compare

Compare linear mixed-effects models

Syntax

results = compare(lme,altlme)

results = compare(___,Name,Value)

[results,siminfo]
= compare(lme,altlme,'NSim',nsim)

[results,siminfo]
= compare(___,Name,Value)

Description

results = compare(lme,altlme) returns the results of a likelihood ratio test that compares the linear mixed-effects models 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:

H₀: Observed response vector is generated by lme.

H₁: 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 fixed-effects 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.

example

results = compare(___,Name,Value) also returns the results of a likelihood ratio test that compares linear mixed-effects models 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.

example

[results,siminfo] = compare(lme,altlme,'NSim',nsim) returns the results of a simulated likelihood ratio test that compares linear mixed-effects models 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 fixed-effects design matrix.

example

[results,siminfo] = compare(___,Name,Value) also returns the results of a simulated likelihood ratio test that compares linear mixed-effects models 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 p-value.

Input Arguments

expand all

`lme` — Linear mixed-effects model
`LinearMixedModel` object

Linear mixed-effects model, specified as a LinearMixedModel object constructed using fitlme or fitlmematrix.

`altlme` — Alternative linear mixed-effects model
`LinearMixedModel` object

Alternative linear mixed-effects 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 mixed-effects 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

Name-Value Pair Arguments

Specify optional comma-separated 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 comma-separated 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 comma-separated pair consisting of 'Options', and a structure created by statset('LinearMixedModel').

These options require Parallel Computing Toolbox™.

compare uses the following fields.

`'UseParallel'`	`False` for serial computation. Default. `True` for parallel computation. You need Parallel Computing Toolbox for parallel computation.
`'UseSubstreams'`	`False` for not using a separate substream of the random number generator for each iteration. Default. `True` for using a separate substream of the random number generator for each iteration. You can only use this option with random stream types that support substreams.
`'Streams'`	If `'UseSubstreams'` is `True`, then `'Streams'` must be a single random number stream, or a scalar cell array containing a single stream. If `'UseSubstreams'` is `False` and `'UseParallel'` is `False`, then `'Streams'` must be a single random number stream, or a scalar cell array containing a single stream. `'UseParallel'` is `True`, then `'Streams'` must be equal to the number of processors used. If a parallel pool is open, then the `'Streams'` is the same length as the size of the parallel pool. If `'UseParallel'` is `True`, a parallel pool might open up for you. But since `'Streams'` must be equal to the number of processors used, it is best to open a pool explicitly using the `parpool` command, before calling `compare` with the`'UseParallel','True'` option.

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 comma-separated 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

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`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` versus `lme`
`deltaDF`	`DF` for `altlme` minus `DF` for `lme`
`pValue`	p-value 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` versus `lme`
`pValue`	p-value 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`	Simulation-based p-value.
`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

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Test for Random Effects

Open Live Script

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 linear-mixed effects model, your data must be in a properly formatted dataset array. To fit a linear mixed-effects 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 mixed-effects model, with a varying intercept and varying slope for each region, grouped by Date.

altlme = fitlme(flu2,'FluRate ~ 1 + Region + (1 + Region|Date)');

Fit a linear mixed-effects model with fixed effects for the region and a random intercept that varies by Date.

lme = fitlme(flu2,'FluRate ~ 1 + Region + (1|Date)');

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

Open Live Script

Load the sample data.

load('fertilizer.mat');

The dataset array includes data from a split-plot 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 mixed-effects model, where Fertilizer and Tomato are the fixed-effects 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 + (1|Soil) + (1|Soil:Tomato)');

Refit the model after removing the interaction term Tomato:Fertilizer and the random-effects term (1 | Soil).

lmeSmall = fitlme(ds,'Yield ~ Fertilizer + Tomato + (1|Soil: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 random-effect terms. Note that both models are fit using the default fitting method, ML. That’s why, there is no restriction on the fixed-effects design matrices. If you use restricted maximum likelihood (REML) method, both models must have identical fixed-effects 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

Open Script

Load the sample data.

load carbig

Fit a linear mixed-effects 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    0.094905    0.077462


    Upper  
           
    0.11477

The high $p$ -value indicates that lme2 is not a significantly better fit than lme.

Simulated Likelihood Ratio Test Using Parallel Computing

This example uses:

Open Live Script

Load the sample data.

load('fertilizer.mat')

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 mixed-effects model, where Fertilizer and Tomato are the fixed-effects 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 + (1|Soil) + (1|Soil:Tomato)');

Refit the model after removing the interaction term Tomato:Fertilizer and the random-effects term (1|Soil).

lme2 = fitlme(tbl,'Yield ~ Fertilizer + Tomato + (1|Soil:Tomato)');

Create the options structure for LinearMixedModel.

opt = statset('LinearMixedModel')

opt = struct with fields:
          Display: 'off'
      MaxFunEvals: []
          MaxIter: 10000
           TolBnd: []
           TolFun: 1.0000e-06
       TolTypeFun: []
             TolX: 1.0000e-12
         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

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Likelihood Ratio Test

Under the null hypothesis H₀, the observed likelihood ratio test statistic has an approximate chi-squared reference distribution with degrees of freedom deltaDF. When comparing two models, compare computes the p-value for the likelihood ratio test by comparing the observed likelihood ratio test statistic with this chi-squared reference distribution.

