6  Generalized linear mixed models for binary responses

Attach the packages to be used in this chapter

Code
using AlgebraOfGraphics
using CairoMakie
using DataFrames         # only used for `describe`
using EmbraceUncertainty: dataset
using FreqTables
using MixedModels
using MixedModelsMakie
using StatsBase
using StatsModels

and defined some constants, if not already defined.

Code
@isdefined(contrasts) || const contrasts = Dict{Symbol,Any}()
@isdefined(nAGQ) || const nAGQ = 9
@isdefined(progress) || const progress = false

In this chapter we consider mixed-effects models for data sets in which the response is binary, representing yes/no or true/false or correct/incorrect responses.

Because the response must be one of only two possible values we adapt our models to predict the probability of the positive response. As for linear models and linear mixed-effects models, the mean response, \({\boldsymbol{\mu}}\), is determined by a linear predictor, \[ {\boldsymbol{\eta}}={\mathbf{X}}{\boldsymbol{\beta}}+{\mathbf{Z}}{\mathbf{b}} \tag{6.1}\] depending on the fixed-effects parameters, \({\boldsymbol{\beta}}\), the random effects, \({\mathbf{b}}\), and the model matrices, \({\mathbf{X}}\) and \({\mathbf{Z}}\). For a linear model the mean response, \({\boldsymbol{\mu}}\), is equal to the linear predictor, \({\boldsymbol{\eta}}\). But for a generalized linear model \({\boldsymbol{\eta}}\) determines \({\boldsymbol{\mu}}\) according to a link function, \(g\). For historical reasons it is the function taking an element of \({\boldsymbol{\mu}}\) to the corresponding element of \({\boldsymbol{\eta}}\) that is called the link. The transformation in the opposite direction, from \({\boldsymbol{\eta}}\) to \({\boldsymbol{\mu}}\), is called the inverse link.

As in previous chapters, we will begin with an example to help illustrate these ideas.

6.1 Artificial contraception use in regions of Bangladesh

One of the test data sets from the Center for Multilevel Modelling, University of Bristol is derived from the 1989 Bangladesh Fertility Survey, (Huq & Cleland, 1990). The data are a subsample of 1934 women selected from 60 of the 64 political districts or zila, available as the contra data set in the MixedModels package.

contra = Table(dataset(:contra))
Table with 5 columns and 1934 rows:
      dist  urban  livch  age     use
    ┌────────────────────────────────
 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
 ⋮  │  ⋮      ⋮      ⋮      ⋮      ⋮

with summary

Code
let df = DataFrame(contra)
  describe(df, :mean, :min, :median, :max, :nunique, :eltype)
end
5×7 DataFrame
Row variable mean min median max nunique eltype
Symbol Union… Any Union… Any Union… DataType
1 dist D01 D61 60 String
2 urban N Y 2 String
3 livch 0 3+ 4 String
4 age 0.00204757 -13.56 -1.56 19.44 Float64
5 use N Y 2 String

The response of interest is use, whether the woman chooses to use artificial contraception, which is a binary response with only two possible values, N and Y. The covariates include the district (dist) in which the woman resides, the number of live children she currently has (livch, coded as 0, 1, 2, and 3+), her age, and urban, also coded as N and Y, indicating rural or urban.

Note that the mean of these age values is close to zero but not exactly zero. This occurs when the values have been centered about the sample mean then rounded. In this case it appears that the ages were recorded as a whole number of years and the mean was rounded to two decimal places ending in .56. Thus, all the negative values end in .56 and the positive values end in .44. The mean of these rounded, centered values is not exactly zero because of the rounding after centering.

Centering the ages allows for meaningful interpretation of intercepts and fixed effects for other covariates, because they refer to an age within the range of the observed ages. Regretably, the information on what the centering age (i.e. the original mean age) was does not seem to be available.

6.1.1 Plotting the binary response

Producing informative graphical displays of a binary response as it relates to covariates is somewhat more challenging that the corresponding plots for responses on a continuous scale. If we were to plot the 1934 responses as 0/1 values versus, for example, the woman’s centered age, we would end up with a rather uninformative plot, because all the points would fall on one of two horizontal lines, at \(y=0\) and \(y=1\).

