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Topic: Testing the significance of difference in set means at different times
Replies: 9   Last Post: Nov 7, 2012 7:53 PM

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Ray Koopman

Posts: 3,383
Registered: 12/7/04
Re: Testing the significance of difference in set means at different times
Posted: Nov 7, 2012 5:06 AM
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On Nov 6, 8:45 pm, wrote:
> On Friday, 2 November 2012 19:30:02 UTC+11, Ray Koopman wrote:
>> On Nov 1, 3:10 pm, wrote:
>>> [...]
>>> Sets A and B are cross-sectional (representative) samples of two different populations. The members of the sets at time 1 and 2 are different, though still representative. The principal measure of interest is the mean value (of a rating given by) the respective sets.
>>> I have used one-way ANOVA to explore the significance of the difference in mean score between A1 and A2, B1 and B2, A1 and B1 (ie, [Ma1-Mb1]), and, A2 and B2 (ie, [Ma2-Mb2]).
>>> My interest/question was about testing the significance of the 'difference of the differences' (Ma1-Mb1)-(Ma2-Mb2).
>>> I had considered a two-way ANOVA using all of the data and looking at the significance of interaction term, but was unsure. I will look at this.
>>> Thanks again.

>> If you have the following:
>> Sample Sizes: Na1 Nb1 Na2 Nb2
>> Means: Ma1 Mb1 Ma2 Mb2
>> Variances: Va1 Vb1 Va2 Vb2
>> then calculate
>> (Ma1 - Mb1) - (Ma2 - Mb2)
>> z = -------------------------------------------,
>> sqrt(Va1/Na1 + Vb1/Nb1 + Va2/Na2 + Vb2/Nb2)
>> and refer it to the standard normal distribution in the usual way.
>> (Actually, what you have is not strictly a z, but an approximate t
>> whose degrees of freedom are at least in the hundreds, and possibly
>> in the thousands, so there is little lost by treating it as a z.)

> Hi Ray,
> Sorry to Pester.
> If I understand correctly, this is essentially Welch's t-test
> with the composite variance extended for more than two groups.
> Do you have a reference for this approach?
> Regards, Stuart Palmer.

Research Design and Statistical Analysis
Jerome L. Myers and Arnold D. Well
1st ed (HarperCollins, 1991): sec 6.8, p 187
2nd ed (Erlbaum, 2003): sec 9.3.2, pp 239-241

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