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RE: st: Tukey's HSD test from summary statistics
From
Maria Niarchou <[email protected]>
To
<[email protected]>
Subject
RE: st: Tukey's HSD test from summary statistics
Date
Wed, 8 Feb 2012 23:30:50 +0200
Dear Jeff,
This was very helpful. Thank you very much for your assistance.
Best wishes,
Maria
> From: [email protected]
> To: [email protected]
> Subject: Re: st: Tukey's HSD test from summary statistics
> Date: Wed, 8 Feb 2012 13:46:23 -0600
>
> Maria Niarchou <[email protected]> asks
>
> > Is there a way to calculate Tukey's HSD test in Stata when only sample
> > sizes, means and standard deviations are available?
>
> The short answer is: Yes.
>
> New in Stata 12 are the functions -tukeyprob()- and -invtukeyprob()- that
> compute cumulative probabilities and quantiles from Tukey's studentized range
> distribution.
>
> -----------------------------------------------------------------------------
>
> Here is the longer answer with some formulas, followed by an example.
>
> Suppose we have k means to compare, where mean m_i and standard deviation s_i
> were computed from group i having sample size n_i.
>
> Our first problem is to determine how to estimate the standard error of a
> given difference, say
>
> SE(m_1-m_2) = ?
>
> Assuming a common variance between the k groups, we can pool the sample
> variance estimates to get
>
> MSE = (1/df) sum_i (n_i-1)*s_i^2
>
> where
>
> df = sum_i (n_i - 1)
>
> So the HSD test statistic, assuming equal variances, becomes
>
> q = abs(m_1 - m_2)/sqrt(MSE*(1/n_1 + 1/n_2)/2)
>
> The extra divisor 2 in the square root comes from the fact that we are looking
> as the absolute difference between m_1 and m_2.
>
> A 5% critical value can be computed using the -invtukeyprob()- function.
>
> crit = invtukeyprob(k, df, .95)
>
> The corresponding p-value can be computed using the -tukeyprob()- function.
>
> p = 1 - tukeyprob(k, df, q)
>
> If we can't assume unequal variances, then the test statistic becomes
>
> q = (m_1 - m_2)/sqrt((s_1^2/n_1 + s_2^2/n_2)/2)
>
> -----------------------------------------------------------------------------
>
> Example 6 in -[R] ttest- performs an unpaired ttest assuming equal variances
>
> ***** BEGIN:
> . ttesti 20 20 5 32 15 4
>
> Two-sample t test with equal variances
> ------------------------------------------------------------------------------
> | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
> ---------+--------------------------------------------------------------------
> x | 20 20 1.118034 5 17.65993 22.34007
> y | 32 15 .7071068 4 13.55785 16.44215
> ---------+--------------------------------------------------------------------
> combined | 52 16.92308 .6943785 5.007235 15.52905 18.3171
> ---------+--------------------------------------------------------------------
> diff | 5 1.256135 2.476979 7.523021
> ------------------------------------------------------------------------------
> diff = mean(x) - mean(y) t = 3.9805
> Ho: diff = 0 degrees of freedom = 50
>
> Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
> Pr(T < t) = 0.9999 Pr(|T| > |t|) = 0.0002 Pr(T > t) = 0.0001
> ***** END:
>
> Suppose this test represents only 1 comparison among 5 means, and lets pretend
> that sqrt(MSE) is the same as the Std. Dev. for the combined means above.
> Also, let's assume the total degrees of freedom is df = 100.
>
> The HSD test statistic is
>
> q = (20 - 15)/(5.007235*sqrt((1/20 + 1/15)/2))
> = 4.1344109
>
> The 5% critical value is
>
> crit = invtukeyprob(k, df, .95)
> = 3.9289372
>
> The p-value is
>
> p = 1 - tukeyprob(k, df, q)
> = .03400394
>
> For unequal variances, the results from -ttesti- are
>
> ***** BEGIN:
> . ttesti 20 20 5 32 15 4, unequal
>
> Two-sample t test with unequal variances
> ------------------------------------------------------------------------------
> | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]
> ---------+--------------------------------------------------------------------
> x | 20 20 1.118034 5 17.65993 22.34007
> y | 32 15 .7071068 4 13.55785 16.44215
> ---------+--------------------------------------------------------------------
> combined | 52 16.92308 .6943785 5.007235 15.52905 18.3171
> ---------+--------------------------------------------------------------------
> diff | 5 1.322876 2.311343 7.688657
> ------------------------------------------------------------------------------
> diff = mean(x) - mean(y) t = 3.7796
> Ho: diff = 0 Satterthwaite's degrees of freedom = 33.9142
>
> Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
> Pr(T < t) = 0.9997 Pr(|T| > |t|) = 0.0006 Pr(T > t) = 0.0003
> ***** END:
>
> The HSD test statistic is
>
> q = (20 - 15)/sqrt((5^2/20 + 4^2/32)/2)
> = 5.3452248
>
> The 5% critical value is still
>
> crit = invtukeyprob(k, df, .95)
> = 3.9289372
>
> The p-value is
>
> p = 1 - tukeyprob(k, df, q)
> = .00243234
>
> --Jeff
> [email protected]
> *
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