Note: This page has been translated by MathWorks. Click here to see

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

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

**MathWorks Machine Translation**

The automated translation of this page is provided by a general purpose third party translator tool.

MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation.

Fit generalized linear mixed-effects model

`glme = fitglme(tbl,formula)`

`glme = fitglme(tbl,formula,Name,Value)`

returns
a generalized linear mixed-effects model using additional options
specified by one or more `glme`

= fitglme(`tbl`

,`formula`

,`Name,Value`

)`Name,Value`

pair arguments.
For example, you can specify the distribution of the response, the
link function, or the covariance pattern of the random-effects terms.

Load the sample data.

`load mfr`

This simulated data is from a manufacturing company that operates 50 factories across the world, with each factory running a batch process to create a finished product. The company wants to decrease the number of defects in each batch, so it developed a new manufacturing process. To test the effectiveness of the new process, the company selected 20 of its factories at random to participate in an experiment: Ten factories implemented the new process, while the other ten continued to run the old process. In each of the 20 factories, the company ran five batches (for a total of 100 batches) and recorded the following data:

Flag to indicate whether the batch used the new process (

`newprocess`

)Processing time for each batch, in hours (

`time`

)Temperature of the batch, in degrees Celsius (

`temp`

)Categorical variable indicating the supplier of the chemical used in the batch (

`supplier`

)Number of defects in the batch (

`defects`

)

The data also includes `time_dev`

and `temp_dev`

, which represent the absolute deviation of time and temperature, respectively, from the process standard of 3 hours at 20 degrees Celsius.

Fit a generalized linear mixed-effects model using `newprocess`

, `time_dev`

, `temp_dev`

, and `supplier`

as fixed-effects predictors. Include a random-effects term for intercept grouped by `factory`

, to account for quality differences that might exist due to factory-specific variations. The response variable `defects`

has a Poisson distribution, and the appropriate link function for this model is log. Use the Laplace fit method to estimate the coefficients. Specify the dummy variable encoding as `'effects'`

, so the dummy variable coefficients sum to 0.

The number of defects can be modeled using a Poisson distribution

$${\text{defects}}_{ij}\sim \text{Poisson}\left({\mu}_{ij}\right).$$

This corresponds to the generalized linear mixed-effects model

$$\mathrm{log}\left({\mu}_{ij}\right)={\beta}_{0}+{\beta}_{1}{\text{newprocess}}_{ij}+{\beta}_{2}{\text{time}\text{\_}\text{dev}}_{ij}+{\beta}_{3}{\text{temp}\text{\_}\text{dev}}_{ij}+{\beta}_{4}{\text{supplier}\text{\_}\text{C}}_{ij}+{\beta}_{5}{\text{supplier}\text{\_}\text{B}}_{ij}+{b}_{i},$$

where

$${\text{defects}}_{ij}$$ is the number of defects observed in the batch produced by factory $$i$$ during batch $$j$$.

$${\mu}_{ij}$$ is the mean number of defects corresponding to factory $$i$$ (where $$i=1,2,...,20$$) during batch $$j$$ (where $$j=1,2,...,5$$).

$${\text{newprocess}}_{ij}$$, $${\text{time}\text{\_}\text{dev}}_{ij}$$, and $${\text{temp}\text{\_}\text{dev}}_{ij}$$ are the measurements for each variable that correspond to factory $$i$$ during batch $$j$$. For example, $${\text{newprocess}}_{ij}$$ indicates whether the batch produced by factory $$i$$ during batch $$j$$ used the new process.

$${\text{supplier}\text{\_}\text{C}}_{ij}$$ and $${\text{supplier}\text{\_}\text{B}}_{ij}$$ are dummy variables that use effects (sum-to-zero) coding to indicate whether company

`C`

or`B`

, respectively, supplied the process chemicals for the batch produced by factory $$i$$ during batch $$j$$.$${b}_{i}\sim N(0,{\sigma}_{b}^{2})$$ is a random-effects intercept for each factory $$i$$ that accounts for factory-specific variation in quality.

