residuals

Class: GeneralizedLinearMixedModel

Residuals of fitted generalized linear mixed-effects model

Description

r = residuals(glme) returns the raw conditional residuals from a fitted generalized linear mixed-effects model glme.

example

r = residuals(glme,Name,Value) returns the residuals using additional options specified by one or more Name,Value pair arguments. For example, you can specify to return Pearson residuals for the model.

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.

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Indicator for conditional residuals, specified as the comma-separated pair consisting of 'Conditional' and one of the following.

ValueDescription
trueContributions from both fixed effects and random effects (conditional)
falseContribution from only fixed effects (marginal)

Conditional residuals include contributions from both fixed- and random-effects predictors. Marginal residuals include contribution from only fixed effects. To obtain marginal residual values, residuals computes the conditional mean of the response with the empirical Bayes predictor vector of random effects, b, set to 0.

Example: 'Conditional',false

Residual type, specified as the comma-separated pair consisting of 'ResidualType' and one of the following.

Residual TypeConditionalMarginal
'raw'

rci=yig1(xiTβ^+ziTb^+δi)

rmi=yig1(xiTβ^+δi)

'Pearson'

rcipearson=rciσ2^wivi(μi(β^,b^))

rmipearson=rmiσ2^wivi(μi(β^,0))

In each of these equations:

  • yi is the ith element of the n-by-1 response vector, y, where i = 1, ..., n.

  • g-1 is the inverse link function for the model.

  • xiT is the ith row of the fixed-effects design matrix X.

  • ziT is the ith row of the random-effects design matrix Z.

  • δi is the ith offset value.

  • σ2 is the dispersion parameter.

  • wi is the ith observation weight.

  • vi is the variance term for the ith observation.

  • μi is the mean of the response for the ith observation.

  • β^ and b^ are estimated values of β and b.

Raw residuals from a generalized linear mixed-effects model have nonconstant variance. Pearson residuals are expected to have an approximately constant variance, and are generally used for analysis.

Example: 'ResidualType','Pearson'

Output Arguments

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Residuals of the fitted generalized linear mixed-effects model glme returned as an n-by-1 vector, where n is the number of observations.

Examples

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Load the sample data.

load mfr

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

defectsijPoisson(μij)

This corresponds to the generalized linear mixed-effects model

log(μij)=β0+β1newprocessij+β2time_devij+β3temp_devij+β4supplier_Cij+β5supplier_Bij+bi,

where

  • 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 C or B, respectively, supplied the process chemicals for the batch produced by factory i during batch j.

  • biN(0,σb2) 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');

Generate the conditional Pearson residuals and the conditional fitted values from the model.

r = residuals(glme,'ResidualType','Pearson');
mufit = fitted(glme);

Display the first ten rows of the Pearson residuals.

r(1:10)
ans = 10×1

    0.4530
    0.4339
    0.3833
   -0.2653
    0.2811
   -0.0935
   -0.2984
   -0.2509
    1.5547
   -0.3027

Plot the Pearson residuals versus the fitted values, to check for signs of nonconstant variance among the residuals (heteroscedasticity).

figure
scatter(mufit,r)
title('Residuals versus Fitted Values')
xlabel('Fitted Values')
ylabel('Residuals')

The plot does not show a systematic dependence on the fitted values, so there are no signs of nonconstant variance among the residuals.