# ANOVA (One-Way ANOVA) (G Dataflow)

Performs a one-way analysis of variance (ANOVA) and returns the effect of the levels of the factor on the experimental outcome.

## level

Number of levels in the factor. You must specify at least two levels. Otherwise, this node returns an error.

Default: 2

## x

All the observational data. You must make at least one observation per level. Otherwise, this node returns an error.

## index

The level to which the corresponding observation belongs.

This input converts input levels that do not begin with zero or input levels that have nonconsecutive values. For example, if you enter an index that contains the levels 3, 5, and 7, this input converts the levels to an index array with level values of 0, 1, and 2.

## error in

Error conditions that occur before this node runs.

The node responds to this input according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Default: No error

## significance

Probability that a value sampled from the F distribution with dofa and dofe degrees of freedom is greater than fa, where dofa, dofe, and fa are values in summary.

Compare significance with the chosen level of significance to determine whether the level of the factor has an effect on the experimental outcome. A common choice of the chosen level of significance is 0.05. If significance is less than the chosen level of significance, you can conclude that at least one level of the factor has some effect on the experimental outcome.

Algorithm for Calculating significance

This node calculates significance using the following equation:

$\mathbf{s}\mathbf{i}\mathbf{g}\mathbf{n}\mathbf{i}\mathbf{f}\mathbf{i}\mathbf{c}\mathbf{a}\mathbf{n}\mathbf{c}\mathbf{e}=\mathrm{Prob}\left\{{F}_{\mathrm{dofa},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dofe}}>\mathrm{fa}\right\}$

where ${F}_{\mathrm{dofa},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{dofe}}$ is the F distribution with dofa and dofe degrees of freedom.

## summary

A 2-by-4 matrix that displays the obtained values for analysis.

$\mathrm{summary}=\left[\begin{array}{cc}\mathrm{ssa}& \mathrm{dofa}\\ \mathrm{sse}& \mathrm{dofe}\end{array}\phantom{\rule{0ex}{0ex}}\text{\hspace{0.17em}}\phantom{\rule{0ex}{0ex}}\phantom{\square }\begin{array}{cc}\mathrm{msa}& \mathrm{fa}\\ \mathrm{mse}& \mathrm{0.0}\end{array}\right]$

where

• ssa and sse are the sums of squares associated with the factor and residual error, respectively
• dofa and dofe are the respective degrees of freedom
• msa and mse are the respective mean squares
• fa is the F value

Algorithm for Calculating Sums of Squares

This node calculates the sums of squares using the following equations:

$\mathrm{ssa}=\underset{i=0}{\overset{k-1}{\sum }}\underset{m=0}{\overset{{n}_{i}-1}{\sum }}{\left(\stackrel{¯}{{x}_{i\cdot }}-\stackrel{¯}{{x}_{\cdot \cdot }}\right)}^{2}$
$\mathrm{sse}=\underset{i=0}{\overset{k-1}{\sum }}\underset{m=0}{\overset{{n}_{i}-1}{\sum }}{\left({x}_{im}-\stackrel{¯}{{x}_{i\cdot }}\right)}^{2}$

where

• k is the number of levels
• i is the index of each level, starting from 0
• ni is the number of observational data at the ith level
• m is the index of each observational data at a certain level, starting from 0
• $\stackrel{¯}{{x}_{i\cdot }}$ is the mean of all the observational data at the ith level
• $\stackrel{¯}{{x}_{\cdot \cdot }}$ is the mean of all the observational data
• xim is the mth observation data of the ith level

Algorithm for Calculating Degrees of Freedom

This node calculates the degrees of freedom using the following equations:

$\mathrm{dofa}=k-1$
$\mathrm{dofe}=n-k$

where

• k is the number of levels
• n is the number of all the observational data

Algorithm for Calculating Mean Squares

This node calculates the mean squares using the following equations:

$\mathrm{msa}=\frac{\mathrm{ssa}}{\mathrm{dofa}}$
$\mathrm{mse}=\frac{\mathrm{sse}}{\mathrm{dofe}}$

where

• ssa is a measure of variation attributed to the factor
• dofa is the degree of freedom of ssa
• sse is a measure of variation attributed to random fluctuation
• dofe is the degree of freedom of sse

Algorithm for Calculating the F Value

This node calculates the F value using the following equation:

$\mathrm{fa}=\frac{\mathrm{msa}}{\mathrm{mse}}$

where

• msa is the mean square quantity of ssa
• mse is the mean square quantity of sse

The greater msa is relative to mse, which means, the greater fa is, the more significant effect the factor has on the experimental outcome.

## error out

Error information.

The node produces this output according to standard error behavior.

Standard Error Behavior

Many nodes provide an error in input and an error out output so that the node can respond to and communicate errors that occur while code is running. The value of error in specifies whether an error occurred before the node runs. Most nodes respond to values of error in in a standard, predictable way.

error in does not contain an error error in contains an error
If no error occurred before the node runs, the node begins execution normally.

If no error occurs while the node runs, it returns no error. If an error does occur while the node runs, it returns that error information as error out.

If an error occurred before the node runs, the node does not execute. Instead, it returns the error in value as error out.

Using age as a factor, this example demonstrates how to test whether age has an effect on the number of sit-ups a person can do.

The following table defines the levels of age.

 Level 0 6 years old to 10 years old Level 1 11 years old to 15 years old Level 2 16 years old to 20 years old

The following table lists the results of a random sampling of five people. The results are based on a series of observations of how many sit-ups people from different age groups can do.

Note

To perform a one-way analysis of variance, you must make at least one observation per level.

 Person 1 8 years old (Level 0) 10 sit-ups Person 2 12 years old (Level 1) 15 sit-ups Person 3 16 years old (Level 2) 20 sit-ups Person 4 20 years old (Level 2) 25 sit-ups Person 5 13 years old (Level 1) 17 sit-ups

The following table lists the inputs and outputs of this node.

 level 3 x [10, 15, 20, 25, 17] index [0, 1, 2, 2, 1] summary ssa 110.7 sse 14.5 dofa 2 dofe 2 msa 55.35 mse 7.25 fa 7.63448 0.0 0 significance 0.1158

Where This Node Can Run:

Desktop OS: Windows

FPGA: This product does not support FPGA devices