Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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 character vector 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`

and _{j}`g`

can
be categorical, logical, character arrays, or cell arrays of character
vectors._{r}

Then, in a formula of the form, `'y ~ fixed + (random`

,
the term _{1}|g_{1})
+ ... + (random_{R}|g_{R})'`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` ,
..., `X` |

`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 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. |

Value. To learn how value classes affect
copy operations, see Copying Objects in
the MATLAB^{®} documentation.

Was this topic helpful?