disp

Class:LinearMixedModel

Display linear mixed-effects model

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Syntax

display(lme)

Description

example

display(lme)displays the fitted linear mixed-effects modellme.

Input Arguments

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

Linear mixed-effects model, specified as aLinearMixedModelobject constructed usingfitlmeorfitlmematrix.

Examples

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Randomized Block Design

Open Live Script

Load the sample data.

load('shift.mat');

The dataset array shows the absolute deviations from the target quality characteristic measured from the products that five operators manufacture during three shifts, morning, evening, and night. This is a randomized block design, where the operators are the blocks. The experiment is designed to study the impact of the time of shift on the performance. The performance measure is the absolute deviation of the quality characteristics from the target value. This is simulated data.

ShiftandOperatorare nominal variables.

shift.Shift = nominal(shift.Shift); shift.Operator = nominal(shift.Operator);

Fit a linear mixed-effects model with a random intercept grouped by operator to assess if performance significantly differs according to the time of the shift.

lme = fitlme(shift,'QCDev ~ Shift + (1|Operator)');

Display the model.

disp(lme)

Linear mixed-effects model fit by ML Model information: Number of observations 15 Fixed effects coefficients 3 Random effects coefficients 5 Covariance parameters 2 Formula: QCDev ~ 1 + Shift + (1 | Operator) Model fit statistics: AIC BIC LogLikelihood Deviance 59.012 62.552 -24.506 49.012 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue {'(Intercept)' } 3.1196 0.88681 3.5178 12 0.0042407 {'Shift_Morning'} -0.3868 0.48344 -0.80009 12 0.43921 {'Shift_Night' } 1.9856 0.48344 4.1072 12 0.0014535 Lower Upper 1.1874 5.0518 -1.4401 0.66653 0.93227 3.0389 Random effects covariance parameters (95% CIs): Group: Operator (5 Levels) Name1 Name2 Type Estimate {'(Intercept)'} {'(Intercept)'} {'std'} 1.8297 Lower Upper 0.94915 3.5272 Group: Error Name Estimate Lower Upper {'Res Std'} 0.76439 0.49315 1.1848

This display includes the model performance statistics,Akaike and Bayesian Information Criteria,Akaike and Bayesian Information Criteria, loglikelihood, andDeviance.

The fixed-effects coefficients table includes the names and estimates of the coefficients in the first two columns. The third columnSEshows the standard errors of the coefficients. The columntStatincludes the $t$ -statistic values that correspond to each coefficient.DFis the residual degrees of freedom, and thepValueis the $p$ -value that corresponds to the corresponding $t$ -statistic value. The columnsLowerandUpper显示较低和知名的95%的上限ence interval for each fixed-effects coefficient.

The first table for the random effects shows the types and the estimates of the random effects covariance parameters, with the lower and upper limits of a 95% confidence interval for each parameter. The display also shows the name of the grouping variable, operator, and the total number of levels, 5.

The second table for the random effects shows the estimate of the observation error, with the lower and upper limits of a 95% confidence interval.

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Akaike and Bayesian Information Criteria

Akaike information criterion (AIC) isAIC= –2*logL_M+ 2*(nc+p+ 1), where logL_Mis the maximized log likelihood (or maximized restricted log likelihood) of the model, andnc+p+ 1 is the number of parameters estimated in the model.pis the number of fixed-effects coefficients, andncis the total number of parameters in the random-effects covariance excluding the residual variance.

Bayesian information criterion (BIC) isBIC= –2*logL_M+ ln(n_eff)*(nc+p+ 1), where logL_Mis the maximized log likelihood (or maximized restricted log likelihood) of the model,n_effis 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), thenn_eff=n, wherenis the number of observations.
If the fitting method is restricted maximum likelihood (REML), thenn_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

LinearMixedModelcomputes the deviance of modelMas minus two times the loglikelihood of that model. LetL_Mdenote the maximum value of the likelihood function for modelM. Then, the deviance of modelMis

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

A lower value of deviance indicates a better fit. SupposeM₁andM₂are two different models, whereM₁is nested inM₂. Then, the fit of the models can be assessed by comparing the deviancesDev₁andDev₂of 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 freedomvequal 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 M₁and M₂. You can get thep-value for this test using1 – chi2cdf(Dev,V), whereDev=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: $D = (\begin{matrix} D_{11} & 0 \\ 0 & 0 \end{matrix}),$ Dis aq-by-qsymmetric positive semidefinite matrix.

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

That is,H₁states that the last row and column ofDare different from zero. Here, the bigger modelM₂hasq+ 1 parameters and the smaller modelM₁hasqparameters. AndDevhas a 50:50 mixture ofχ²_qandχ²_{(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.

disp

Syntax

Description

Input Arguments

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

Examples

Randomized Block Design

More About

Akaike and Bayesian Information Criteria

Deviance

References

See Also

Statistics and Machine Learning Toolbox Documentation

万博1manbetx

Mastering Machine Learning: A Step-by-Step Guide with MATLAB

disp

Syntax

Description

Input Arguments

lme—Linear mixed-effects modelLinearMixedModelobject

Examples

Randomized Block Design

More About

Akaike and Bayesian Information Criteria

Deviance

References

See Also

Statistics and Machine Learning Toolbox Documentation

万博1manbetx

Mastering Machine Learning: A Step-by-Step Guide with MATLAB

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