2  Models with multiple random-effects terms

Attach the packages to be used in this chapter

Code
using AlgebraOfGraphics
using CairoMakie
CairoMakie.activate!(; type="svg")
using CategoricalArrays
using DataFrameMacros
using DataFrames
using EmbraceUncertainty: dataset
using MixedModels
using MixedModelsMakie
using Random
using RCall
using StatsBase

and define some constants, if not already defined,

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

The mixed models considered in the previous chapter had only one random-effects term, which was a simple, scalar random-effects term, and a single fixed-effects coefficient. Although such models can be useful, it is with the facility to use multiple random-effects terms and to use random-effects terms beyond a simple, scalar term that we can begin to realize the flexibility and versatility of mixed models.

In this chapter we consider models with multiple simple, scalar random-effects terms, showing examples where the grouping factors for these terms are in completely crossed or nested or partially crossed configurations. For ease of description we will refer to the random effects as being crossed or nested although, strictly speaking, the distinction between nested and non-nested refers to the grouping factors, not the random effects.

2.1 A model with crossed random effects

One of the areas in which the methods in the MixedModels.jl package are particularly effective is in fitting models to cross-classified data where several factors have random effects associated with them. For example, in many experiments in psychology a reaction time for each of a group of subjects exposed to some or all of a group of stimuli or items is measured. If the subjects are considered to be a sample from a population of subjects and the items are a sample from a population of items, then it would make sense to associate random effects with both these factors.

In the past it was difficult to fit mixed models with multiple, crossed grouping factors to large, possibly unbalanced, data sets. The methods in MixedModels.jl are able to do this. To introduce the methods let us first consider a small, balanced data set with crossed grouping factors.

2.1.1 The penicillin data

The data are derived from Table 6.6, p. 144 of Davies & Goldsmith (1972) where they are described as coming from an investigation to

assess the variability between samples of penicillin by the B. subtilis method. In this test method a bulk-inoculated nutrient agar medium is poured into a Petri dish of approximately 90 mm. diameter, known as a plate. When the medium has set, six small hollow cylinders or pots (about 4 mm. in diameter) are cemented onto the surface at equally spaced intervals. A few drops of the penicillin solutions to be compared are placed in the respective cylinders, and the whole plate is placed in an incubator for a given time. Penicillin diffuses from the pots into the agar, and this produces a clear circular zone of inhibition of growth of the organisms, which can be readily measured. The diameter of the zone is related in a known way to the concentration of penicillin in the solution.

As with the dyestuff data, we examine the structure

penicillin = dataset(:penicillin)
Arrow.Table with 144 rows, 3 columns, and schema:
 :plate     String
 :sample    String
 :diameter  Int8

and a summary

penicillin = DataFrame(penicillin)
describe(penicillin)
3×7 DataFrame
Row variable mean min median max nmissing eltype
Symbol Union… Any Union… Any Int64 DataType
1 plate a x 0 String
2 sample A F 0 String
3 diameter 22.9722 18 23.0 27 0 Int8

of the data, then plot it

Code
"""
    _meanrespfrm(df, :resp::Symbol, :grps::Symbol; sumryf::Function=mean)

Returns a `DataFrame` created from df with the levels of `grps` reordered according to
`combine(groupby(df, grps), resp => sumryf)` and this summary DataFrame, also with the
levels of `grps` reordered.
"""
function _meanrespfrm(
  df,
  resp::Symbol,
  grps::Symbol;
  sumryf::Function=mean,
)
  # ensure the relevant columns are types that Makie can deal with 
  df = transform(
    df,
    resp => Array,
    grps => CategoricalArray;
    renamecols=false,
  )
  # create a summary table by mean resp
  sumry =
    sort!(combine(groupby(df, grps), resp => sumryf => resp), resp)
  glevs = string.(sumry[!, grps])   # group levels in ascending order of mean resp
  levels!(df[!, grps], glevs)
  levels!(sumry[!, grps], glevs)
  return df, sumry
end

let
  df, _ = _meanrespfrm(penicillin, :diameter, :plate)
  sort!(df, [:plate, :sample])

  mp = mapping(
    :diameter => "Diameter of inhibition zone [mm]",
    :plate => "Plate";
    color=:sample,
  )
  draw(
    data(df) * mp * visual(ScatterLines; marker='○', markersize=12),
  )
end
Figure 2.1: Diameter of inhibition zone by plate and sample. Plates are ordered by increasing mean response.

The variation in the diameter is associated with the plates and with the samples. Because each plate is used only for the six samples shown here we are not interested in the contributions of specific plates as much as we are interested in the variation due to plates, and in assessing the potency of the samples after accounting for this variation. Thus, we will use random effects for the plate factor. We will also use random effects for the sample factor because, as in the dyestuff example, we are more interested in the sample-to-sample variability in the penicillin samples than in the potency of a particular sample.