The p-values obtained using the likelihood ratio test can be conservative when testing for the presence or absence of random-effects terms and anticonservative when testing for the presence or absence of fixed-effects 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 model lme.
Fit the model specified in lme and alternate model altlme to the simulated data ysim.
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 p-value for the simulated likelihood ratio test by comparing the observed likelihood ratio test statistic with the simulated reference distribution. The p-value 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,

$p - v a l u e = \frac{[\sum_{j = 1}^{n s i m} I (T_{H_{0}} (j) \geq T)] + 1}{n s i m + 1} .$

To account for the uncertainty in the simulated reference distribution, compare computes a 100*(1 – α)% confidence interval for the true p-value.

You can use the simulated likelihood ratio test to compare arbitrary linear mixed-effects 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 fixed-effects design matrix.

Nesting Requirements

The 'CheckNesting','True' name-value pair argument checks the following requirements.

For a simulated likelihood ratio test:

You must use the same method to fit both models (lme and altlme). 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 and altlme, then both models must have the same fixed-effects 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 and altlme must be identical.
If you use ML to fit lme and altlme, the fixed-effects design matrix of the bigger model (altlme) must contain that of the smaller model (lme).
The random-effects 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 fixed-effects coefficients, and nc is the total number of parameters in the random-effects 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

$- 2 * \log L_{M} .$

A lower value of deviance indicates a better fit. Suppose M₁ and M₂ are two different models, where M₁ is nested in M₂. Then, the fit of the models can be assessed by comparing the deviances Dev₁ and Dev₂ of these models. The difference of the deviances is

$D e v = D e v_{1} - D e v_{2} = 2 (\log L M_{2} - \log L M_{1}) .$

Usually, the asymptotic distribution of this difference has a chi-square 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 M₁ and M₂. You can get the p-value for this test using 1 – chi2cdf(Dev,V), where Dev = Dev₂ – Dev₁.

However, in mixed-effects 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

H₀: $D = (\begin{matrix} D_{11} & 0 \\ 0 & 0 \end{matrix}),$ D is a q-by-q symmetric positive semidefinite matrix.

H₁: D is a (q+1)-by-(q+1) symmetric positive semidefinite matrix.

That is, H₁ states that the last row and column of D are different from zero. Here, the bigger model M₂ has q + 1 parameters and the smaller model M₁ has q parameters. And Dev has a 50:50 mixture of χ²_q and χ²_{(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 mixed-effects 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, set the 'UseParallel' option to true.

Set the 'UseParallel' field of the options structure to true using statset and specify the 'Options' name-value pair argument in the call to this function.

For example: 'Options',statset('UseParallel',true)

For more information, see the 'Options' name-value pair argument.

For more general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).

Documentation

compare

Syntax

Description

Input Arguments

`lme` — Linear mixed-effects model
`LinearMixedModel` object

`altlme` — Alternative linear mixed-effects model
`LinearMixedModel` object

`nsim` — Number of replications for simulations
positive integer number

Name-Value Pair Arguments

`'Alpha'` — Significance level
0.05 (default) | scalar value in the range 0 to 1

`'Options'` — Options for performing simulated likelihood ratio test
structure

`'CheckNesting'` — Indicator to check nesting between two models
`false` (default) | `true`

Output Arguments

`results` — Results of likelihood ratio test or simulated likelihood ratio test
dataset array

`siminfo` — Simulation output
structure

Examples

Test for Random Effects

Test for Fixed and Random Effects

Models with Correlated and Uncorrelated Random Effects

Simulated Likelihood Ratio Test Using Parallel Computing

More About

Likelihood Ratio Test

Simulated Likelihood Ratio Test

Nesting Requirements

Akaike and Bayesian Information Criteria

Deviance

References

Extended Capabilities

Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

See Also

Statistics and Machine Learning Toolbox Documentation

Support

Documentation

compare

Syntax

Description

Input Arguments

lme — Linear mixed-effects model LinearMixedModel object

altlme — Alternative linear mixed-effects model LinearMixedModel object

nsim — Number of replications for simulations positive integer number

Name-Value Pair Arguments

'Alpha' — Significance level 0.05 (default) | scalar value in the range 0 to 1

'Options' — Options for performing simulated likelihood ratio test structure

'CheckNesting' — Indicator to check nesting between two models false (default) | true

Output Arguments

results — Results of likelihood ratio test or simulated likelihood ratio test dataset array

siminfo — Simulation output structure

Examples

Test for Random Effects

Test for Fixed and Random Effects

Models with Correlated and Uncorrelated Random Effects

Simulated Likelihood Ratio Test Using Parallel Computing

More About

Likelihood Ratio Test

Simulated Likelihood Ratio Test

Nesting Requirements

Akaike and Bayesian Information Criteria

Deviance

References

Extended Capabilities

Automatic Parallel Support Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.

See Also

Statistics and Machine Learning Toolbox Documentation

Support

`lme` — Linear mixed-effects model
`LinearMixedModel` object

`altlme` — Alternative linear mixed-effects model
`LinearMixedModel` object

`nsim` — Number of replications for simulations
positive integer number

`'Alpha'` — Significance level
0.05 (default) | scalar value in the range 0 to 1

`'Options'` — Options for performing simulated likelihood ratio test
structure

`'CheckNesting'` — Indicator to check nesting between two models
`false` (default) | `true`

`results` — Results of likelihood ratio test or simulated likelihood ratio test
dataset array

`siminfo` — Simulation output
structure

Automatic Parallel Support
Accelerate code by automatically running computation in parallel using Parallel Computing Toolbox™.