# anova2

Two-way analysis of variance

## Syntax

## Description

`anova2`

performs two-way analysis
of variance (ANOVA) with balanced designs. To perform two-way ANOVA
with unbalanced designs, see `anovan`

.

returns
the `p`

= anova2(`y`

,`reps`

)*p*-values for a balanced two-way ANOVA for comparing
the means of two or more columns and two or more rows of the observations
in `y`

.

`reps`

is the number of replicates for each
combination of factor groups, which must be constant, indicating a
balanced design. For unbalanced designs, use `anovan`

.
The `anova2`

function tests the main effects for
column and row factors and their interaction effect. To test the interaction
effect, `reps`

must be greater than 1.

`anova2`

also displays the standard ANOVA
table.

enables the ANOVA table display when `p`

= anova2(`y`

,`reps`

,`displayopt`

)`displayopt`

is `'on'`

(default)
and suppresses the display when `displayopt`

is `'off'`

.

`[`

returns a `p`

,`tbl`

,`stats`

]
= anova2(___)`stats`

structure,
which you can use to perform a multiple comparison test. A multiple
comparison test enables you to determine which pairs of group means
are significantly different. To perform this test, use `multcompare`

, providing the `stats`

structure
as input.

## Examples

### Two-Way ANOVA

Load the sample data.

```
load popcorn
popcorn
```

`popcorn = `*6×3*
5.5000 4.5000 3.5000
5.5000 4.5000 4.0000
6.0000 4.0000 3.0000
6.5000 5.0000 4.0000
7.0000 5.5000 5.0000
7.0000 5.0000 4.5000

The data is from a study of popcorn brands and popper types (Hogg 1987). The columns of the matrix `popcorn`

are brands, Gourmet, National, and Generic, respectively. The rows are popper types, oil and air. In the study, researchers popped a batch of each brand three times with each popper, that is, the number of replications is 3. The first three rows correspond to the oil popper, and the last three rows correspond to the air popper. The response values are the yield in cups of popped popcorn.

Perform a two-way ANOVA. Save the ANOVA table in the cell array `tbl`

for easy access to results.

[p,tbl] = anova2(popcorn,3);

The column `Prob>F`

shows the *p*-values for the three brands of popcorn (0.0000), the two popper types (0.0001), and the interaction between brand and popper type (0.7462). These values indicate that popcorn brand and popper type affect the yield of popcorn, but there is no evidence of an interaction effect of the two.

Display the cell array containing the ANOVA table.

tbl

`tbl=`*6×6 cell array*
Columns 1 through 5
{'Source' } {'SS' } {'df'} {'MS' } {'F' }
{'Columns' } {[15.7500]} {[ 2]} {[ 7.8750]} {[ 56.7000]}
{'Rows' } {[ 4.5000]} {[ 1]} {[ 4.5000]} {[ 32.4000]}
{'Interaction'} {[ 0.0833]} {[ 2]} {[ 0.0417]} {[ 0.3000]}
{'Error' } {[ 1.6667]} {[12]} {[ 0.1389]} {0x0 double}
{'Total' } {[ 22]} {[17]} {0x0 double} {0x0 double}
Column 6
{'Prob>F' }
{[7.6790e-07]}
{[1.0037e-04]}
{[ 0.7462]}
{0x0 double }
{0x0 double }

Store the *F*-statistic for the factors and factor interaction in separate variables.

Fbrands = tbl{2,5}

Fbrands = 56.7000

Fpoppertype = tbl{3,5}

Fpoppertype = 32.4000

Finteraction = tbl{4,5}

Finteraction = 0.3000

### Multiple Comparisons for Two-Way ANOVA

Load the sample data.

```
load popcorn
popcorn
```

`popcorn = `*6×3*
5.5000 4.5000 3.5000
5.5000 4.5000 4.0000
6.0000 4.0000 3.0000
6.5000 5.0000 4.0000
7.0000 5.5000 5.0000
7.0000 5.0000 4.5000

The data is from a study of popcorn brands and popper types (Hogg 1987). The columns of the matrix `popcorn`

are brands (Gourmet, National, and Generic). The rows are popper types oil and air. The first three rows correspond to the oil popper, and the last three rows correspond to the air popper. In the study, researchers popped a batch of each brand three times with each popper. The values are the yield in cups of popped popcorn.