In this experiment each sample is used on each plate. We say that the plate and sample factors are crossed, as opposed to nested factors, which we will describe in the next section. By itself, the designation “crossed” just means that the factors are not nested. If we wish to be more specific, we could describe these factors as being completely crossed, which means that we have at least one observation for each combination of a level of plate and a level of sample. We can see this in Figure 2.1. Alternatively, because there are moderate numbers of levels in these factors, we could also check by a cross-tabulation of these factors.

Like the dyestuff data, the factors in the penicillin data are balanced. That is, there are exactly the same number of observations on each plate and for each sample and, furthermore, there is the same number of observations on each combination of levels. In this case there is exactly one observation for each combination of sample and plate. We would describe the configuration of these two factors as an unreplicated, completely balanced, crossed design.

In general, balance is a desirable but precarious property of a data set. We may be able to impose balance in a designed experiment but we typically cannot expect that data from an observation study will be balanced. Also, as anyone who analyzes real data soon finds out, expecting that balance in the design of an experiment will produce a balanced data set is contrary to Murphy’s law. That’s why statisticians allow for missing data. Even when we apply each of the six samples to each of the 24 plates, something could go wrong for one of the samples on one of the plates, leaving us without a measurement for that combination of levels and thus an unbalanced data set.

2.1.2 A model for the penicillin data

A model incorporating random effects for both the plate and the sample is straightforward to specify — we include simple, scalar random effects terms for both these factors.

pnm01 = let f = @formula diameter ~ 1 + (1 | plate) + (1 | sample)
  fit(MixedModel, f, penicillin; contrasts, progress)
end
Est. SE z p σ_plate σ_sample
(Intercept) 22.9722 0.7446 30.85 <1e-99 0.8456 1.7706
Residual 0.5499

This model display indicates that the sample-to-sample variability has the greatest contribution, then plate-to-plate variability and finally the “residual” variability that cannot be attributed to either the sample or the plate. These conclusions are consistent with what we see in the data plot (Figure 2.1).

Code
caterpillar!(
  Figure(; resolution=(800, 480)),
  ranefinfo(pnm01, :plate),
)
Figure 2.2: Conditional modes and 95% prediction intervals of random effects for plate in model pnm01
Code
caterpillar!(
  Figure(; resolution=(800, 160)),
  ranefinfo(pnm01, :sample),
)
Figure 2.3: Conditional modes and 95% prediction intervals of random effects for sample in model pnm01

The prediction intervals on the random effects (Figure 2.2 and Figure 2.3) confirm that the conditional distribution of the random effects for plate displays much less variability than does the conditional distribution of the random effects for sample. (Note that the horizontal scales on these two plots are different.) However, the conditional distribution of the random effect for a particular sample, say sample F, has less variability than the conditional distribution of the random effect for a particular plate, say plate m. That is, the lines in Figure 2.3 are wider than the lines in Figure 2.2, even after taking the different axis scales into account. This is because the conditional distribution of the random effect for a particular sample depends on 24 responses while the conditional distribution of the random effect for a particular plate depends on only 6 responses.

In Chapter 1 we saw that a model with a single, simple, scalar random-effects term generated a random-effects model matrix, \({\mathbf{Z}}\), that is the matrix of indicators of the levels of the grouping factor. When we have multiple, simple, scalar random-effects terms, as in model pnm01, each term generates a matrix of indicator columns and these sets of indicators are concatenated to form the model matrix \({\mathbf{Z}}\) whose transpose is

Listing 2.1: Pattern of non-zeros in the transpose of the model matrix Z for model pnm01
vcat(transpose.(sparse.(pnm01.reterms))...)
30×144 SparseArrays.SparseMatrixCSC{Float64, Int32} with 288 stored entries:
⎡⠉⠙⠒⠒⠒⠤⠤⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠒⠒⠦⠤⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠢⠤⠤⠤⣀⣀⣀⡀⠀⎥
⎣⠑⢌⠢⡜⢦⠓⢌⠢⡘⢦⠓⢌⠢⡘⢦⠳⢌⠢⡑⢦⠳⢌⠢⡑⢦⠳⡌⠢⡑⢤⠳⡌⠢⡑⢤⠳⡜⠢⡙⢍⎦

The relative covariance factor, \({\boldsymbol{\Lambda}}_{{\boldsymbol{\theta}}}\), for this model is block diagonal, with two blocks, one of size 24 and one of size 6, each of which is a multiple of the identity. The diagonal elements of the two blocks are \(\theta_1\) and \(\theta_2\), respectively. The numeric values of these parameters can be obtained as

pnm01.θ'
1×2 adjoint(::Vector{Float64}) with eltype Float64:
 1.53758  3.21975

The first parameter is the relative standard deviation of the random effects for plate, which has the value \(0.845565/0.549933=1.53758\) at convergence, and the second is the relative standard deviation of the sample random effects (\(1.770648/0.549933=3.21975\)).

