anomheI

One-way Analysis of Means with Heteroscedasticity.

Analysis of Means (ANOM) is a statistical procedure for troubleshooting industrial processes and analyzing the results of experimental designs with factors at fixed levels. ANOM is an alternative to ANOVA for a fixed effects model. Unlike ANOVA only determines if there is a significant difference between the treatment means, ANOM identifies the means that are significantly different. For it is considered a class of multiple comparasions procedure. ANOM not only answers the question of whether or not there are any differences among the factor levels, but when there are differences, it also tell us which levels are better and which are worst.

It compares the absolute deviations of group means from their overall mean, an approach that was initially studied by Laplace in 1827. Halperin et al. (1955) derived a version of this method in the form of a multiple significance test in 1955. Ott (1967, 1975) developed a graphical representation for the test and introduced the term 'analysis of means'. Nelson (1982) and Nelson (1983) provided exact critical values for ANOM
when the groups have equal sample sizes (balanced).

According to Nelson et al. (2005), basically, an ANOM is the generation of a decison chart similar in appearance to a control chart. It has a centerline, located at the overall mean, rate or proportion, and upper and lower decision limits. The group means, rate or proportions are plotted, and those that fall beyond the decision limits are said to be significantly diferent from the overall value. These differences are statistical diferences, if they exist. Then, ANOM allows one to easily evaluate the practical differences.

This m-file considers, according to Nelson and Dudewicz (2002) and Dudewics and Nelson (2003), the heteroscedastic situation where different processes do not necessarily have equal variances. These new results allow an experimenter to set a goal of detecting differences among the k treatment means when two of them differ by at least a specified amound d, which does not depend on the possibly different) standard eviations of the processes. In order to review the statistic fundamentals of this procedure, see Nelson and Dudewicz (2002) and Dudewics and Nelson (2003).

It is considered the errors of the treatments being compared are normally distributed with unequal variances, and all the observations are independent.

This statistical procedure it is also known in the literature as HANOM.

You can find the ANOM balanced (ANOMBAL) and unbalanced (ANOMUNBAL) m-files in this same FEX page.

--We deeply thank Dr. Edward J. Dudewicz for sent us the hard copies of some valuable papers here used.--

Syntax: function anomheI(X,no,d,a)

Inputs:
X - data matrix (Size of matrix must be n-by-2; sample=column 1, data=column 2)
no - initial sample size for the first-stage
d - amount that any two treatment means differing by it will lead to rejection of the null hypothesis
a - significance level (default = 0.05)

Outputs:
- Complete one-way heteroscedastic Analysis of Means
- ANOMHEI chart