# fitglme

Fit generalized linear mixed-effects model

## Syntax

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

## Description

example

````glme = fitglme(tbl,formula)` returns a generalized linear mixed-effects model, `glme`. The model is specified by `formula` and fitted to the predictor variables in the table or dataset array, `tbl`.```
````glme = fitglme(tbl,formula,Name,Value)` returns a generalized linear mixed-effects model using additional options specified by one or more `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.```

## Examples

collapse all

`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 $j$.

• ${b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)$ 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`.

## Input Arguments

collapse all

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 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)'`

### Name-Value Pair Arguments

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`.

Example: `'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.

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`

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`

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'`

Pattern 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 rth random-effects term. The options for each element follow.

ValueDescription
`'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 σ21 is the common variance of the random-effects terms.

`'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 σ2b1 is the common variance of the random-effects terms and σb1,b2 is the common covariance between any two random-effects term .

`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`

Indicator to compute dispersion parameter for `'binomial'` and `'poisson'` distributions, specified as the comma-separated pair consisting of `'DispersionFlag'` and one of the following.

ValueDescription
`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 of the response variable, specified as the comma-separated pair consisting of `'Distribution'` and one of the following.

ValueDescription
`'Normal'`Normal distribution
`'Binomial'`Binomial distribution
`'Poisson'`Poisson distribution
`'Gamma'`Gamma distribution
`'InverseGaussian'`Inverse Gaussian distribution

Example: `'Distribution','Binomial'`

Coding to use for dummy variables created from the categorical variables, specified as the comma-separated pair consisting of `'DummyVarCoding'` and one of the variables in this table.

ValueDescription
`'reference'` (default)`fitglme` creates dummy variables with a reference group. This scheme treats the first category as a reference group and creates one less dummy variables than the number of categories. You can check the category order of a categorical variable by using the `categories` function, and change the order by using the `reordercats` function.
`'effects'``fitglme` creates dummy variables using effects coding. This scheme uses –1 to represent the last category. This scheme creates one less dummy variables than the number of categories.
`'full'``fitglme` creates full dummy variables. This scheme creates one dummy variable for each category.

For more details about creating dummy variables, see Automatic Creation of Dummy Variables.

Example: `'DummyVarCoding','effects'`

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'`

Options for empirical Bayes optimization, specified as the comma-separated pair consisting of `'EBOptions'` and a structure containing the following.

ValueDescription
`'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`

Indices 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`

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'`

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`

Starting value for conditional mean, specified as the comma-separated pair consisting of `'MuStart'` and a scalar value. Valid values are as follows.

Response DistributionValid Values
`'Normal'``(-Inf,Inf)`
`'Binomial'``(0,1)`
`'Poisson'``(0,Inf)`
`'Gamma'``(0,Inf)`
`'InverseGaussian'``(0,Inf)`

Data Types: `single` | `double`

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`

Optimization algorithm, specified as the comma-separated pair consisting of `'Optimizer'` and either of the following.

ValueDescription
`'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'`

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')`.

Relative tolerance on the gradient of the objective function, specified as a positive scalar value.

Absolute tolerance on the step size, specified as a positive scalar value.

Maximum number of iterations allowed, specified as a positive scalar value.

Level of display, specified as one of `'off'`, `'iter'`, or `'final'`.

Maximum 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`

Relative 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`

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

ValueDescription
`'default'`An internally defined default value
`'random'`A random initial value

Example: `'StartMethod','random'`

, 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`

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`

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`

## Output Arguments

collapse all

Generalized linear mixed-effects model, specified as a `GeneralizedLinearMixedModel` object. For properties and methods of this object, see `GeneralizedLinearMixedModel`.

collapse all

### Formula

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, `Xj`, which can be continuous or grouping variables

• Grouping variables, `g1`, `g2`, ..., `gR`,

where the grouping variables in `Xj` and `gr` can be categorical, logical, character arrays, string arrays, or cell arrays of character vectors.

Then, in a formula of the form, ```'y ~ fixed + (random1|g1) + ... + (randomR|gR)'```, 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 NotationFactors in Standard Notation
`1`Constant (intercept) term
`X^k`, where `k` is a positive integer`X`, `X2`, ..., `Xk`
`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:

`'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.