Scheffé's method
Encyclopedia
In statistics
, Scheffé's method, named after 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 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
s among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.
Let μ1, ..., μr be the mean
s 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.)
We estimate C by
for which the estimated variance is
It can be shown that the probability is 1 − α that all confidence limits of the type
are correct simultaneously.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Scheffé's method, named after Henry Scheffé
Henry Scheffé
Henry Scheffé was an American statistician. He is known for the Lehmann–Scheffé theorem and Scheffé's method.- External links :...
, is a method for adjusting significance levels
Statistical significance
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....
in a linear regression
Linear regression
In statistics, linear regression is an approach to modeling the relationship between a scalar variable y and one or more explanatory variables denoted X. The case of one explanatory variable is called simple regression...
analysis to account for multiple comparisons
Multiple comparisons
In statistics, the multiple comparisons or multiple testing problem occurs when one considers a set of statistical inferences simultaneously. Errors in inference, including confidence intervals that fail to include their corresponding population parameters or hypothesis tests that incorrectly...
. It is particularly useful in analysis of variance
Analysis of variance
In statistics, analysis of variance is a collection of statistical models, and their associated procedures, in which the observed variance in a particular variable is partitioned into components attributable to different sources of variation...
, and in constructing simultaneous confidence band
Confidence band
A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Confidence bands are often used as part of the graphical presentation of results in a statistical analysis...
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
Contrast (statistics)
In statistics, particularly analysis of variance, a contrast is a linear combination of two or more factor level means whose coefficients add up to zero. A simple contrast is the difference between two means...
s among the factor level means, not just the pairwise differences considered by the Tukey–Kramer method.
Let μ1, ..., μr be the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
s 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.)
We estimate C by
for which the estimated variance is
It can be shown that the probability is 1 − α that all confidence limits of the type
are correct simultaneously.