Navigate to the folder containing the sample data. Load the sample data.

cd(matlabroot) cd('help/toolbox/stats/examples') loadmfr

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 of the chemical used in the batch (supplier)

• Number of defects in the batch (defects)

The data also includestime_devandtemp_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 usingnewprocess,time_dev,temp_dev, andsupplieras fixed-effects predictors. Include a random-effects term for intercept grouped byfactory, to account for quality differences that might exist due to factory-specific variations. The response variabledefectshas 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 factoryduring batch.

• is the mean number of defects corresponding to factory(where) during batch(where).

• ,, andare the measurements for each variable that correspond to factoryduring batch. For example,indicates whether the batch produced by factoryduring batchused the new process.

• andare dummy variables that use effects (sum-to-zero) coding to indicate whether companyCorB, respectively, supplied the process chemicals for the batch produced by factoryduring batch.

• is a random-effects intercept for each factorythat 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');

Display the model.

disp(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 416.35 434.58 -201.17 402.35 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 1.4689 0.15988 9.1875 94 9.8194e-15 'newprocess' -0.36766 0.17755 -2.0708 94 0.041122 'time_dev' -0.094521 0.82849 -0.11409 94 0.90941 'temp_dev' -0.28317 0.9617 -0.29444 94 0.76907 'supplier_C' -0.071868 0.078024 -0.9211 94 0.35936 'supplier_B' 0.071072 0.07739 0.91836 94 0.36078 Lower Upper 1.1515 1.7864 -0.72019 -0.015134 -1.7395 1.5505 -2.1926 1.6263 -0.22679 0.083051 -0.082588 0.22473 Random effects covariance parameters: Group: factory (20 Levels) Name1 Name2 Type Estimate '(Intercept)' '(Intercept)' 'std' 0.31381 Group: Error Name Estimate 'sqrt(Dispersion)' 1

TheModel informationtable displays the total number of observations in the sample data (100), the number of fixed- and random-effects coefficients (6 and 20, respectively), and the number of covariance parameters (1). It also indicates that the response variable has aPoissondistribution, the link function isLog, and the fit method isLaplace.

Formulaindicates the model specification using Wilkinson's notation.

TheModel fit statisticstable displays statistics used to assess the goodness of fit of the model. This includes the Akaike information criterion (AIC), Bayesian information criterion (BIC) values, log likelihood (LogLikelihood), and deviance (Deviance) values.

TheFixed effects coefficientstable indicates thatfitglmereturned 95% confidence intervals. It contains one row for each fixed-effects predictor, and each column contains statistics corresponding to that predictor. Column 1 (Name) contains the name of each fixed-effects coefficient, column 2 (Estimate) contains its estimated value, and column 3 (SE) contains the standard error of the coefficient. Column 4 (tStat) contains the-statistic for a hypothesis test that the coefficient is equal to 0. Column 5 (DF) and column 6 (pValue) contain the degrees of freedom and-value that correspond to the-statistic, respectively. The last two columns (LowerandUpper) display the lower and upper limits, respectively, of the 95% confidence interval for each fixed-effects coefficient.

Random effects covariance parametersdisplays a table for each grouping variable (here, onlyfactory),包括水平(20)的总数,和the type and estimate of the covariance parameter. Here,stdindicates thatfitglmereturns the standard deviation of the random effect associated with the factory predictor, which has an estimated value of 0.31381. It also displays a table containing the error parameter type (here, the square root of the dispersion parameter), and its estimated value of 1.

The standard display generated byfitglmedoes not provide confidence intervals for the random-effects parameters. To compute and display these values, usecovarianceParameters.