One approach to illustrating the structure of the data more effectively is to add scatterplot smoother lines to the plot, as in Figure 6.1,

Code
draw(
  data(contra) *
  mapping(
    :age => "Centered age (yr)",
    :use => ==("Y") => "Frequency of contraceptive use";
    col=:urban => renamer(["N" => "Rural", "Y" => "Urban"]),
    color=:livch,
  ) *
  smooth();
  figure=(; size=(650, 400)),
)
Figure 6.1: Smoothed relative frequency of contraception use versus centered age for women in the 1989 Bangladesh Fertility Survey

to show the trend in the response with respect to the covariate. Once we have the smoother lines in such a plot we can omit the data points themselves, as we did here, because they add very little information.

The first thing to notice about the plot is that the proportion of women using contraception is not linear in age, which, on reflection, makes sense. A woman in the middle of this age range (probably corresponding to an age around 25) is more likely to use artificial contraception than is a girl in her early teens or a woman in her mid-forties. We also see that women in an urban setting are more likely to use contraception than those in a rural setting and that women with no live children are less likely to use contraception than are women who already have children. There do not seem to be strong differences between women who have 1, 2 or 3 or more children compared to the differences between women with children and those without children.

Interestingly, the quadratic pattern with respect to age does not seem to have been noticed in other analyses of these data. Comparisons of model fits through different software systems, as provided by the Center for Multilevel Modelling, incorporate only a linear term in age, even though the pattern is clearly nonlinear. The lesson here is similar to what we have seen in other examples; careful plotting of the data should, whenever possible, precede attempts to fit models to the data.

6.1.2 Initial GLMM fit to the contraception data

Fitting a generalized linear mixed model (GLMM) is very similar to fitting a linear mixed model (LMM). We call the fit function with the first three arguments being the model type, MixedModel, then a formula specifying the response, fixed-effects terms and random-effects terms, then a data table. For GLMMs we also specify a fourth positional argument which is a distribution - in this case Bernoulli().

Establishing the contrasts and fitting a preliminary model with random effects for dist, main effects for livch, age, age^2, and urban, plus interaction terms for age & urban and age^2 & urban can be done as

contrasts[:livch] = EffectsCoding(; base="0")
contrasts[:urban] = HelmertCoding()

com01 =
  let d = contra,
    ds = Bernoulli(),
    f = @formula(use ~
      1 + livch + (age + abs2(age)) * urban + (1 | dist))

  fit(MixedModel, f, d, ds; contrasts, nAGQ, progress)
end
Est. SE z p σ_dist
(Intercept) -0.0126 0.1058 -0.12 0.9052 0.4787
livch: 1 0.1532 0.0982 1.56 0.1188
livch: 2 0.2561 0.1045 2.45 0.0142
livch: 3+ 0.2590 0.1022 2.53 0.0113
age 0.0006 0.0095 0.07 0.9466
abs2(age) -0.0047 0.0008 -6.13 <1e-09
urban: Y 0.3770 0.0804 4.69 <1e-05
age & urban: Y -0.0067 0.0069 -0.98 0.3261
abs2(age) & urban: Y -0.0004 0.0007 -0.51 0.6075

Notice that in the formula language defined by the StatsModels package, an interaction term is written with the & operator. Crossing of terms, which generates main effects and interactions, is written with the * operator (as in the formula language in R). An interaction of a numeric variable with itself is performed by multiplication so the coefficient labelled age & age in the table is the quadratic term in age. (Notice that, in this formula language, age * age, which may easily be interpreted as age^2, expands to age + age^2.) Thus, what is written in the formula as a three-way interaction age * age * urban becomes an urban contrast plus linear and quadratic terms for age and each of their interactions with the urban contrast, which will be coded as -1 for N and +1 for Y.

A fifth positional argument can be used to specify the link function, described in Section 6.2, but in most cases the canonical link function for the distribution is used. In the case of the Bernoulli distribution the canonical link is the logit link.

As for LMMs, the named argument contrasts specifies the contrasts to apply to some of the covariates in a key-value dictionary. Another named argument, nAGQ, specifies the number of quadrature points to use in an adaptive Gauss-Hermite quadrature rule for evaluating the deviance (see Section C.6 for details). A small, odd number, such as nAGQ=9 defined in the first code block of this chapter, is the preferred choice.

The interpretation of the coefficients in this model is somewhat different from the linear mixed models coefficients that we examined previously, but many of the model-building steps are similar. A rough assessment of the utility of a particular term in the fixed-effects part of the model can be obtained from examining the estimates of the coefficients associated with it and their standard errors, which are the basis for the z (z-statistic) and p (p-value) columns in the table. However, these p-values are even more approximate than those provided for LMMs. To perform a more accurate test of whether a particular term is useful we omit it from the model, refit and compare the reduced model fit to the original according to the change in deviance.

