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:
:error
: error on encountering unobserved levels:population
: use population values (i.e. only the fixed effects) for observations with unobserved levels:missing
: returnmissing
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
⋮
279.92212396847816
290.38940992807414
300.8566958876701
311.32398184726617
321.79126780686215
332.25855376645814
342.7258397260541
353.1931256856501
363.66041164524614
predict(slpm, slp2; new_re_levels=:population)
180-element Vector{Float64}:
251.40510484848508
261.8723908080811
272.3396767676771
282.80696272727306
293.2742486868691
303.7415346464651
314.20882060606107
324.67610656565705
335.14339252525303
345.6106784848491
⋮
279.92212396847816
290.38940992807414
300.8566958876701
311.32398184726617
321.79126780686215
332.25855376645814
342.7258397260541
353.1931256856501
363.66041164524614
Note
Currently, we do not support predicting based on a subset of the random effects.
Note
predict
is deterministic (within the constraints of floating point) and never adds noise to the result. If you want to construct prediction intervals, then simulate
will generate new data with noise (including new values of the random effects).
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}:
252.6752171388051
223.3096173669927
248.6842723713382
255.64305141427212
255.08087250538287
286.58862952243004
309.5056915487952
286.88987784213936
281.3989535401713
282.98845110151944
⋮
250.81655745160242
271.2543649019418
248.48273283810676
305.9014182133385
260.66238230893254
298.11865200662567
360.1816309547196
353.9285987403432
382.89445970316854
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
-895.6915 1791.3830 1799.3830 1799.6116 1812.1549
Variance components:
Column Variance Std.Dev.
subj (Intercept) 912.8712 30.2138
Residual 972.8494 31.1905
Number of obs: 180; levels of grouping factors: 18
Fixed-effects parameters:
───────────────────────────────────────────────────
Coef. Std. Error z Pr(>|z|)
───────────────────────────────────────────────────
(Intercept) 248.336 8.32982 29.81 <1e-99
days 8.00599 0.809393 9.89 <1e-22
───────────────────────────────────────────────────
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}:
255.00432150851708
225.63872173670472
251.0133767410502
257.9721557839841
257.40997687509486
288.91773389214205
311.83479591850715
289.21898221185137
283.72805790988326
285.31755547123146
⋮
197.4028748844943
138.22165310892973
187.19029860225433
215.87081467989628
220.4557805243008
267.6032837398289
245.3979923049997
304.0437000517243
253.65615200259157
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
-996.8296 1993.6592 2001.6592 2001.8643 2014.8524
Variance components:
Column Variance Std.Dev.
subj (Intercept) 1292.4331 35.9504
Residual 956.1199 30.9212
Number of obs: 200; levels of grouping factors: 20
Fixed-effects parameters:
───────────────────────────────────────────────────
Coef. Std. Error z Pr(>|z|)
───────────────────────────────────────────────────
(Intercept) 250.284 9.32369 26.84 <1e-99
days 8.08794 0.761227 10.62 <1e-25
───────────────────────────────────────────────────
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.
Warning
Simulating the model response in place may not yield the same result as simulating into a pre-allocated or new vector, depending on choice of pseudorandom number generator. Random number generation in Julia allows optimization based on type, and the internal storage type of the model response (currently a view into a matrix storing the concatenated fixed-effects model matrix and the response) may not match the type of a pre-allocated or new vector. See also discussion here.
Note
All the methods that take new data as a table construct an additional MixedModel
behind the scenes, even when the new data is exactly the same as the data that the model was fitted to. For the simulation methods in particular, these thus form a convenience wrapper for constructing a new model and calling simulate
without new data on that model with the parameters from the original model.