**Class: **LinearMixedModel

Display linear mixed-effects model

`lme`

— Linear mixed-effects model`LinearMixedModel`

objectLinear mixed-effects model, specified as a `LinearMixedModel`

object constructed using `fitlme`

or `fitlmematrix`

.

Load the sample data.

load(fullfile(matlabroot,'examples','stats','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.

`Shift`

and `Operator`

are 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, and Deviance.

The fixed-effects coefficients table includes the names and estimates of the coefficients in the first two columns. The third column `SE`

shows the standard errors of the coefficients. The column `tStat`

includes the $$t$$-statistic values that correspond to each coefficient. `DF`

is the residual degrees of freedom, and the `pValue`

is the $$p$$-value that corresponds to the corresponding $$t$$-statistic value. The columns `Lower`

and `Upper`

display the lower and upper limits of a 95% confidence 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.

Akaike information criterion (AIC) is *AIC* =
–2*log*L*_{M} +
2*(*nc* + *p* + 1), where log*L*_{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*log*L*_{M} +
ln(*n _{eff}*)*(

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.

`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 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 M_{1} and
M_{2}. You can get the *p*-value for this test
using `1 – chi2cdf(Dev,V)`

, where *Dev* =
*Dev*_{2} –
*Dev*_{1}.

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

*H*_{0}: $$D=\left(\begin{array}{cc}{D}_{11}& 0\\ 0& 0\end{array}\right),$$ *D* is a *q*-by-*q* 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).

[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.

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