LinearMixedModel class

Linear mixed-effects model class

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.

Construction

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).

Input Arguments

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`tbl` — Input data
table | `dataset` array

Input data, which includes the response variable, predictor variables, and grouping variables, specified as a table or dataset array. The predictor variables can be continuous or grouping variables (see Grouping Variables). You must specify the model for the variables using formula.

Data Types: table

`formula` — Formula for model specification
character vector or string scalar of the form `'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'`

Formula for model specification, specified as a character vector or string scalar of the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'. For a full description, see Formula.

Example: 'y ~ treatment +(1|block)'

`X` — Fixed-effects design matrix
n-by-p matrix

Fixed-effects design matrix, specified as an n-by-p matrix, where n is the number of observations, and p is the number of fixed-effects predictor variables. Each row of X corresponds to one observation, and each column of X corresponds to one variable.

Data Types: single | double

`y` — Response values
n-by-1 vector

Response values, specified as an n-by-1 vector, where n is the number of observations.

Data Types: single | double

`Z` — Random-effects design
n-by-q matrix | cell array of R n-by-q(r) matrices, r = 1, 2, ..., R

Random-effects design, specified as either of the following.

If there is one random-effects term in the model, then Z must be an n-by-q matrix, where n is the number of observations and q is the number of variables in the random-effects term.
If there are R random-effects terms, then Z must be a cell array of length R. Each cell of Z contains an n-by-q(r) design matrix Z{r}, r = 1, 2, ..., R, corresponding to each random-effects term. Here, q(r) is the number of random effects term in the rth random effects design matrix, Z{r}.

Data Types: single | double | cell

`G` — Grouping variable or variables
n-by-1 vector | cell array of R n-by-1 vectors

Grouping variable or variables, specified as either of the following.

If there is one random-effects term, then G must be an n-by-1 vector corresponding to a single grouping variable with M levels or groups.
G can be a categorical vector, logical vector, numeric vector, character array, string array, or cell array of character vectors.
If there are multiple random-effects terms, then G must be a cell array of length R. Each cell of G contains a grouping variable G{r}, r = 1, 2, ..., R, with M(r) levels.
G{r} can be a categorical vector, logical vector, numeric vector, character array, string array, or cell array of character vectors.

Properties

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`Coefficients` — Fixed-effects coefficient estimates
dataset array

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

`Name`	Name of the term.
`Estimate`	Estimated value of the coefficient.
`SE`	Standard error of the coefficient.
`tStat`	t-statistics for testing the null hypothesis that the coefficient is equal to zero.
`DF`	Degrees 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'`.
`pValue`	p-value for the t-test.
`Lower`	Lower limit of the confidence interval for coefficient. `Coefficients` always uses the 95% confidence level, i.e.`'alpha'` is 0.05.
`Upper`	Upper 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.

`CoefficientCovariance` — Covariance of the estimated fixed-effects coefficients
p-by-p matrix

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

`CoefficientNames` — Names of the fixed-effects coefficients
1-by-p cell array of character vectors

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

`DFE` — Residual degrees of freedom
positive integer value

Residual degrees of freedom, stored as a positive integer value. DFE = n – p, 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

`FitMethod` — Method used to fit the linear mixed-effects model
`ML` | `REML`

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

`Formula` — Specification of the fixed- and random-effects terms, and grouping variables
object

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.

`LogLikelihood` — Maximized log or restricted log likelihood
scalar value

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

`ModelCriterion` — Model criterion
dataset array

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

`AIC`	Akaike Information Criterion
`BIC`	Bayesian Information Criterion
`Loglikelihood`	Log 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)
- n – p, when the fitting method is restricted maximum likelihood (REML)

`MSE` — ML or REML estimate
positive scalar value

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

`NumCoefficients` — Number of fixed-effects coefficients
positive integer value

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

Data Types: double

`NumEstimatedCoefficients` — Number of estimated fixed-effects coefficients
positive integer value

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

Data Types: double

`NumObservations` — Number of observations
positive integer value

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

`NumPredictors` — Number of predictors
positive integer value

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

Data Types: double

`NumVariables` — Total number of variables
positive integer value

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

`ObservationInfo` — Information about the observations
table

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

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

`Weights`	The value of the weighted variable for that observation. Default value is 1.
`Excluded`	`true`, 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.
`Subset`	`true`, if the observation was used in the fit, `false`, if it was not used because it is missing or excluded.

