# Hypothesis Tests for Bernoulli Experiments: Ordering the Sample Space by Bayes Factors and Using Adaptive Significance Levels for Decisions

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## Abstract

**:**

## 1. Introduction

## 2. Blending Bayesian and Classical Concepts

#### 2.1. Statistical Model

**,**and the posterior density function $g\left(\theta |x\right).$ In order to be appropriate, the family of likelihood functions indexed by $x$, $\left\{L\left(\theta |x\right)=f\left(x|\theta \right);x\in \mathit{X}\right\},$ must be measurable in the prior σ-algebra.

- positive probabilities of the hypotheses, $\mathsf{\pi}\left(H\right)>0\mathrm{e}\mathsf{\pi}\left(A\right)=1-\mathsf{\pi}\left(H\right)0;$ and
- a density on the subset that has the smaller dimension. The choice of this density should be coherent with the original prior density over the global parameter space$\Theta $.

#### 2.2. Significance Index

**H**. We begin this section by stating a generalization of the Neyman–Pearson Lemma, as presented by DeGroot [19]. Cox [22,23] also considers the classical $p$-value as an evidence measure, and Evans [24] considers evidence measures in general, outlines the relative belief theory developed in the references of that paper, and suggests that the associated evidence measure could have advantages over other measures of evidence and be the basis of a complete approach to estimation and hypothesis-assessment problems. The classical $p$-value is the most widely used significance index across diverse fields of study, including almost all scientific areas. In the present work, we present a replacement for the classical $p$-value has a number of advantages that will be described here and in future work. The conceptual and operational similarity between classical hypothesis tests as currently used and the new tests could potentially help researchers accept and use the new tests.

**H**. The test function is

**Generalized**

**Neyman–Pearson Lemma:**

**H**in favor of A if $A{f}_{H}\left(x\right)<B{f}_{A}\left(x\right)$, does not reject

**H**if $A{f}_{H}\left(x\right)>B{f}_{A}\left(x\right)$, and is indifferent if $A{f}_{H}\left(x\right)=B{f}_{A}\left(x\right).$ Then, for any other test $\delta ,$

- Define a prior density $g\left(\theta \right)$ over the entire parameter space$\Theta $. This function can be chosen either objectively of subjectively.
- Clearly define the hypotheses to be tested,
**H**and**A**. - Obtain the predictive functions under the two alternative hypotheses. In the case for which the parametric subspaces defined by the hypotheses are of different dimensionalities, the definition of a prior density under the subset of smaller dimension, say
**H**, is obtained from the following expression, subject to the condition (on the parameter space as a whole and the hypotheses) that the integral in the denominator can be defined:$$g\left(\theta |H\right)=\{\begin{array}{cc}0& if\theta \notin {\Theta}_{A}\\ \frac{g\left(\theta \right)}{{{\displaystyle \oint}}_{{\Theta}_{H}}g\left(y\right)dy}& if\theta \in {\Theta}_{H}\end{array}.$$

**H**is the true hypothesis. Figure 1 illustrates how $g\left(\theta |H\right)$ is obtained from the prior $g\left(\theta \right)$ over the full parameter space$\Theta $.

- 4.
- Define the loss function, considering mainly the relative importance of the hypotheses and of the two types of error—consider, for example, a governor who is concerned more with the budget than with public health and who will strongly prefer the hypothesis that the apparent wave of meningitis cases in his state do not represent an epidemic.
- 5.
- Use the Bayes factor to order the sample space: $\left\{BF\left(x\right):x\in X\right\}\subset \mathcal{R}$ establishes the order of each$x\in \mathit{X}$. This ordering can be used independently of the dimensionalities of the spaces $\mathit{X}\mathrm{and}\Theta $.
- 6.
- Using the theorem above, compute the optimal averaged error probabilities and use the value of $\alpha \left({\delta}^{*}\right)$ as the adaptive level of significance, which will depend on the loss function, the probability densities, the prior probability$\pi $, and especially on the sample size.
- 7.
- Calculate the significance index, the $P$-value, as follows: if ${x}_{0}$ is the observed value of a statistic and ${C}_{0}=\left\{x;BF\left(x\right)\le BF\left({x}_{0}\right)\right\}$ is the observed tail under the new ordering, the $P$-value is calculated using the expression ${P}_{0}={{\displaystyle \int}}_{{C}_{0}}{f}_{H}\left(x\right)dx$. Clearly, this may be a single or a multiple integral or sum.
- 8.
- Compare the value ${P}_{0}$ with the value of $\alpha \left({\delta}^{*}\right).$Reject (do not reject)
**H**if ${P}_{0}\begin{array}{c}<\\ (>)\end{array}\alpha \left({\delta}^{*}\right)$. In the case of equality, take either decision without prejudice to optimality. - 9.
- Finally, if a value of $\alpha \left({\delta}^{*}\right)$ is specified a priori, calculate the sample size needed to make this fixed value as close as possible to optimal according to the generalized Neyman–Pearson Lemma.

