residuals

Class: GeneralizedLinearMixedModel

Residuals of fitted generalized linear mixed-effects model

Syntax

r = residuals(glme)

r = residuals(glme,Name,Value)

Description

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

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

glme — Generalized linear mixed-effects model

GeneralizedLinearMixedModel object

Generalized linear mixed-effects model, specified as a GeneralizedLinearMixedModel object. For properties and methods of this object, see GeneralizedLinearMixedModel.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is 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 residuals

true (default) | false

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

Value	Description
true	Contributions from both fixed effects and random effects (conditional)
false	Contribution 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

ResidualType — Residual type

'raw' (default) | 'Pearson'

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

Residual Type	Conditional	Marginal
'raw'	$$r_{ci}=y_i-g^{-1}\left(x_i^T\widehat{\beta }+z_i^T\widehat{b}+{\delta }_i\right)$$	$$r_{mi}=y_i-g^{-1}\left(x_i^T\widehat{\beta }+{\delta }_i\right)$$
'Pearson'	$$r_{ci}^{pearson}=\frac{r_{ci}}{\sqrt{\frac{\widehat{{\sigma }^2}}{w_i}{v}_i\left({\mu }_i\left(\widehat{\beta },\widehat{b}\right)\right)}}$$	$$r_{mi}^{pearson}=\frac{r_{mi}}{\sqrt{\frac{\widehat{{\sigma }^2}}{w_i}{v}_i\left({\mu }_i\left(\widehat{\beta },0\right)\right)}}$$

In each of these equations:

• y_i 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.

• x_i^T is the ith row of the fixed-effects design matrix X.

• z_i^T is the ith row of the random-effects design matrix Z.

• δ_i is the ith offset value.

• σ² is the dispersion parameter.

• w_i is the ith observation weight.

• v_i is the variance term for the ith observation.

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

• $$\widehat{\beta }$$ and $$\widehat{b}$$ 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

r — Residuals

n-by-1 vector

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

Plot Residuals Versus Fitted Values

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

$${\text{defects}}_{ij}\sim \text{Poisson}({\mu }_{ij})$$

This corresponds to the generalized linear mixed-effects model

$$\log ({\mu }_{ij})={\beta }_0+{\beta }_1{\text{newprocess}}_{ij}+{\beta }_2{\text{time}\text{\_}\text{dev}}_{ij}+{\beta }_3{\text{temp}\text{\_}\text{dev}}_{ij}+{\beta }_4{\text{supplier}\text{\_}\text{C}}_{ij}+{\beta }_5{\text{supplier}\text{\_}\text{B}}_{ij}+b_i,$$

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 $$j$$.

• $$b_i\sim N(0,{\sigma }_b^2)$$ 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.