Documentation |
Linear mixed-effects model class
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.
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).
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 |
disp | Display linear mixed-effects model |
fit | Fit linear mixed-effects model using tables |
fitmatrix | Fit linear mixed-effects model using design matrices |
fitted | Fitted responses from a linear mixed-effects model |
fixedEffects | Estimates of fixed effects and related statistics |
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 |
In general, a formula for model specification is a string 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, g_{1}, g_{2}, ..., g_{R},
where the grouping variables in X_{j} and g_{r} can be categorical, logical, character arrays, or cell arrays of strings.
Then, in a formula of the form, 'y ~ fixed + (random_{1}|g_{1}) + ... + (random_{R}|g_{R})', the term fixed corresponds to a specification of the fixed-effects design matrix X, random_{1} is a specification of the random-effects design matrix Z_{1} corresponding to grouping variable g_{1}, 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, X^{2}, ..., 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 |
X1*X2*X3 - X1:X2:X3 | X1, X2, X3, X1*X2, X1*X3, X2*X3 |
X1*(X2 + X3) | X1, X2, X3, X1*X2, X1*X3 |
Statistics 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. |
Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB^{®} documentation.