# Alternative model to binomial

Consider the following protocol. We want to test metallic bars for brittleness. The bars were molded under similar conditions and tested at 5 places. Under the assumption that bars have uniform composition, the $\#$ of breaks/bar should follow $\text{Bin}(5,p)$, with probability of failure $p$ unknown. If all bars have uniform strength, $p$ should be the across bars (under the previous assumption it is also the same within a bar). If they are of different strength the failure probability should vary.

I have tested the null hypothesis (using given data which are not important for my question -- the data are basically the number of bars out of 280 that exhibited $i$ breaks/bar, where $0 \le i \le 5$) that $p$ is the same for all bars and found that the binomial model is unsuccessful in modeling this protocol. I am trying to find a model that better represents the physical mechanism that generates the data.

Since the Binomial model does not really fit the data well, I am assuming that modeling the process as a Bernoulli process where we have $5\times 280$ trials with equal probability of failure would yield the same results. Poisson is usually used for modeling rare events so my intuition tells me to avoid it here. A normal distribution seems to be unlikely, too, since the number of breaks is not a continuous quantity. But this pretty much exhausts the discrete distributions I am aware of, for modeling the process. Am I missing something fundamental?

• The normal distribution is not discrete. – Michael R. Chernick Nov 28 '16 at 21:47
• Can you categorize the bars into say k types where each type has a binomial distribution n=5 and p=p_i? – Michael R. Chernick Nov 28 '16 at 21:51
• @MichaelChernick No, unfortunately this is all the information I have from the problem. – Orest Xherija Nov 28 '16 at 22:59
• Could you give us some more details and/or an example? What exactly is your data (5-element vectors of 0s and 1s)? What do you want to test? What is your alternative hypothesis? Would you expect that if something went wrong that you should notice that on some position there is noticeably greater probability of failure, or some other patterns? Why not conducting a permutation test? – Tim Nov 29 '16 at 12:17

It seems you do not have individual result for each of the five sites within bars, so estimating the individual $$p_i$$'s do not seem possible. Then, maybe try a distribution for the sum which is accommodating that the $$p_i$$'s may differ. One such distribution is the beta-binomial, which arises if one assumes that the probabilities $$p_i$$ are themself drawn from a beta distribution, $$\text{Beta}(\alpha, \beta)$$. You would then estimate the two parameters $$\alpha, \beta$$ from the data. This could also accommodate probabilities varying among the bars. See https://en.wikipedia.org/wiki/Beta-binomial_distribution