glme = fitglme(mfr,'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)', ... 'Distribution','Poisson','Link','log','FitMethod','Laplace', ... 'DummyVarCoding','effects');

Display the model.

disp(glme)

Generalized linear mixed-effects model fit by ML Model information: Number of observations 100 Fixed effects coefficients 6 Random effects coefficients 20 Covariance parameters 1 Distribution Poisson Link Log FitMethod Laplace Formula: defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1 | factory) Model fit statistics: AIC BIC LogLikelihood Deviance 416.35 434.58 -201.17 402.35 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 1.4689 0.15988 9.1875 94 9.8194e-15 'newprocess' -0.36766 0.17755 -2.0708 94 0.041122 'time_dev' -0.094521 0.82849 -0.11409 94 0.90941 'temp_dev' -0.28317 0.9617 -0.29444 94 0.76907 'supplier_C' -0.071868 0.078024 -0.9211 94 0.35936 'supplier_B' 0.071072 0.07739 0.91836 94 0.36078 Lower Upper 1.1515 1.7864 -0.72019 -0.015134 -1.7395 1.5505 -2.1926 1.6263 -0.22679 0.083051 -0.082588 0.22473 Random effects covariance parameters: Group: factory (20 Levels) Name1 Name2 Type Estimate '(Intercept)' '(Intercept)' 'std' 0.31381 Group: Error Name Estimate 'sqrt(Dispersion)' 1

The `Model information`

table displays the total number of observations in the sample data (100), the number of fixed- and random-effects coefficients (6 and 20, respectively), and the number of covariance parameters (1). It also indicates that the response variable has a `Poisson`

distribution, the link function is `Log`

, and the fit method is `Laplace`

.

`Formula`

indicates the model specification using Wilkinson’s notation.

The `Model fit statistics`

table displays statistics used to assess the goodness of fit of the model. This includes the Akaike information criterion (`AIC`

), Bayesian information criterion (`BIC`

) values, log likelihood (`LogLikelihood`

), and deviance (`Deviance`

) values.

The `Fixed effects coefficients`

table indicates that `fitglme`

returned 95% confidence intervals. It contains one row for each fixed-effects predictor, and each column contains statistics corresponding to that predictor. Column 1 (`Name`

) contains the name of each fixed-effects coefficient, column 2 (`Estimate`

) contains its estimated value, and column 3 (`SE`

) contains the standard error of the coefficient. Column 4 (`tStat`

) contains the $$t$$-statistic for a hypothesis test that the coefficient is equal to 0. Column 5 (`DF`

) and column 6 (`pValue`

) contain the degrees of freedom and $$p$$-value that correspond to the $$t$$-statistic, respectively. The last two columns (`Lower`

and `Upper`

) display the lower and upper limits, respectively, of the 95% confidence interval for each fixed-effects coefficient.

`Random effects covariance parameters`

displays a table for each grouping variable (here, only `factory`

), including its total number of levels (20), and the type and estimate of the covariance parameter. Here, `std`

indicates that `fitglme`

returns the standard deviation of the random effect associated with the factory predictor, which has an estimated value of 0.31381. It also displays a table containing the error parameter type (here, the square root of the dispersion parameter), and its estimated value of 1.

The standard display generated by `fitglme`

does not provide confidence intervals for the random-effects parameters. To compute and display these values, use `covarianceParameters`

.

`tbl`

— Input datatable | 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`

.

`formula`

— Formula for model specificationcharacter 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)'`

. The
formula is case sensitive. For a full description, see Formula.

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

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects'`

specifies
the response variable distribution as Poisson, the link function as
log, the fit method as Laplace, and dummy variable coding where the
coefficients sum to 0.`'BinomialSize'`

— Number of trials for binomial distribution1 (default) | scalar value | vector | variable name

Number of trials for binomial distribution, that is the sample
size, specified as the comma-separated pair consisting of a scalar
value, a vector of the same length as the response, or the name of
a variable in the input table. If you specify the name of a variable,
then the variable must be of the same length as the response. `BinomialSize`

applies
only when the `Distribution`

parameter is `'binomial'`

.

