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 StatsBase
# use a DataFrame to make it easier to change things later
slp = DataFrame(MixedModels.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)  1296.8692 36.0121
Residual               954.5279 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  Float64  
─────┼────────────────────────
   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
   ⋮
 279.9221239684782
 290.3894099280742
 300.8566958876702
 311.32398184726617
 321.7912678068622
 332.2585537664582
 342.7258397260542
 353.19312568565016
 363.66041164524614

predict(slpm, slp2; new_re_levels=:population)

180-element Vector{Float64}:
 251.40510484848508
 261.8723908080811
 272.3396767676771
 282.80696272727306
 293.27424868686904
 303.7415346464651
 314.20882060606107
 324.67610656565705
 335.14339252525303
 345.610678484849
   ⋮
 279.9221239684782
 290.3894099280742
 300.8566958876702
 311.32398184726617
 321.7912678068622
 332.2585537664582
 342.7258397260542
 353.19312568565016
 363.66041164524614

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(MixedModels.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}:
 283.00861574005904
 296.92517678007596
 321.9608558229577
 322.3364745700326
 396.9842436751096
 317.15150692050014
 348.5132411307083
 378.2824298353154
 302.3001342626326
 425.3905773601467
   ⋮
 314.6855844434403
 350.0319264778643
 293.6498076642398
 375.40194314495807
 324.582361638989
 311.1027683741317
 338.76222094115917
 345.35559219294737
 301.28717914641953

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)  2420.4286 49.1979
Residual               964.3709 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}:
 262.9357707048248
 276.8523317448417
 301.8880107877235
 302.2636295347983
 376.91139863987536
 297.0786618852659
 328.44039609547406
 358.20958480008113
 282.2272892273984
 405.31773232491247
   ⋮
 336.4096634873374
 274.01980819379816
 232.79273026239147
 281.7738595392708
 296.84052804670836
 379.1660196692867
 305.03738411734776
 358.384456848015
 381.72368137904283

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)   663.5397 25.7593
Residual              1012.9884 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.