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# refit

Refit generalized linear mixed-effects model

## Syntax

``glmenew = refit(glme,ynew)``

## Description

example

````glmenew = refit(glme,ynew)` returns a refitted generalized linear mixed-effects model, `glmenew`, based on the input model `glme`, using a new response vector, `ynew`.```

## Input Arguments

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

New response vector, specified as an n-by-1 vector of scalar values, where n is the number of observations used to fit `glme`.

For an observation i with prior weights wip and binomial size ni (when applicable), the response values yi contained in `ynew` can have the following values.

DistributionPermitted ValuesNotes
`Binomial`

`$\left\{0,\frac{1}{{w}_{i}^{p}{n}_{i}},\frac{2}{{w}_{i}^{p}{n}_{i}},.\dots ,1\right\}$`

wip and ni are integer values > 0
`Poisson`

`$\left\{0,\frac{1}{{w}_{i}^{p}},\frac{2}{{w}_{i}^{p}},\cdots ,1\right\}$`

wip is an integer value > 0
`Gamma`(0,∞)wip ≥ 0
`InverseGaussian`(0,∞)wip ≥ 0
`Normal`(–∞,∞)wip ≥ 0

You can access the prior weights property wip using dot notation.

`glme.ObservationInfo.Weights`

Data Types: `single` | `double`

## Output Arguments

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Generalized linear mixed-effects model, returned as a `GeneralizedLinearMixedModel` object. `glmenew` is an updated version of the generalized linear mixed-effects model `glme`, refit to the values in the response vector `ynew`.

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

## Examples

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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

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

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');`

Use `random` to simulate a new response vector from the fitted model.

```rng(0,'twister'); % For reproducibility ynew = random(glme);```

Refit the model using the new response vector.

`glme = refit(glme,ynew)`
```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 469.24 487.48 -227.62 455.24 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue {'(Intercept)'} 1.5738 0.18674 8.4276 94 4.0158e-13 {'newprocess' } -0.21089 0.2306 -0.91455 94 0.36277 {'time_dev' } -0.13769 0.77477 -0.17772 94 0.85933 {'temp_dev' } 0.24339 0.84657 0.2875 94 0.77436 {'supplier_C' } -0.12102 0.07323 -1.6526 94 0.10175 {'supplier_B' } 0.098254 0.066943 1.4677 94 0.14551 Lower Upper 1.203 1.9445 -0.66875 0.24696 -1.676 1.4006 -1.4375 1.9243 -0.26642 0.024381 -0.034662 0.23117 Random effects covariance parameters: Group: factory (20 Levels) Name1 Name2 Type Estimate {'(Intercept)'} {'(Intercept)'} {'std'} 0.46587 Group: Error Name Estimate {'sqrt(Dispersion)'} 1 ```

## Tips

• You can use `refit` and `random` to conduct a simulated likelihood ratio test or parametric bootstrap.

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