If `BinomialSize`

is a scalar value, that means
all observations have the same number of trials.

**Data Types: **`single`

| `double`

`'CheckHessian'`

— Indicator to check positive definiteness of Hessian`false`

(default) | `true`

Indicator to check the positive definiteness of the Hessian
of the objective function with respect to unconstrained parameters
at convergence, specified as the comma-separated pair consisting of `'CheckHessian'`

and
either `false`

or `true`

. Default
is `false`

.

Specify `'CheckHessian'`

as `true`

to
verify optimality of the solution or to determine if the model is
overparameterized in the number of covariance parameters.

If you specify `'FitMethod'`

as `'MPL'`

or `'REMPL'`

,
then the covariance of the fixed effects and the covariance parameters
is based on the fitted linear mixed-effects model from the final pseudo
likelihood iteration.

**Example: **`'CheckHessian',true`

`'CovarianceMethod'`

— Method to compute covariance of estimated parameters`'conditional'`

(default) | `'JointHessian'`

Method to compute covariance of estimated parameters, specified
as the comma-separated pair consisting of `'CovarianceMethod'`

and
either `'conditional'`

or `'JointHessian'`

.
If you specify `'conditional'`

, then `fitglme`

computes
a fast approximation to the covariance of fixed effects given the
estimated covariance parameters. It does not compute the covariance
of covariance parameters. If you specify `'JointHessian'`

,
then `fitglme`

computes the joint covariance of
fixed effects and covariance parameters via the observed information
matrix using the Laplacian loglikelihood.

If you specify `'FitMethod'`

as `'MPL'`

or `'REMPL'`

,
then the covariance of the fixed effects and the covariance parameters
is based on the fitted linear mixed-effects model from the final pseudo
likelihood iteration.

**Example: **`'CovarianceMethod','JointHessian'`

`'CovariancePattern'`

— Pattern of covariance matrix`'FullCholesky'`

| `'Isotropic'`

| `'Full'`

| `'Diagonal'`

| `'CompSymm'`

| square symmetric logical matrix | string array | cell array of character vectors or logical matricesPattern of the covariance matrix of the random effects, specified as the comma-separated pair
consisting of `'CovariancePattern'`

and
`'FullCholesky'`

, `'Isotropic'`

,
`'Full'`

, `'Diagonal'`

,
`'CompSymm'`

, a square symmetric logical matrix, a string array, or
a cell array containing character vectors or logical matrices.

If there are *R* random-effects terms, then the value of
`'CovariancePattern'`

must be a string array or cell array of
length *R*, where each element *r* of the array
specifies the pattern of the covariance matrix of the random-effects vector associated
with the *r*th random-effects term. The options for each element
follow.

Value | Description |
---|---|

`'FullCholesky'` | Full covariance matrix using the Cholesky parameterization. `fitglme` estimates
all elements of the covariance matrix. |

`'Isotropic'` |
Diagonal covariance matrix with equal variances. That is, off-diagonal elements of the covariance matrix are constrained to be 0, and the diagonal elements are constrained to be equal. For example, if there are three random-effects terms with an isotropic covariance structure, this covariance matrix looks like $$\left(\begin{array}{ccc}{\sigma}_{b}^{2}& 0& 0\\ 0& {\sigma}_{b}^{2}& 0\\ 0& 0& {\sigma}_{b}^{2}\end{array}\right)$$ where
σ |

`'Full'` | Full covariance matrix, using the log-Cholesky parameterization. `fitlme` estimates
all elements of the covariance matrix. |

`'Diagonal'` |
Diagonal covariance matrix. That is, off-diagonal elements of the covariance matrix are constrained to be 0. $$\left(\begin{array}{ccc}{\sigma}_{b1}^{2}& 0& 0\\ 0& {\sigma}_{b2}^{2}& 0\\ 0& 0& {\sigma}_{b3}^{2}\end{array}\right)$$ |

