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From |
jpitblado@stata.com (Jeff Pitblado, StataCorp LP) |

To |
statalist@hsphsun2.harvard.edu |

Subject |
Re: st: Tukey's HSD test from summary statistics |

Date |
Wed, 08 Feb 2012 13:46:23 -0600 |

Maria Niarchou <m.niarchou@hotmail.com> 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 jpitblado@stata.com * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/statalist/faq * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:**Re: st: Tukey's HSD test from summary statistics***From:*Muhammad Anees <anees@aneconomist.com>

**RE: st: Tukey's HSD test from summary statistics***From:*Maria Niarchou <m.niarchou@hotmail.com>

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