## 3. Illustrative Examples

#### 3.1. Example 1—Comparing Two Proportions

^{2}$p$-value is 0.106, changed to 0.281 with the Yates continuity correction applied, and Fisher’s exact $p$-value is 0.282. Traditional analysts would conclude that there were no statistically significant differences between the two treatments, using any of the canonical significance levels. Note that these procedures were for testing a sharp hypothesis against a composite alternative: $H:{\theta}_{0}={\theta}_{1}\mathrm{and}A:{\theta}_{0}\ne {\theta}_{1},$ comparing the proportion of success of the two treatments. In what follows, we calculate the proposed $P$-value and use the optimal significance level $\alpha \left({\delta}^{*}\right)$ to make the decision of choosing one of the hypotheses.

**H**) in ${\mathsf{\Psi}}_{obs}$:

#### 3.2. Example 2—Two Proportions, Varying Sample Sizes

^{2}$p$-value is$p=0.0467$, indicating rejection of the null hypothesis at the canonical $5\%$level of significance. This agrees with our decision of rejecting the null hypothesis since again$P\alpha \left({\delta}^{*}\right)$. It is interesting to see the relative distance between the index and the level of significance. For the χ

^{2}test, we have $1-\frac{0.0467}{0.05}=0.07$ and the adaptive case obtains $1-\frac{0.029}{0.0995}=0.71$.

_{1}= n

_{2}= 20 (Table 2), despite the unbalanced sample having a total size of $70$ and the balanced sample just 40.

#### 3.3. Example 3—Test for One Proportion and the Likelihood Principle

- for a (positive) binomial,$${f}_{H}\left(x\right)=\left(\begin{array}{c}x+y\\ x\end{array}\right){\left(\frac{1}{2}\right)}^{x+y}\mathrm{and}{f}_{A}\left(x\right)={\left(x+y+1\right)}^{-1}$$
- for a negative binomial,$${f}_{H}\left(x\right)=\left(\begin{array}{c}x+y-1\\ x\end{array}\right){\left(\frac{1}{2}\right)}^{x+y}\mathrm{and}{f}_{A}\left(x\right)=y{\left[\left(x+y\right)\left(x+y+1\right)\right]}^{-1}.$$

#### 3.4. Example 4

## 4. Final Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**A prior$g\left(p,q\right)$ made of independent $Beta\left(2,4\right)$ and $Beta\left(4,2\right)$ distributions in a two-dimensional parameter space is cut along the line $p=q$ and one of the pieces moved away to show the resulting prior on the lower-dimensional set$p=q$.

x | y | Sum | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ||

0 | 4.765 | 2.382 | 1.112 | 0.476 | 0.183 | 0.061 | 0.017 | 0.003 | 4e^{-04} | 9 |

1 | 2.382 | 2.541 | 1.906 | 1.173 | 0.611 | 0.267 | 0.093 | 0.024 | 0.003 | 9 |

2 | 1.112 | 1.906 | 2.052 | 1.710 | 1.166 | 0.653 | 0.290 | 0.093 | 0.017 | 9 |

3 | 0.476 | 1.173 | 1.710 | 1.866 | 1.633 | 1.161 | 0.653 | 0.267 | 0.061 | 9 |

4 | 0.183 | 0.611 | 1.166 | 1.633 | 1.814 | 1.633 | 1.166 | 0.611 | 0.183 | 9 |

5 | 0.061 | 0.267 | 0.653 | 1.161 | 1.633 | 1.866 | 1.710 | 1.173 | 0.476 | 9 |

6 | 0.017 | 0.093 | 0.290 | 0.653 | 1.166 | 1.710 | 2.052 | 1.906 | 1.112 | 9 |

7 | 0.003 | 0.024 | 0.093 | 0.267 | 0.611 | 1.173 | 1.906 | 2.541 | 2.382 | 9 |

8 | 4e^{-04} | 0.003 | 0.017 | 0.061 | 0.183 | 0.476 | 1.112 | 2.382 | 4.765 | 9 |

Sum | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 9 | 81 |

**Table 2.**Optimal levels of significance ($\alpha $ ) and Type-II error probabilities ($\beta )$ for two proportions: Two independent binomial likelihoods and various sample sizes.

