## Documentation Center |

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`, which can be continuous or grouping variables_{j}Grouping variables,

`g`,_{1}`g`, ...,_{2}`g`,_{R}

where the grouping variables in `X _{j}` and

Then, in a formula of the form, `'y ~ fixed + (random _{1}|g_{1})
+ ... + (random_{R}|g_{R})'`,
the term

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.

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