Load the sample data.

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.

Use `random`

to simulate a new response vector from the fitted model.

Refit the model using the new response vector.

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