Because \({\boldsymbol{\Lambda}}_{{\boldsymbol{\theta}}}\) is diagonal, the pattern of non-zeros in \({\boldsymbol{\Lambda}}_{\boldsymbol{\theta}}'{\mathbf{Z}}'{\mathbf{Z}}{\boldsymbol{\Lambda}}_{\boldsymbol{\theta}}+{\mathbf{I}}\) will be the same as that in \({\mathbf{Z}}'{\mathbf{Z}}\). The sparse Cholesky factor, \({\mathbf{L}}\), is lower triangular and has non-zero elements in the lower right hand corner in positions where \({\mathbf{Z}}'{\mathbf{Z}}\) has systematic zeros.

sparseL(pnm01)
30×30 SparseArrays.SparseMatrixCSC{Float64, Int32} with 189 stored entries:
⎡⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⎥
⎢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⎥
⎣⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⎦

We say that “fill-in” has occurred when forming the sparse Cholesky decomposition. In this case there is a relatively minor amount of fill but in other cases there can be a substantial amount of fill. The computational methods are tuned to reduce the amount of fill.

2.1.3 Precision of parameter estimates in the Pencillin model

A parametric bootstrap sample of the parameter estimates

Code
bsrng = Random.seed!(9876789)
pnm01samp = parametricbootstrap(bsrng, 10_000, pnm01; progress)
pnm01pars = DataFrame(pnm01samp.allpars);

can be used to create shortest 95% coverage intervals for the parameters in the model.

DataFrame(shortestcovint(pnm01samp))
4×5 DataFrame
Row type group names lower upper
String String? String? Float64 Float64
1 β missing (Intercept) 21.4993 24.4402
2 σ plate (Intercept) 0.586766 1.0985
3 σ sample (Intercept) 0.627596 2.55106
4 σ residual missing 0.475331 0.61701

As for model dsm01 the bootstrap parameter estimates of the fixed-effects parameter have approximately a “normal” or Gaussian shape, as shown in the kernel density plot (Figure 2.4)

Code
draw(
  data(@subset(pnm01pars, :type == "β")) *
  mapping(
    :value => "Bootstrap samples of β";
    color=(:names => "Names"),
  ) *
  AlgebraOfGraphics.density();
  figure=(; resolution=(800, 450)),
)
Figure 2.4: Parametric bootstrap estimates of fixed-effects parameters in model pnm01

and the shortest coverage interval on this parameter is close to the Wald interval

Code
show(only(pnm01.beta) .+ [-2, 2] * only(pnm01.stderror))
[21.483030339365666, 24.461414105078195]

The densities of the variance-components, on the scale of the standard deviation parameters, are diffuse but do not exhibit point masses at zero.

Code
draw(
  data(@subset(pnm01pars, :type == "σ")) *
  mapping(
    :value => "Bootstrap samples of σ";
    color=(:group => "Group"),
  ) *
  AlgebraOfGraphics.density();
  figure=(; resolution=(800, 450)),
)
Figure 2.5: Parametric bootstrap estimates of variance components in model pnm01

The lack of precision in the estimate of \(\sigma_2\), the standard deviation of the random effects for sample, is a consequence of only having 6 distinct levels of the sample factor. The plate factor, on the other hand, has 24 distinct levels. In general it is more difficult to estimate a measure of spread, such as the standard deviation, than to estimate a measure of location, such as a mean, especially when the number of levels of the factor is small. Six levels are about the minimum number required for obtaining sensible estimates of standard deviations for simple, scalar random effects terms.

2.2 A model with nested random effects

In this section we again consider a simple example, this time fitting a model with nested grouping factors for the random effects.

2.2.1 The pastes data

The third example from Davies & Goldsmith (1972, Table 6.5, p. 138) is described as coming from

deliveries of a chemical paste product contained in casks where, in addition to sampling and testing errors, there are variations in quality between deliveries …As a routine, three casks selected at random from each delivery were sampled and the samples were kept for reference. …Ten of the delivery batches were sampled at random and two analytical tests carried out on each of the 30 samples.

The structure and summary of the data object are

pastes = dataset(:pastes)
Arrow.Table with 60 rows, 3 columns, and schema:
 :batch     String
 :cask      String
 :strength  Float64
pastes =
  @transform(DataFrame(pastes), :sample = string(:batch, :cask))
describe(pastes)
4×7 DataFrame
Row variable mean min median max nmissing eltype
Symbol Union… Any Union… Any Int64 DataType
1 batch A J 0 String
2 cask a c 0 String
3 strength 60.0533 54.2 59.3 66.0 0 Float64
4 sample Aa Jc 0 String

As stated in the description in Davies & Goldsmith (1972), there are 30 samples, three from each of the 10 delivery batches. We have created a sample factor by concatenating the label of the batch factor with ‘a’, ‘b’ or ‘c’ to distinguish the three samples taken from that batch.

