Introduction
The proposed fisher’s exact test
Class 1 | Class 2 | Total | |
---|---|---|---|
Sample 1 | \({a}_{L}+{a}_{U}{I}_{N}\) | \({b}_{L}+{b}_{U}{I}_{N}\) | \(\left({a}_{L}+{b}_{L}\right)+\left({a}_{U}+{b}_{U}\right){I}_{N}\) |
Sample 2 | \({c}_{L}+{c}_{U}{I}_{N}\) | \({d}_{L}+{d}_{U}{I}_{N}\) | \(\left({c}_{L}+{d}_{L}\right)+\left({c}_{U}+{d}_{U}\right){I}_{N}\) |
Total | \(\left({a}_{L}+{b}_{L}\right)+\left({a}_{U}+{c}_{U}\right){I}_{N}\) | \(\left({b}_{L}+{d}_{L}\right)+\left({b}_{U}+{d}_{U}\right){I}_{N}\) | \({N}_{L}+{N}_{U}{I}_{N}\) |
Application using industrial data
Machines | \({A}_{1}\) | \({A}_{2}\) | Total |
---|---|---|---|
Production time | [1, 1] | [1, 1] | [2, 2] |
Number of defective | [10, 15] | [26, 32] | [36, 47] |
Total | [11, 16] | [27, 33] | [38, 49] |
[0,0] | [2,2] | [2,2] |
[11,16] | [25,31] | [36,47] |
[11,16] | [27,33] | [38,49] |
Advantages based on industrial data
Simulation study
\(\sum {p}_{N}\epsilon \left[\sum {p}_{L},\sum {p}_{U}\right]\) | \(\sum {p}_{L}+\sum {p}_{U}{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[{I}_{{p}_{L}},{I}_{{p}_{U}}\right]\) | Decision |
---|---|---|
\(\sum {p}_{N}\epsilon\)[0.04, 0.01] | \(0.04-0.01{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,3}\right]\) | Reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.09, 0.05] | \(0.09-0.05{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.80}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.14, 0.10] | \(0.14-0.10{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.40}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.19, 0.15] | \(0.19-0.15{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.27}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.24, 0.20] | \(0.24-0.20{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.20}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.29, 0.25] | \(0.29-0.25{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.16}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.34, 0.30] | \(0.34-0.30{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.13}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.39, 0.35] | \(0.39-0.35{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.11}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.44, 0.40] | \(0.44-0.40{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.10}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.49, 0.45] | \(0.49-0.45{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.09}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.54, 0.50] | \(0.54-0.50{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.08}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.59, 0.55] | \(0.59-0.55{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.07}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.64, 0.60] | \(0.64-0.60{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.07}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.69, 0.65] | \(0.69-0.65{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.06}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.74, 0.70] | \(0.74-0.70{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.06}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.79, 0.75] | \(0.79-0.75{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.05}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.84, 0.80] | \(0.84-0.80{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.05}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.89, 0.85] | \(0.89-0.85{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.05}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.94, 0.90] | \(0.94-0.90{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.04}\right]\) | Do not reject \({H}_{0}\) |
\(\sum {p}_{N}\epsilon\)[0.99, 0.95] | \(0.99-0.95{I}_{{p}_{N}};{I}_{{p}_{N}}\epsilon \left[\mathrm{0,0.04}\right]\) | Do not reject \({H}_{0}\) |
Sensitivity analysis
Power of the test
-
Step-1: Generate a set of 10,000 random samples of the test statistic \(\sum {p}_{N}\)
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Step-2: Compare the values of \(\sum {p}_{N}\) with the level of significance and record whether the null hypothesis \({H}_{0}\) is rejected or accepted.
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Step-3: Determine the values of \(\beta\) (Type II error rate) by the ratio of the number of erroneous conclusion to the total number of replications.
\(\alpha\) | \(\left(1-\beta \right)\) |
---|---|
0.01 | [0.9895, 0.9418] |
0.02 | [0.9789, 0.8994] |
0.04 | [0.9589, 0.8284] |
0.05 | [0.951, 0.801] |
0.08 | [0.9183, 0.7184] |
0.10 | [0.8978, 0.6705] |
0.15 | [0.8527, 0.569] |
0.20 | [0.8036, 0.4878] |
Effect of indeterminacy on level of significance
-
Step-1: Generate a set of 10,000 random samples of the test statistic \(\sum {p}_{N}\)
-
Step-2: Compare the values of \(\sum {p}_{N}\) with the level of significance and record whether the null hypothesis \({H}_{0}\) is rejected or accepted.
-
Step-3: Determine the values of \({\widehat{\alpha }}_{N}\epsilon \left[{\widehat{\alpha }}_{L},{\widehat{\alpha }}_{U}\right]\) (Type I error rate) by the ratio of the number of rejection conclusions to the total number of replications.
\({\alpha }_{0}\) | \({\widehat{\alpha }}_{N}\epsilon \left[{\widehat{\alpha }}_{L},{\widehat{\alpha }}_{U}\right]\) |
---|---|
0.01 | [0.01, 0.05] |
0.02 | [0.02, 0.10] |
0.04 | [0.04, 0.17] |
0.05 | [0.05, 0.20] |
0.08 | [0.08, 0.28] |
0.10 | [0.09, 0.33] |
0.15 | [0.15, 0.43] |
0.20 | [0.20, 0.52] |