`'CompSymm'` |
Compound symmetry structure. That is, common variance along diagonals and equal correlation between all random effects. For example, if there are three random-effects terms with a covariance matrix having a compound symmetry structure, this covariance matrix looks like $$\left(\begin{array}{ccc}{\sigma}_{b1}^{2}& {\sigma}_{b1,b2}& {\sigma}_{b1,b2}\\ {\sigma}_{b1,b2}& {\sigma}_{b1}^{2}& {\sigma}_{b1,b2}\\ {\sigma}_{b1,b2}& {\sigma}_{b1,b2}& {\sigma}_{b1}^{2}\end{array}\right)$$ where
σ |

`PAT` | Square symmetric logical matrix. If `'CovariancePattern'` is
defined by the matrix `PAT` , and if ```
PAT(a,b)
= false
``` , then the `(a,b)` element of the
corresponding covariance matrix is constrained to be 0. |

For scalar random-effects terms, the default is `'Isotropic'`

.
Otherwise, the default is `'FullCholesky'`

.

**Example: **`'CovariancePattern','Diagonal'`

**Example: **`'CovariancePattern',{'Full','Diagonal'}`

**Data Types: **`char`

| `string`

| `logical`

| `cell`

`'DispersionFlag'`

— Indicator to compute dispersion parameter`false`

for `'binomial'`

and `'poisson'`

distributions (default) | `true`

Indicator to compute dispersion parameter for `'binomial'`

and `'poisson'`

distributions,
specified as the comma-separated pair consisting of `'DispersionFlag'`

and
one of the following.

Value | Description |
---|---|

`true` | Estimate a dispersion parameter when computing standard errors |

`false` | Use the theoretical value of `1.0` when computing
standard errors |

`'DispersionFlag'`

only applies if `'FitMethod'`

is `'MPL'`

or `'REMPL'`

.

The fitting function always estimates the dispersion for other distributions.

**Example: **`'DispersionFlag',true`

`'Distribution'`

— Distribution of the response variable`'Normal'`

(default) | `'Binomial'`

| `'Poisson'`

| `'Gamma'`

| `'InverseGaussian'`

Distribution of the response variable, specified as the comma-separated
pair consisting of `'Distribution'`

and one of the
following.

Value | Description |
---|---|

`'Normal'` | Normal distribution |

`'Binomial'` | Binomial distribution |

`'Poisson'` | Poisson distribution |

`'Gamma'` | Gamma distribution |

`'InverseGaussian'` | Inverse Gaussian distribution |

**Example: **`'Distribution','Binomial'`

`'DummyVarCoding'`

— Coding to use for dummy variables`'reference'`

(default) | `'effects'`

| `'full'`

Coding to use for dummy variables created from the categorical
variables, specified as the comma-separated pair consisting of `'DummyVarCoding'`

and
one of the following.

Value | Description |
---|---|

`'reference'` | Default. Coefficient for first category set to 0. |

`'effects'` | Coefficients sum to 0. |

`'full'` | One dummy variable for each category. |

**Example: **`'DummyVarCoding','effects'`

`'EBMethod'`

— Method used to approximate empirical Bayes estimates of random effects`'Auto'`

(default) | `'LineSearchNewton'`

| `'TrustRegion2D'`

| `'fsolve'`

Method used to approximate empirical Bayes estimates of random
effects, specified as the comma-separated pair consisting of `'EBMethod'`

and
one of the following.

`'Auto'`

`'LineSearchNewton'`

`'TrustRegion2D'`

`'fsolve'`

`'Auto'`

is similar to `'LineSearchNewton'`

but
uses a different convergence criterion and does not display iterative
progress. `'Auto'`

and `'LineSearchNewton'`

may
fail for non-canonical link functions. For non-canonical link functions, `'TrustRegion2D'`

or `'fsolve'`

are
recommended. You must have Optimization
Toolbox™ to use `'fsolve'`

.

**Example: **`'EBMethod','LineSearchNewton'`

`'EBOptions'`

— Options for empirical Bayes optimizationstructure

Options for empirical Bayes optimization, specified as the comma-separated
pair consisting of `'EBOptions'`

and a structure
containing the following.

Value | Description |
---|---|

