Adaptive Standard Error
DESCRIPTION:
Returns an estimate of the adaptive standard error (ASE) of estimated effects from a fractional factorial design and the cumulative probabilities and quantiles for the distribution of the ASE based statistic described below.
USAGE: 
ase(x)
pase(q,k)
qase(p,k)
REQUIRED ARGUMENTS:
x:
vector of estimated effects. Missing values (NAs) are not allowed.
p:
vector of probabilities. Missing values (NAs) are allowed.
q:
vector of quantiles. Missing values (NAs) are allowed.
k:
number of estimated effects. k must be between 5 and 31, inclusive.
VALUE:
ase returns the estimated adaptive standard error of a set of estimated effects. pase and qase return probability and quantile vectors for the distribution of the ASE based statistic.
DETAILS:
Elements of q or p that are missing will cause the corresponding elements of the result to be missing. Values of the pase and qase are available for k = 5 to 31. The quantiles of the empirical cdf are stored in a matrix cdf.ase for the values .70 to .99 by .01. Thus the the quantile associated with p between .70 and .99 is found in the kth row and floor(p*100)-69 th column of cdf.ase.
BACKGROUND:
Due to cost and time restrictions, industrial experimentation is often geared toward the use of highly fractionated, unreplicated factorial designs. These designs typically allow no degrees of freedom for the estimation of error and are referred to by Box and Meyer (1986) as effect saturated designs. Because there is no independent estimate of the error, identification of important effects lies outside the range of classical methods (Haaland and O'Connell 1994).

An approach to this problem which motivates the use of robust estimators of scale is as follows: think of the estimated effects as a sample from a zero mean normal distribution (the null effects) contaminated by the non-null effects. Use robust methods to find an estimate of the scale of the null effects that is insensitive to the non-null effects. Then the estimated effects that are large compared to this scale estimate correspond to the non-null effects.

Dong (1993) proposes the adaptive standard error (ASE) as a robust scale estimator for this problem. Haaland and O'Connell (1994) studied the properties of this and several related tests. The ASE based test is recommended when there is a priori reason to believe that there will be only a few significant effects, say 0 to 3. However, the PSE (pseudo standard error -- Lenth, 1989) is recommended as an all around good test for identifying significant effects in a saturated fractional factorial design.

The value of the ASE is included in the fac.aov object created in the standard analysis of a fractional factorial design in S+DOX. The reference distribution is used to provide approximate p-values in the summary procedure and to draw a cut-off line for significant effects on the pareto and half-normal plots. The estimated ASE is equal to 1/slope of the line through the null effects on the half-normal plot. Tests based on the ASE are also used in the empirical Bayes plot.

REFERENCES:
Box, G.E.P. and Meyer, R.D. (1986). "An Analysis for Unreplicated Fractional Factorials." Technometrics 28, 1-18.

Dong, F. (1993). "On the Idendification of Active Contrasts in Unreplicated Fractional Factorials." Statistica Sinica 3, 209-217.

Haaland, P. D. and M. A. O'Connell (1994), "Inference for Effect Saturated Fractional Factorials", to apear in Technometrics.

Lenth, R. V. (1989), "Quick and Easy Analysis of Unreplicated Fractional Factorials", Technometrics, 31, 469-473.

Zahn, D. A. (1975). "An Empirical Study of the Half-Normal Plot." Technometrics 17, 201-211.

SEE ALSO:
pse , tse , fac.aov , summary.fac.aov , pareto.fac.aov , qqnorm.fac.aov , ebplot .
EXAMPLES: 
buffer.fac <- fac.aov(buffer.df)
buffer.fac$ase$
ase(buffer.fac$feffects$)
qase(.95,15)
summary(buffer.fac,method="ase")
pareto(buffer.fac,method="ase")
qqnorm(buffer.fac,method="ase")
ebplot(buffer.fac,method="ase")