-3.56	N	1
21	D01	Y	0	-4.56	N	0
22	D01	Y	0	-9.56	N	0
23	D01	Y	3+	2.44	N	1
24	D01	Y	2	2.44	Y	1
25	D01	Y	1	-4.56	Y	1
26	D01	Y	3+	14.44	N	1
27	D01	Y	0	-6.56	Y	0
28	D01	Y	1	-3.56	Y	1
29	D01	Y	1	-5.56	Y	1
30	D01	Y	1	-1.56	Y	1
⋮	⋮	⋮	⋮	⋮	⋮	⋮

gm2 = let
  form = @formula(use ~ 1 + age * children + abs2(age) + urban + (1 | dist))
  fit(MixedModel, form, contra, dist; nAGQ, contrasts)
end

Minimizing 133   Time: 0:00:00 ( 0.91 ms/it)

	Est.	SE	z	p	σ_dist
(Intercept)	-0.3614	0.1275	-2.83	0.0046	0.4756
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

Various model-fit statistics

Code

let
  mods = [gm2, gm1]
  DataFrame(;
    model = [:gm2, :gm1],
    npar = dof.(mods),
    deviance = deviance.(mods),
    AIC = aic.(mods),
    BIC = bic.(mods),
    AICc = aicc.(mods),
  )
end

2 rows × 6 columns

	model	npar	deviance	AIC	BIC	AICc
	Symbol	Int64	Float64	Float64	Float64	Float64
1	gm2	7	2364.92	2379.18	2418.15	2379.24
2	gm1	8	2372.46	2388.73	2433.27	2388.81

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.

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) + urban + (1 | urban & dist)
  )
  fit(MixedModel, form, contra, dist; nAGQ, contrasts)
end

Minimizing 158   Time: 0:00:00 ( 0.92 ms/it)

	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	0.6064	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

Code

let
  mods = [gm3, gm2, gm1]
  DataFrame(;
    model = [:gm3, :gm2, :gm1],
    npar = dof.(mods),
    deviance = deviance.(mods),
    AIC = aic.(mods),
    BIC = bic.(mods),
    AICc = aicc.(mods),
  )
end

3 rows × 6 columns

	model	npar	deviance	AIC	BIC	AICc
	Symbol	Int64	Float64	Float64	Float64	Float64
1	gm3	7	2353.82	2368.48	2407.46	2368.54
2	gm2	7	2364.92	2379.18	2418.15	2379.24
3	gm1	8	2372.46	2388.73	2433.27	2388.81

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. 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	-2.83	0.0046	0.4756
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	0.6064	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.

Summarizing the results

From the data plot we can see a quadratic trend in the probability by age.
The patterns for women with children are similar and we do not need to distinguish between 1, 2, and 3+ children.
We do distinguish between those women who do not have children and those with children. This shows up in a signficant age & children interaction term.

Matrix notation for the sleepstudy model

Linear predictors in LMMs and GLMMs

Generalized Linear Mixed Models

An example of a Bernoulli GLMM

Convert livch to a binary factor

Using urban&dist as a grouping factor

Summarizing the results

Convert `livch` to a binary factor

Using `urban&dist` as a grouping factor