`'TolFun'` | Relative tolerance on the gradient norm. Default is 1e-6. |

`'TolX'` | Absolute tolerance on the step size. Default is 1e-8. |

`'MaxIter'` | Maximum number of iterations. Default is 100. |

`'Display'` | `'off'` , `'iter'` , or `'final'` .
Default is `'off'` . |

If `EBMethod`

is `'Auto'`

and `'FitMethod'`

is `'Laplace'`

, `TolFun`

is
the relative tolerance on the linear predictor of the model, and the `'Display'`

option
does not apply.

If `'EBMethod'`

is `'fsolve'`

,
then `'EBOptions'`

must be specified as an object
created by `optimoptions('fsolve')`

.

**Data Types: **`struct`

`'Exclude'`

— Indices for rows to excludeuse all rows without

`NaNs`

(default) | vector of integer or logical valuesIndices for rows to exclude from the generalized linear mixed-effects
model in the data, specified as the comma-separated pair consisting
of `'Exclude'`

and a vector of integer or logical
values.

For example, you can exclude the 13th and 67th rows from the fit as follows.

**Example: **`'Exclude',[13,67]`

**Data Types: **`single`

| `double`

| `logical`

`'FitMethod'`

— Method for estimating model parameters`'MPL'`

(default) | `'REMPL'`

| `'Laplace'`

| `'ApproximateLaplace`

Method for estimating model parameters, specified as the comma-separated
pair consisting of `'FitMethod'`

and one of the following.

`'MPL'`

— Maximum pseudo likelihood`'REMPL'`

— Restricted maximum pseudo likelihood`'Laplace'`

— Maximum likelihood using Laplace approximation`'ApproximateLaplace'`

— Maximum likelihood using approximate Laplace approximation with fixed effects profiled out

**Example: **`'FitMethod','REMPL'`

`'InitPLIterations'`

— Initial number of pseudo likelihood iterations10 (default) | integer value in the range [1,∞)

Initial number of pseudo likelihood iterations used to initialize
parameters for `ApproximateLaplace`

and `Laplace`

fit
methods, specified as the comma-separated pair consisting of `'InitPLIterations'`

and
an integer value greater than or equal to 1.

**Data Types: **`single`

| `double`

`'Link'`

— Link function`'identity'`

| `'log'`

| `'logit'`

| `'probit'`

| `'comploglog'`

| `'reciprocal'`

| scalar value | structureLink function, specified as the comma-separated pair consisting
of `'Link'`

and one of the following.

Value | Description |
---|---|

`'identity'` |
This is the default for the normal distribution. |

`'log'` |
This is the default for the Poisson distribution. |

`'logit'` |
This is the default for the binomial distribution. |

`'loglog'` | `g(mu) = log(-log(mu))` |

`'probit'` | `g(mu) = norminv(mu)` |

`'comploglog'` | `g(mu) = log(-log(1-mu))` |

`'reciprocal'` | `g(mu) = mu.^(-1)` |

Scalar value `P` | `g(mu) = mu.^P` |

Structure `S` | A structure containing four fields whose values are function handles with the following names: `S.Link` — Link function`S.Derivative` — Derivative`S.SecondDerivative` — Second derivative`S.Inverse` — Inverse of link
Specification of |

The default link function used by `fitglme`

is
the canonical link that depends on the distribution of the response.

Response Distribution | Canonical Link Function |
---|---|

`'Normal'` | `'identity'` |

`'Binomial'` | `'logit'` |

`'Poisson'` | `'log'` |

`'Gamma'` | `-1` |

`'InverseGaussian'` | `-2` |

**Example: **`'Link','log'`

**Data Types: **`char`

| `string`

| `single`

| `double`

| `struct`

`'MuStart'`

— Starting value for conditional meanscalar value

Starting value for conditional mean, specified as the comma-separated
pair consisting of `'MuStart'`

and a scalar value.
Valid values are as follows.

Response Distribution | Valid Values |
---|---|

`'Normal'` | `(-Inf,Inf)` |

`'Binomial'` | `(0,1)` |

`'Poisson'` | `(0,Inf)` |

`'Gamma'` | `(0,Inf)` |

`'InverseGaussian'` | `(0,Inf)` |

**Data Types: **`single`

| `double`