Data Types: table

`ObservationNames` — Names of observations
cell array of character vectors

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

`PredictorNames` — Names of predictors
cell array of character vectors

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

`ResponseName` — Names of response variable
character vector

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

Data Types: char

`Rsquared` — Proportion of variability in the response explained by the fitted model
structure

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.

Ordinary R-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

`SSE` — Error sum of squares
positive scalar value

Error sum of squares, that is, sum of the squared conditional residuals, stored as a positive scalar value.

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.

Data Types: double

`SSR` — Regression sum of squares
positive scalar value

Regression sum of squares, that is, the sum of squares explained by the linear mixed-effects regression, stored as a positive scalar value. It is the sum of squared deviations of the conditional fitted values from their mean.

SSR = sum((F – mean(F)).^2), where F is the fitted conditional response of the linear mixed-effects model. The conditional model has contributions from both fixed and random effects.

Data Types: double

`SST` — Total sum of squares
positive scalar value

Total sum of squares, that is, the sum of the squared deviations of the observed response values from their mean, stored as a positive scalar value.

SST = sum((y – mean(y)).^2) = SSR + SSE, where y is the response vector.

Data Types: double

`Variables` — Variables
table

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

`VariableInfo` — Information about the variables
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.

`Class`	Class 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

`VariableNames` — Names of the variables
cell array of character vectors

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

Object Functions

`anova`	Analysis of variance for linear mixed-effects model
`coefCI`	Confidence intervals for coefficients of linear mixed-effects model
`coefTest`	Hypothesis test on fixed and random effects of linear mixed-effects model
`compare`	Compare linear mixed-effects models
`covarianceParameters`	Extract covariance parameters of linear mixed-effects model
`designMatrix`	Fixed- and random-effects design matrices
`fitted`	Fitted responses from a linear mixed-effects model
`fixedEffects`	Estimates of fixed effects and related statistics
`partialDependence`	Compute partial dependence
`plotPartialDependence`	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
`plotResiduals`	Plot 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
`residuals`	Residuals of fitted linear mixed-effects model
`response`	Response vector of the linear mixed-effects model

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects.

Examples

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Random Intercept Model with Categorical Predictor

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 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, $I [M i d A t l]$ is the dummy variable representing the region MidAtl. For details, see Dummy Variables.

The corresponding model is

$\begin{array}{l} y_{i m} = β_{0} + β_{1} I {[M i d A t l]}_{i} + β_{2} I {[E N C e n t r a l]}_{i} + β_{3} I {[W N C e n t r a l]}_{i} + β_{4} I {[S A t l]}_{i} \\ + β_{5} I {[E S C e n t r a l]}_{i} + β_{6} I {[W S C e n t r a l]}_{i} + β_{7} I {[M t n]}_{i} + β_{8} I {[P a c]}_{i} + b_{0 m} + ε_{i m}, m = 1, 2, . . ., 52, \end{array}$

where $y_{i m}$ is the observation $i$ for level $m$ of grouping variable Date, $β_{j}$ , $j$ = 0, 1, ..., 8, are the fixed-effects coefficients, $b_{0 m}$ is the random effect for level $m$ of the grouping variable Date, and $ε_{i m}$ is the observation error for observation $i$ . The random effect has the prior distribution, $b_{0 m} \sim N (0, σ_{b}^{2})$ and the error term has the distribution, $ε_{i m} \sim N (0, σ^{2})$ .

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 
    {'(Intercept)'     }          1.2233    0.096678     12.654    459
    {'Region_MidAtl'   }        0.010192    0.052221    0.19518    459
    {'Region_ENCentral'}        0.051923    0.052221     0.9943    459
    {'Region_WNCentral'}         0.23687    0.052221     4.5359    459
    {'Region_SAtl'     }        0.075481    0.052221     1.4454    459
    {'Region_ESCentral'}         0.33917    0.052221      6.495    459
    {'Region_WSCentral'}           0.069    0.052221     1.3213    459
    {'Region_Mtn'      }        0.046673    0.052221    0.89377    459
    {'Region_Pac'      }        -0.16013    0.052221    -3.0665    459


    pValue        Lower        Upper    
     1.085e-31       1.0334       1.4133
       0.84534    -0.092429      0.11281
        0.3206    -0.050698      0.15454
    7.3324e-06      0.13424      0.33949
       0.14902     -0.02714       0.1781
    2.1623e-10      0.23655      0.44179
       0.18705    -0.033621      0.17162
       0.37191    -0.055948      0.14929
     0.0022936     -0.26276    -0.057514

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


    Lower     Upper  
    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, $σ_{b}$ , 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

