**Class: **GeneralizedLinearMixedModel

Fixed- and random-effects design matrices

`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`

.

`gnumbers`

— Grouping variable numbersarray of integer values

Grouping variable numbers, specified as an array of integer
values containing elements in the range [1,*R*],
where *R* is the length of the cell array that contains
the grouping variables for the generalized linear mixed-effects model `glme`

.

For example, you can specify the grouping variables g_{1},
g_{3}, and g_{r} as `[1,3,r]`

.

**Data Types: **`single`

| `double`

`D`

— Design matrixmatrix

Design matrix of a generalized linear mixed-effects model `glme`

returned
as one of the following:

Fixed-effects design matrix —

*n*-by-*p*matrix consisting of the fixed-effects design matrix of`glme`

, where*n*is the number of observations and*p*is the number of fixed-effects terms.Random-effects design matrix —

*n*-by-*k*matrix, consisting of the random-effects design matrix of`glme`

. Here,*k*is equal to`length(B)`

, where`B`

is the random-effects coefficients vector of generalized linear mixed-effects model`glme`

. The random-effects design matrix is returned as a sparse matrix. For more information, see Sparse Matrices (MATLAB).If

`glme`

has*R*grouping variables g_{1}, g_{2}, ..., g_{R}, with levels*m*_{1},*m*_{2}, ...,*m*_{R}, respectively, and if*q*_{1},*q*_{2}, ...,*q*_{R}are the lengths of the random-effects vectors that are associated with g_{1}, g_{2}, ..., g_{R}, respectively, then`B`

is a column vector of length*q*_{1}**m*_{1}+*q*_{2}**m*_{2}+ ... +*q*_{R}**m*_{R}.`B`

is made by concatenating the empirical Bayes predictors of random effects vectors corresponding to each level of each grouping variable as`[g`

._{1}level_{1}; g_{1}level_{2}; ...; g_{1}level_{m1}; g_{2}level_{1}; g_{2}level_{2}; ...; g_{2}level_{m2}; ...; g_{R}level_{1}; g_{R}level_{2}; ...; g_{R}level_{mR}]'

**Data Types: **`single`

| `double`

`Dsub`

— Submatrix of random-effects design matrixmatrix

Submatrix of random-effects design matrix that corresponds to
the grouping variables specified by `gnumbers`

,
returned as an *n*-by-*k* matrix,
where *k* is length of the column vector `Bsub`

.

`Bsub`

contains the concatenated empirical
Bayes predictors of random-effects vectors, corresponding to each
level of the grouping variables, specified by `gnumbers`

.

If, for example, `gnumbers`

is `[1,3,r]`

,
this corresponds to the grouping variables g_{1},
g_{3}, and g_{r}.
Then, `Bsub`

contains the empirical Bayes predictors
of random-effects vectors corresponding to each level of the grouping
variables g_{1}, g_{3}, and
g* _{r}*, such as

`[g`

. _{1}level_{1};
g_{1}level_{2}; ...; g_{1}level_{m}_{1};
g_{3}level_{1}; g_{3}level_{2};
...; g_{3}level_{m}_{3};
g* _{r}*level

Thus, `Dsub*Bsub`

represents the contribution
of all random effects corresponding to grouping variables g_{1},
g_{3}, and g* _{r}* to
the response of

`glme`

.If `gnumbers`

is empty, then `Dsub`

is
the full random-effects design matrix.

**Data Types: **`single`

| `double`

`gnames`

— Names of grouping variablesNames of grouping variables corresponding to the integers in `gnumbers`

if
the design type is `'Random'`

, returned as a *k*-by-1
cell array. If the design type is `'Fixed'`

, then `gnames`

is
an empty matrix `[]`

.

**Data Types: **`cell`

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 (

`A`

,`B`

, or`C`

) 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

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

This corresponds to the generalized linear mixed-effects model

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

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');

Extract the fixed-effects design matrix and display rows 1 through 10.

```
Dfe = designMatrix(glme,'Fixed');
disp(Dfe(1:10,:))
```

1.0000 0 0.1834 0.2259 1.0000 0 1.0000 0 0.3035 0.0725 0 1.0000 1.0000 0 0.0717 0.1630 1.0000 0 1.0000 0 0.1069 0.0809 -1.0000 -1.0000 1.0000 0 0.0241 0.0319 1.0000 0 1.0000 0 0.1214 0.1114 0 1.0000 1.0000 0 0.0033 0.0553 1.0000 0 1.0000 0 0.2350 0.0616 1.0000 0 1.0000 0 0.0488 0.0177 0 1.0000 1.0000 0 0.1148 0.0105 1.0000 0

Column 1 of the fixed-effects design matrix `Dfe`

contains the constant term. Column 2, 3, and 4 contain the `newprocess`

, `time_dev`

, and `temp_dev`

terms, respectively. Columns 5 and 6 contain dummy variables for `supplier_C`

and `supplier_B`

, respectively.

Extract the random-effects design matrix and display rows 1 through 10.

```
Dre = designMatrix(glme,'Random');
disp(Dre(1:10,:))
```

(1,1) 1 (2,1) 1 (3,1) 1 (4,1) 1 (5,1) 1 (6,2) 1 (7,2) 1 (8,2) 1 (9,2) 1 (10,2) 1

Convert the sparse matrix `Dre`

to a full matrix and display rows 1 through 10.

full(Dre(1:10,:))

`ans = `*10×20*
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Each column corresponds to a level of the grouping variable `factory`

.

`GeneralizedLinearMixedModel`

| `fitglme`

| `fitted`

| `residuals`

| `response`

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