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RE: Anova results explanation

From: Ted Harding
Subject: RE: Anova results explanation
Date: Wed, 12 May 2004 18:27:25 +0100 (BST)

On 12-May-04 DrakeGis wrote:
> Hi,
>    Perhaps is a silly question, but is my first time working with ANOVA
> and I don't understand the meaning of the output of ANOVA function in
>    I'm running ANOVA test on some data and I have the following
> results...
>    What is the meaning of p-value ??? Is Octave taking 95% of
> confidence,
> or 97.5% ? What if p-value is greater or equal to 1 ???
>    Thanks
>     Drake.
> Here are some results....
> One-way ANOVA Table:
> Source of Variation   Sum of Squares    df  Empirical Var
> *********************************************************
> Between Groups               50.4100     1        50.4100
> Within Groups               372.5800    98         3.8018
> ---------------------------------------------------------
> Total                       422.9900    99
> Test Statistic f             13.2594
> p-value                       0.0004

In this case, the p-value is the probability that a variate distributed
as F with 1:98 degrees of freedom should exceed the value 13.2594. This
probability cannot exceed 1 (and if software said it did, then there
would be a bug somewhere).

As such it provided the means to test the null hypothesis that there
is no difference between the means of the two groups, since (subject
to normal distribution, independent data, and homogeneity of variance,
the ratio of Between/Within Groups sums of squares, each divided by
its degrees of freedom, will have this F distribution.

On the alternative hypothesis that there is a difference of means this
ratio will have a non-central F distribution in which the probability
of exceeding this (or any other) value will be greater than if the
means are equal.

So a sufficiently small p-value is evidence against the null hypothesis,
the more so, the smaller the p-value is. If you want to decide whether
or not to reject the NH, then you can set a threshold value (significance
level), e.g. 0.05 or 0.01 or ... (suit yourself).

There is nothing of "95% confidence" or "97.5% confidence" here, except
insofar as you have adopted a threshold (e.g. 5%) for the decision to
reject, in which case if p < 0.05 then the case "means equal" is outside
a 95% confidence set for the means of the groups (confidence set defined
by the non-centrality parameter).

Octave itself, in this function, makes no suppositions about what
confidence level or significance level one may be working with (if any).
Once you know the p-value, you can choose what you like.

Best wishes,

E-Mail: (Ted Harding) <address@hidden>
Fax-to-email: +44 (0)870 167 1972
Date: 12-May-04                                       Time: 18:27:25
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