When plotting the strength versus batch and cask in the data we should remember that we have two strength measurements on each of the 30 samples. It is tempting to use the cask designation (‘a’, ‘b’ and ‘c’) to determine, say, the plotting symbol within a batch. It would be fine to do this within a batch but the plot would be misleading if we used the same symbol for cask ‘a’ in different batches. There is no relationship between cask ‘a’ in batch ‘A’ and cask ‘a’ in batch ‘B’. The labels ‘a’, ‘b’ and ‘c’ are used only to distinguish the three samples within a batch; they do not have a meaning across batches.

Code
RCall.ijulia_setdevice(MIME("image/svg+xml"); width=8, height=8)
R"""
require("lattice", quietly=TRUE)
pp <- $pastes
pp <- within(pp, bb <- reorder(batch, strength))
print(
  dotplot(sample ~ strength | bb, pp, pch = 21, strip = FALSE,
    strip.left = TRUE, layout = c(1, 10),
    scales = list(y = list(relation = "free")),
    ylab = "Sample within batch", type = c("p", "a"),
    xlab = "Paste strength", jitter.y = TRUE)
)
""";
Figure 2.6: Strength of paste preparations according to sample within batch

In Figure 2.6 we plot the two strength measurements on each of the samples within each of the batches and join up the average strength for each sample. The perceptive reader will have noticed that the levels of the factors on the vertical axis in this figure, and in Figure 1.1, and Figure 2.1, have been reordered according to increasing average response. In all these cases there is no inherent ordering of the levels of the covariate such as batch or plate. Rather than confuse our interpretation of the plot by determining the vertical displacement of points according to a random ordering, we impose an ordering according to increasing mean response. This allows us to more easily check for structure in the data, including undesirable characteristics like increasing variability of the response with increasing mean level of the response.

In Figure 2.6 we order the samples within each batch separately then order the batches according to increasing mean strength.

Figure 2.6 shows considerable variability in strength between samples relative to the variability within samples. There is some indication of variability between batches, in addition to the variability induced by the samples, but not a strong indication of a batch effect. For example, batches I and D, with low mean strength relative to the other batches, each contained one sample (I:b and D:c, respectively) that had high mean strength relative to the other samples. Also, batches H and C, with comparatively high mean batch strength, contain samples H:a and C:a with comparatively low mean sample strength. In Section 2.2.4 we will examine the need for incorporating batch-to-batch variability, in addition to sample-to-sample variability, in the statistical model.

2.2.2 Nested factors

Because each level of sample occurs with one and only one level of batch we say that sample is nested within batch. Some presentations of mixed-effects models, especially those related to multilevel modeling (Rasbash et al., 2000) or hierarchical linear models (Raudenbush & Bryk, 2002), leave the impression that one can only define random effects with respect to factors that are nested. This is the origin of the terms “multilevel”, referring to multiple, nested levels of variability, and “hierarchical”, also invoking the concept of a hierarchy of levels. To be fair, both those references do describe the use of models with random effects associated with non-nested factors, but such models tend to be treated as a special case.

The blurring of mixed-effects models with the concept of multiple, hierarchical levels of variation results in an unwarranted emphasis on “levels” when defining a model and leads to considerable confusion. It is perfectly legitimate to define models having random effects associated with non-nested factors. The reasons for the emphasis on defining random effects with respect to nested factors only are that such cases do occur frequently in practice and that some of the computational methods for estimating the parameters in the models can only be easily applied to nested factors.

This is not the case for the methods used MixedModels.jl. Indeed there is nothing special done for models with random effects for nested factors. When random effects are associated with multiple factors exactly the same computational methods are used whether the factors form a nested sequence or are partially crossed or are completely crossed.

There is, however, one aspect of nested grouping factors that we should emphasize, which is the possibility of a factor that is implicitly nested within another factor. Suppose, for example, that the factor sample had been defined as having three levels instead of 30 with the implicit assumption that sample is nested within batch. It may seem silly to try to distinguish 30 different batches with only three levels of a factor but, unfortunately, data are frequently organized and presented like this, especially in text books. The factor cask in the data is exactly such an implicitly nested factor. If we cross-tabulate cask and batch we get the impression that these factors are crossed, not nested. If we know that the cask should be considered as nested within the batch then we should create a new categorical variable giving the batch-cask combination, which is exactly what the sample factor is.

In a small data set like we can quickly detect a factor being implicitly nested within another factor and take appropriate action. In a large data set, such as from a multi-center study, it is often assumed that, say, subject identifiers are unique to each center. Frequently this is not the case. Especially when dealing with large data sets, assumptions about nested identifiers should be checked carefully.

