compare
Class: LinearMixedModel
Compare linear mixedeffects models
Syntax
Description
returns the results of a likelihood ratio
test that compares the linear mixedeffects models
results
= compare(lme
,altlme
)lme
and altlme
. Both models must use
the same response vector in the fit and lme
must be nested in
altlme
for a valid theoretical likelihood ratio test.
Always input the smaller model first, and the larger model second.
compare
tests the following null and alternate
hypotheses:
H_{0}: Observed response vector is
generated by lme
.
H_{1}: Observed response vector is
generated by model altlme
.
It is recommended that you fit lme
and
altlme
using the maximum likelihood (ML) method prior to
model comparison. If you use the restricted maximum likelihood (REML) method,
then both models must have the same fixedeffects design matrix.
To test for fixed effects, use compare
with the simulated likelihood ratio
test when lme
and altlme
are
fit using ML or use the fixedEffects
,
anova
, or coefTest
methods.
also returns the results of a likelihood ratio test that compares linear
mixedeffects models results
= compare(___,Name,Value
)lme
and altlme
with
additional options specified by one or more Name,Value
pair
arguments.
For example, you can check if the first input model is nested in the second input model.
[
returns the results of a simulated likelihood ratio test that compares linear
mixedeffects models results
,siminfo
]
= compare(lme
,altlme
,'NSim',nsim
)lme
and
altlme
.
You can fit lme
and altlme
using ML or
REML. Also, lme
does not have to be nested in
altlme
. If you use the restricted maximum likelihood
(REML) method to fit the models, then both models must have the same
fixedeffects design matrix.
[
also returns the results of a simulated likelihood ratio test that compares
linear mixedeffects models results
,siminfo
]
= compare(___,Name,Value
)lme
and altlme
with additional options specified by one or more Name,Value
pair arguments.
For example, you can change the options for performing the simulated likelihood ratio test, or change the confidence level of the confidence interval for the pvalue.
Input Arguments
lme
— Linear mixedeffects model
LinearMixedModel
object
Linear mixedeffects model, specified as a LinearMixedModel
object constructed using fitlme
or fitlmematrix
.
altlme
— Alternative linear mixedeffects model
LinearMixedModel
object
Alternative linear mixedeffects model fit to the same response vector but
with different model specifications, specified as a
LinearMixedModel
object. lme
must be nested in altlme
, that is, lme
should be obtained from altlme
by setting some parameters
to fixed values, such as 0. You can create a linear mixedeffects object
using fitlme
or fitlmematrix
.
nsim
— Number of replications for simulations
positive integer number
Number of replications for simulations in the simulated likelihood ratio
test, specified as a positive integer number. You must specify
nsim
to do a simulated likelihood ratio test.
Example: 'NSim',1000
Data Types: double
 single
NameValue Arguments
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
Alpha
— Significance level
0.05 (default)  scalar value in the range 0 to 1
Significance level, specified as the commaseparated pair consisting of
'Alpha'
and a scalar value in the range 0 to 1. For a value α,
the confidence level is 100*(1–α)%.
For example, for 99% confidence intervals, you can specify the confidence level as follows.
Example: 'Alpha',0.01
Data Types: single
 double
Options
— Options for performing simulated likelihood ratio test
structure
Options for performing the simulated likelihood ratio test in
parallel, specified as the commaseparated pair consisting of
'Options'
, and a structure created by
statset('LinearMixedModel')
.
These options require Parallel Computing Toolbox™.
compare
uses the following fields.
'UseParallel' 
You need Parallel Computing Toolbox for parallel computation. 
'UseSubstreams' 

'Streams' 

For information on parallel statistical computing at the command line, enter
help parallelstats
Data Types: struct
CheckNesting
— Indicator to check nesting between two models
false
(default)  true
Indicator to check
nesting between two models, specified as the commaseparated
pair consisting of 'CheckNesting'
and one of the
following.
false  Default. No checks. 
true  compare checks if the smaller
model lme is nested in the bigger
model altlme . 
lme
must be nested in the alternate model
altlme
for a valid theoretical likelihood ratio
test. compare
returns an error message if
the nesting requirements are not satisfied.
Although valid for both tests, the nesting requirements are weaker for the simulated likelihood ratio test.
