![Sample our Mathematics & Statistics journals, sign in here to start your FREE access for 14 days]({"type":"image" , "src" : "/sda/5943/TOC-Subject-Sample-2021-Mathematics-Statistics.jpg"})
ABSTRACT
Binomial reliability demonstration tests are a way to demonstrate the product or process capability where binary data are available. It is easy to compute the sample size for a binomial reliability demonstration test (BDRT) when the assumption of independent samples holds. However, the assumption of independence may not always be valid; thus, it is desirable to account for this dependence in a sample size calculation. We show how a Markov dependence model can be used to calculate the sample size. We provide tables of sample sizes and give an example for which a BRDT can be performed with dependent data.
APPENDIX: ADDITIONAL SAMPLE SIZE CALCULATIONS
This Appendix describes how sample sizes can be determined for values of . Klotz (Citation1973, eq. [3.1]) gave the joint distribution in terms of the random variables
,
, and
, which is
where ,
,
, and
.
is the value of the joint distribution when all of the values of
are equal to zero as shown earlier in Eq. [Equation2
[2] ]. The values of
,
, and
were defined earlier in the section on the estimators of
. This joint distribution in terms of
and
is given by summing the joint distribution across the possible combinations of
,
and
that result in values of
and
.
For some value of , the BRDT will be successful when
. Thus, we have to sum the probabilities for all values of
in that range. The possible values of
are limited by the value of
. For example, if there are
failures in a data set, then possible values of
are 0 or 1. The possible values of
are also limited by the values of
. For example, if there is a single failure (
) in a data set, then possible values of
are 0 or 1. Regardless of the value of
, the maximum value of
is 2, because there are only two total locations at the beginning and end. We also add the restriction that
to avoid challenges in computation when
and the impossible situation where
.
Given these limitations, we can now define the joint distribution across all values of ,
, and
for a given value for
and
. This joint distribution is given as
where as defined in [Equation2
[2] ] for the case where there are no failures and where
for the impossible case where
and
.
To illustrate this joint distribution, if the maximum allowable number of failures is , then this joint distribution simplifies to
.
This joint distribution can then be used to find the sample size. For a given value of ,
, and
, we find the value of
such that
This equation was used to calculate the sample sizes shown in – for ,
, and
, respectively. The format is similar to that of shown earlier for
.
TABLE A1 Required Sample Sizes for BRDT with Dependent Samples for Different Values of Where
TABLE A2 Required Sample Sizes for BRDT with Dependent Samples for Different Values of Where
TABLE A3 Required Sample Sizes for BRDT with Dependent Samples for Different Values of Where