Prediction and simulation in Mixed-Effects Models

We recommend the MixedModelsSim.jl package and associated documentation for useful tools in constructing designs to simulate. For now, we'll use the sleep study data as a starting point.

using DataFrames
using MixedModels
using MixedModelsDatasets
using StatsBase
# use a DataFrame to make it easier to change things later
slp = DataFrame(MixedModelsDatasets.dataset(:sleepstudy))
slpm = fit(MixedModel, @formula(reaction ~ 1 + days + (1|subj)), slp)

Linear mixed model fit by maximum likelihood
 reaction ~ 1 + days + (1 | subj)
   logLik   -2 logLik     AIC       AICc        BIC    
  -897.0393  1794.0786  1802.0786  1802.3072  1814.8505

Variance components:
            Column    Variance Std.Dev.
subj     (Intercept)  1296.8699 36.0121
Residual               954.5278 30.8954
 Number of obs: 180; levels of grouping factors: 18

  Fixed-effects parameters:
──────────────────────────────────────────────────
                Coef.  Std. Error      z  Pr(>|z|)
──────────────────────────────────────────────────
(Intercept)  251.405     9.50618   26.45    <1e-99
days          10.4673    0.801735  13.06    <1e-38
──────────────────────────────────────────────────

Prediction

The simplest form of prediction are the fitted values from the model: they are indeed the model's predictions for the observed data.

predict(slpm) ≈ fitted(slpm)

true

When generalizing to new data, we need to consider what happens if there are new, previously unobserved levels of the grouping variable(s). MixedModels.jl provides three options:

1. :error: error on encountering unobserved levels
2. :population: use population values (i.e. only the fixed effects) for observations with unobserved levels
3. :missing: return missing for observations with unobserved levels.

Providing either no prediction (:error, :missing) or providing the population-level values seem to be the most reasonable ways for predicting new values. For simulating new values based on previous estimates of the variance components, use simulate.

In the case where there are no new levels of the grouping variable, all three of these methods provide the same results:

predict(slpm, slp; new_re_levels=:population) ≈ fitted(slpm)

true

predict(slpm, slp; new_re_levels=:missing) ≈ fitted(slpm)

true

predict(slpm, slp; new_re_levels=:error) ≈ fitted(slpm)

true

In the case where there are new levels of the grouping variable, these methods differ.

# create a new level
slp2 = transform(slp, :subj => ByRow(x -> (x == "S308" ? "NEW" : x)) => :subj)

180×3 DataFrame
 Row │ subj    days  reaction 
     │ String  Int8  Float32  
─────┼────────────────────────
   1 │ NEW        0   249.56
   2 │ NEW        1   258.705
   3 │ NEW        2   250.801
   4 │ NEW        3   321.44
   5 │ NEW        4   356.852
   6 │ NEW        5   414.69
   7 │ NEW        6   382.204
   8 │ NEW        7   290.149
  ⋮  │   ⋮      ⋮       ⋮
 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
              165 rows omitted

try
  predict(slpm, slp2; new_re_levels=:error)
catch e
  show(e)
end

ArgumentError("New level encountered in subj")

predict(slpm, slp2; new_re_levels=:missing)

180-element Vector{Union{Missing, Float64}}:
    missing
    missing
    missing
    missing
    missing
    missing
    missing
    missing
    missing
    missing
   ⋮
 279.9221289683628
 290.38941447397224
 300.8566999795817
 311.3239854851912
 321.7912709908007
 332.2585564964101
 342.7258420020196
 353.1931275076291
 363.66041301323855

predict(slpm, slp2; new_re_levels=:population)

180-element Vector{Float64}:
 251.405106053206
 261.8723915588155
 272.3396770644249
 282.8069625700344
 293.27424807564387
 303.74153358125335
 314.2088190868628
 324.67610459247226
 335.14339009808174
 345.6106756036912
   ⋮
 279.9221289683628
 290.38941447397224
 300.8566999795817
 311.3239854851912
 321.7912709908007
 332.2585564964101
 342.7258420020196
 353.1931275076291
 363.66041301323855

For generalized linear mixed models, there is an additional keyword argument to predict: type specifies whether the predictions are returned on the scale of the linear predictor (:linpred) or on the level of the response (:response) (i.e. the level at which the values were originally observed).

cbpp = DataFrame(MixedModelsDatasets.dataset(:cbpp))
cbpp.rate = cbpp.incid ./ cbpp.hsz
gm = fit(MixedModel, @formula(rate ~ 1 + period + (1|herd)), cbpp, Binomial(), wts=float(cbpp.hsz))
predict(gm, cbpp; type=:response) ≈ fitted(gm)

false

logit(x) = log(x / (1 - x))
predict(gm, cbpp; type=:linpred) ≈ logit.(fitted(gm))

false

Simulation

In contrast to predict, simulate and simulate! introduce randomness. This randomness occurs both at the level of the observation-level (residual) variance and at the level of the random effects, where new conditional modes are sampled based on the specified covariance parameter (θ; see Details of the parameter estimation), which defaults to the estimated value of the model. For reproducibility, we specify a pseudorandom generator here; if none is provided, the global PRNG is taken as the default.