Example: 'CheckNesting',true
Data Types: single
 double
Output Arguments
results
— Results of likelihood ratio test or simulated likelihood ratio test
dataset array
Results of the likelihood ratio test or simulated likelihood ratio test,
returned as a dataset array with two rows. The first row is for
lme
, and the second row is for
altlme
. The columns of results
depend on whether the test is a likelihood ratio or a simulated likelihood
ratio test.
If you use the likelihood ratio test, then
results
contains the following columns.Model
Name of the model DF
Degrees of freedom, that is, the number of free parameters in the model AIC
Akaike information criterion for the model BIC
Bayesian information criterion for the model LogLik
Maximized log likelihood for the model LRStat
Likelihood ratio test statistic for comparing altlme
versuslme
deltaDF
DF
foraltlme
minusDF
forlme
pValue
pvalue for the likelihood ratio test If you use the simulated likelihood ratio test, then
results
contains the following columns.Model
Name of the model DF
Degrees of freedom, that is, the number of free parameters in the model LogLik
Maximized log likelihood for the model LRStat
Likelihood ratio test statistic for comparing altlme
versuslme
pValue
pvalue for the likelihood ratio test Lower
Lower limit of the confidence interval for pValue
Upper
Upper limit of the confidence interval for pValue
siminfo
— Simulation output
structure
Simulation output, returned as a structure with the following fields.
nsim  Value set for nsim . 
alpha  Value set for 'Alpha' . 
pValueSim  Simulationbased pvalue. 
pValueSimCI  Confidence interval for pValueSim . The
first element of the vector is the lower limit and the
second element of the vector contains the upper
limit. 
deltaDF  The number of free parameters in
altlme minus the number of free
parameters in lme . DF
for altlme minus DF
for lme . 
THO  A vector of simulated likelihood ratio test statistics
under the null hypothesis that the model
lme generated the observed response
vector y . 
Examples
Test for Random Effects
Load the sample data.
load flu
The flu
dataset array has a Date
variable, and 10 variables containing estimated influenza rates (in 9 different regions, estimated from Google® searches, plus a nationwide estimate from the CDC).
To fit a linearmixed effects model, your data must be in a properly formatted dataset array. To fit a linear mixedeffects model with the influenza rates as the responses and region as the predictor variable, combine the nine columns corresponding to the regions into an array. The new dataset array, flu2
, must have the response variable, FluRate
, the nominal variable, Region
, that shows which region each estimate is from, and the grouping variable Date
.
flu2 = stack(flu,2:10,'NewDataVarName','FluRate',... 'IndVarName','Region'); flu2.Date = nominal(flu2.Date);
Fit a linear mixedeffects model, with a varying intercept and varying slope for each region, grouped by Date
.
altlme = fitlme(flu2,'FluRate ~ 1 + Region + (1 + RegionDate)');
Fit a linear mixedeffects model with fixed effects for the region and a random intercept that varies by Date
.
lme = fitlme(flu2,'FluRate ~ 1 + Region + (1Date)');
Compare the two models. Also check if lme2
is nested in lme
.
compare(lme,altlme,'CheckNesting',true)
ans = Theoretical Likelihood Ratio Test Model DF AIC BIC LogLik LRStat deltaDF pValue lme 11 318.71 364.35 148.36 altlme 55 305.51 77.346 207.76 712.22 44 0
The small $$p$$value of 0 indicates that model altlme
is significantly better than the simpler model lme
.
Test for Fixed and Random Effects
Load the sample data.
load('fertilizer.mat');
The dataset array includes data from a splitplot experiment, where soil is divided into three blocks based on the soil type: sandy, silty, and loamy. Each block is divided into five plots, where five different types of tomato plants (cherry, heirloom, grape, vine, and plum) are randomly assigned to these plots. The tomato plants in the plots are then divided into subplots, where each subplot is treated by one of four fertilizers. This is simulated data.
