# Statistical Power Analysis in Reliability Demonstration Testing: The Probability of Test Success
## 3 Calculation of the Probability of Test Success
### Exact Calculation for the Success Run Test  
While the aforementioned procedure for calculating \(P_{\text{Is}}\) can be used for all test types, the SR test enables an exact calculation in a closed form and no bootstrap procedure is required. As an SR test does not yield an estimation of the lifetime, rather, a confidence distribution of the reliability at the required lifetime \(t_{\text{r}}\) in the form of a binomial distribution, the hypothesis in (1) and (2) can be reformulated as follows:  
\[H_{0}:\ R(t_{\text{r}})<R_{\text{r}}(t_{\text{r}}) \tag{13}\]  
Figure 2: General procedure for calculating the Probability of Test Success, \(P_{\text{Is}}\), of an EoL test. \[H_{1}:\ R(t_{\rm r})\geq R_{\rm r}(t_{\rm r}) \tag{14}\]  
Here, the demonstrated reliability, \(R(t_{\rm r})\), at the required lifetime, \(t_{\rm r}\), is the deciding value for a successful demonstration (\(H_{1}\)) or a failed one (\(H_{0}\)). With a sample size \(n\) of the SR test, the achieved confidence level \(C\) of the required reliability \(R_{\rm r}\) can be calculated using the following binomial distribution [10]:  
\[C=1-\sum_{i=0}^{k}\binom{n}{i}\cdot(R_{\rm r})^{n-i}\cdot(1-R_{\rm r})^{i} \tag{15}\]  
That is, if during the test time equal to the required lifetime \(t_{\rm r}\) a maximum of \(k\) specimens fail. The binomial approach assumes that the parameter of the binomial distribution's success rate is equal to the complement of the required reliability, \(R_{\rm r}\), which corresponds to the null hypothesis of (13) (marginal case of \(s=0\)). Analogous to (15) as well as to the integrals of (6) and (7), the \(P_{\rm ts}\) of an SR test can be calculated analytically using the binomial distribution:  
\[P_{\rm ts}=\sum_{i=0}^{k}\binom{n}{i}\cdot\left(R_{\rm p}\right)^{n-i}\cdot \left(1-R_{\rm p}\right)^{i} \tag{16}\]  
Instead of the required reliability, the reliability at the required lifetime stemming from prior knowledge \(R_{\rm p}(t_{\rm r})=1-F(t_{\rm r})\) is used as the complement of the success rate of the binomial distribution. If \(R_{\rm p}(t_{\rm r})>R_{\rm r}(t_{\rm r})\), this corresponds to the alternative hypothesis of (14). Due to the relationship between the binomial distribution and the beta distribution (see e.g., [54]), Equations (15) and (16) can be written in terms of the beta distributions as follows:  
\[C =\int_{R_{\rm r}}^{1}\frac{R^{n-k-1}\cdot(1-R)^{k}}{\beta(n-k,k+1 )}\mathrm{d}R \tag{17}\] \[P_{\rm ts} =\int_{0}^{R_{\rm p}}\frac{R^{n-k-1}\cdot(1-R)^{k}}{\beta(n-k,k+1 )}\mathrm{d}R \tag{18}\]  
where \(\beta(A,B)\) is the Euler beta function [55] and the resulting beta distributions both have the parameters \(A=n-k\) and \(B=k+1\), as the resulting confidence distribution of the SR test is determined only by the number of surviving \(n-k\) and the number of failing specimens \(k\). Because it is the same distribution, it is immediately apparent that \(P_{\rm ts}\) is the complement of the confidence as the product's reliability approaches the required level (\(P_{\rm ts}\to 1-C\) as \(R_{\rm p}\to R_{\rm r}\) or \(s\to 0\)). When planning the SR test, (15) is used to calculate the required sample size to meet the reliability requirement; hence, the sample size and the resulting distribution of reliability (beta distribution in (17) and (18)) are fixed by design. Because of this, the SR test can only achieve acceptable values for \(P_{\rm ts}\) (e.g., \(P_{\rm ts}>>50\%\)) if the product is over-sized to a great extent in terms of its reliability performance. The relevant distributions (see (17) and (18)) and parameters are shown in Figure 3.  
Figure 3: Beta distribution of an SR test and respective integrals of the confidence level, \(C\), and the Probability of Test Success, \(P_{\rm ts}\).