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anova

Class: GeneralizedLinearMixedModel

Analysis of variance for generalized linear mixed-effects model

Description

example

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

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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.

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.

ValueDescription
'residual'The degrees of freedom are assumed to be constant and equal to np, 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

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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 NameDescription
TermName of the fixed-effects term
FStatF-statistic for the term
DF1Numerator degrees of freedom for the F-statistic
DF2Denominator degrees of freedom for the F-statistic
pValuep-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

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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

defectij Poisson(μij)

This corresponds to the generalized linear mixed-effects model

logμij=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')
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       

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.