Documentation

disp

Display generalized linear mixed-effects model

Description

example

disp(glme) displays fitted generalized linear mixed-effects model glme.

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.

Examples

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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 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');

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
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

The Model information table 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 a Poisson distribution, the link function is Log, and the fit method is Laplace.

Formula indicates the model specification using Wilkinson's notation.

The Model fit statistics table 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.

The Fixed effects coefficients table indicates that fitglme returned 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 $t$-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 $p$-value that correspond to the $t$-statistic, respectively. The last two columns (Lower and Upper) display the lower and upper limits, respectively, of the 95% confidence interval for each fixed-effects coefficient.

Random effects covariance parameters displays a table for each grouping variable (here, only factory), including its total number of levels (20), and the type and estimate of the covariance parameter. Here, std indicates that fitglme returns 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 by fitglme does not provide confidence intervals for the random-effects parameters. To compute and display these values, use covarianceParameters.