$\begin{aligned} {y_{}^{ˆ}}_{W N C e n t r a l, 10 / 9 / 2005} & = {β_{}^{ˆ}}_{0} + {β_{}^{ˆ}}_{3} I [W N C e n t r a l] + {b_{}^{ˆ}}_{10 / 9 / 2005} \\ = 1.2233 + 0.23687 - 0.1718 \\ = 1.28837 . \end{aligned}$

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

Linear Mixed-Effects Model with a Random Slope

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

${M P G}_{i m} = β_{0} + β_{1} {A c c}_{i} + β_{2} H P + b_{0 m} + {b_{1}}_{m} {A c c}_{i m} + ε_{i m}, m = 1, 2, 3,$

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

$b_{m} = (\begin{array}{l} b_{0 m} \\ b_{1 m} \end{array}) \sim N (0, (\begin{array}{cccccccccccccccccccc} σ_{0}^{2} & σ_{0, 1} \\ σ_{0, 1} & σ_{1}^{2} \end{array})),$

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:
    Linear Mixed Formula with 4 predictors.

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 
    {'Intercept'   }          50.064       2.3176     21.602    389
    {'Acceleration'}        -0.57897      0.13843    -4.1825    389
    {'Horsepower'  }        -0.16958    0.0073242    -23.153    389


    pValue        Lower       Upper   
    1.4185e-68      45.507       54.62
    3.5654e-05    -0.85112    -0.30681
    3.5289e-75    -0.18398    -0.15518

Random effects covariance parameters (95% CIs):
Group: Model_Year (13 Levels)
    Name1                   Name2                   Type            Estimate
    {'Intercept'   }        {'Intercept'   }        {'std' }           3.72 
    {'Acceleration'}        {'Intercept'   }        {'corr'}        -0.8769 
    {'Acceleration'}        {'Acceleration'}        {'std' }         0.3593 


    Lower       Upper   
      1.5215      9.0954
    -0.98275    -0.33845
     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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Formula

In general, a formula for model specification is a character vector or string scalar of the form 'y ~ terms'. For the linear mixed-effects models, this formula is in the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)', where fixed and random contain the fixed-effects and the random-effects terms.

Suppose a table tbl contains the following:

A response variable, y
Predictor variables, X_j, which can be continuous or grouping variables
Grouping variables, g₁, g₂, ..., g_R,

where the grouping variables in X_j and g_r can be categorical, logical, character arrays, string arrays, or cell arrays of character vectors.

Then, in a formula of the form, 'y ~ fixed + (random₁|g₁) + ... + (random_R|g_R)', the term fixed corresponds to a specification of the fixed-effects design matrix X, random₁ is a specification of the random-effects design matrix Z₁ corresponding to grouping variable g₁, and similarly random_R is a specification of the random-effects design matrix Z_R corresponding to grouping variable g_R. You can express the fixed and random terms using Wilkinson notation.

Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.

Wilkinson Notation	Factors in Standard Notation
`1`	Constant (intercept) term
`X^k`, where `k` is a positive integer	`X`, `X²`, ..., `X^k`
`X1 + X2`	`X1`, `X2`
`X1*X2`	`X1`, `X2`, `X1.*X2 (elementwise multiplication of X1 and X2)`
`X1:X2`	`X1.*X2` only
`- X2`	Do not include `X2`
`X1*X2 + X3`	`X1`, `X2`, `X3`, `X1*X2`
`X1 + X2 + X3 + X1:X2`	`X1`, `X2`, `X3`, `X1*X2`
`X1X2X3 - X1:X2:X3`	`X1`, `X2`, `X3`, `X1X2`, `X1X3`, `X2*X3`
`X1*(X2 + X3)`	`X1`, `X2`, `X3`, `X1X2`, `X1X3`

Statistics and Machine Learning Toolbox™ notation always includes a constant term unless you explicitly remove the term using -1. Here are some examples for linear mixed-effects model specification.

Examples:

Formula	Description
`'y ~ X1 + X2'`	Fixed effects for the intercept, `X1` and `X2`. This is equivalent to `'y ~ 1 + X1 + X2'`.
`'y ~ -1 + X1 + X2'`	No intercept and fixed effects for `X1` and `X2`. The implicit intercept term is suppressed by including `-1`.
`'y ~ 1 + (1 \| g1)'`	Fixed effects for the intercept plus random effect for the intercept for each level of the grouping variable `g1`.
`'y ~ X1 + (1 \| g1)'`	Random intercept model with a fixed slope.
`'y ~ X1 + (X1 \| g1)'`	Random intercept and slope, with possible correlation between them. This is equivalent to `'y ~ 1 + X1 + (1 + X1\|g1)'`.
`'y ~ X1 + (1 \| g1) + (-1 + X1 \| g1)'`	Independent random effects terms for intercept and slope.
`'y ~ 1 + (1 \| g1) + (1 \| g2) + (1 \| g1:g2)'`	Random intercept model with independent main effects for `g1` and `g2`, plus an independent interaction effect.

