This is the help file for beta binomial calculations.

The program gives an output for a probability associated with a certain number of positives from a consignment. In the context of forensic science this would be used where a scientist is given a consignment of units (up to 50 for this program) and needs to make some sort of statement about how many units are positive based on a sample.

This is used in situations where a consignment of potientially illicit units has been seized, the whole consignment consisting of many smalller sub-units. The scientist needs to say how many of the units are illegal, but cannot look in detail at all the units, so has to take a sample.

The vertical sliders on the central panel are used to enter information about the number of units in the consignment (N), the number of units from the consignment which have been examined (sample size), and, the number of units in the sample found to contain the illicit substance (positives).

For example: a polythene bag has been found in the pocession of a suspect. The bag contains 30 badly constructed tin foil envelopes containing an off-white powder. The suspicion is that the white powder is amphetamine sulphide. The drugs investigator selects 5 at random, and subjects them to a pharmceutical analysis. All 5 are found analitcaly to test positive for amphetamine. The next stage is to make some estimate based on the sample of how many more from the consignment contain amphetamine.

The investigator adjusts the first slider (N) in the central panel to indicate 30, the second slider (sample size) gets set to 5, and the third (positives) to 5. 

(note: if the number of positives cannot exceed the sample size, so the sample size is automatically adjusted up if the positives slider is brought above the sample size. Also the sample size cannot exceed the number of units in the consignment, so N is automatically adjusted to compensate if the sample size slider is brought above that of N.) 

The output is in the bottom frame. It reads `there is a 95 percent probability that there are more than 14 additional illegal units in the consignment', which means just what it says. In this case there would be a 95% probability that the consignment contained 19 illegal units (14 + 5 already sampled and found to be positive)

The process can also be seen in the graphics window with the two graphs. The top is the prior (see below), the bottom the posterior probability density functions. It is the bottom one from which the results are calculated.

If the investigator requires a different level of probability, then the horizontal slider can be adjusted. If in the example above the level is taken up to 99 percent then it will be found that at that level there is a 99 percent probability that 10, or more additional illegal units are present in the consignment. So from that the investigator could say that there is a 99% probability the consignment contains 15 or more units of amphetamine.

Intuitively the results make some sense. On the basis of the evidence it is possible to say that there is a 95% probability of there being 19 units of amphetimine, and a 99% probability of there being 14 units in total.

The level can be reduced to fulfil the requirements of lower magnitudes of proof. For instance using the horizontal slider to reduce the level to 50% it is possible to say that there is a 50% probability that there are 27 or more units in the consignment.

The top frame is really only for advanced users, and allows the setting up of different priors. The default prior is uniform. That is there is no opinion about what proportion of the consignment contains the illicit substance. This can be changed to reflect diffreent prior opinion by moving the sliders in the top frame. The horizontal one controls the mean of the prior density, the vertical one the `pointyness' of it. Any adjustments made are reflected in the top graph of the graphics window.

The posterior probability density function from which the summary statistics are calculated is not to be confused with a confidence region. If asked about `how confident' you are about that result the answer is totally. The analysis here is fundamentally different to more standard estimates in that it doesn't give an estimate and confidence interval for the estimate, but a posterior probability distribution (see Aitken 1995 for more details).

On *nix systems if the graphics window is for any reason covered by another window then it will not redraw, and will look blank. To rectify this force the function to redraw the graph by hitting Ret - Windows systems seem to be immune to this.

References.

Aitken, C.G.G. (1999) Sampling - How big a sample? Journal of Forensic Sciences; 44(4), 750-760.

Aitken, C.G.G. (1995) Statistics and the Evaluation of Evidence for Forensic Scientists. John Wiley and Sons. London.
