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LinearMixedModel

Linear mixed-effects model

Description

A LinearMixedModel object represents a model of a response variable with fixed and random effects. It comprises data, a model description, fitted coefficients, covariance parameters, design matrices, residuals, residual plots, and other diagnostic information for a linear mixed-effects model. You can predict model responses with the predict function and generate random data at new design points using the random function.

Creation

Create a LinearMixedModel model using fitlme or fitlmematrix. You can fit a linear mixed-effects model using fitlme(tbl,formula) if your data is in a table or dataset array. Alternatively, if your model is not easily described using a formula, you can create matrices to define the fixed and random effects, and fit the model using fitlmematrix(X,y,Z,G)

Properties

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Coefficient Estimates

Fixed-effects coefficient estimates and related statistics, stored as a dataset array containing the following fields.

NameName of the term.
EstimateEstimated value of the coefficient.
SEStandard error of the coefficient.
tStatt-statistics for testing the null hypothesis that the coefficient is equal to zero.
DFDegrees of freedom for the t-test. Method to compute DF is specified by the 'DFMethod' name-value pair argument. Coefficients always uses the 'Residual' method for 'DFMethod'.
pValuep-value for the t-test.
LowerLower limit of the confidence interval for coefficient. Coefficients always uses the 95% confidence level, i.e.'alpha' is 0.05.
UpperUpper limit of confidence interval for coefficient. Coefficients always uses the 95% confidence level, i.e.'alpha' is 0.05.

You can change 'DFMethod' and 'alpha' while computing confidence intervals for or testing hypotheses involving fixed- and random-effects, using the coefCI and coefTest methods.

Covariance of the estimated fixed-effects coefficients of the linear mixed-effects model, stored as a p-by-p matrix, where p is the number of fixed-effects coefficients.

You can display the covariance parameters associated with the random effects using the covarianceParameters method.

Data Types: double

Names of the fixed-effects coefficients of a linear mixed-effects model, stored as a 1-by-p cell array of character vectors.

Data Types: cell

Number of fixed-effects coefficients in the fitted linear mixed-effects model, stored as a positive integer value.

Data Types: double

Number of estimated fixed-effects coefficients in the fitted linear mixed-effects model, stored as a positive integer value.

Data Types: double

Fitting Method

Method used to fit the linear mixed-effects model, stored as either of the following.

  • ML, if the fitting method is maximum likelihood

  • REML, if the fitting method is restricted maximum likelihood

Data Types: char

Input Data

Specification of the fixed-effects terms, random-effects terms, and grouping variables that define the linear mixed-effects model, stored as an object.

For more information on how to specify the model to fit using a formula, see Formula.

Number of observations used in the fit, stored as a positive integer value. This is the number of rows in the table or dataset array, or the design matrices minus the excluded rows or rows with NaN values.

Data Types: double

Number of variables used as predictors in the linear mixed-effects model, stored as a positive integer value.

Data Types: double

Total number of variables including the response and predictors, stored as a positive integer value.

  • If the sample data is in a table or dataset array tbl, NumVariables is the total number of variables in tbl including the response variable.

  • If the fit is based on matrix input, NumVariables is the total number of columns in the predictor matrix or matrices, and response vector.

NumVariables includes variables, if there are any, that are not used as predictors or as the response.

Data Types: double

Information about the observations used in the fit, stored as a table.

ObservationInfo has one row for each observation and the following four columns.

WeightsThe value of the weighted variable for that observation. Default value is 1.
Excludedtrue, if the observation was excluded from the fit using the 'Exclude' name-value pair argument, false, otherwise. 1 stands for true and 0 stands for false.
Missing

true, if the observation was excluded from the fit because any response or predictor value is missing, false, otherwise.

Missing values include NaN for numeric variables, empty cells for cell arrays, blank rows for character arrays, and the <undefined> value for categorical arrays.

Subsettrue, if the observation was used in the fit, false, if it was not used because it is missing or excluded.

Data Types: table

Names of observations used in the fit, stored as a cell array of character vectors.

  • If the data is in a table or dataset array, tbl, containing observation names, ObservationNames has those names.

  • If the data is provided in matrices, or a table or dataset array without observation names, then ObservationNames is an empty cell array.

Data Types: cell

Names of the variables that you use as predictors in the fit, stored as a cell array of character vectors that has the same length as NumPredictors.

Data Types: cell

Name of the variable used as the response variable in the fit, stored as a character vector.