We will examine the terms in the model first and discuss the interpretation of the coefficients in Section 6.2.

6.1.3 Model building for the contra data

We noted that Figure 6.1 shows similar patterns for women with children, whether they have one, two, or three or more children. We have set the contrasts for the livch factor to be offsets relative to the reference level, in this case women who do not have any live children. Although the coefficients labeled livch: 1, livch: 2, and livch: 3+ are all large relative to their standard errors, they are reasonably close to each other. This confirms our earlier impression that the main distinction is between women with children and those without and, for those who do have children, the number of children is not an important distinction.

After incorporating a new variable ch — an indicator of whether the woman has any children — in the data,

contrasts[:ch] = HelmertCoding();
contra = Table(contra; ch=contra.livch .!= "0")
Table with 6 columns and 1934 rows:
      dist  urban  livch  age     use  ch
    ┌───────────────────────────────────────
 1  │ D01   Y      3+     18.44   N    true
 2  │ D01   Y      0      -5.56   N    false
 3  │ D01   Y      2      1.44    N    true
 4  │ D01   Y      3+     8.44    N    true
 5  │ D01   Y      0      -13.56  N    false
 6  │ D01   Y      0      -11.56  N    false
 7  │ D01   Y      3+     18.44   N    true
 8  │ D01   Y      3+     -3.56   N    true
 9  │ D01   Y      1      -5.56   N    true
 10 │ D01   Y      3+     1.44    N    true
 11 │ D01   Y      0      -11.56  Y    false
 12 │ D01   Y      0      -2.56   N    false
 13 │ D01   Y      1      -4.56   N    true
 14 │ D01   Y      3+     5.44    N    true
 15 │ D01   Y      3+     -0.56   N    true
 16 │ D01   Y      3+     4.44    Y    true
 17 │ D01   Y      0      -5.56   N    false
 ⋮  │  ⋮      ⋮      ⋮      ⋮      ⋮     ⋮
Code
let df = DataFrame(contra)
  describe(df, :mean, :min, :median, :max, :nunique, :eltype)
end
6×7 DataFrame
Row variable mean min median max nunique eltype
Symbol Union… Any Union… Any Union… DataType
1 dist D01 D61 60 String
2 urban N Y 2 String
3 livch 0 3+ 4 String
4 age 0.00204757 -13.56 -1.56 19.44 Float64
5 use N Y 2 String
6 ch 0.725957 false 1.0 true Bool

we fit a reduced model.

com02 =
  let d = contra,
    ds = Bernoulli(),
    f = @formula(use ~ 1 + ch + age * age * urban + (1 | dist))

    fit(MixedModel, f, d, ds; contrasts, nAGQ, progress)
  end
Est. SE z p σ_dist
(Intercept) -0.2194 0.1155 -1.90 0.0576 0.4773
ch: true 0.4342 0.0741 5.86 <1e-08
age 0.0035 0.0082 0.43 0.6674
urban: Y 0.3734 0.0801 4.66 <1e-05
age & age -0.0048 0.0008 -6.27 <1e-09
age & urban: Y -0.0068 0.0069 -0.99 0.3231
age & age & urban: Y -0.0004 0.0007 -0.49 0.6261

Comparing this model to the previous model

MixedModels.likelihoodratiotest(com02, com01)
model-dof deviance χ² χ²-dof P(>χ²)
use ~ 1 + ch + age + urban + age & age + age & urban + age & age & urban + (1 | dist) 8 2371
use ~ 1 + livch + age + :(abs2(age)) + urban + age & urban + :(abs2(age)) & urban + (1 | dist) 10 2371 0 2 0.7823

indicates that the reduced model is adequate.

Apparently neither the second-order interaction age & urban nor the third-order interaction age & age & urban is significant and we fit a model without these terms.

com03 =
  let f = @formula(use ~ 1 + urban + ch + age * age + (1 | dist)),
    d = contra,
    ds = Bernoulli()

    fit(MixedModel, f, d, ds; contrasts, nAGQ, progress)
  end
Est. SE z p σ_dist
(Intercept) -0.2302 0.1140 -2.02 0.0434 0.4774
urban: Y 0.3461 0.0599 5.78 <1e-08
ch: true 0.4303 0.0737 5.84 <1e-08
age 0.0063 0.0078 0.80 0.4249
age & age -0.0046 0.0007 -6.47 <1e-10

A likelihood ratio test

MixedModels.likelihoodratiotest(com03, com02)
model-dof deviance χ² χ²-dof P(>χ²)
use ~ 1 + urban + ch + age + age & age + (1 | dist) 6 2373
use ~ 1 + ch + age + urban + age & age + age & urban + age & age & urban + (1 | dist) 8 2371 2 2 0.4119

indicates that these terms can safely be eliminated.