`'Offset'`

— Offset`zeros(n,1)`

(default) | Offset, specified as the comma-separated pair consisting of `'Offset'`

and
an *n*-by-1 vector of scalar values, where *n* is
the length of the response vector. You can also specify the variable
name of an *n*-by-1 vector of scalar values. `'Offset'`

is
used as an additional predictor that has a coefficient value fixed
at `1.0`

.

**Data Types: **`single`

| `double`

`'Optimizer'`

— Optimization algorithm`'quasinewton'`

(default) | `'fminsearch'`

| `'fminunc'`

Optimization algorithm, specified as the comma-separated pair
consisting of `'Optimizer'`

and either of the following.

Value | Description |
---|---|

`'quasinewton'` | Uses a trust region based quasi-Newton optimizer. You can change
the options of the algorithm using `statset('fitglme')` .
If you do not specify the options, then `fitglme` uses
the default options of `statset('fitglme')` . |

`'fminsearch'` | Uses a derivative-free Nelder-Mead method. You can change the
options of the algorithm using `optimset('fminsearch')` .
If you do not specify the options, then `fitglme` uses
the default options of `optimset('fminsearch')` . |

`'fminunc'` | Uses a line search-based quasi-Newton method. You must have Optimization
Toolbox to
specify this option. You can change the options of the algorithm using `optimoptions('fminunc')` .
If you do not specify the options, then `fitglme` uses
the default options of `optimoptions('fminunc')` with `'Algorithm'` set
to `'quasi-newton'` . |

**Example: **`'Optimizer','fminsearch'`

`'OptimizerOptions'`

— Options for optimization algorithmstructure returned by

`statset`

| structure returned by `optimset`

| object returned by `optimoptions`

Options for the optimization algorithm, specified as the comma-separated
pair consisting of `'OptimizerOptions'`

and a structure
returned by `statset('fitglme')`

, a structure created
by `optimset('fminsearch')`

, or an object returned
by `optimoptions('fminunc')`

.

If

`'Optimizer'`

is`'fminsearch'`

, then use`optimset('fminsearch')`

to change the options of the algorithm. If`'Optimizer'`

is`'fminsearch'`

and you do not supply`'OptimizerOptions'`

, then the defaults used in`fitglme`

are the default options created by`optimset('fminsearch')`

.If

`'Optimizer'`

is`'fminunc'`

, then use`optimoptions('fminunc')`

to change the options of the optimization algorithm. See`optimoptions`

for the options`'fminunc'`

uses. If`'Optimizer'`

is`'fminunc'`

and you do not supply`'OptimizerOptions'`

, then the defaults used in`fitglme`

are the default options created by`optimoptions('fminunc')`

with`'Algorithm'`

set to`'quasi-newton'`

.If

`'Optimizer'`

is`'quasinewton'`

, then use`statset('fitglme')`

to change the optimization parameters. If`'Optimizer'`

is`'quasinewton'`

and you do not change the optimization parameters using`statset`

, then`fitglme`

uses the default options created by`statset('fitglme')`

.

The `'quasinewton'`

optimizer uses the following
fields in the structure created by `statset('fitglme')`

.

`TolFun`

— Relative tolerance on gradient of objective function`1e-6`

(default) | positive scalar valueRelative tolerance on the gradient of the objective function, specified as a positive scalar value.

`TolX`

— Absolute tolerance on step size`1e-12`

(default) | positive scalar valueAbsolute tolerance on the step size, specified as a positive scalar value.

`MaxIter`

— Maximum number of iterations allowed`10000`

(default) | positive scalar valueMaximum number of iterations allowed, specified as a positive scalar value.

`Display`

— Level of display`'off'`

(default) | `'iter'`

| `'final'`

Level of display, specified as one of `'off'`

, `'iter'`

,
or `'final'`

.

`'PLIterations'`

— Maximum number of pseudo likelihood iterations`100`