Data Types: char

Variables, stored as a table.

  • If the fit is based on a table or dataset array tbl, then Variables is identical to tbl.

  • If the fit is based on matrix input, then Variables is a table containing all the variables in the predictor matrix or matrices, and response variable.

Data Types: table

Information about the variables used in the fit, stored as a table.

VariableInfo has one row for each variable and contains the following four columns.

ClassClass of the variable ('double', 'cell', 'nominal', and so on).
Range

Value range of the variable.

  • For a numerical variable, it is a two-element vector of the form [min,max].

  • For a cell or categorical variable, it is a cell or categorical array containing all unique values of the variable.

InModel

true, if the variable is a predictor in the fitted model.

false, if the variable is not in the fitted model.

IsCategorical

true, if the variable has a type that is treated as a categorical predictor, such as cell, logical, or categorical, or if it is specified as categorical by the 'Categorical' name-value pair argument of the fit method.

false, if it is a continuous predictor.

Data Types: table

Names of the variables used in the fit, stored as a cell array of character vectors.

  • If sample data is in a table or dataset array tbl, VariableNames contains the names of the variables in tbl.

  • If sample data is in matrix format, then VariableInfo includes variable names you supply while fitting the model. If you do not supply the variable names, then VariableInfo contains the default names.

Data Types: cell

Summary Statistics

Residual degrees of freedom, stored as a positive integer value. DFE = np, where n is the number of observations, and p is the number of fixed-effects coefficients.

This corresponds to the 'Residual' method of calculating degrees of freedom in the fixedEffects and randomEffects methods.

Data Types: double

Maximized log likelihood or maximized restricted log likelihood of the fitted linear mixed-effects model depending on the fitting method you choose, stored as a scalar value.

Data Types: double

Model criterion to compare fitted linear mixed-effects models, stored as a dataset array with the following columns.

AICAkaike Information Criterion
BICBayesian Information Criterion
LoglikelihoodLog likelihood value of the model
Deviance–2 times the log likelihood of the model

If n is the number of observations used in fitting the model, and p is the number of fixed-effects coefficients, then for calculating AIC and BIC,

  • The total number of parameters is nc + p + 1, where nc is the total number of parameters in the random-effects covariance excluding the residual variance

  • The effective number of observations is

    • n, when the fitting method is maximum likelihood (ML)

    • np, when the fitting method is restricted maximum likelihood (REML)

ML or REML estimate, based on the fitting method used for estimating σ2, stored as a positive scalar value. σ2 is the residual variance or variance of the observation error term of the linear mixed-effects model.

Data Types: double

Proportion of variability in the response explained by the fitted model, stored as a structure. It is the multiple correlation coefficient or R-squared. Rsquared has two fields.

OrdinaryR-squared value, stored as a scalar value in a structure. Rsquared.Ordinary = 1 – SSE./SST
Adjusted

R-squared value adjusted for the number of fixed-effects coefficients, stored as a scalar value in a structure.

Rsquared.Adjusted = 1 – (SSE./SST)*(DFT./DFE),

where DFE = n – p, DFT = n – 1, and n is the total number of observations, p is the number of fixed-effects coefficients.

Data Types: struct

Error sum of squares, specified as a positive scalar. SSE is equal to the squared conditional residuals, that is

SSE = sum((y – F).^2),

where y is the response vector and F is the fitted conditional response of the linear mixed-effects model. The conditional model has contributions from both fixed and random effects.

If the model was trained with observation weights, the sum of squares in the SSE calculation is the weighted sum of squares.

Data Types: double

Regression sum of squares, specified as a positive scalar. SSR is the sum of squares explained by the linear mixed-effects regression, and is equal to the sum of the squared deviations between the fitted values and the mean of the response.

SSR = sum((F – mean(y)).^2),

where F is the fitted conditional response of the linear mixed-effects model and y is the response vector. The conditional model has contributions from both fixed and random effects.

If the model was trained with observation weights, the sum of squares in the SSR calculation is the weighted sum of squares.

Data Types: double

Total sum of squares, specified as a positive scalar.

For a linear mixed-effects model with an intercept, SST is calculated as

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

For a linear mixed-effects model without an intercept, SST is calculated as the sum of the squared deviations of the observed response values from their mean, that is

SST = sum((y – mean(y)).^2),

where y is the response vector.

If the model was trained with observation weights, the sum of squares in the SST calculation is the weighted sum of squares.

