# varianceComponent

Variance component estimates for analysis of variance (ANOVA)

Since R2022b

## Syntax

``v = varianceComponent(aov)``
``v = varianceComponent(aov,Alpha=alpha)``

## Description

example

````v = varianceComponent(aov)` returns a table of variance component estimates of the random factors and error for an `anova` object at the 95% confidence level.```

example

````v = varianceComponent(aov,Alpha=alpha)` returns the variance component estimates with $100\left(1-\alpha \right)%$ confidence intervals.```

## Examples

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`load carsmall`

Data for the country of origin, model year, and mileage is stored in the variables `Origin`, `Model_Year`, and `MPG`, respectively.

Perform a two-way ANOVA to test the null hypothesis that mean mileage is not affected by the country of origin or model year. The factors `Origin` and `Year` are random because the data was sampled from a larger population.

`aov = anova({Origin, Model_Year},MPG,RandomFactors=[1 2],FactorNames=["Origin" "Year"])`
```aov = 2-way anova, constrained (Type III) sums of squares. Y ~ 1 + Origin + Year SumOfSquares DF MeanSquares F pValue ____________ __ ___________ ______ __________ Origin 1078.1 5 215.62 10.675 5.3303e-08 Year 2638.4 2 1319.2 65.312 5.5975e-18 Error 1737 86 20.198 Total 6005.3 93 Properties, Methods ```

The p-values for `Origin` and `Year` indicate that the country of origin and model year have statistically significant effects on mileage.

Display the variance component estimates for the error and random factors with confidence intervals. Use the default confidence level of 95%.

`vtbl = varianceComponent(aov)`
```vtbl=3×3 table VarianceComponent VarianceComponentLower VarianceComponentUpper _________________ ______________________ ______________________ Origin 21.337 6.1257 139.94 Year 44.031 11.176 1765.7 Error 20.198 15.298 27.909 ```

The variance components for `Origin` and `Year` are due to the random sampling of the data. The variance of `MPG` is the sum of the variance components for `Origin`, `Year`, and `Error`. The table output shows that the variance components for `Origin` and `Year` are responsible for the majority of the variance in `MPG`.

`load carsmall`

Data for the model year and mileage is stored in the variables `Model_Year` and `MPG`, respectively.

Perform a two-way ANOVA to test the null hypothesis that mean mileage is not affected by the model year. `Year` is a random factor because it contains a randomly selected subset of all possible model years.

`aov = anova(Model_Year, MPG, RandomFactors=,FactorNames=["Year"])`
```aov = 1-way anova, constrained (Type III) sums of squares. Y ~ 1 + Year SumOfSquares DF MeanSquares F pValue ____________ __ ___________ _____ __________ Year 3190.1 2 1595.1 51.56 1.0694e-15 Error 2815.2 91 30.936 Total 6005.3 93 Properties, Methods ```

Display the variance component estimates for `Year` and the error with confidence intervals. Specify a confidence level of 99%.

`vtbl = varianceComponent(aov,Alpha=0.01)`
```vtbl=2×3 table VarianceComponent VarianceComponentLower VarianceComponentUpper _________________ ______________________ ______________________ Year 50.026 8.1282 10177 Error 30.936 21.74 46.915 ```

The output shows that `Year` contributes more to the sample variance than `Error`.

## Input Arguments

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ANOVA results, specified as an `anova` object. The properties of `aov` contain the factors and response data used by `varianceComponent` to compute the variance component estimates and their confidence intervals.

Significance level for the estimates, specified as a scalar value in the range (0,1). The confidence level of the confidence intervals is $100\left(1-\alpha \right)%$. The default value for `alpha` is `0.05`, which returns 95% confidence intervals for the estimates.

Example: `Alpha=0.01`

Data Types: `single` | `double`

## Output Arguments

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Variance component estimates and their confidence intervals, returned as a table. The varianceComponent function assumes that coefficients for dummy variables corresponding to the same random factor have equal variance. The table `v` has rows for the error term and for each of the random terms in `aov.Formula`. The columns of `v` correspond to the following variables:

• `VarianceComponent` — The estimated variance component.

• `VarianceComponentLower` — A lower confidence bound of the variance component. You can specify the confidence level using `alpha`.

• `VarianceComponentUpper` — An upper confidence bound of the variance component.

You can use the variance component estimates to determine if the random sampling has a significant effect on the mean squares of a term.

Data Types: `table`

 Dunn, O. J., and V. A. Clark. Applied Statistics: Analysis of Variance and Regression. New York: Wiley, 1974.

 Goodnight, J. H., and F. M. Speed. Computing Expected Mean Squares. Cary, NC: SAS Institute, 1978.

 Seber, G. A. F., and A. J. Lee. Linear Regression Analysis. 2nd ed. Hoboken, NJ: Wiley-Interscience, 2003.