This example shows how to fit a generalized linear mixed-effects model (GLME) to sample data.
Load the sample data.
A manufacturing company operates 50 factories across the world, and each runs 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. However, the company wants to test the new process in select factories to ensure that it is effective before rolling it out to all 50 locations.
To test whether the new process significantly reduces the number of defects in each batch, the company selected 20 of its factories at random to participate in an experiment. Ten factories implemented the new process, while the other ten used the old process.
In each of the 20 factories (i = 1, 2, ..., 20), the company ran five batches (j = 1, 2, ..., 5) and recorded the following data in the table
Flag to indicate use of the new process:
If the batch used the new process, then
If the batch used the old process, then
Processing time for the batch, in hours
Temperature of the batch, in degrees Celsius
Supplier of the chemical used in the batch
supplier is a categorical variable with
C, where each level represents one of the
Number of defects in the batch (defects)
The data also includes
temp_dev, which represent the absolute deviation of time
and temperature, respectively, from the process standard of 3 hours and 20
degrees Celsius. The response variable
defects has a Poisson
distribution. This is simulated data.
The company wants to determine whether the new process significantly reduces the number of defects in each batch, while accounting for quality differences that might exist due to factory-specific variations in time, temperature, and supplier. The number of defects per batch can be modeled using a Poison distribution:
Use a generalized linear mixed-effects model to model the number of defects per batch:
defectsij is the number of defects observed in the batch produced by factory i during batch j.
μij is the mean number of defects corresponding to factory i (where i = 1, 2, ..., 20) during batch j (where j = 1, 2, ..., 5).
newprocessij, time_devij, and temp_devij are the measurements for each variable that correspond to factory i during batch j. For example, newprocessij indicates whether the batch produced by factory i during batch j used the new process.
supplier_Cij and supplier_Bij are dummy variables that use effects (sum-to-zero)
coding to indicate whether company
B, respectively, supplied the process chemicals
for the batch produced by factory i during batch
bi ~ N(0,σb2) is a random-effects intercept for each factory i that accounts for factory-specific variation in quality.
Fit a generalized linear mixed-effects model using
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
glme = fitglme(mfr,... 'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)',... 'Distribution','Poisson','Link','log','FitMethod','Laplace',... 'DummyVarCoding','effects')
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
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
Formula indicates the model specification using Wilkinson’s
Model fit statistics table displays statistics used to
assess the goodness of fit of the model. This includes the Akaike information
AIC), Bayesian information criterion
BIC) values, log likelihood
LogLikelihood), and deviance
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 (
contains its estimated value, and column 3 (
SE) contains the
standard error of the coefficient. Column 4 (
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
Upper) display the lower
and upper limits, respectively, of the 95% confidence interval for each
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
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
To determine whether the random-effects intercept grouped by
factory is statistically significant, compute the
confidence intervals for the estimated covariance parameter.
[psi,dispersion,stats] = covarianceParameters(glme);
covarianceParameters returns the estimated covariance
psi, the estimated dispersion parameter
dispersion, and a cell array of related statistics
stats. The first cell of
contains statistics for
factory, while the second cell
contains statistics for the dispersion parameter.
Display the first cell of
stats to see the confidence
intervals for the estimated covariance parameter for
ans = Covariance Type: Isotropic Group Name1 Name2 Type factory '(Intercept)' '(Intercept)' 'std' Estimate Lower Upper 0.31381 0.19253 0.51148
Upper display the
default 95% confidence interval for the estimated covariance parameter for
factory. Because the interval [0.19253,0.51148] does not
contain 0, the random-effects intercept is significant at the 5% significance
level. Therefore, the random effect due to factory-specific variation must be
considered before drawing any conclusions about the effectiveness of the new
Compare the mixed-effects model that includes a random-effects intercept
factory with a model that does not include the
random effect, to determine which model is a better fit for the data. Fit the
FEglme, using only the fixed-effects predictors
supplier. Fit the second
glme, using these same fixed-effects predictors, but
also including a random-effects intercept grouped by
FEglme = fitglme(mfr,... 'defects ~ 1 + newprocess + time_dev + temp_dev + supplier',... 'Distribution','Poisson','Link','log','FitMethod','Laplace'); glme = fitglme(mfr,... 'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)',... 'Distribution','Poisson','Link','log','FitMethod','Laplace');
Compare the two models using a likelihood ratio test. Specify
compare returns a warning if the nesting
requirements are not satisfied.
results = compare(FEglme,glme,'CheckNesting',true)
results = Theoretical Likelihood Ratio Test Model DF AIC BIC LogLik LRStat deltaDF FEglme 6 431.02 446.65 -209.51 glme 7 416.35 434.58 -201.17 16.672 1 pValue 4.4435e-05
compare returns the degrees of freedom
DF), the Akaike information criterion
AIC), Bayesian information criterion
BIC), and log likelihood values for each model.
glme has smaller AIC, BIC, and log likelihood values than
FEglme, which indicates that
model containing the random-effects term for intercept grouped by factory) is
the better-fitting model for this data. Additionally, the small
p-value indicates that
compare rejects the null hypothesis that the response vector was
generated by the fixed-effects-only model
FEglme, in favor of
the alternative that the response vector was generated by the mixed-effects
Generate the fitted conditional mean values for the model.
mufit = fitted(glme);
Plot the observed response values versus the fitted response values.
figure scatter(mfr.defects,mufit) title('Observed Values versus Fitted Values') xlabel('Fitted Values') ylabel('Observed Values')
Create diagnostic plots using conditional Pearson residuals to test model assumptions. Since raw residuals for generalized linear mixed-effects models do not have a constant variance across observations, use the conditional Pearson residuals instead.
Plot a histogram to visually confirm that the mean of the Pearson residuals is equal to 0. If the model is correct, we expect the Pearson residuals to be centered at 0.
The histogram shows that the Pearson residuals are centered at 0.
Plot the Pearson residuals versus the fitted values, to check for signs of nonconstant variance among the residuals (heteroscedasticity). We expect the conditional Pearson residuals to have a constant variance. Therefore, a plot of conditional Pearson residuals versus conditional fitted values should not reveal any systematic dependence on the conditional fitted values.
The plot does not show a systematic dependence on the fitted values, so there are no signs of nonconstant variance among the residuals.
Plot the Pearson residuals versus lagged residuals, to check for correlation among the residuals. The conditional independence assumption in GLME implies that the conditional Pearson residuals are approximately uncorrelated.
There is no pattern to the plot, so there are no signs of correlation among the residuals.