Store the data in a dataset array called ds
, for practical purposes, and define Tomato
, Soil
, and Fertilizer
as categorical variables.
ds = fertilizer; ds.Tomato = nominal(ds.Tomato); ds.Soil = nominal(ds.Soil); ds.Fertilizer = nominal(ds.Fertilizer);
Fit a linear mixedeffects model, where Fertilizer
and Tomato
are the fixedeffects variables, and the mean yield varies by the block (soil type) and the plots within blocks (tomato types within soil types) independently.
lmeBig = fitlme(ds,'Yield ~ Fertilizer * Tomato + (1Soil) + (1Soil:Tomato)');
Refit the model after removing the interaction term Tomato:Fertilizer
and the randomeffects term (1  Soil)
.
lmeSmall = fitlme(ds,'Yield ~ Fertilizer + Tomato + (1Soil:Tomato)');
Compare the two models using the simulated likelihood ratio test with 1000 replications. You must use this test to test for both fixed and randomeffect terms. Note that both models are fit using the default fitting method, ML. That’s why, there is no restriction on the fixedeffects design matrices. If you use restricted maximum likelihood (REML) method, both models must have identical fixedeffects design matrices.
[table,siminfo] = compare(lmeSmall,lmeBig,'nsim',1000)
table = Simulated Likelihood Ratio Test: Nsim = 1000, Alpha = 0.05 Model DF AIC BIC LogLik LRStat pValue lmeSmall 10 511.06 532 245.53 lmeBig 23 522.57 570.74 238.29 14.491 0.57343 Lower Upper 0.54211 0.60431
siminfo = struct with fields:
nsim: 1000
alpha: 0.0500
pvalueSim: 0.5734
pvalueSimCI: [0.5421 0.6043]
deltaDF: 13
TH0: [1000x1 double]
The high $$p$$value suggests that the larger model, lme
is not significantly better than the smaller model, lme2
. The smaller values of Akaike and Bayesian Information Criteria for lme2
also support this.
Models with Correlated and Uncorrelated Random Effects
Load the sample data.
load carbig
Fit a linear mixedeffects model for miles per gallon (MPG), with fixed effects for acceleration, horsepower, and the cylinders, and potentially correlated random effects for intercept and acceleration grouped by model year.
First, prepare the design matrices.
X = [ones(406,1) Acceleration Horsepower]; Z = [ones(406,1) Acceleration]; Model_Year = nominal(Model_Year); G = Model_Year;
Now, fit the model using fitlmematrix
with the defined design matrices and grouping variables.
lme = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',.... {'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',... {{'Intercept','Acceleration'}},'RandomEffectGroups',{'Model_Year'});
Refit the model with uncorrelated random effects for intercept and acceleration. First prepare the random effects design and the random effects grouping variables.
Z = {ones(406,1),Acceleration}; G = {Model_Year,Model_Year}; lme2 = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',.... {'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',... {{'Intercept'},{'Acceleration'}},'RandomEffectGroups',... {'Model_Year','Model_Year'});
Compare lme
and lme2
using the simulated likelihood ratio test.
compare(lme2,lme,'CheckNesting',true,'NSim',1000)
ans = SIMULATED LIKELIHOOD RATIO TEST: NSIM = 1000, ALPHA = 0.05 Model DF AIC BIC LogLik LRStat pValue Lower lme2 6 2194.5 2218.3 1091.3 lme 7 2193.5 2221.3 1089.7 3.0323 0.095904 0.078373 Upper 0.11585
The high value indicates that lme2
is not a significantly better fit than lme
.
Simulated Likelihood Ratio Test Using Parallel Computing
Load the sample data.
load('fertilizer.mat')
The dataset array includes data from a splitplot experiment, where soil is divided into three blocks based on the soil type: sandy, silty, and loamy. Each block is divided into five plots, where five different types of tomato plants (cherry, heirloom, grape, vine, and plum) are randomly assigned to these plots. The tomato plants in the plots are then divided into subplots, where each subplot is treated by one of four fertilizers. This is simulated data.
Store the data in a table called tbl
, and define Tomato
, Soil
, and Fertilizer
as categorical variables.
tbl = dataset2table(fertilizer); tbl.Tomato = categorical(tbl.Tomato); tbl.Soil = categorical(tbl.Soil); tbl.Fertilizer = categorical(tbl.Fertilizer);
Fit a linear mixedeffects model, where Fertilizer
and Tomato
are the fixedeffects variables, and the mean yield varies by the block (soil type), and the plots within blocks (tomato types within soil types) independently.
lme = fitlme(tbl,'Yield ~ Fertilizer * Tomato + (1Soil) + (1Soil:Tomato)');
Refit the model after removing the interaction term Tomato:Fertilizer
and the randomeffects term (1Soil)
.
