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The equivalence of t.test and the hypothesis testing of one way ANOVA

Hi, 

Student's T-test is a test that can be used to test ONE SINGLE linear restriction - which serves the as alternative hypothesis - on a linear model - which serves as the null hypothesis - AT THE SAME TIME. 

Fisher's F test is an extension of the T test. The F test can be used to test ONE OR MORE linear restriction(s) on a linear model AT THE SAME TIME. 

So, to test a single restriction on a linear model one can use both the F test and the T test. When multiple restrictions are tested at the same time one needs to apply the F test. 

Both the F and the T test actually require equal variances. However, using a transformation matrix one can transform a model assuming unequal variances into an equivalent model assuming equal variances. On such a transformed model the F test or T test can be applied. The untransformed models are usually called general linear models. In R they can be handled using the glm() function. (See ?glm) 

A (one-way) Anova model is a specific type of general linear model (glm). So hypotheses on an Anova model are tested in exactly the same way as any other restrictions on a glm should be tested. 

Best wishes, 

Guido
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