Data Types: double

Object Functions

anovaAnalysis of variance for linear mixed-effects model
coefCI Confidence intervals for coefficients of linear mixed-effects model
coefTestHypothesis test on fixed and random effects of linear mixed-effects model
compareCompare linear mixed-effects models
covarianceParametersExtract covariance parameters of linear mixed-effects model
designMatrixFixed- and random-effects design matrices
fittedFitted responses from a linear mixed-effects model
fixedEffectsEstimates of fixed effects and related statistics
partialDependenceCompute partial dependence
plotPartialDependenceCreate partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotResidualsPlot residuals of linear mixed-effects model
predict Predict response of linear mixed-effects model
random Generate random responses from fitted linear mixed-effects model
randomEffects Estimates of random effects and related statistics
residualsResiduals of fitted linear mixed-effects model
responseResponse vector of the linear mixed-effects model

Examples

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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 Center for Disease Control and Prevention, 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 fixed effects for region and a random intercept that varies by Date.

Because region is a nominal variable, fitlme takes the first region, NE, as the reference and creates eight dummy variables representing the other eight regions. For example, I[MidAtl] is the dummy variable representing the region MidAtl. For details, see Dummy Variables.

The corresponding model is

yim=β0+β1I[MidAtl]i+β2I[ENCentral]i+β3I[WNCentral]i+β4I[SAtl]i+β5I[ESCentral]i+β6I[WSCentral]i+β7I[Mtn]i+β8I[Pac]i+b0m+εim,m=1,2,...,52,

where yim is the observation i for level m of grouping variable Date, βj, j = 0, 1, ..., 8, are the fixed-effects coefficients, b0m is the random effect for level m of the grouping variable Date, and εim is the observation error for observation i. The random effect has the prior distribution, b0mN(0,σb2) and the error term has the distribution, εimN(0,σ2).

lme = fitlme(flu2,'FluRate ~ 1 + Region + (1|Date)')
lme = 
Linear mixed-effects model fit by ML

Model information:
    Number of observations             468
    Fixed effects coefficients           9
    Random effects coefficients         52
    Covariance parameters                2

Formula:
    FluRate ~ 1 + Region + (1 | Date)

Model fit statistics:
    AIC       BIC       LogLikelihood    Deviance
    318.71    364.35    -148.36          296.71  

Fixed effects coefficients (95% CIs):
    Name                        Estimate    SE          tStat      DF     pValue        Lower        Upper    
    {'(Intercept)'     }          1.2233    0.096678     12.654    459     1.085e-31       1.0334       1.4133
    {'Region_MidAtl'   }        0.010192    0.052221    0.19518    459       0.84534    -0.092429      0.11281
    {'Region_ENCentral'}        0.051923    0.052221     0.9943    459        0.3206    -0.050698      0.15454
    {'Region_WNCentral'}         0.23687    0.052221     4.5359    459    7.3324e-06      0.13424      0.33949
    {'Region_SAtl'     }        0.075481    0.052221     1.4454    459       0.14902     -0.02714       0.1781
    {'Region_ESCentral'}         0.33917    0.052221      6.495    459    2.1623e-10      0.23655      0.44179
    {'Region_WSCentral'}           0.069    0.052221     1.3213    459       0.18705    -0.033621      0.17162
    {'Region_Mtn'      }        0.046673    0.052221    0.89377    459       0.37191    -0.055948      0.14929
    {'Region_Pac'      }        -0.16013    0.052221    -3.0665    459     0.0022936     -0.26276    -0.057514

Random effects covariance parameters (95% CIs):
Group: Date (52 Levels)
    Name1                  Name2                  Type           Estimate    Lower     Upper  
    {'(Intercept)'}        {'(Intercept)'}        {'std'}        0.6443      0.5297    0.78368

Group: Error
    Name               Estimate    Lower      Upper
    {'Res Std'}        0.26627     0.24878    0.285

The p-values 7.3324e-06 and 2.1623e-10 respectively show that the fixed effects of the flu rates in regions WNCentral and ESCentral are significantly different relative to the flu rates in region NE.

The confidence limits for the standard deviation of the random-effects term, σb, do not include 0 (0.5297, 0.78368), which indicates that the random-effects term is significant. You can also test the significance of the random-effects terms using the compare method.

