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In [[statistics]], '''Scheffé's method''', named after the [[United States|American]] [[statistician]] [[Henry Scheffé]], is a method for adjusting [[statistical significance|significance levels]] in a [[linear regression]] analysis to account for [[multiple comparisons]]. It is particularly useful in [[analysis of variance]], and in constructing simultaneous [[confidence band]]s for regressions involving [[basis functions]]. | |||
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible [[contrast (statistics)|contrast]]s among the factor level means, not just the pairwise differences considered by the [[Tukey–Kramer method]]. | |||
== The method == | |||
Let ''μ''<sub>1</sub>, ..., ''μ''<sub>''r''</sub> be the [[mean]]s of some variable in ''r'' disjoint populations. | |||
An arbitrary contrast is defined by | |||
:<math>C = \sum_{i=1}^r c_i\mu_i</math> | |||
where | |||
:<math>\sum_{i=1}^r c_i = 0.</math> | |||
If ''μ''<sub>1</sub>, ..., ''μ''<sub>''r''</sub> are all equal to each other, then all contrasts among them are 0. Otherwise, some contrasts differ from 0. | |||
Technically there are infinitely many contrasts. The simultaneous confidence coefficient is exactly 1 − α, whether the factor level sample sizes are equal or unequal. (Usually only a finite number of comparisons are of interest. In this case, Scheffé's method is typically quite conservative, and the experimental error rate will generally be much smaller than α.)<ref name="MaxwellDelaney">Scott E. Maxwell and Harold D. Delaney. ''Designing Experiments and Analyzing Data: A Model Comparison''. Lawrence Erlbaum Associates, 2004, ISBN 0-8058-3718-3, pp. 217–218.</ref><ref name="MillikenJohnson">George A. Milliken and Dallas E. Johnson. ''Analysis of Messy Data''. CRC Press, 1993, ISBN 0-412-99081-4, pp. 35–36.</ref> | |||
We estimate ''C'' by | |||
:<math>\hat{C} = \sum_{i=1}^r c_i\bar{Y}_i</math> | |||
for which the estimated variance is | |||
:<math>s_{\hat{C}}^2 = \hat{\sigma}_e^2\sum_{i=1}^r \frac{c_i^2}{n_i},</math> | |||
where | |||
* ''n''<sub>''i''</sub> is the size of the sample taken from the ''i''th population (the one whose mean is ''μ''<sub>''i''</sub>), and | |||
* <math>\hat{\sigma}_e^2</math> is the estimated variance of the [[errors and residuals in statistics|errors]]. | |||
It can be shown that the probability is 1 − α that all confidence limits of the type | |||
:<math>\hat{C}\pm\,s_\hat{C}\sqrt{\left(r-1\right)F_{\alpha;r-1;N-r}} </math> | |||
are simultaneously correct, where as usual N is the size of the whole population. | |||
==Denoting Scheffé significance in a table== | |||
Frequently, superscript letters are used to indicate which values are significantly different using the Scheffé method. For example, when mean values of variables that have been analyzed using an [[ANOVA]] are presented in a table, they are assigned a different letter superscript based on a Scheffé contrast. Values that are not significantly different based on the post-hoc Scheffé contrast will have the same superscript and values that are significantly different will have different superscripts (i.e. 15a, 17a, 34b would mean that the first and second variables both differ from the third variable but not each other because they are both assigned the superscript "a").{{citation needed|date=August 2012}} | |||
==Comparison with the Tukey–Kramer method== | |||
If only pairwise comparisons are to be made, the [[Tukey–Kramer method]] will result in a narrower confidence limit, which is preferable. In the general case when many or all contrasts might be of interest, the Scheffé method tends to give narrower confidence limits and is therefore the preferred method. | |||
==References== | |||
{{reflist}} | |||
*Bohrer, Robert (1967) "On Sharpening Scheffé Bounds", ''[[Journal of the Royal Statistical Society]], Series B'', 29 (1), 110–114 {{jstor|2984571}} | |||
*Scheffé, H. (1959). ''The Analysis of Variance''. Wiley, New York. (reprinted 1999, ISBN 0-471-34505-9) | |||
==External links== | |||
*[http://www.itl.nist.gov/div898/handbook/prc/section4/prc472.htm Scheffé's method] | |||
{{NIST-PD}} | |||
{{DEFAULTSORT:Scheffe's method}} | |||
[[Category:Multiple comparisons]] | |||
Latest revision as of 22:28, 7 April 2013
In statistics, Scheffé's method, named after the American statistician Henry Scheffé, is a method for adjusting significance levels in a linear regression analysis to account for multiple comparisons. It is particularly useful in analysis of variance, and in constructing simultaneous confidence bands for regressions involving basis functions.
Scheffé's method is a single-step multiple comparison procedure which applies to the set of estimates of all possible contrasts among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.
The method
Let μ1, ..., μr be the means of some variable in r disjoint populations.
An arbitrary contrast is defined by
where
If μ1, ..., μr are all equal to each other, then all contrasts among them are 0. Otherwise, some contrasts differ from 0.
Technically there are infinitely many contrasts. The simultaneous confidence coefficient is exactly 1 − α, whether the factor level sample sizes are equal or unequal. (Usually only a finite number of comparisons are of interest. In this case, Scheffé's method is typically quite conservative, and the experimental error rate will generally be much smaller than α.)[1][2]
We estimate C by
for which the estimated variance is
where
- ni is the size of the sample taken from the ith population (the one whose mean is μi), and
- is the estimated variance of the errors.
It can be shown that the probability is 1 − α that all confidence limits of the type
are simultaneously correct, where as usual N is the size of the whole population.
Denoting Scheffé significance in a table
Frequently, superscript letters are used to indicate which values are significantly different using the Scheffé method. For example, when mean values of variables that have been analyzed using an ANOVA are presented in a table, they are assigned a different letter superscript based on a Scheffé contrast. Values that are not significantly different based on the post-hoc Scheffé contrast will have the same superscript and values that are significantly different will have different superscripts (i.e. 15a, 17a, 34b would mean that the first and second variables both differ from the third variable but not each other because they are both assigned the superscript "a").Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.
Comparison with the Tukey–Kramer method
If only pairwise comparisons are to be made, the Tukey–Kramer method will result in a narrower confidence limit, which is preferable. In the general case when many or all contrasts might be of interest, the Scheffé method tends to give narrower confidence limits and is therefore the preferred method.
References
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- Bohrer, Robert (1967) "On Sharpening Scheffé Bounds", Journal of the Royal Statistical Society, Series B, 29 (1), 110–114 Template:Jstor
- Scheffé, H. (1959). The Analysis of Variance. Wiley, New York. (reprinted 1999, ISBN 0-471-34505-9)