BASIS TOLERANCE LIMITS

Generates A-basis and B-basis tolerance intervals for the
Weibull, normal, and lognormal distributions.

Description:

Standard tolerance intervals calculate a confidence interval
that contains a fixed percentage (or proportion) of the data.
This is related to, but distinct from, the confidence interval
for the mean.

There are two numbers for the tolerance interval:

The coverage probability is the fixed percentage of
the data to be covered.

The confidence level.

Standard tolerance limits are given by

\( \bar{X} \pm k*s \)

where \( \bar{X} \) is the sample mean, s is the sample
standard deviation, and k is determined so that one can
state with (1- α)% confidence that at least
Φ% of the data fall within the given limits. The values
for k, assuming a normal distribution, have been numerically
tabulated.

This is commonly stated as something like "a 95% confidence
interval for 90% coverage".

A and B basis values are a special case of this. Specifically,
the B basis value is a 95% lower confidence bound on the
tenth percentile of a specified population of measurements
and the A basis value is a 95% lower confidence bound of
the first percentile. Alternatively, this can be stated
as the B basis value is a 95% lower tolerance bound for the
upper 90% of a specified population and the A basis value is
a 95% lower tolerance bound for the upper 99% of a
specified population.

Note that the A and B basis values are one sided intervals
(the standard tolerance limits are two sided). Also, the
standard tolerance limits are typically based on a
normality assumption while the A and B basis values can
be computed for Weibull, normal, or lognormal distributions
or they can be computed non-parametrically if none of
these distributions provide an adequate fit.

A and B basis values were added to support the MIL-17
Handbook standard (see the Reference section below).
The mathematics of computing these basis values are given
in the MIL-17 Handbook and are not given here.

A and B basis values are used for the case where the data
can be considered unstructured. That is, the data are
either univariate to start with or the Anderson-Darling
k-sample test has determined that the data can be treated
as coming from a common sample. Also, the appropriate
distribution should be determined first. The MIL-17
Handbook recommends using the Anderson-Darling goodness
of fit test. It also recommends trying the Weibull, then
the lognormal, then the normal. If all of these fail, then
the non-parametric case can be used.
Syntax 1:

BBASIS <dist> TOLERANCE LIMITS <y>
<SUBSET/EXCEPT/FOR qualification>
where <dist> is WEIBULL, NORMAL, LOGNORMAL, or NONPARAMETRIC;
<y> is the response variable,
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax computes B basis values.

Syntax 2:

ABASIS <dist> TOLERANCE LIMITS <y>
<SUBSET/EXCEPT/FOR qualification>
where <dist> is WEIBULL, NORMAL, LOGNORMAL, or NONPARAMETRIC;
<y> is the response variable,
and where the <SUBSET/EXCEPT/FOR qualification> is optional.