One Way Layout

Overview


A one way lalyout is an experiment where independent measurements are made of several groups, each of which has been treated with a different treatment.

Setup


For this analysis, we assume that we {% I %} groups, each with {% J %} samples. Then we define {% Y_{ij} %} to be the {% j^{th} %} observation of the {% i^{th} %} treatment. We assume an equation of the following form.
{% Y_{ij} = \mu + \alpha_i + \epsilon_{ij} %}
  • {% \mu %} - the overall mean
  • {% \alpha_i %} - the incremental effect of the {% i^{th} %} treatment
  • {% \epsilon_{ij} %} - the error, which is assumed to be normally distributed with zero mean and variance {% \sigma^2 %}
{% \sum_{i=1}^I \sum_{j=1}^J (Y_{ij} - \bar{Y})^2 = \sum_{i=1}^I \sum_{j=1}^J (Y_{ij} - \bar{Y}_i)^2 + J \sum_{i=1}^I (\bar{Y}_i - \bar{Y})^2 %}
This equation is abbreviated as
{% SS_{TOT} = S_W + SS_B %}
That is the total variation is the sum of the variation within groups plus the variation between groups.

F Test


The F-statistic, defined as
{% F = \frac{SS_B/(I-1)}{SS_W}/[I(J-1)] %}
under the null hypothesis
{% H_0 : \alpha_1 = \alpha_2 = ... = \alpha_n = 0 %}
is distributed as an F distribution with {% (I-1) %} and {% I(J-1) %} degrees of freedom.

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