${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{n}}_{1}$ | ${\mathit{n}}_{2}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 10 | 0.1639 | 0.4050 | 50 | 50 | 0.0667 | 0.2718 | 80 | 10 | 0.1130 | 0.3648 | 90 | 70 | 0.0529 | 0.2323 |

20 | 10 | 0.1318 | 0.3939 | 60 | 10 | 0.1097 | 0.3741 | 80 | 20 | 0.0834 | 0.3122 | 90 | 80 | 0.0493 | 0.2281 |

20 | 20 | 0.0995 | 0.3651 | 60 | 20 | 0.0860 | 0.3193 | 80 | 30 | 0.0704 | 0.2847 | 90 | 90 | 0.0468 | 0.2240 |

30 | 10 | 0.1159 | 0.3900 | 60 | 30 | 0.0765 | 0.2903 | 80 | 40 | 0.0634 | 0.2671 | 100 | 10 | 0.1111 | 0.3627 |

30 | 20 | 0.1045 | 0.3333 | 60 | 40 | 0.0689 | 0.2747 | 80 | 50 | 0.0603 | 0.2530 | 100 | 20 | 0.0818 | 0.3079 |

30 | 30 | 0.0997 | 0.3070 | 60 | 50 | 0.0626 | 0.2652 | 80 | 60 | 0.0553 | 0.2455 | 100 | 30 | 0.0684 | 0.2795 |

40 | 10 | 0.1250 | 0.3703 | 60 | 60 | 0.0591 | 0.2572 | 80 | 70 | 0.0531 | 0.2380 | 100 | 40 | 0.0617 | 0.2601 |

40 | 20 | 0.0868 | 0.3357 | 70 | 10 | 0.1130 | 0.3675 | 80 | 80 | 0.0508 | 0.2327 | 100 | 50 | 0.0559 | 0.2479 |

40 | 30 | 0.0850 | 0.3029 | 70 | 20 | 0.0865 | 0.3132 | 90 | 10 | 0.1131 | 0.3626 | 100 | 60 | 0.0538 | 0.2368 |

40 | 40 | 0.0706 | 0.2968 | 70 | 30 | 0.0727 | 0.2876 | 90 | 20 | 0.0810 | 0.3114 | 100 | 70 | 0.0512 | 0.2291 |

50 | 10 | 0.1126 | 0.3761 | 70 | 40 | 0.0645 | 0.2717 | 90 | 30 | 0.0707 | 0.2804 | 100 | 80 | 0.0483 | 0.2238 |

50 | 20 | 0.0883 | 0.3240 | 70 | 50 | 0.0603 | 0.2593 | 90 | 40 | 0.0648 | 0.2608 | 100 | 90 | 0.0467 | 0.2188 |

50 | 30 | 0.0767 | 0.2992 | 70 | 60 | 0.0575 | 0.2501 | 90 | 50 | 0.0575 | 0.2506 | 100 | 100 | 0.0449 | 0.2150 |

50 | 40 | 0.0718 | 0.2817 | 70 | 70 | 0.0539 | 0.2446 | 90 | 60 | 0.0550 | 0.2401 |

Hypotheses | Predictive Densities under H ^{1} |
---|---|

H: $\mathit{\theta}$ = $\mathit{\theta}$_{0} | $\mathit{C}\left(\mathit{x},\mathit{y}\right){\mathit{\theta}}_{\mathbf{0}}^{\mathit{x}}{\left(\mathbf{1}-{\mathit{\theta}}_{\mathbf{0}}\right)}^{\mathit{y}}$ |

H: $\mathit{\theta}\ne \mathit{\theta}$_{0} | $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left(\mathit{U},\mathit{V}\right)}{\mathit{B}\left(\mathit{u},\mathit{v}\right)}$ |