2.2.3 Fitting a model with nested random effects

Fitting a model with simple, scalar random effects for nested factors is done in exactly the same way as fitting a model with random effects for crossed grouping factors. We include random-effects terms for each factor, as in

psm01 = let f = @formula strength ~ 1 + (1 | sample) + (1 | batch)
  fit(MixedModel, f, pastes; contrasts, progress)
end
Est. SE z p σ_sample σ_batch
(Intercept) 60.0533 0.6421 93.52 <1e-99 2.9041 1.0951
Residual 0.8234

Not only is the model specification similar for nested and crossed factors, the internal calculations are performed according to the methods described in for each model type. Comparing the patterns in the matrices \({\boldsymbol{\Lambda}}\), \({\mathbf{Z}}'{\mathbf{Z}}\) and \({\mathbf{L}}\) and the block structure

sparseL(psm01; full=true)
42×42 SparseArrays.SparseMatrixCSC{Float64, Int32} with 153 stored entries:
⎡⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⠀⠀⠀⠑⢄⠀⎥
⎣⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⎦
BlockDescription(psm01)
rows sample batch fixed
30 Diagonal
10 Sparse Diagonal
2 Dense Dense Dense

of psm01 to that of pnm01 Listing 2.1 shows that models with nested factors produce simple repeated structures along the diagonal of the sparse Cholesky factor, \({\mathbf{L}}\). This type of structure has the desirable property that there is no “fill-in” during calculation of the Cholesky factor. In other words, the number of non-zeros in \({\mathbf{L}}\) is the same as the number of non-zeros in the lower triangle of the matrix being factored, \({\boldsymbol{\Lambda}}'{\mathbf{Z}}'{\mathbf{Z}}{\boldsymbol{\Lambda}}+{\mathbf{I}}\) (which, because \({\boldsymbol{\Lambda}}\) is diagonal, has the same structure as \({\mathbf{Z}}'{\mathbf{Z}}\)).

Fill-in of the Cholesky factor is not an important issue when we have a few dozen random effects, as we do here. It is an important issue when we have millions of random effects in complex configurations, as has been the case in some of the models that have been fit using MixedModels.

2.2.4 Assessing parameter estimates in psm01

The parameter estimates are: \(\widehat{\sigma_1}\), the standard deviation of the random effects for sample; \(\widehat{\sigma_2}\), the standard deviation of the random effects for batch; \(\widehat{\sigma}\), the standard deviation of the residual noise term; and \(\widehat{\beta_1}\), the overall mean response, which is labeled (Intercept) in these models.

The estimated standard deviation for sample is nearly three times as large as that for batch, which confirms what we saw in Figure 2.6. Indeed our conclusion from Figure 2.6 was that there may not be a significant batch-to-batch variability in addition to the sample-to-sample variability.

Plots of the prediction intervals of the random effects (Figure 2.7)

Code
caterpillar!(
  Figure(resolution=(800, 300)),
  ranefinfo(psm01, :batch),
)
Figure 2.7: Plot of batch prediction intervals from psm01

confirm this impression in that all the prediction intervals for the random effects for contain zero.

Furthermore, kernel density estimates from a parametric bootstrap sample of the estimated standard deviations of the random effects and residuals

Random.seed!(4567654)
psm01samp = parametricbootstrap(10_000, psm01; progress)
psm01pars = DataFrame(psm01samp.allpars);
Code
draw(
  data(@subset(psm01pars, :type == "σ")) *
  mapping(
    :value => "Bootstrap samples of σ";
    color=(:group => "Group"),
  ) *
  AlgebraOfGraphics.density();
  figure=(; resolution=(800, 450)),
)
Figure 2.8: Kernel density plots of bootstrap estimates of σ for model psm01

Because there are several indications that \(\sigma_2\) could reasonably be zero, resulting in a simpler model incorporating random effects for only sample, we perform a statistical test of this hypothesis.

2.2.5 Testing \(H_0:\sigma_2=0\) versus \(H_a:\sigma_2>0\)

One of the many famous statements attributed to Albert Einstein is “Everything should be made as simple as possible, but not simpler.” In statistical modeling this principal of parsimony is embodied in hypothesis tests comparing two models, one of which contains the other as a special case. Typically, one or more of the parameters in the more general model, which we call the alternative hypothesis, is constrained in some way, resulting in the restricted model, which we call the null hypothesis. Although we phrase the hypothesis test in terms of the parameter restriction, it is important to realize that we are comparing the quality of fits obtained with two nested models. That is, we are not assessing parameter values per se; we are comparing the model fit obtainable with some constraints on parameter values to that without the constraints. Because the more general model, \(H_a\), must provide a fit that is at least as good as the restricted model, \(H_0\), our purpose is to determine whether the change in the quality of the fit is sufficient to justify the greater complexity of model \(H_a\). This comparison is often reduced to a p-value, which is the probability of seeing a difference in the model fits as large as we did, or even larger, when, in fact, \(H_0\) is adequate. Like all probabilities, a p-value must be between 0 and 1. When the p-value for a test is small (close to zero) we prefer the more complex model, saying that we “reject \(H_0\) in favor of \(H_a\)”. On the other hand, when the p-value is not small we “fail to reject \(H_0\)”, arguing that there is a non-negligible probability that the observed difference in the model fits could reasonably be the result of random chance, not the inherent superiority of the model \(H_a\). Under these circumstances we prefer the simpler model, \(H_0\), according to the principal of parsimony.

