Trimmed Standard Error
DESCRIPTION:
Returns an estimate of the trimmed standard error (TSE) of estimated effects from a fractional factorial design and the cumulative probabilities and quantiles for the distribution of the TSE based statistic described below.
USAGE: 
tse(x)
ptse(q,k)
qtse(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:
tse returns the estimated trimmed standard error of a set of estimated effects. ptse and qtse return probabilities and quantiles for the distribution of the TSE 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 ptse and qtse are available for k = 5 to 31. The quantiles of the empirical cdf are stored in a matrix cdf.tse 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.tse.
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.

Berk and Picard (1991) proposes the trimmed standard error (TSE) as a robust scale estimator for this problem. Haaland and O'Connell (1994) studied the properties of this and several related tests. The TSE based test is not as powerful as the PSE (pseudo standard error, Lenth, 1989) based test but is an acceptable alternative as long as there aren't too many non-null effects.

The value of the TSE 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 TSE is equal to 1/slope of the line through the null effects on the half-normal plot. Tests based on the tse are also used in the empirical bayes plot.

REFERENCES:
Berk, K.N. and Picard, R.R. (1991). "Significance Tests for Saturated Orthogonal Arrays." Journal of Quality Technology 23, 174-178.

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

Haaland, P. D. and M. A. O'Connell (1994), "Inference for Effect Saturated Fractional Factorials." to appear 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:
ase , pse , fac.aov , summary.fac.aov , pareto.fac.aov , qqnorm.fac.aov , ebplot
EXAMPLES: 
buffer.fac <- fac.aov(buffer.df)
buffer.fac$tse
tse(buffer.fac$feffects)
qtse(.95,15)
summary(buffer.fac,method="tse")
pareto(buffer.fac,method="tse")
qqnorm(buffer.fac,method="tse")
ebplot(buffer.fac,method="tse")