# fitted

**Class: **GeneralizedLinearMixedModel

Fitted responses from generalized linear mixed-effects model

## Description

returns
the fitted response with additional options specified by one or more
name-value pair arguments. For example, you can specify to compute
the marginal fitted response.`mufit`

= fitted(`glme`

,`Name,Value`

)

## Input Arguments

`glme`

— Generalized linear mixed-effects model

`GeneralizedLinearMixedModel`

object

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

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

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`Conditional`

— Indicator for conditional response

`true`

(default) | `false`

Indicator for conditional response, specified as the comma-separated
pair consisting of `'Conditional'`

and one of the
following.

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

`true` | Contributions from both fixed effects and random effects (conditional) |

`false` | Contribution from only fixed effects (marginal) |

To obtain fitted marginal response values, `fitted`

computes
the conditional mean of the response with the empirical Bayes predictor
vector of random effects *b* set equal to 0. For
more information, see Conditional and Marginal Response

**Example: **`'Conditional',false`

## Output Arguments

`mufit`

— Fitted response values

*n*-by-1 vector

Fitted response values, returned as an *n*-by-1
vector, where *n* is the number of observations.

## Examples

### Plot Observed Versus Fitted Values

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');

Generate the fitted conditional mean values for the model.

mufit = fitted(glme);

Create a scatterplot of the observed values versus fitted values.

figure scatter(mfr.defects,mufit) title('Residuals versus Fitted Values') xlabel('Fitted Values') ylabel('Residuals')

## More About

### Conditional and Marginal Response

A *conditional response* includes
contributions from both fixed- and random-effects predictors. A *marginal
response* includes contribution from only fixed effects.

Suppose the generalized linear mixed-effects model `glme`

has
an *n*-by-*p* fixed-effects design
matrix `X`

and an *n*-by-*q* random-effects
design matrix `Z`

. Also, suppose the estimated *p*-by-1
fixed-effects vector is $$\widehat{\beta}$$,
and the *q*-by-1 empirical Bayes predictor vector
of random effects is $$\widehat{b}$$.

The fitted conditional response corresponds to the `'Conditional',true`

name-value
pair argument, and is defined as

$${\widehat{\mu}}_{cond}={g}^{-1}\left({\widehat{\eta}}_{ME}\right)\text{\hspace{0.05em}},$$

where $${\widehat{\eta}}_{ME}$$ is the linear predictor including the fixed- and random-effects of the generalized linear mixed-effects model

$${\widehat{\eta}}_{ME}=X\widehat{\beta}+Z\widehat{b}+\delta \text{\hspace{0.17em}}.$$

The fitted marginal response corresponds to the `'Conditional',false`

name-value
pair argument, and is defined as

$${\widehat{\mu}}_{mar}={g}^{-1}\left({\widehat{\eta}}_{FE}\right)\text{\hspace{0.05em}},$$

where$${\widehat{\eta}}_{FE}$$ is the linear predictor including only the fixed-effects portion of the generalized linear mixed-effects model

$${\widehat{\eta}}_{FE}=X\widehat{\beta}+\delta \text{\hspace{0.17em}}.$$

## See Also

`GeneralizedLinearMixedModel`

| `fitglme`

| `residuals`

| `response`

| `designMatrix`

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