Generalized linear mixed models

Author

Phillip Alday, Douglas Bates, and Reinhold Kliegl

Published

2024-09-09

Load the packages to be used

Code

using AlgebraOfGraphics
using CairoMakie
using DataFrameMacros
using DataFrames
using MixedModels
using MixedModelsMakie
using SMLP2024: dataset

const progress=false

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.

1 Matrix notation for the sleepstudy model

sleepstudy = DataFrame(dataset(:sleepstudy))

180×3 DataFrame

155 rows omitted

Row	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
⋮	⋮	⋮	⋮
169	S371	8	350.781
170	S371	9	369.469
171	S372	0	269.412
172	S372	1	273.474
173	S372	2	297.597
174	S372	3	310.632
175	S372	4	287.173
176	S372	5	329.608
177	S372	6	334.482
178	S372	7	343.22
179	S372	8	369.142
180	S372	9	364.124

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

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)  565.51065 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
   ⋮
 273.474
 297.5968
 310.6316
 287.1726
 329.6076
 334.4818
 343.2199
 369.1417
 364.1236

m1.β

2-element Vector{Float64}:
 251.4051048484828
  10.467285959596277

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  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}}:
 [2.8158177107740383 -40.04844073875996 … 0.7232622004879082 12.118907715900336; 9.075511973765797 -8.64407958138207 … -0.9710526648887464 1.3106980698541022]

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, 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.(first(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  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}:
 254.22092255925685
 273.7637204926189
 293.306518425981
 312.84931635934305
 332.3921142927051
 351.9349122260672
 371.4777101594293
 391.0205080927913
 410.56330602615344
 430.1061039595155
   ⋮
 275.30199659383356
 287.0799806232839
 298.85796465273427
 310.6359486821847
 322.413932711635
 334.19191674108544
 345.9699007705358
 357.7478847999862
 369.52586882943655

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

180-element Vector{Float64}:
 254.22092255925685
 273.7637204926189
 293.306518425981
 312.84931635934305
 332.3921142927051
 351.9349122260672
 371.4777101594293
 391.0205080927913
 410.56330602615344
 430.1061039595155
   ⋮
 275.3019965938335
 287.0799806232839
 298.85796465273427
 310.6359486821846
 322.413932711635
 334.19191674108544
 345.96990077053573
 357.7478847999862
 369.52586882943655

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 \(\widehat{\sigma^2}\).

m1.θ

3-element Vector{Float64}:
 0.9292213025841999
 0.018168360086059557
 0.22264488361408383

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)	565.51065	23.78047
	days	32.68212	5.71683	+0.08
Residual		654.94145	25.59182

2 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, \(\boldsymbol\eta = \bf{X\boldsymbol\beta+Zb}\).

3 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 possibly 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.”)

3.1 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(dataset(:contra))

1934×5 DataFrame

1909 rows omitted

Row	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
⋮	⋮	⋮	⋮	⋮	⋮
1923	D61	N	0	-11.56	Y
1924	D61	N	3+	1.44	N
1925	D61	N	1	-5.56	N
1926	D61	N	3+	14.44	N
1927	D61	N	3+	19.44	N
1928	D61	N	2	-9.56	Y
1929	D61	N	2	-2.56	N
1930	D61	N	3+	14.44	N
1931	D61	N	2	-4.56	N
1932	D61	N	3+	14.44	N
1933	D61	N	0	-13.56	N
1934	D61	N	3+	10.44	N

combine(groupby(contra, :dist), nrow)

60×2 DataFrame

35 rows omitted

Row	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
⋮	⋮	⋮
49	D49	4
50	D50	19
51	D51	37
52	D52	61
53	D53	19
54	D55	6
55	D56	45
56	D57	27
57	D58	33
58	D59	10
59	D60	32
60	D61	42

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)),
)

┌ Warning: Found `resolution` in the theme when creating a `Scene`. The `resolution` keyword for `Scene`s and `Figure`s has been deprecated. Use `Figure(; size = ...` or `Scene(; size = ...)` instead, which better reflects that this is a unitless size and not a pixel resolution. The key could also come from `set_theme!` calls or related theming functions.
└ @ Makie ~/.julia/packages/Makie/WgbrE/src/scenes.jl:227

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(),
  :livch => DummyCoding(), # default, but no harm in being explicit
)
nAGQ = 9
dist = Bernoulli()
gm1 = let
  form = @formula(
    use ~ 1 + age + abs2(age) + urban + livch + (1 | dist)
  )
  fit(MixedModel, form, contra, dist; nAGQ, contrasts, progress)
end

	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+ children 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)	0.229095	0.478638

Notice that there is no “residual” variance being estimated. This is because the Bernoulli distribution doesn’t have a scale parameter.

3.2 Convert `livch` to a binary factor

@transform!(contra, :children = :livch ≠ "0")
# add the associated contrast specifier
contrasts[: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, progress)
end

	Est.	SE	z	p	σ_dist
(Intercept)	-0.3614	0.1275	-2.84	0.0046	0.4757
age	-0.0131	0.0110	-1.19	0.2351
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

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

2×6 DataFrame

Row	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.

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.

3.3 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, progress)
end

	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

3×6 DataFrame

Row	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. 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	0.2351
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.

3.4 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:

using Effects
design = Dict(
  :children => [true, false], :urban => ["Y", "N"], :age => [0.0]
)
preds = effects(design, gm3; invlink=AutoInvLink())

4×7 DataFrame

Row	children	age	urban	use: Y	err	lower	upper
	Bool	Float64	String	Float64	Float64	Float64	Float64
1	true	0.0	Y	0.65894	0.0338299	0.62511	0.69277
2	false	0.0	Y	0.364863	0.0534101	0.311452	0.418273
3	true	0.0	N	0.467889	0.0281378	0.439751	0.496026
4	false	0.0	N	0.20726	0.0364055	0.170855	0.243666

4 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 significant age & children interaction term.

1 Matrix notation for the sleepstudy model

2 Linear predictors in LMMs and GLMMs

3 Generalized Linear Mixed Models

3.1 An example of a Bernoulli GLMM

3.2 Convert livch to a binary factor

3.3 Using urban&dist as a grouping factor

3.4 Predictions for some subgroups

4 Summarizing the results

3.2 Convert `livch` to a binary factor

3.3 Using `urban&dist` as a grouping factor