A plot of the smoothed observed proportions versus centered age by urban and ch, Figure 6.2,

Code
draw(
  data(contra) *
  mapping(
    :age => "Centered age (yr)",
    :use => ==("Y") => "Frequency of contraceptive use";
    col=:urban => renamer(["N" => "Rural", "Y" => "Urban"]),
    color=:ch => "Children",
  ) *
  smooth();
  figure=(; size=(650, 400)),
)
Figure 6.2: Smoothed relative frequency of contraception use versus centered age for women in the 1989 Bangladesh Fertility Survey. The livch factor has been collapsed to children/nochildren.

indicates that all four groups have a quadratic trend with respect to age but the location of the peak proportion is shifted for those without children relative to those with children. Incorporating an interaction of age and ch allows for such a shift.

com04 =
  let f =
      @formula(use ~ 1 + urban + ch * age + abs2(age) + (1 | dist)),
    d = contra,
    ds = Bernoulli()

    fit(MixedModel, f, d, ds; contrasts, nAGQ, progress)
  end
Est. SE z p σ_dist
(Intercept) -0.3614 0.1275 -2.84 0.0046 0.4757
urban: Y 0.3567 0.0602 5.93 <1e-08
ch: true 0.6054 0.1035 5.85 <1e-08
age -0.0131 0.0110 -1.19 0.2351
abs2(age) -0.0058 0.0008 -6.89 <1e-11
ch: true & age 0.0342 0.0127 2.69 0.0072

Comparing this fitted model to the previous one

MixedModels.likelihoodratiotest(com03, com04)
model-dof deviance χ² χ²-dof P(>χ²)
use ~ 1 + urban + ch + age + age & age + (1 | dist) 6 2373
use ~ 1 + urban + ch + age + :(abs2(age)) + ch & age + (1 | dist) 7 2365 8 1 0.0047

confirms the usefulness of this term.

A series of such model fits led to a model with random effects for the combinations of dist and urban, because differences between urban and rural women in the same district were comparable to differences between districts, even after accounting for an effect of urban at the fixed-effects (or population) level.

com05 =
  let f = @formula(use ~
      1 + urban + ch * age + abs2(age) + (1 | dist & urban)),
    d = contra,
    ds = Bernoulli()

    fit(MixedModel, f, d, ds; contrasts, nAGQ, progress)
  end
Est. SE z p σ_dist & urban
(Intercept) -0.3415 0.1269 -2.69 0.0071 0.5761
urban: Y 0.3936 0.0859 4.58 <1e-05
ch: true 0.6065 0.1045 5.80 <1e-08
age -0.0129 0.0112 -1.16 0.2470
abs2(age) -0.0056 0.0008 -6.66 <1e-10
ch: true & age 0.0332 0.0128 2.59 0.0096

In more detail,

Code
print(com05)
Generalized Linear Mixed Model fit by maximum likelihood (nAGQ = 9)
  use ~ 1 + urban + ch + age + :(abs2(age)) + ch & age + (1 | dist & urban)
  Distribution: Distributions.Bernoulli{Float64}
  Link: GLM.LogitLink()

   logLik    deviance     AIC       AICc        BIC    
 -1177.2418  2353.8242  2368.4836  2368.5418  2407.4550

Variance components:
                Column   Variance Std.Dev. 
dist & urban (Intercept)  0.331927 0.576131

 Number of obs: 1934; levels of grouping factors: 102

Fixed-effects parameters:
─────────────────────────────────────────────────────────
                      Coef.   Std. Error      z  Pr(>|z|)
─────────────────────────────────────────────────────────
(Intercept)     -0.341486    0.126906     -2.69    0.0071
urban: Y         0.393595    0.0859086     4.58    <1e-05
ch: true         0.606464    0.104531      5.80    <1e-08
age             -0.0129123   0.0111549    -1.16    0.2470
abs2(age)       -0.00562457  0.000844101  -6.66    <1e-10
ch: true & age   0.0332117   0.0128228     2.59    0.0096
─────────────────────────────────────────────────────────