These are the general principles of statistical hypothesis tests. To perform a test in practice we must specify the criterion for comparing the model fits, the method for calculating the p-value from an observed value of the criterion, and the standard by which we will determine if the p-value is “small” or not. The criterion is called the test statistic, the p-value is calculated from a reference distribution for the test statistic, and the standard for small p-values is called the level of the test.

In Section 1.3.2 we referred to likelihood ratio tests (LRTs) for which the test statistic is the difference in the deviance. That is, the LRT statistic is \(d_0-d_a\) where \(d_a\) is the deviance in the more general (\(H_a\)) model fit and \(d_0\) is the deviance in the constrained (\(H_0\)) model. An approximate reference distribution for an LRT statistic is the \(\chi^2_\nu\) distribution where \(\nu\), the degrees of freedom, is determined by the number of constraints imposed on the parameters of \(H_a\) to produce \(H_0\).

The restricted model fit

psm02 = let f = @formula strength ~ 1 + (1 | sample)
  fit(MixedModel, f, pastes; contrasts, progress)
end
Est. SE z p σ_sample
(Intercept) 60.0533 0.5765 104.16 <1e-99 3.1037
Residual 0.8234

is compared to model psm01 as

MixedModels.likelihoodratiotest(psm02, psm01)
model-dof deviance χ² χ²-dof P(>χ²)
strength ~ 1 + (1 | sample) 3 248
strength ~ 1 + (1 | sample) + (1 | batch) 4 248 0 1 0.5234

which provides a p-value of 52%. Because typical standards for “small” p-values are 5% or 1%, a p-value over 50% would not be considered significant at any reasonable level.

We do need to be cautious in quoting this p-value, however, because the parameter value being tested, \(\sigma_2=0\), is on the boundary of set of possible values, \(\sigma_2\ge 0\), for this parameter. The argument for using a \(\chi^2_1\) distribution to calculate a p-value for the change in the deviance does not apply when the parameter value being tested is on the boundary. As shown in Pinheiro & Bates (2000, Sect. 2.5), the p-value from the \(\chi^2_1\) distribution will be “conservative” in the sense that it is larger than a simulation-based p-value would be. In the worst-case scenario the \(\chi^2\)-based p-value will be twice as large as it should be but, even if that were true, an effective p-value of 26% would not cause us to reject \(H_0\) in favor of \(H_a\).

2.2.6 Assessing the reduced model, psm02

Comparing the coverage intervals for models psm01 and psm02

Code
DataFrame(shortestcovint(psm01samp))
4×5 DataFrame
Row type group names lower upper
String String? String? Float64 Float64
1 β missing (Intercept) 58.7732 61.3025
2 σ sample (Intercept) 1.93501 3.61949
3 σ batch (Intercept) 0.0 2.0894
4 σ residual missing 0.608905 1.02399
Code
psm02samp = parametricbootstrap(
  Random.seed!(9753579),
  10_000,
  psm02;
  progress,
)
DataFrame(shortestcovint(psm02samp))
3×5 DataFrame
Row type group names lower upper
String String? String? Float64 Float64
1 β missing (Intercept) 58.9201 61.1886
2 σ sample (Intercept) 2.24908 3.83207
3 σ residual missing 0.61145 1.02324

The confidence intervals on \(\sigma\) and \(\beta_0\) are similar for the two models. The confidence interval on \(\sigma_1\) is slightly wider and incorporates larger values in model psm02 than in model psm01, because the variability that is attributed to batch in psm01 is incorporated into the variability due to sample in psm02.

2.3 A model with partially crossed random effects

Especially in observational studies with multiple grouping factors, the configuration of the factors frequently ends up neither nested nor completely crossed. We describe such situations as having partially crossed grouping factors for the random effects.

Studies in education, in which test scores for students over time are also associated with teachers and schools, usually result in partially crossed grouping factors. If students with scores in multiple years have different teachers for the different years, the student factor cannot be nested within the teacher factor. Conversely, student and teacher factors are not expected to be completely crossed. To have complete crossing of the student and teacher factors it would be necessary for each student to be observed with each teacher, which would be unusual. A longitudinal study of thousands of students with hundreds of different teachers inevitably ends up partially crossed.

In this section we consider an example with thousands of students and instructors where the response is the student’s evaluation of the instructor’s effectiveness. These data, like those from most large observational studies, are quite unbalanced.

2.3.1 The insteval data

The data are from a special evaluation of lecturers by students at the Swiss Federal Institute for Technology–Zürich (ETH–Zürich), to determine who should receive the “best-liked professor” award. These data have been slightly simplified and identifying labels have been removed, so as to preserve anonymity.

