# refit

Refit generalized linear mixed-effects model

## 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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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
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.