The estimated value of an observation is the sum of the fixed effects and the random-effect value at the grouping variable level corresponding to that observation. For example, the estimated best linear unbiased predictor (BLUP) of the flu rate for region WNCentral in week 10/9/2005 is

yˆWNCentral,10/9/2005=βˆ0+βˆ3I[WNCentral]+bˆ10/9/2005=1.2233+0.23687-0.1718=1.28837.

This is the fitted conditional response, since it includes contribution to the estimate from both the fixed and random effects. You can compute this value as follows.

beta = fixedEffects(lme);
[~,~,STATS] = randomEffects(lme); % Compute the random-effects statistics (STATS)
STATS.Level = nominal(STATS.Level);
y_hat = beta(1) + beta(4) + STATS.Estimate(STATS.Level=='10/9/2005')
y_hat = 1.2884

You can simply display the fitted value using the fitted method.

F = fitted(lme);
F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')
ans = 1.2884

Compute the fitted marginal response for region WNCentral in week 10/9/2005.

F = fitted(lme,'Conditional',false);
F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')
ans = 1.4602

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 uncorrelated random-effect for intercept and acceleration grouped by the model year. This model corresponds to

MPGim=β0+β1Acci+β2HP+b0m+b1mAccim+εim,m=1,2,3,

with the random-effects terms having the following prior distributions:

bm=(b0mb1m)N(0,(σ02σ0,1σ0,1σ12)),

where m represents the model year.

First, prepare the design matrices for fitting the linear mixed-effects model.

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. Use the 'fminunc' optimization algorithm.

lme = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',....
{'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',...
{{'Intercept','Acceleration'}},'RandomEffectGroups',{'Model_Year'},...
'FitMethod','REML')
lme = 
Linear mixed-effects model fit by REML

Model information:
    Number of observations             392
    Fixed effects coefficients           3
    Random effects coefficients         26
    Covariance parameters                4

Formula:
    y ~ Intercept + Acceleration + Horsepower + (Intercept + Acceleration | Model_Year)

Model fit statistics:
    AIC       BIC       LogLikelihood    Deviance
    2202.9    2230.7    -1094.5          2188.9  

Fixed effects coefficients (95% CIs):
    Name                    Estimate    SE           tStat      DF     pValue        Lower       Upper   
    {'Intercept'   }          50.064       2.3176     21.602    389    1.4185e-68      45.507       54.62
    {'Acceleration'}        -0.57897      0.13843    -4.1825    389    3.5654e-05    -0.85112    -0.30681
    {'Horsepower'  }        -0.16958    0.0073242    -23.153    389    3.5289e-75    -0.18398    -0.15518

Random effects covariance parameters (95% CIs):
Group: Model_Year (13 Levels)
    Name1                   Name2                   Type            Estimate    Lower       Upper   
    {'Intercept'   }        {'Intercept'   }        {'std' }           3.72       1.5215      9.0954
    {'Acceleration'}        {'Intercept'   }        {'corr'}        -0.8769     -0.98274    -0.33846
    {'Acceleration'}        {'Acceleration'}        {'std' }         0.3593      0.19418     0.66483

Group: Error
    Name               Estimate    Lower     Upper 
    {'Res Std'}        3.6913      3.4331    3.9688

The fixed effects coefficients display includes the estimate, standard errors (SE), and the 95% confidence interval limits (Lower and Upper). The p-values for (pValue) indicate that all three fixed-effects coefficients are significant.

The confidence intervals for the standard deviations and the correlation between the random effects for intercept and acceleration do not include zeros, hence they seem significant. Use the compare method to test for the random effects.

Display the covariance matrix of the estimated fixed-effects coefficients.

lme.CoefficientCovariance
ans = 3×3

    5.3711   -0.2809   -0.0126
   -0.2809    0.0192    0.0005
   -0.0126    0.0005    0.0001

The diagonal elements show the variances of the fixed-effects coefficient estimates. For example, the variance of the estimate of the intercept is 5.3711. Note that the standard errors of the estimates are the square roots of the variances. For example, the standard error of the intercept is 2.3176, which is sqrt(5.3711).

The off-diagonal elements show the correlation between the fixed-effects coefficient estimates. For example, the correlation between the intercept and acceleration is –0.2809 and the correlation between acceleration and horsepower is 0.0005.

Display the coefficient of determination for the model.

lme.Rsquared
ans = struct with fields:
    Ordinary: 0.7866
    Adjusted: 0.7855

The adjusted value is the R-squared value adjusted for the number of predictors in the model.

More About

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Version History

Introduced in R2013b