The variables

insteval = dataset(:insteval)
Arrow.Table with 73421 rows, 7 columns, and schema:
 :s        String
 :d        String
 :dept     String
 :studage  String
 :lectage  String
 :service  String
 :y        Int8
describe(DataFrame(insteval))
7×7 DataFrame
Row variable mean min median max nmissing eltype
Symbol Union… Any Union… Any Int64 DataType
1 s S0001 S2972 0 String
2 d I0001 I2160 0 String
3 dept D01 D15 0 String
4 studage 2 8 0 String
5 lectage 1 6 0 String
6 service N Y 0 String
7 y 3.20574 1 3.0 5 0 Int8

have somewhat cryptic names. Factor s designates the student and d the instructor. The factor dept is the department for the course and service indicates whether the course was a service course taught to students from other departments.

Although the response, y, is on a scale of 1 to 5,

Code
draw(
  data(Table(response=1:5, count=counts(insteval.y))) *
  mapping(:response => "Rating", :count => "Count") *
  visual(BarPlot);
  figure=(; resolution=(800, 450)),
)
Figure 2.9: Histogram of instructor ratings in the insteval data

it is sufficiently diffuse to warrant treating it as if it were a continuous response.

At this point we will fit models that have random effects for student, instructor, and department (or the combination of department and service) to these data.

iem01 = let f = @formula y ~ 1 + (1 | s) + (1 | d) + (1 | dept)
  fit(MixedModel, f, insteval; contrasts, progress)
end
Est. SE z p σ_s σ_d σ_dept
(Intercept) 3.2519 0.0279 116.69 <1e-99 0.3264 0.5173 0.0773
Residual 1.1777

All three estimated standard deviations of the random effects are less than \(\widehat{\sigma}\), with \(\widehat{\sigma}_3\), the estimated standard deviation of the random effects for the dept, less than one-tenth the estimated residual standard deviation.

It is not surprising that zero is within most of the prediction intervals on the random effects for this factor (Figure 2.10).

Code
caterpillar!(Figure(resolution=(800, 400)), ranefinfo(iem01, :dept))
Figure 2.10: Prediction intervals on random effects for department in model iem01

However, the p-value for the LRT of \(H_0:\sigma_3=0\) versus \(H_a:\sigma_3>0\)

iem02 = let f = @formula y ~ 1 + (1 | s) + (1 | d)
  fit(MixedModel, f, insteval; contrasts, progress)
end
MixedModels.likelihoodratiotest(iem02, iem01)
model-dof deviance χ² χ²-dof P(>χ²)
y ~ 1 + (1 | s) + (1 | d) 4 237778
y ~ 1 + (1 | s) + (1 | d) + (1 | dept) 5 237770 8 1 0.0043

is highly significant. That is, we have very strong evidence that we should reject \(H_0\) in favor of \(H_a\).

The seeming inconsistency of these conclusions is due to the large sample size (\(n=73421\)). When a model is fit to a large sample even the most subtle of differences can be highly “statistically significant”. The researcher or data analyst must then decide if these terms have practical significance, beyond the apparent statistical significance.

The large sample size also helps to assure that the parameters have good normal approximations. We could profile this model fit but doing so would take a very long time and, in this particular case, the analysts are more interested in a model that uses fixed-effects parameters for the instructors.

2.3.2 Structure of L for model iem01

Before leaving this model we examine the sparse Cholesky factor, \({\mathbf{L}}\), which is of size \(4116\times4116\).

L = sparseL(iem01; full=true)
4116×4116 SparseArrays.SparseMatrixCSC{Float64, Int32} with 741328 stored entries:
⎡⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎤
⎢⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⎥
⎢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⎥
⎢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⎥
⎢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⎥
⎣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⎦

Even as a sparse matrix L requires a considerable amount of memory, about 9 MB,

(Base.summarysize(L), Base.summarysize(collect(L)))
(8912564, 135531688)

but as a triangular dense matrix it requires over 10 times as much (about 135 MB). There are \(4116^2\) elements in the triangular matrix, each of which requires 8 bytes of storage.

2.3.3 Effect of service courses

It is sometimes felt that it is more difficult to achieve favorable ratings from students in a service course (i.e. a course taught to students majoring in another program) as opposed to students taking a course in their major.

There are several ways in which service can be incorporated in a model like this. The simplest approach is to add service to the fixed-effects specification

iem03 =
  let f = @formula y ~ 1 + service + (1 | s) + (1 | d) + (1 | dept)
    fit(MixedModel, f, insteval; contrasts, progress)
  end
Est. SE z p σ_s σ_d σ_dept
(Intercept) 3.2826 0.0284 115.54 <1e-99 0.3255 0.5150 0.0785
service: Y -0.0926 0.0134 -6.92 <1e-11
Residual 1.1775

In model iem03 the effect of service is considered to be constant across departments and is modeled with a single fixed-effects parameter, which is the difference in a typical rating in a service course to a non-service course. This parameter also affects the interpretation of the (Intercept) coefficient. With the service term in the model the (Intercept) becomes a typical rating at the reference level (i.e. non-service or service: N) because the default coding for the service term is zero at the reference level and one for service: Y.

