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
This corresponds to the generalized linear mixed-effects model
where
is the number of defects observed in the batch produced by factory during batch .
is the mean number of defects corresponding to factory (where ) during batch (where ).
, , and are the measurements for each variable that correspond to factory during batch . For example, indicates whether the batch produced by factory during batch used the new process.
and 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 during batch .
is a random-effects intercept for each factory that accounts for factory-specific variation in quality.
Compute and display the estimated fixed-effects coefficient values and related statistics.
stats =
Fixed effect coefficients: DFMethod = 'residual', Alpha = 0.05
Name Estimate SE tStat DF pValue Lower Upper
{'(Intercept)'} 1.4689 0.15988 9.1875 94 9.8194e-15 1.1515 1.7864
{'newprocess' } -0.36766 0.17755 -2.0708 94 0.041122 -0.72019 -0.015134
{'time_dev' } -0.094521 0.82849 -0.11409 94 0.90941 -1.7395 1.5505
{'temp_dev' } -0.28317 0.9617 -0.29444 94 0.76907 -2.1926 1.6263
{'supplier_C' } -0.071868 0.078024 -0.9211 94 0.35936 -0.22679 0.083051
{'supplier_B' } 0.071072 0.07739 0.91836 94 0.36078 -0.082588 0.22473
The returned results indicate, for example, that the estimated coefficient for temp_dev
is –0.28317. Its large -value, 0.76907, indicates that it is not a statistically significant predictor at the 5% significance level. Additionally, the confidence interval boundaries Lower
and Upper
indicate that the 95% confidence interval for the coefficient for temp_dev
is [-2.1926 , 1.6263]. This interval contains 0, which supports the conclusion that temp_dev
is not statistically significant at the 5% significance level.