Perform a two-way ANOVA. Also compute the statistics that you need to perform a multiple comparison test on the main effects.

`[~,~,stats] = anova2(popcorn,3,"off")`

`stats = `*struct with fields:*
source: 'anova2'
sigmasq: 0.1389
colmeans: [6.2500 4.7500 4]
coln: 6
rowmeans: [4.5000 5.5000]
rown: 9
inter: 1
pval: 0.7462
df: 12

The `stats`

structure includes

The mean squared error (

`sigmasq`

)The estimates of the mean yield for each popcorn brand (

`colmeans`

)The number of observations for each popcorn brand (

`coln`

)The estimate of the mean yield for each popper type (

`rowmeans`

)The number of observations for each popper type (

`rown`

)The number of interactions (

`inter`

)The

*p*-value that shows the significance level of the interaction term (`pval`

)The error degrees of freedom (

`df`

).

Perform a multiple comparison test to see if the popcorn yield differs between pairs of popcorn brands (columns).

c1 = multcompare(stats);

Note: Your model includes an interaction term. A test of main effects can be difficult to interpret when the model includes interactions.

The figure shows the multiple comparisons of the means. By default, the group 1 mean is highlighted and the comparison interval is in blue. Because the comparison intervals for the other two groups do not intersect with the intervals for the group 1 mean, they are highlighted in red. This lack of intersection indicates that both means are different than group 1 mean. Select other group means to confirm that all group means are significantly different from each other.

Display the multiple comparison results in a table.

tbl1 = array2table(c1,"VariableNames", ... ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"])

`tbl1=`*3×6 table*
Group A Group B Lower Limit A-B Upper Limit P-value
_______ _______ ___________ ____ ___________ __________
1 2 0.92597 1.5 2.074 4.1188e-05
1 3 1.676 2.25 2.824 6.1588e-07
2 3 0.17597 0.75 1.324 0.011591

The first two columns of `c1`

show the groups that are compared. The fourth column shows the difference between the estimated group means. The third and fifth columns show the lower and upper limits for 95% confidence intervals for the true mean difference. The sixth column contains the *p*-value for a hypothesis test that the corresponding mean difference is equal to zero. All *p*-values are very small, which indicates that the popcorn yield differs across all three brands.

Perform a multiple comparison test to see the popcorn yield differs between the two popper types (rows).

c2 = multcompare(stats,"Estimate","row");

Note: Your model includes an interaction term. A test of main effects can be difficult to interpret when the model includes interactions.

tbl2 = array2table(c2,"VariableNames", ... ["Group A","Group B","Lower Limit","A-B","Upper Limit","P-value"])

`tbl2=`*1×6 table*
Group A Group B Lower Limit A-B Upper Limit P-value
_______ _______ ___________ ___ ___________ __________
1 2 -1.3828 -1 -0.61722 0.00010037

The small *p*-value indicates that the popcorn yield differs between the two popper types (air and oil). The figure shows the same results. The disjoint comparison intervals indicate that the group means are significantly different from each other.

## Input Arguments

`y`

— Sample data

matrix

Sample data, specified as a matrix. The columns correspond to
groups of one factor, and the rows correspond to the groups of the
other factor and the replications. Replications are the measurements
or observations for each combination of groups (levels) of the row
and column factor. For example, in the following data the row factor *A* has
three levels, column factor *B* has two levels, and
there are two replications (`reps = 2`

). The subscripts
indicate row, column, and replication, respectively.

$$\begin{array}{c}\begin{array}{cc}B=1& B=2\end{array}\\ \left[\begin{array}{cc}{y}_{111}& {y}_{121}\\ {y}_{112}& {y}_{122}\\ {y}_{211}& {y}_{221}\\ {y}_{212}& {y}_{222}\\ {y}_{311}& {y}_{321}\\ {y}_{312}& {y}_{322}\end{array}\right]\end{array}\begin{array}{c}\\ \begin{array}{c}\begin{array}{c}\\ \end{array}\}A=1\\ \begin{array}{c}\\ \end{array}\}A=2\\ \begin{array}{c}\\ \end{array}\}A=3\end{array}\end{array}$$

**Data Types: **`single`

| `double`

`reps`

— Number of replications

1 (default) | an integer number

Number of replications for each combination of groups, specified
as an integer number. For example, the following data has two replications
(`reps = 2`