Notice that, although there are 60 distinct districts, there are only 102 distinct combinations of dist and urban represented in the data. In 15 of the 60 districts there are no rural women in the sample and in 3 districts there are no urban women in the sample, as shown in a frequency table

Code
freqtable(contra, :urban, :dist)
2×60 Named Matrix{Int64}
urban ╲ dist │ D01  D02  D03  D04  D05  D06  …  D56  D57  D58  D59  D60  D61
─────────────┼──────────────────────────────────────────────────────────────
N            │  54   20    0   19   37   58  …   24   23   20   10   22   31
Y            │  63    0    2   11    2    7  …   21    4   13    0   10   11

6.3 Interpretation of random effects

We should also be aware that the random effects are defined on the linear predictor scale and not on the probability scale. A normal probability plot of the conditional modes of the random effects for model com05, Figure 6.6

Code
qqcaterpillar!(Figure(; size=(650, 450)), com05)
Figure 6.6: Caterpillar plot of the conditional modes of the random-effects for model com05

shows that the smallest random effects are approximately -1 and the largest are approximately 1.

The numerical values and the identifier of the combination of dist and urban for these extreme values can be obtained from the first few rows and the last few rows of the sorted random-effects table.

srtdre = let retbl = only(raneftables(com05))
  retbl[sortperm(last.(retbl))]
end
first(srtdre, 6)
Table with 2 columns and 6 rows:
     dist & urban  (Intercept)
   ┌──────────────────────────
 1 │ ("D01", "N")  -0.957366
 2 │ ("D11", "N")  -0.93198
 3 │ ("D24", "N")  -0.603418
 4 │ ("D01", "Y")  -0.580203
 5 │ ("D27", "N")  -0.576647
 6 │ ("D55", "Y")  -0.548261
last(srtdre, 6)
Table with 2 columns and 6 rows:
     dist & urban  (Intercept)
   ┌──────────────────────────
 1 │ ("D43", "N")  0.64408
 2 │ ("D04", "Y")  0.644669
 3 │ ("D42", "N")  0.665114
 4 │ ("D58", "N")  0.677418
 5 │ ("D14", "Y")  0.695322
 6 │ ("D34", "N")  1.08591

The largest random effect is for rural settings in D34. There were 26 women in the sample from rural D34

D34N = filter(r -> (r.dist == "D34") & (r.urban == "N"), contra)
Table with 6 columns and 26 rows:
      dist  urban  livch  age     use  ch
    ┌───────────────────────────────────────
 1  │ D34   N      0      -11.56  Y    false
 2  │ D34   N      1      3.44    Y    true
 3  │ D34   N      2      4.44    Y    true
 4  │ D34   N      3+     -3.56   Y    true
 5  │ D34   N      0      -8.56   Y    false
 6  │ D34   N      2      8.44    N    true
 7  │ D34   N      3+     8.44    Y    true
 8  │ D34   N      3+     4.44    Y    true
 9  │ D34   N      2      -3.56   Y    true
 10 │ D34   N      3+     8.44    Y    true
 11 │ D34   N      2      -3.56   N    true
 12 │ D34   N      1      -9.56   N    true
 13 │ D34   N      3+     -2.56   Y    true
 14 │ D34   N      3+     2.44    Y    true
 15 │ D34   N      2      0.44    Y    true
 16 │ D34   N      1      4.44    N    true
 17 │ D34   N      2      -7.56   Y    true
 ⋮  │  ⋮      ⋮      ⋮      ⋮      ⋮     ⋮

of whom 20 used contraception, an unusually large proportion for a rural setting.

But this happens when you have relatively small survey sizes - we expect considerable variation.

Also there is considerable variability in the lengths of the prediction intervals in Figure 6.6 because the data are unbalanced with respect to district and rural/urban districts.

Consider the cross-tabulation of counts of interviewees by district and urban/rural status presented at the end of Section 6.1.3. The data contains responses from 54 rural women in district D01 but only 21 rural women from D11. Thus the bottom line in Figure 6.6, for ("D21", "N"), and based on 54 responses, is shorter than the line second from the bottom, for ("D11", "N") and based on 21 women only.

6.3.1 Conversion of random effects to relative odds

The exponential of the random effect is the relative odds of a woman in a particular urban/district combination using artificial birth control compared to her counterpart (same age, same with/without children status, same urban/rural status) in a typical district. The thus, the relative odds of a rural woman in district D01 using artificial contraception relative to the general population of women her age is

exp(last(first(srtdre)))
0.38390287360272524

or about 40%.

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