Fitted responses from generalized linear mixed-effects model
the fitted response with additional options specified by one or more
name-value pair arguments. For example, you can specify to compute
the marginal fitted response.
mufit = fitted(
Specify optional pairs of arguments as
the argument name and
Value is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name in quotes.
Conditional — Indicator for conditional response
true (default) |
Indicator for conditional response, specified as the comma-separated
pair consisting of
'Conditional' and one of the
|Contributions from both fixed effects and random effects (conditional)|
|Contribution from only fixed effects (marginal)|
To obtain fitted marginal response values,
the conditional mean of the response with the empirical Bayes predictor
vector of random effects b set equal to 0. For
more information, see Conditional and Marginal Response
mufit — Fitted response values
Fitted response values, returned as an n-by-1 vector, where n is the number of observations.
Plot Observed Versus Fitted Values
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 (
Processing time for each batch, in hours (
Temperature of the batch, in degrees Celsius (
Categorical variable indicating the supplier (
C) of the chemical used in the batch (
Number of defects in the batch (
The data also includes
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
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
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
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.
glme = fitglme(mfr,'defects ~ 1 + newprocess + time_dev + temp_dev + supplier + (1|factory)', ... 'Distribution','Poisson','Link','log','FitMethod','Laplace','DummyVarCoding','effects');
Generate the fitted conditional mean values for the model.
mufit = fitted(glme);
Create a scatterplot of the observed values versus fitted values.
figure scatter(mfr.defects,mufit) title('Residuals versus Fitted Values') xlabel('Fitted Values') ylabel('Residuals')
Conditional and Marginal Response
A conditional response includes contributions from both fixed- and random-effects predictors. A marginal response includes contribution from only fixed effects.
Suppose the generalized linear mixed-effects model
an n-by-p fixed-effects design
X and an n-by-q random-effects
Z. Also, suppose the estimated p-by-1
fixed-effects vector is ,
and the q-by-1 empirical Bayes predictor vector
of random effects is .
The fitted conditional response corresponds to the
pair argument, and is defined as
where is the linear predictor including the fixed- and random-effects of the generalized linear mixed-effects model
The fitted marginal response corresponds to the
pair argument, and is defined as
where is the linear predictor including only the fixed-effects portion of the generalized linear mixed-effects model