Documentation

LinearMixedModel class

Description

Construction

Input Arguments

tbl — Input data table | dataset array

formula — Formula for model specification character vector or string scalar of the form 'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'

X — Fixed-effects design matrix n-by-p matrix

y — Response values n-by-1 vector

Z — Random-effects design n-by-q matrix | cell array of R n-by-q(r) matrices, r = 1, 2, ..., R

G — Grouping variable or variables n-by-1 vector | cell array of R n-by-1 vectors

Properties

Coefficients — Fixed-effects coefficient estimates dataset array

CoefficientCovariance — Covariance of the estimated fixed-effects coefficients p-by-p matrix

CoefficientNames — Names of the fixed-effects coefficients 1-by-p cell array of character vectors

DFE — Residual degrees of freedom positive integer value

FitMethod — Method used to fit the linear mixed-effects model ML | REML

Formula — Specification of the fixed- and random-effects terms, and grouping variables object

LogLikelihood — Maximized log or restricted log likelihood scalar value

ModelCriterion — Model criterion dataset array

MSE — ML or REML estimate positive scalar value

NumCoefficients — Number of fixed-effects coefficients positive integer value

NumEstimatedCoefficients — Number of estimated fixed-effects coefficients positive integer value

NumObservations — Number of observations positive integer value

NumPredictors — Number of predictors positive integer value

NumVariables — Total number of variables positive integer value

ObservationInfo — Information about the observations table

ObservationNames — Names of observations cell array of character vectors

PredictorNames — Names of predictors cell array of character vectors

ResponseName — Names of response variable character vector

Rsquared — Proportion of variability in the response explained by the fitted model structure

SSE — Error sum of squares positive scalar value

SSR — Regression sum of squares positive scalar value

SST — Total sum of squares positive scalar value

Variables — Variables table

VariableInfo — Information about the variables table

VariableNames — Names of the variables cell array of character vectors

Object Functions

Copy Semantics

Examples

Random Intercept Model with Categorical Predictor

Linear Mixed-Effects Model with a Random Slope

More About

Formula

See Also

Statistics and Machine Learning Toolbox Documentation

Support

`tbl` — Input data
table | `dataset` array

`formula` — Formula for model specification
character vector or string scalar of the form `'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'`

`X` — Fixed-effects design matrix
n-by-p matrix

`y` — Response values
n-by-1 vector

`Z` — Random-effects design
n-by-q matrix | cell array of R n-by-q(r) matrices, r = 1, 2, ..., R

`G` — Grouping variable or variables
n-by-1 vector | cell array of R n-by-1 vectors

`Coefficients` — Fixed-effects coefficient estimates
dataset array

`CoefficientCovariance` — Covariance of the estimated fixed-effects coefficients
p-by-p matrix

`CoefficientNames` — Names of the fixed-effects coefficients
1-by-p cell array of character vectors

`DFE` — Residual degrees of freedom
positive integer value

`FitMethod` — Method used to fit the linear mixed-effects model
`ML` | `REML`

`Formula` — Specification of the fixed- and random-effects terms, and grouping variables
object

`LogLikelihood` — Maximized log or restricted log likelihood
scalar value

`ModelCriterion` — Model criterion
dataset array

`MSE` — ML or REML estimate
positive scalar value

`NumCoefficients` — Number of fixed-effects coefficients
positive integer value

`NumEstimatedCoefficients` — Number of estimated fixed-effects coefficients
positive integer value

`NumObservations` — Number of observations
positive integer value

`NumPredictors` — Number of predictors
positive integer value

`NumVariables` — Total number of variables
positive integer value

`ObservationInfo` — Information about the observations
table

`ObservationNames` — Names of observations
cell array of character vectors

`PredictorNames` — Names of predictors
cell array of character vectors

`ResponseName` — Names of response variable
character vector

`Rsquared` — Proportion of variability in the response explained by the fitted model
structure

`SSE` — Error sum of squares
positive scalar value

`SSR` — Regression sum of squares
positive scalar value

`SST` — Total sum of squares
positive scalar value

`Variables` — Variables
table

`VariableInfo` — Information about the variables
table

`VariableNames` — Names of the variables
cell array of character vectors