The simplest example of simulate takes a fitted model and generates a new response vector based on the existing model matrices combined with noise.

using Random
ynew = simulate(MersenneTwister(42), slpm)

180-element Vector{Float64}:
 283.008632396066
 296.925192842634
 321.960870842494
 322.3364895436127
 396.98425559973197
 317.151522042183
 348.5132549535918
 378.28244242379003
 302.3001498924851
 425.39058798238773
   ⋮
 314.68558961527066
 350.0319301897881
 293.6498136250591
 375.40194576957305
 324.5823662875929
 311.1027735369789
 338.7622249548992
 345.3555959093326
 301.28718461382755

The simulated response can also be placed in a pre-allocated vector:

ynew2 = zeros(nrow(slp))
simulate!(MersenneTwister(42), ynew2, slpm)
ynew2 ≈ ynew

true

Or even directly replace the previous response vector in a model, at which point the model must be refit to the new values:

slpm2 = deepcopy(slpm)
refit!(simulate!(MersenneTwister(42), slpm2))

Linear mixed model fit by maximum likelihood
 reaction ~ 1 + days + (1 | subj)
   logLik   -2 logLik     AIC       AICc        BIC    
  -903.1987  1806.3974  1814.3974  1814.6259  1827.1692

Variance components:
            Column    Variance Std.Dev.
subj     (Intercept)  2420.4298 49.1979
Residual               964.3708 31.0543
 Number of obs: 180; levels of grouping factors: 18

  Fixed-effects parameters:
───────────────────────────────────────────────────
                 Coef.  Std. Error      z  Pr(>|z|)
───────────────────────────────────────────────────
(Intercept)  263.59      12.3684    21.31    <1e-99
days           9.35638    0.805859  11.61    <1e-30
───────────────────────────────────────────────────

This inplace simulation actually forms the basis of parametricbootstrap.

Finally, we can also simulate the response from entirely new data.

df = DataFrame(days = repeat(1:10, outer=20), subj=repeat(1:20, inner=10))
df[!, :subj] = string.("S", lpad.(df.subj, 2, "0"))
df[!, :reaction] .= 0
df

200×3 DataFrame
 Row │ days   subj    reaction 
     │ Int64  String  Int64    
─────┼─────────────────────────
   1 │     1  S01            0
   2 │     2  S01            0
   3 │     3  S01            0
   4 │     4  S01            0
   5 │     5  S01            0
   6 │     6  S01            0
   7 │     7  S01            0
   8 │     8  S01            0
  ⋮  │   ⋮      ⋮        ⋮
 194 │     4  S20            0
 195 │     5  S20            0
 196 │     6  S20            0
 197 │     7  S20            0
 198 │     8  S20            0
 199 │     9  S20            0
 200 │    10  S20            0
               185 rows omitted

ysim = simulate(MersenneTwister(42), slpm, df)

200-element Vector{Float64}:
 262.9357776683969
 276.8523381149648
 301.8880161148249
 302.2636348159436
 376.91140087206287
 297.07866731451384
 328.4404002259227
 358.2095876961209
 282.2272951648159
 405.31773325471863
   ⋮
 336.4096599492376
 274.0198071475012
 232.79273085223534
 281.7738581179209
 296.84052598540785
 379.1660142485997
 305.0373816630906
 358.38445220603853
 381.7236757626364

Note that this is a convenience method for creating a new model and then using the parameters from the old model to call simulate on that model. In other words, this method incurs the cost of constructing a new model and then discarding it. If you could re-use that model (e.g., fitting that model as part of a simulation study), it often makes sense to do these steps to perform these steps explicitly and avoid the unnecessary construction and discarding of an intermediate model:

msim = LinearMixedModel(@formula(reaction ~ 1 + days + (1|subj)), df)
simulate!(MersenneTwister(42), msim; θ=slpm.θ, β=slpm.β, σ=slpm.σ)
response(msim) ≈ ysim

true

fit!(msim)

Linear mixed model fit by maximum likelihood
 reaction ~ 1 + days + (1 | subj)
   logLik   -2 logLik     AIC       AICc        BIC    
  -996.0696  1992.1392  2000.1392  2000.3443  2013.3325

Variance components:
            Column    Variance Std.Dev.
subj     (Intercept)   663.5398 25.7593
Residual              1012.9883 31.8275
 Number of obs: 200; levels of grouping factors: 20

  Fixed-effects parameters:
───────────────────────────────────────────────────
                 Coef.  Std. Error      z  Pr(>|z|)
───────────────────────────────────────────────────
(Intercept)  259.607      7.53747   34.44    <1e-99
days           9.46755    0.783538  12.08    <1e-32
───────────────────────────────────────────────────

For simulating from generalized linear mixed models, there is no type option because the observation-level always occurs at the level of the response and not of the linear predictor.