(default) | positive integer valueMaximum number of pseudo likelihood (PL) iterations, specified
as the comma-separated pair consisting of `'PLIterations'`

and
a positive integer value. PL is used for fitting the model if `'FitMethod'`

is `'MPL'`

or `'REMPL'`

.
For other `'FitMethod'`

values, PL iterations are
used to initialize parameters for subsequent optimization.

**Example: **`'PLIterations',200`

**Data Types: **`single`

| `double`

`'PLTolerance'`

— Relative tolerance factor for pseudo likelihood iterations`1e–08`

(default) | positive scalar valueRelative tolerance factor for pseudo likelihood iterations,
specified as the comma-separated pair consisting of `'PLTolerance'`

and
a positive scalar value.

**Example: **`'PLTolerance',1e-06`

**Data Types: **`single`

| `double`

`'StartMethod'`

— Method to start iterative optimization`'default'`

(default) | `'random'`

Method to start iterative optimization, specified as the comma-separated
pair consisting of `'StartMethod'`

and either of
the following.

Value | Description |
---|---|

`'default'` | An internally defined default value |

`'random'` | A random initial value |

**Example: **`'StartMethod','random'`

`'UseSequentialFitting'`

— Initial fitting type`false`

(default) | `true`

, specified as the comma-separated pair consisting of `'UseSequentialFitting'`

and
either `false`

or `true`

. If `'UseSequentialFitting'`

is `false`

,
all maximum likelihood methods are initialized using one or more pseudo
likelihood iterations. If `'UseSequentialFitting'`

is `true`

,
the initial values from pseudo likelihood iterations are refined using `'ApproximateLaplace'`

for `'Laplace'`

fitting.

**Example: **`'UseSequentialFitting',true`

`'Verbose'`

— Indicator to display optimization process on screen`0`

(default) | `1`

| `2`

Indicator to display the optimization process on screen, specified
as the comma-separated pair consisting of `'Verbose'`

and `0`

, `1`

,
or `2`

. If `'Verbose'`

is specified
as `1`

or `2`

, then `fitglme`

displays
the progress of the iterative model-fitting process. Specifying `'Verbose'`

as `2`

displays
iterative optimization information from the individual pseudo likelihood
iterations. Specifying `'Verbose'`

as `1`

omits
this display.

The setting for `'Verbose'`

overrides the field `'Display'`

in `'OptimizerOptions'`

.

**Example: **`'Verbose',1`

`'Weights'`

— Observation weightsvector of nonnegative scalar values

Observation weights, specified as the comma-separated pair consisting
of `'Weights'`

and an *n*-by-1 vector
of nonnegative scalar values, where *n* is the number
of observations. If the response distribution is binomial or Poisson,
then `'Weights'`

must be a vector of positive integers.

**Data Types: **`single`

| `double`

`glme`

— Generalized linear mixed-effects model`GeneralizedLinearMixedModel`

objectGeneralized linear mixed-effects model, specified as a `GeneralizedLinearMixedModel`

object.
For properties and methods of this object, see `GeneralizedLinearMixedModel`

.

In general, a formula for model specification is a character vector or string
scalar of the form `'y ~ terms'`

. For the generalized 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, string 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 generalized 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. |

You clicked a link that corresponds to this MATLAB command:

Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.

Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .

Select web siteYou can also select a web site from the following list:

Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.

- América Latina (Español)
- Canada (English)
- United States (English)

- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)

- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)