H: $\mathit{\theta}$
≤$\mathit{\theta}$_{0} | $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left({\mathit{\theta}}_{\mathbf{0}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left({\mathit{\theta}}_{\mathbf{0}};\mathit{u},\mathit{v}\right)}$ |

H: $\mathit{\theta}$ > $\mathit{\theta}$_{0} | $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left(\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{0}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left(\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{0}};\mathit{u},\mathit{v}\right)}$ |

H: $\mathit{\theta}$_{1} ≤ $\mathit{\theta}$ ≤ $\mathit{\theta}$_{2} | $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{u},\mathit{v}\right)}$ |

H: ($\mathit{\theta}$ < $\mathit{\theta}$_{1})∪($\mathit{\theta}$ > $\mathit{\theta}$_{2})
| $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left(\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{U},\mathit{V}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left(\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{u},\mathit{v}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{u},\mathit{v}\right)}$ |

H: ($\mathit{\theta}$_{1} ≤ $\mathit{\theta}$ ≤ $\mathit{\theta}$_{2})∪($\mathit{\theta}$_{3} ≤ $\mathit{\theta}$ ≤ $\mathit{\theta}$_{4})
| $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{U},\mathit{V}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{4}};\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{3}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{u},\mathit{v}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{4}};\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{3}};\mathit{u},\mathit{v}\right)}$ |

H: ($\mathit{\theta}$ < $\mathit{\theta}$_{1})∪$(\mathit{\theta}$_{2} < $\mathit{\theta}$ < $\mathit{\theta}$_{3})∪($\mathit{\theta}$ > $\mathit{\theta}$_{4})
| $\mathit{C}\left(\mathit{x},\mathit{y}\right)\frac{\mathit{B}\left(\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{U},\mathit{V}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{U},\mathit{V}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{4}};\mathit{U},\mathit{V}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{3}};\mathit{U},\mathit{V}\right)}{\mathit{B}\left(\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{2}};\mathit{u},\mathit{v}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{1}};\mathit{u},\mathit{v}\right)-\mathit{B}\left({\mathit{\theta}}_{\mathbf{4}};\mathit{u},\mathit{v}\right)+\mathit{B}\left({\mathit{\theta}}_{\mathbf{3}};\mathit{u},\mathit{v}\right)}$ |

^{1}Prior distribution for $\theta $: $\theta ~Beta\left(u,v\right)$; $U=u+x$; $V=v+y$; $C\left(x,y\right)=\left(\begin{array}{c}x+y\\ x\end{array}\right)$ for positive binomial or $C\left(x,y\right)=\left(\begin{array}{c}x+y-1\\ x\end{array}\right)$ for negative binomial; $B\left(r,s\right)={{\displaystyle \int}}_{0}^{1}{z}^{r-1}{\left(1-z\right)}^{s-1}dz$ is the beta functions; and $B\left(p;r,s\right)={{\displaystyle \int}}_{0}^{p}{z}^{r-1}{\left(1-z\right)}^{s-1}dz$ is the incomplete beta function.

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**MDPI and ACS Style**

Pereira, C.A.d.B.; Nakano, E.Y.; Fossaluza, V.; Esteves, L.G.; Gannon, M.A.; Polpo, A.
Hypothesis Tests for Bernoulli Experiments: Ordering the Sample Space by Bayes Factors and Using Adaptive Significance Levels for Decisions. *Entropy* **2017**, *19*, 696.
https://doi.org/10.3390/e19120696

**AMA Style**

Pereira CAdB, Nakano EY, Fossaluza V, Esteves LG, Gannon MA, Polpo A.
Hypothesis Tests for Bernoulli Experiments: Ordering the Sample Space by Bayes Factors and Using Adaptive Significance Levels for Decisions. *Entropy*. 2017; 19(12):696.
https://doi.org/10.3390/e19120696

**Chicago/Turabian Style**

Pereira, Carlos A. de B., Eduardo Y. Nakano, Victor Fossaluza, Luís Gustavo Esteves, Mark A. Gannon, and Adriano Polpo.
2017. "Hypothesis Tests for Bernoulli Experiments: Ordering the Sample Space by Bayes Factors and Using Adaptive Significance Levels for Decisions" *Entropy* 19, no. 12: 696.
https://doi.org/10.3390/e19120696