) for each group combination of row factor *A* and
column factor *B*.

$$\begin{array}{c}\begin{array}{cc}B=1& B=2\end{array}\\ \left[\begin{array}{cc}{y}_{111}& {y}_{121}\\ {y}_{112}& {y}_{122}\\ {y}_{211}& {y}_{221}\\ {y}_{212}& {y}_{222}\\ {y}_{311}& {y}_{321}\\ {y}_{312}& {y}_{322}\end{array}\right]\end{array}\begin{array}{c}\\ \begin{array}{c}\begin{array}{c}\\ \end{array}\}A=1\\ \begin{array}{c}\\ \end{array}\}A=2\\ \begin{array}{c}\\ \end{array}\}A=3\end{array}\end{array}$$

When

`reps`

is`1`

(default),`anova2`

returns two*p*-values in vector`p`

:The

*p*-value for the null hypothesis that all samples from factor*B*(i.e., all column samples in`y`

) are drawn from the same population.The

*p*-value for the null hypothesis, that all samples from factor*A*(i.e., all row samples in`y`

) are drawn from the same population.

When

`reps`

is greater than`1`

,`anova2`

also returns the*p*-value for the null hypothesis that factors*A*and*B*have no interaction (i.e., the effects due to factors*A*and*B*are*additive*).

**Example: **`p = anova(y,3)`

specifies that each
combination of groups (levels) has three replications.

**Data Types: **`single`

| `double`

`displayopt`

— Indicator to display the ANOVA table

`'on'`

(default) | `'off'`

Indicator to display the ANOVA table as a figure, specified
as `'on'`

or `'off'`

.

## Output Arguments

`p`

— *p*-value

scalar value

*p*-value for the *F*-test,
returned as a scalar value. A small *p*-value indicates
that the results are statistically significant. Common significance
levels are 0.05 or 0.01. For example:

A sufficiently small

*p*-value for the null hypothesis for group means of row factor*A*suggests that at least one row-sample mean is significantly different from the other row-sample means; i.e., there is a main effect due to factor*A*A sufficiently small

*p*-value for the null hypothesis for group (level) means of column factor*B*suggests that at least one column-sample mean is significantly different from the other column-sample means; i.e., there is a main effect due to factor*B*.A sufficiently small

*p*-value for combinations of groups (levels) of factors*A*and*B*suggests that there is an interaction between factors*A*and*B*.

`tbl`

— ANOVA table

cell array

ANOVA table, returned as a cell array. `tbl`

has
six columns.

Column name | Definition |
---|---|

`source` | Source of the variability. |

`SS` | Sum of squares due to each source. |

`df` | Degrees of freedom associated with each source. |

`MS` | Mean squares for each source, which is the ratio `SS/df` . |

`F` | F-statistic, which is the ratio of the mean
squares. |

`Prob>F` | p-value, which is the probability that the F-statistic
can take a value larger than the computed test-statistic
value. `anova2` derives this probability
from the cdf of the
F-distribution. |

The rows of the ANOVA table show the variability in the data,
divided by the source into three or four parts, depending on the value
of `reps`

.

Row | Definition |
---|---|

`Columns` | Variability due to the differences among the column means |

`Rows` | Variability due to the differences among the row means |

`Interaction` | Variability due to the interaction between rows and columns
(if |

`Error` | Remaining variability not explained by any systematic source |

**Data Types: **`cell`

`stats`

— Statistics for multiple comparison test

structure

Statistics for multiple comparisons tests, returned
as a structure. Use `multcompare`

to
perform multiple comparison tests, supplying `stats`

as
an input argument. `stats`

has nine fields.

Field | Definition |
---|---|

`source` | Source of the `stats` output |

`sigmasq` | Mean squared error |

`colmeans` | Estimated values of the column means |

`coln` | Number of observations for each group in columns |

`rowmeans` | Estimated values of the row means |

`rown` | Number of observations for each group in rows |

`inter` | Number of interactions |

`pval` | p-value for the interaction term |

`df` | Error degrees of freedom (reps —
1)*r*c where reps is
the number of replications and c and r are
the number of groups in factors, respectively. |

**Data Types: **`struct`

## References

[1] Hogg, R. V., and J. Ledolter. *Engineering
Statistics*. New York: MacMillan, 1987.

## Version History

**Introduced before R2006a**

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