Accelerating the pace of engineering and science

# compare

Class: LinearMixedModel

Compare linear mixed-effects models

## Syntax

• results = compare(lme,altlme)
• results = compare(___,Name,Value) example
• [results,siminfo] = compare(lme,altlme,'NSim',nsim) example
• [results,siminfo] = compare(___,Name,Value) example

## 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 modelLinearMixedModel object

Linear mixed-effects model, returned as a LinearMixedModel object.

For properties and methods of this object, see LinearMixedModel.

### altlme — Alternative linear mixed-effects modelLinearMixedModel 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 simulationspositive 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 single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

### 'Alpha' — Confidence level0.05 (default) | scalar value in the range 0 to 1

Confidence 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 teststructure

Options for performing the simulated likelihood ratio test, specified as the comma-separated pair consisting of 'Options', and a structure created by statset('LinearMixedModel').

compare uses the following fields.

 'UseParallel' False for serial computation. Default.True 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 modelsfalse (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

expand all

### results — Results of likelihood ratio test or simulated likelihood ratio testdataset 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 outputstructure

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

expand all

### Test for Random Effects

`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 a tall 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

Navigate to a folder containing sample data.

```cd(matlabroot)
cd('help/toolbox/stats/examples')```

`load fertilizer`

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     Lower      Upper
lme2     10    511.06       532    -245.53
lme      23    522.57    570.74    -238.29    14.491    0.54845    0.51702    0.5796

siminfo =

nsim: 1000
alpha: 0.0500
pvalueSim: 0.5485
pvalueSimCI: [0.5170 0.5796]
TH0: [1000x1 double]```

The high p-value 0.5485 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.

### Models with Correlated and Uncorrelated Random Effects

`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       Upper
lme2     6     2194.5    2218.3    -1091.3
lme      7     2193.5    2221.3    -1089.7    3.0323    0.095904    0.078373    0.11585```

The high p-value of 0.095904 indicates that lme2 is not a significantly better fit than lme.

### Simulated Likelihood Ratio Test Using Parallel Computing

Navigate to a folder containing sample data.

```cd(matlabroot)
cd('help/toolbox/stats/examples')```

`load fertilizer`

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.

`lme = 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).

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

Create the options structure for LinearMixedModel.

`opt = statset('LinearMixedModel')`
```opt =

Display: 'off'
MaxFunEvals: []
MaxIter: 10000
TolBnd: []
TolFun: 1.0000e-06
TolTypeFun: []
TolX: 1.0000e-12
TolTypeX: []
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 parpool using the 'local' profile ... connected to 2 workers.

mypool =

Pool with properties:

AttachedFiles: {0x1 cell}
NumWorkers: 2
Cluster: [1x1 parallel.cluster.Local]
SpmdEnabled: 1```

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    0.54845    0.51702    0.5796```

The high p-value, 0.5485 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.

## Definitions

### 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 TH0. Then,

$p-value=\frac{\left[\sum _{j=1}^{nsim}I\left({T}_{{H}_{0}}\left(j\right)\ge T\right)\right]+1}{nsim+1}.$

To account for the uncertainty in the simulated reference distribution, compare computes a 100*(1 – α)% confidence interval for the true 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*logLM + 2*(nc + p + 1), where logLM 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*logLM + ln(neff)*(nc + p + 1), where logLM is the maximized log likelihood (or maximized restricted log likelihood) of the model, neff 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 neff = n, where n is the number of observations.

• If the fitting method is restricted maximum likelihood (REML), then neff = np.

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 LM 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 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

$Dev=De{v}_{1}-De{v}_{2}=2\left(\mathrm{log}L{M}_{2}-\mathrm{log}L{M}_{1}\right).$

Usually, the asymptotic distribution of this difference has a 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 = Dev2Dev1.

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: $D=\left(\begin{array}{cc}{D}_{11}& 0\\ 0& 0\end{array}\right),$ 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 χ2q 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.