# groupmeans

Mean response estimates for analysis of variance (ANOVA)

Since R2022b

## Syntax

``means = groupmeans(aov)``
``means = groupmeans(aov,factors)``
``means = groupmeans(___,Alpha=alpha)``

## Description

example

````means = groupmeans(aov)` returns a table of mean response estimates, standard errors, and 95% confidence intervals for each value of the factor in a one-way ANOVA.```
````means = groupmeans(aov,factors)` returns `means` with information for each unique combination of values of the factors specified in `factors`.```

example

````means = groupmeans(___,Alpha=alpha)` specifies the confidence level of the confidence intervals to be $100\left(1-\alpha \right)%$.```

## Examples

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`load carbig.mat`

Create a table that has variables for acceleration and horsepower category. Obtain the horsepower categories by sorting the variable `Horsepower` into three horsepower ranges.

```tbl = table(Acceleration); tbl.HorsepowerCats = discretize(Horsepower,[0 100 200 300])```
```tbl=406×2 table Acceleration HorsepowerCats ____________ ______________ 12 2 11.5 2 11 2 12 2 10.5 2 10 2 9 3 8.5 3 10 3 8.5 2 17.5 2 11.5 2 11 2 10.5 2 11 2 10 2 ⋮ ```

Perform a one-way ANOVA to test the null hypothesis that the mean acceleration time is the same across the three horsepower ranges.

`aov = anova(tbl,"Acceleration")`
```aov = 1-way anova, constrained (Type III) sums of squares. Acceleration ~ 1 + HorsepowerCats SumOfSquares DF MeanSquares F pValue ____________ ___ ___________ ______ __________ HorsepowerCats 975.93 2 487.96 89.571 7.8471e-33 Error 2162.8 397 5.4478 Total 3138.7 399 Properties, Methods ```

The small p-value indicates that the mean acceleration time is different for at least one of the horsepower categories. Investigate which horsepower ranges have different mean acceleration times by inspecting the means of the horsepower categories.

`means = groupmeans(aov)`
```means=3×5 table HorsepowerCats Mean SE MeanLower MeanUpper ______________ ______ _______ _________ _________ 1 16.804 0.15526 16.498 17.11 2 13.969 0.18282 13.608 14.33 3 11.136 0.70374 9.5683 12.704 ```

The table `means` shows that each category has a mean that is outside the 95% confidence intervals of the mean estimates for the other categories. Therefore, the mean acceleration time is significantly different for all three horsepower categories.

Load the car mileage sample data.

`load mileage.mat`

The columns of the 6-by-3 matrix `mileage` contain mileage data for three car models. The first three rows contain data for cars built at one factory, and the last three rows contain data for cars built at another factory.

Convert `mileage` to a vector.

`mileage = mileage(:);`

Create string arrays of factor values for the factory and car model factors using the function `repmat`.

```factory = repmat(["factory1";"factory1";"factory1";... "factory2";"factory2";"factory2"], [3, 1]); model = [repmat("model1",6,1);... repmat("model2",6,1);repmat("model3",6,1)]; factors = {factory,model};```

Perform a two-way ANOVA to test the null hypothesis that car mileage is not affected by the factory or car model factors.

`aov = anova(factors,mileage,FactorNames=["Factory","Model"])`
```aov = 2-way anova, constrained (Type III) sums of squares. Y ~ 1 + Factory + Model SumOfSquares DF MeanSquares F pValue ____________ __ ___________ ______ __________ Factory 1.445 1 1.445 14.382 0.0019807 Model 53.351 2 26.676 265.49 7.3827e-12 Error 1.4067 14 0.10048 Total 56.203 17 Properties, Methods ```

The small p-values indicate that the model of a car has a more significant effect on car mileage than the factory in which the car was manufactured.

To investigate which car models have different mileages at the 99% confidence level, inspect the group means.

`means = groupmeans(aov,"Model",Alpha=0.01)`
```means=3×5 table Model Mean SE MeanLower MeanUpper ________ ______ _______ _________ _________ "model1" 32.95 0.12941 32.428 33.472 "model2" 34.017 0.12941 33.495 34.538 "model3" 37.017 0.12941 36.495 37.538 ```

The table shows that the 99% confidence intervals of all car models do not overlap. Therefore, all three models have statistically significant differences in mean car mileage at the 99% confidence level.

## 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 `groupmeans` to compute the mean responses.

Factors used to group the response data, specified as a string vector or cell array of character vectors. The `groupmeans` function groups the response data by the combinations of values for the factors in `factors`. The `factors` argument must be one or more of the names in `aov.FactorNames`.

Example: `["g1","g2"]`

Data Types: `string` | `cell`

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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Mean response estimates, standard errors, and confidence intervals, returned as a table. The table `means` has one row per unique combination of factor values. If `aov` is a one-way `anova` object, `means` has a column corresponding to the single factor. If `aov` is a two- or N-way `anova` object, `means` contains a column for each factor specified in `factors`. In addition to the factor columns, `means` contains the following:

• `Mean` — Estimate of the mean response of the factor value

• `SE` — Standard error of the mean estimate

• `MeanLower` — 95% lower confidence bound of the mean estimate

• `MeanUpper` — 95% upper confidence bound of the mean estimate

## Version History

Introduced in R2022b