# anova

Analysis of variance for generalized linear mixed-effects model

## Syntax

``stats = anova(glme)``
``stats = anova(glme,Name,Value)``

## 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 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
`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

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`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

This corresponds to the generalized linear mixed-effects model

`$\mathrm{log}{\mu }_{ij}={\beta }_{0}+{\beta }_{1}{\text{newprocess}}_{ij}+{\beta }_{2}{\text{time}\text{_}\text{dev}}_{ij}+{\beta }_{3}{\text{temp}\text{_}\text{dev}}_{ij}+{\beta }_{4}{\text{supplier}\text{_}\text{C}}_{ij}+{\beta }_{5}{\text{supplier}\text{_}\text{B}}_{ij}+{b}_{i},$`

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 $j$.

• ${b}_{i}\sim N\left(0,{\sigma }_{b}^{2}\right)$ 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.