anova

Class: GeneralizedLinearMixedModel

Analysis of variance for generalized linear mixed-effects model

Syntax

stats = anova(glme)

stats = anova(glme,Name,Value)

Description

stats = anova(glme) returns a table, stats, that contains the results of F-tests to determine if all coefficients representing each fixed-effects term in the generalized linear mixed-effects model glme are equal to 0.

stats = anova(glme,Name,Value) returns a table, stats, using additional options specified by one or more Name,Value pair arguments. For example, you can specify the method used to compute the approximate denominator degrees of freedom for the F-tests.

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.

DFMethod — Method for computing approximate denominator degrees of freedom

'residual' (default) | 'none'

Method for computing approximate denominator degrees of freedom to use in the F-test, specified as the comma-separated pair consisting of 'DFMethod' and one of the following.

Value	Description
'residual'	The degrees of freedom are assumed to be constant and equal to n – p, where n is the number of observations and p is the number of fixed effects.
'none'	All degrees of freedom are set to infinity.

The denominator degrees of freedom for the F-statistic correspond to the column DF2 in the output structure stats.

Example: 'DFMethod','none'

Output Arguments

stats — Results of F-tests for fixed-effects terms

table

Results of F-tests for fixed-effects terms, returned as a table with one row for each fixed-effects term in glme and the following columns.

Column Name	Description
Term	Name of the fixed-effects term
FStat	F-statistic for the term
DF1	Numerator degrees of freedom for the F-statistic
DF2	Denominator degrees of freedom for the F-statistic
pValue	p-value for the term

Each fixed-effects term is a continuous variable, a grouping variable, or an interaction between two or more continuous or grouping variables. For each fixed-effects term, anova performs an F-test (marginal test) to determine if all coefficients representing the fixed-effects term are equal to 0.

To perform tests for the type III hypothesis, when fitting the generalized linear mixed-effects model fitglme, you must use the 'effects' contrasts for the 'DummyVarCoding' name-value pair argument.

Examples

F-Tests for Fixed Effects

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

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

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')

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        Lower        Upper    
    {'(Intercept)'}           1.4689     0.15988      9.1875    94    9.8194e-15       1.1515       1.7864
    {'newprocess' }         -0.36766     0.17755     -2.0708    94      0.041122     -0.72019    -0.015134
    {'time_dev'   }        -0.094521     0.82849    -0.11409    94       0.90941      -1.7395       1.5505
    {'temp_dev'   }         -0.28317      0.9617    -0.29444    94       0.76907      -2.1926       1.6263
    {'supplier_C' }        -0.071868    0.078024     -0.9211    94       0.35936     -0.22679     0.083051
    {'supplier_B' }         0.071072     0.07739     0.91836    94       0.36078    -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       

Perform an $$F$$-test to determine if all fixed-effects coefficients are equal to 0.

stats = anova(glme)

stats = 
    ANOVA MARGINAL TESTS: DFMETHOD = 'RESIDUAL'

    Term                   FStat       DF1    DF2    pValue    
    {'(Intercept)'}           84.41    1      94     9.8194e-15
    {'newprocess' }          4.2881    1      94       0.041122
    {'time_dev'   }        0.013016    1      94        0.90941
    {'temp_dev'   }        0.086696    1      94        0.76907
    {'supplier'   }         0.59212    2      94         0.5552

The $$p$$-values for the intercept, newprocess, time_dev, and temp_dev are the same as in the coefficient table of the glme display. The small $$p$$-values for the intercept and newprocess indicate that these are significant predictors at the 5% significance level. The large $$p$$-values for time_dev and temp_dev indicate that these are not significant predictors at this level.

The $$p$$-value of 0.5552 for supplier measures the combined significance for both coefficients representing the categorical variable supplier. This includes the dummy variables supplier_C and supplier_B as shown in the coefficient table of the glme display. The large $$p$$-value indicates that supplier is not a significant predictor at the 5% significance level.

Tips

• For each fixed-effects term, anova performs an F-test (marginal test) to determine if all coefficients representing the fixed-effects term are equal to 0.

  When fitting a generalized linear mixed-effects (GLME) model using fitglme and one of the maximum likelihood fit methods ('Laplace' or 'ApproximateLaplace'):

  • If you specify the 'CovarianceMethod' name-value pair argument as 'conditional', then the F-tests are conditional on the estimated covariance parameters.

  • If you specify the 'CovarianceMethod' name-value pair as 'JointHessian', then the F-tests account for the uncertainty in estimation of covariance parameters.

  When fitting a GLME model using fitglme and one of the pseudo likelihood fit methods ('MPL' or 'REMPL'), anova uses the fitted linear mixed effects model from the final pseudo likelihood iteration for inference on fixed effects.