Fit generalized linear mixed-effects model

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. |

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