lme2 = fitlme(tbl,'Yield ~ Fertilizer + Tomato + (1Soil:Tomato)');
Create the options structure for LinearMixedModel
.
opt = statset('LinearMixedModel')
opt = struct with fields:
Display: 'off'
MaxFunEvals: []
MaxIter: 10000
TolBnd: []
TolFun: 1.0000e06
TolTypeFun: []
TolX: 1.0000e12
TolTypeX: []
GradObj: []
Jacobian: []
DerivStep: []
FunValCheck: []
Robust: []
RobustWgtFun: []
WgtFun: []
Tune: []
UseParallel: []
UseSubstreams: []
Streams: {}
OutputFcn: []
Change the options for parallel testing.
opt.UseParallel = true;
Start a parallel environment.
mypool = parpool();
Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6).
Compare lme2
and lme
using the simulated likelihood ratio test with 1000 replications and parallel computing.
compare(lme2,lme,'nsim',1000,'Options',opt)
ans = Simulated Likelihood Ratio Test: Nsim = 1000, Alpha = 0.05 Model DF AIC BIC LogLik LRStat pValue Lower Upper lme2 10 511.06 532 245.53 lme 23 522.57 570.74 238.29 14.491 0.53447 0.503 0.56573
The high $$p$$value suggests that the larger model, lme
is not significantly better than the smaller model, lme2
. The smaller values of AIC
and BIC
for lme2
also support this.
More About
Likelihood Ratio Test
Under the null hypothesis
H_{0}, the observed likelihood ratio test
statistic has an approximate chisquared reference distribution with degrees of
freedom deltaDF
. When comparing two models,
compare
computes the pvalue for the
likelihood ratio test by comparing the observed likelihood ratio test statistic with
this chisquared reference distribution.
The pvalues obtained using the likelihood ratio test can be
conservative when testing for the presence or absence of randomeffects terms and
anticonservative when testing for the presence or absence of fixedeffects terms.
Hence, use the fixedEffects
, anova
, or
coefTest
method or the simulated likelihood ratio test while
testing for fixed effects.
Simulated Likelihood Ratio Test
To perform the simulated likelihood ratio test,
compare
first generates the reference distribution of the
likelihood ratio test statistic under the null hypothesis. Then, it assesses the
statistical significance of the alternate model by comparing the observed likelihood
ratio test statistic to this reference distribution.
compare
produces the simulated reference distribution of the
likelihood ratio test statistic under the null hypothesis as follows:
Generate random data
ysim
from the fitted modellme
.Fit the model specified in
lme
and alternate modelaltlme
to the simulated dataysim
.Calculate the likelihood ratio test statistic using results from step 2 and store the value.
Repeat step 1 to 3
nsim
times.
Then, compare
computes the pvalue for the
simulated likelihood ratio test by comparing the observed likelihood ratio test
statistic with the simulated reference distribution. The pvalue
estimate is the ratio of the number of times the simulated likelihood ratio test
statistic is equal to or exceeds the observed value plus one, to the number of
replications plus one.
Suppose the observed likelihood ratio statistic is T, and the simulated reference distribution is stored in vector T_{H0}. Then,
$$pvalue=\frac{\left[{\displaystyle \sum _{j=1}^{nsim}I\left({T}_{{H}_{0}}\left(j\right)\ge T\right)}\right]+1}{nsim+1}.$$
To account for the uncertainty in the simulated reference distribution,
compare
computes a 100*(1 – α)% confidence interval for the
true pvalue.
You can use the simulated likelihood ratio test to compare arbitrary linear
mixedeffects models. That is, when you are using the simulated likelihood ratio
test, lme
does not have to be nested within
altlme
, and you can fit lme
and
altlme
using either maximum likelihood (ML) or restricted
maximum likelihood (REML) methods. If you use the restricted maximum likelihood
(REML) method to fit the models, then both models must have the same fixedeffects
design matrix.
Nesting Requirements
The 'CheckNesting','True'
namevalue pair argument checks the
following requirements.
For a simulated likelihood ratio test:
You must use the same method to fit both models (
lme
andaltlme
).compare
cannot compare a model fit using ML to a model fit using REML.You must fit both models to the same response vector.
If you use REML to fit
lme
andaltlme
, then both models must have the same fixedeffects design matrix.The maximized log likelihood or restricted log likelihood of the bigger model (
altlme
) must be greater than or equal to that of the smaller model (lme
).
