Utente:Grasso Luigi/sandbox3/metodo di Scheffé

In statistica, il metodo di Scheffé, che prende il nome dallo statistico americano Henry Scheffé, è un metodo per impostare i livelli di significatività in un'analisi della regressione lineare che tiene conto di confronti multipli. È particolarmente utile nell'analisi della varianza (un caso speciale di analisi della regressione), e nella costruzione simultanea di bande di confidenza per regressioni che coinvolgono funzioni di base.

Il metodo di Scheffé è una procedura di confronto multipo a passo-singolo che si applica all'insieme di stime di tutti i possibili contrasti tra le medie a livello di fattore, non solo le differenze a coppie considerate dal metodo Tukey–Kramer. Il metodo si basa su principi simili alla procedura di Working–Hotelling per stimare le risposte medie in regressione, che si applica a l'insieme di tutti i possibili livelli di fattore.

Teoria

Let μ₁, ..., μ_r be the means of some variable in r disjoint populations.

An arbitrary contrast is defined by

$C=\sum _{i=1}^{r}c_{i}\mu _{i}$

where

$\sum _{i=1}^{r}c_{i}=0.$

If μ₁, ..., μ_r 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 family-wise error rate (experimental error rate) will generally be much smaller than α.)

We estimate C by

${\hat {C}}=\sum _{i=1}^{r}c_{i}{\bar {Y}}_{i}$

for which the estimated variance is

$s_{\hat {C}}^{2}={\hat {\sigma }}_{e}^{2}\sum _{i=1}^{r}{\frac {c_{i}^{2}}{n_{i}}},$

where

• n_i is the size of the sample taken from the ith population (the one whose mean is μ_i), and
• ${\hat {\sigma }}_{e}^{2}$ is the estimated variance of the errors.

It can be shown that the probability is 1 − α that all confidence limits of the type

${\hat {C}}\pm \,s_{\hat {C}}{\sqrt {\left(r-1\right)F_{\alpha ;r-1;N-r}}}$

are simultaneously correct, where as usual N is the size of the whole population. Draper and Smith, in their 'Applied Regression Analysis' (see references), indicate that 'r' should be in the equation in place of 'r-1'. The slip with 'r-1' is a result of failing to allow for the additional effect of the constant term in many regressions. That the result based on 'r-1' is wrong is readily seen by considering r = 2, as in a standard simple linear regression. That formula would then reduce to one with the usual t distribution, which is appropriate for predicting/estimating for a single value of the independent variable, not for constructing a confidence band for a range of values of the independent value. Also note that the formula is for dealing with the mean values for a range of independent values, not for comparing with individual values such as individual observed data values.

Denotare i livelli significatività di Scheffé in una tabella

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").

Confronto col metodo Tukey–Kramer

If only a fixed number of pairwise comparisons are to be made, the Tukey–Kramer method will result in a more precise confidence interval. In the general case when many or all contrasts might be of interest, the Scheffé method is more appropriate and will give narrower confidence intervals in the case of a large number of comparisons.

Note

• Robert Bohrer, On Sharpening Scheffé Bounds, in Journal of the Royal Statistical Society, vol. 29, n. 1, 1967, pp. 110–114.
• H. Scheffé, The Analysis of Variance, New York, Wiley, 1999, ISBN 0-471-34505-9.

Collegamenti esterni

• Metodo di Scheffé