1D ANOVA VI

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Updated2025-07-30
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Takes an array, X, of experimental observations made at various levels of a factor, with at least one observation per level, and performs a one-way analysis of variance in the fixed effect model.

Inputs/Outputs

X —

X contains all the observational data.

Index —

Index contains the level to which the corresponding observation belongs.

# of levels —

# of levels is the total number of levels.

f —

f is a ratio where f = msa/mse.

ssa —

ssa is a measure of variation attributed to the factor.

sse —

sse is a measure of variation attributed to random fluctuation.

mse —

mse is the mean square quantity associated with sse. It is calculated by dividing sse by its own degree of freedom.

msa —

msa is the mean square quantity associated with ssa. It is calculated by dividing ssa by its degree of freedom.

tss —

tss is the total sum of squares, which is a measure of the total variation of the data from the overall population mean. It is calculated using tss = ssa + sse.

error —

error returns any error or warning from the VI. You can wire error to the Error Cluster From Error Code VI to convert the error code or warning into an error cluster.

sig A —

sig A, given a particular f, is the probability that you will get a value larger than f when sampling from an F distribution.

In the one-way analysis of variance, the VI tests whether the level of the factor has an effect on the experimental outcome.

1D ANOVA Factors and Levels

A factor is a basis for categorizing data. For example, if you count the number of sit-ups individuals can do, one basis of categorization is age. For age, you might have the following levels.


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

Now, suppose that you make a series of observations to see how many sit-ups people can do. If you take a random sampling of five people, you might find the following results.


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

Notice that you have made at least one observation per level. To perform an analysis of variance, you must make at least one observation per level.

To perform the analysis of variance, you specify an array X of observations, with values 10, 15, 20, 25, and 17. The array Index specifies the level (or category) to which each observation applies. In this case, Index has the values 0, 1, 2, 2, and 1. Finally, there are three possible levels, so you pass in a value of 3 for the # of levels parameter.

1D ANOVA Statistical Model

Performing the analysis of variance, you express each experimental outcome as the sum of three parts. Let x_im be the m^th observation from the i^th level. Then each observation is written

x_im = µ + α_i + ε_im

where µ is a standard effect, called the overall mean.

α_i is the effect of the i^th level of the factor, which is the difference between the mean of the i^th level α_i and the overall mean

µ(µ_i) = µ + α_i

and ε_im is a random fluctuation.

1D ANOVA Hypothesis

This VI tests the hypothesis that α_i = 0 for i = 0, 1, …, k – 1, where k is # of levels. In other words, this hypothesis, referred to as the null hypothesis, states that no level affects the experimental outcome and then looks for evidence to the contrary.

1D ANOVA Assumptions

Assume that the populations of measurements at each level are Normally distributed with mean µ_i and variance σ_A², and assume that α_i sum to zero. Finally, assume that for each i and m, ε_im is Normally distributed with mean 0 and variance σ_A².

1D ANOVA General Method

This VI computes the total sum of squares, tss, which is a measure of the total variation of the data from the overall population mean.

tss consists of two parts: ssa, a measure of variation attributed to the factor, and sse, a measure of variation attributed to random fluctuation. In other words,

tss = ssa + sse.

The VI computes the two mean square quantities msa and mse from ssa and sse by dividing ssa and sse by their own degrees of freedom. The larger msa is relative to mse, the more significant effect the factor has on the experimental outcome.

In particular, if the null hypothesis is true, then the ratio f, f = msa/mse, is taken from an F distribution with k – 1 and n – k degrees of freedom, from which you can calculate probabilities. Given a particular f, sigA is the probability that you get a value larger than f when sampling from this distribution.

Testing the 1D ANOVA Hypothesis

Determine when to reject the null hypothesis by deciding how likely you want it to be that you mistakenly reject the null hypothesis. This is the level of significance, a common choice is 0.05. The output sigA is compared to the chosen level of significance to determine whether to accept or reject the null hypothesis. If sigA is less than the chosen level of significance, reject the null hypothesis. If you reject the null hypothesis, you must acknowledge that at least one level has some effect on the experimental outcome.

1D ANOVA Formulas

Let x_im = the m^th observation made at the i^th level for m = 0, 1, …, n_i – 1 and i = 0, 1, …, k – 1, where n_i is the number of observations at the i^th level and k = # of levels.