For a theoretical test, 'CheckNesting','True'
checks all the
requirements listed for a simulated likelihood ratio test and the following:
Weight vectors you use to fit
lme
andaltlme
must be identical.If you use ML to fit
lme
andaltlme
, the fixedeffects design matrix of the bigger model (altlme
) must contain that of the smaller model (lme
).The randomeffects design matrix of the bigger model (
altlme
) must contain that of the smaller model (lme
).
Akaike and Bayesian Information Criteria
Akaike information criterion (AIC) is AIC = –2*logL_{M} + 2*(nc + p + 1), where logL_{M} is the maximized log likelihood (or maximized restricted log likelihood) of the model, and nc + p + 1 is the number of parameters estimated in the model. p is the number of fixedeffects coefficients, and nc is the total number of parameters in the randomeffects covariance excluding the residual variance.
Bayesian information criterion (BIC) is BIC = –2*logL_{M} + ln(n_{eff})*(nc + p + 1), where logL_{M} is the maximized log likelihood (or maximized restricted log likelihood) of the model, n_{eff} is the effective number of observations, and (nc + p + 1) is the number of parameters estimated in the model.
If the fitting method is maximum likelihood (ML), then n_{eff} = n, where n is the number of observations.
If the fitting method is restricted maximum likelihood (REML), then n_{eff} = n – p.
A lower value of deviance indicates a better fit. As the value of deviance decreases, both AIC and BIC tend to decrease. Both AIC and BIC also include penalty terms based on the number of parameters estimated, p. So, when the number of parameters increase, the values of AIC and BIC tend to increase as well. When comparing different models, the model with the lowest AIC or BIC value is considered as the best fitting model.
Deviance
LinearMixedModel
computes the deviance of
model M as minus two times the loglikelihood of
that model. Let L_{M} denote
the maximum value of the likelihood function for model M.
Then, the deviance of model M is
$$2*\mathrm{log}{L}_{M}.$$
A lower value of deviance indicates a better fit. Suppose M_{1} and M_{2} are two different models, where M_{1} is nested in M_{2}. Then, the fit of the models can be assessed by comparing the deviances Dev_{1} and Dev_{2} of these models. The difference of the deviances is
$$Dev=De{v}_{1}De{v}_{2}=2\left(\mathrm{log}L{M}_{2}\mathrm{log}L{M}_{1}\right).$$
Usually, the asymptotic distribution of this difference has a chisquare distribution with
degrees of freedom v equal to the number of parameters that are estimated
in one model but fixed (typically at 0) in the other. That is, it is equal to the difference
in the number of parameters estimated in M_{1} and
M_{2}. You can get the pvalue for this test
using 1 – chi2cdf(Dev,V)
, where Dev =
Dev_{2} –
Dev_{1}.
However, in mixedeffects models, when some variance components fall on the boundary of the parameter space, the asymptotic distribution of this difference is more complicated. For example, consider the hypotheses
H_{0}: $$D=\left(\begin{array}{cc}{D}_{11}& 0\\ 0& 0\end{array}\right),$$ D is a qbyq symmetric positive semidefinite matrix.
H_{1}: D is a (q+1)by(q+1) symmetric positive semidefinite matrix.
That is, H_{1} states that the last row and column of D are different from zero. Here, the bigger model M_{2} has q + 1 parameters and the smaller model M_{1} has q parameters. And Dev has a 50:50 mixture of χ^{2}_{q} and χ^{2}_{(q + 1)} distributions (Stram and Lee, 1994).
References
[1] Hox, J. Multilevel Analysis, Techniques and Applications. Lawrence Erlbaum Associates, Inc., 2002.
[2] Stram D. O. and J. W. Lee. “Variance components testing in the longitudinal mixedeffects model”. Biometrics, Vol. 50, 4, 1994, pp. 1171–1177.
Extended Capabilities
Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.
To run in parallel, specify the 'Options'
namevalue argument in the call
to this function and set the 'UseParallel'
field of the options
structure to true
using statset
.
For example: 'Options',statset('UseParallel',true)
For more information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
See Also
LinearMixedModel
 fitlme
 fitlmematrix
 anova
 fixedEffects
 randomEffects
 covarianceParameters
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