Methods and formulas for the model in One-Way ANOVA

Select the method or formula of your choice.

Notation

The notation is critical to understanding ANOVA models. Listed below is the notation used for one-way analysis of variance.

TermDescription
r number of levels of the factor, i = 1 ...r
i a given factor level
j a given case for a particular factor level, j = 1 ...n i
yij j th observation of the response for the i th factor level
ni number of observations for the i th factor level
n T total number of cases
μi true mean of observations at the i th factor level
yi. total of the observations at the i th factor level
mean of the response for the ith factor

Model fitting

Formula

The one-way analysis of variance model can be specified in several ways. The cell means model is:

All observations for the factor level have the same expectation, μ i . Because μi is a constant, all observations have the same variance, regardless of factor level.

In analysis of variance, least squares estimation is used to fit the model and provide estimates for the parameters, μi .

The hypothesis test for one-way analysis of variance is:

H0: μ 1 = μ 2= … = μ r

H1: At least one mean is not equal to the others

Notation

TermDescription
μ i parameters or the true mean of observations at the i th factor level
ε ij error that is independently and normally distributed with mean 0 and constant variance σ 2
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