The coding can be changed by specifying a non-default contrast for service. For example, the EffectsCoding and HelmerCoding contrasts will both assign -1 to the first level (N in this case) and +1 to the second level (Y).

2.3.4 “Best-liked”

A qqcaterpillar plot of the instructor (i.e. factor d in the data) random effects, Figure 2.11,

Code
qqcaterpillar!(
  Figure(; resolution=(800, 650)),
  ranefinfo(iem01, :d),
)
Figure 2.11: Caterpillar plot of the instructor BLUPS for model iem01 versus standard normal quantiles

shows the differences in precision due to different numbers of observations for different instructors.

Code
draw(
  data(combine(groupby(DataFrame(insteval), :d), nrow => :n)) *
  mapping(:n => "Number of observations") *
  AlgebraOfGraphics.histogram(; bins=410);
  figure=(; resolution=(800, 600)),
)
Figure 2.12: Histogram of the number of observations per instructor in the insteval data

The precision of the conditional distributions of the random effects, as measured by the width of the intervals, varies considerably between instructors.

We can determine that instructor I1258 has the largest mean of the conditional distributions of the random effects

last(
  sort(DataFrame(ranefinfotable(ranefinfo(iem01, :d))), :cmode),
  5,
)
5×4 DataFrame
Row name level cmode cstddev
String String Float64 Float64
1 (Intercept) I0066 1.03151 0.105473
2 (Intercept) I0193 1.04027 0.190826
3 (Intercept) I0844 1.05502 0.169914
4 (Intercept) I1866 1.06625 0.123322
5 (Intercept) I1258 1.17205 0.188496

but the conditional distribution of this random effect clearly overlaps significantly with others.

2.4 Chapter summary

A simple, scalar random effects term in an model formula is of the form (1|f), where f is an expression whose value is the grouping factor for the set of random effects generated by this term. Typically, f is simply the name of a factor, as in most of the examples in this chapter. However, the grouping factor can be the value of an expression.

A model formula can include several such random effects terms. Because configurations such as nested or crossed or partially crossed grouping factors are a property of the data, the specification in the model formula does not depend on the configuration. We simply include multiple random effects terms in the formula specifying the model.

One apparent exception to this rule occurs with implicitly nested factors, in which the levels of one factor are only meaningful within a particular level of the other factor. In the pastes data, levels of the factor cask are only meaningful within a particular level of the factor batch. A model formula of strength ~ 1 + (1|batch) + (1|cask) would result in a fitted model that did not appropriately reflect the sources of variability in the data. Following the simple rule that the factor should be defined so that distinct experimental or observational units correspond to distinct levels of the factor will avoid such ambiguity.

For convenience, multiple, nested random-effects terms can be specified using a slash to separate grouping factors, e.g. (1 | batch / cask), which internally is re-expressed as (1 | batch & cask) + (1 | batch)

print(
  let f = @formula(strength ~ 1 + (1 | batch / cask))
    fit(MixedModel, f, pastes; progress)
  end
)
Linear mixed model fit by maximum likelihood
 strength ~ 1 + (1 | batch) + (1 | batch & cask)
   logLik   -2 logLik     AIC       AICc        BIC    
  -123.9972   247.9945   255.9945   256.7217   264.3718

Variance components:
                Column   Variance Std.Dev. 
batch & cask (Intercept)  8.433616 2.904069
batch        (Intercept)  1.199180 1.095071
Residual                  0.678002 0.823409
 Number of obs: 60; levels of grouping factors: 30, 10

  Fixed-effects parameters:
─────────────────────────────────────────────────
               Coef.  Std. Error      z  Pr(>|z|)
─────────────────────────────────────────────────
(Intercept)  60.0533    0.642136  93.52    <1e-99
─────────────────────────────────────────────────

We will avoid terms of this form, preferring instead an explicit specification with simple, scalar terms based on unambiguous grouping factors.

The insteval data, described in Section 2.3.1, illustrate some of the characteristics of the real data to which mixed-effects models are now fit. There is a large number of observations associated with several grouping factors; two of which, student and instructor, have a large number of levels and are partially crossed. Such data are common in sociological and educational studies but until recently it has been very difficult to fit models that appropriately reflect such a structure. Much of the literature on mixed-effects models leaves the impression that multiple random effects terms can only be associated with nested grouping factors. The resulting emphasis on hierarchical or multilevel configurations is an artifact of the computational methods used to fit the models, not the models themselves.

The parameters of the models fit to small data sets have properties similar to those for the models in the previous chapter. That is, profile-based confidence intervals on the fixed-effects parameter, \(\beta_0\), are symmetric about the estimate but overdispersed relative to those that would be calculated from a normal distribution.

Another observation from the last example is that, for data sets with a very large numbers of observations, a term in a model may be “statistically significant” even when its practical significance is questionable.

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