Binomial Probabilities Under the Null Hypothesis
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| Format: | Recurso digital |
| Langue: | anglais |
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2026
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| _version_ | 1866901154203435008 |
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| author | Barukčić, Ilija |
| author_facet | Barukčić, Ilija |
| contents | <p><strong>Background.</strong><br>In hypothesis testing involving binomial proportions, the conventional \( p \)-value calculation often requires summation over many terms. However, when the null probability \( p \) is extremely close to \( 1 \) and the observed number of failures \( k = n - X_{\text{obs}} \) is very small, the right-tailed \( p \)-value simplifies considerably.</p> <p><strong>Material and methods.</strong></p> <p>We consider a binomial setting with \( n \) independent trials under the null hypothesis \( H_0: p \approx 1 \). Let \( X \) denote the number of successes, with observed value \( X_{\text{obs}} = n - k \), where \( k \) is small. The exact right-tailed \( p \)-value is given by<br>\(<br>p\text{-value} = \sum_{j=0}^{k} \binom{n}{j} p^{\,n-j} (1-p)^{j}.<br>\)<br>For \( p \) near \( 1 \) and small \( k \), we analyze the dominant term in this sum.</p> <p><br><strong>Results.</strong><br>The dominant contribution arises from the term with \( j = 0 \) (no failures), yielding<br>\(<br>p\text{-value} \approx p^{\,n}.<br>\)<br>Expressing this in terms of the observed failures \( k \) and the observed success rate \( \hat{p} = k/n \approx 1 \), we obtain the approximation<br>\(<br>p\text{-value} \approx \hat{p}^{\,k}.<br>\)<br>This approximation becomes increasingly accurate as \( p \) approaches \( 1 \) and \( k \) remains small.</p> <p><strong>Conclusions.</strong><br>When testing a null hypothesis with \( p \) very close to \( 1 \) and observing only a few failures (\( k \) small), the right-tailed \( p \)-value can be reliably approximated by \( \hat{p}^{\,k} \). This simple formula provides a practical alternative to full binomial summation, facilitating rapid interpretation in high-reliability contexts such as pharmacovigilance, quality control, and rare event analysis.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19225554 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Binomial Probabilities Under the Null Hypothesis Barukčić, Ilija Binomial distribution Cause Effect Cause and effect Causation Causality Ilija Barukcic Barukcic, Ilija Barukcic Ilija Barukčić Barukčić, Ilija Barukčić <p><strong>Background.</strong><br>In hypothesis testing involving binomial proportions, the conventional \( p \)-value calculation often requires summation over many terms. However, when the null probability \( p \) is extremely close to \( 1 \) and the observed number of failures \( k = n - X_{\text{obs}} \) is very small, the right-tailed \( p \)-value simplifies considerably.</p> <p><strong>Material and methods.</strong></p> <p>We consider a binomial setting with \( n \) independent trials under the null hypothesis \( H_0: p \approx 1 \). Let \( X \) denote the number of successes, with observed value \( X_{\text{obs}} = n - k \), where \( k \) is small. The exact right-tailed \( p \)-value is given by<br>\(<br>p\text{-value} = \sum_{j=0}^{k} \binom{n}{j} p^{\,n-j} (1-p)^{j}.<br>\)<br>For \( p \) near \( 1 \) and small \( k \), we analyze the dominant term in this sum.</p> <p><br><strong>Results.</strong><br>The dominant contribution arises from the term with \( j = 0 \) (no failures), yielding<br>\(<br>p\text{-value} \approx p^{\,n}.<br>\)<br>Expressing this in terms of the observed failures \( k \) and the observed success rate \( \hat{p} = k/n \approx 1 \), we obtain the approximation<br>\(<br>p\text{-value} \approx \hat{p}^{\,k}.<br>\)<br>This approximation becomes increasingly accurate as \( p \) approaches \( 1 \) and \( k \) remains small.</p> <p><strong>Conclusions.</strong><br>When testing a null hypothesis with \( p \) very close to \( 1 \) and observing only a few failures (\( k \) small), the right-tailed \( p \)-value can be reliably approximated by \( \hat{p}^{\,k} \). This simple formula provides a practical alternative to full binomial summation, facilitating rapid interpretation in high-reliability contexts such as pharmacovigilance, quality control, and rare event analysis.</p> |
| title | Binomial Probabilities Under the Null Hypothesis |
| topic | Binomial distribution Cause Effect Cause and effect Causation Causality Ilija Barukcic Barukcic, Ilija Barukcic Ilija Barukčić Barukčić, Ilija Barukčić |
| url | https://doi.org/10.5281/zenodo.19225554 |