Statistical analysis

Here are some introductory discussions to the statistical analysis used to determine find rates.

Approximate find rate

Suppose that we have performed N searches resulting in S successes (and F = N-S failures).

Our best guess for the find percentage p is given by:

p = S/N

Example: 371 searches on the beach yield 25 pieces of driftwood. N=371, S=25, p = 6.7%. The chance of finding driftwood on the beach is around 6.7%, or 1 find in 15 AP. (May 16, 2006)

General 95% confidence interval

How accurate is our experimentally estimated find rate? Usually, the underlying true find rate will be speculated to be within a range, such as 6.7% +/-5.1% (or 1.6%-11.8%), with an associated confidence (95% certainty). There are several ways to compute the maximum error E.

A simple 95% confidence interval around our determined probability can be estimated by approximating with a normal distribution, using the central limit theorem. For N trials, the margin of error around our experimental mean is +/- 0.98/sqrt(N) with 95% confidence. Loosely speaking, we can be "95% certain" that the true probability is somewhere within +/- 0.98/sqrt(N) of our experimental probability.

E = 0.98/sqrt(N)

(See http://en.wikipedia.org/wiki/Margin_of_error and http://en.wikipedia.org/wiki/Checking_if_a_coin_is_fair true probability section.)

Example: 371 searches yield a maximum error of E = 0.051 = +/-5.1%.

Narrower 95% confidence interval

E = 1.9599*sqrt(p*(1-p)/N)

The previous formula uses the worst possible error bar, by replacing sqrt( p*(1-p)), where p is the probability of success, by 1/2. (This assumes that the p = 0.50, to obtain the maximum error bar.) This is great when testing a fair coin, but in our case the find rates are much lower, so generic error bars are too large. As long as the number of successes in our experimental data is large enough (most sources require at least 5 or 10 successes), then we can use our experimental p in the above formula for E.

Example: 371 searches and 25 finds is p = 6.7%, yielding a maximum error of E = 2.6%. This suggests a driftwood find range of 4.1%-9.3% (with 95% confidence).

Exact binomial tests

When the number of successes S = n*p in our data is smaller than 5 (or 10 if you're feeling conservative), then our data is too skewed to use a normal distribution approximation. (Of course, there also need to be at least 5 (or 10) failures in our data, but in all of our examples successes are much rarer than failures.) In this case, it makes the most sense to compare to an exact binomial distribution for N trials of success probability p. In fact, it always makes the most sense to analyze our results using an exact binomial test, since that is exactly what is happening.

Our experimental data directly supports the hypothesis that the true find rate p = 6.7%. But how well does our data support the hypothesis that true p = 9.8%? Well, assuming a true find of 9.8%, we usually will have at least 36 successes. The chance of 25 or less successes out of 371 searches is the area under the lower tail of the binomial distribution. We can compute this probability in Excel using the BINOMDIST formula:

A- = BINOMDIST(25,371,0.098,TRUE) = 2.45%

When the true value of p is 9.8% (or more), our experimental data is extremely aberrant and unlikely – occurring in less than 1 in 40 collections.

Similarly, for true p = 4.4%, we usually have at most 17 successes. The chance of 25 or more successes is the area under the upper tail:

A+ = 1 - BINOMDIST(24,371,0.044,TRUE) = 2.45%

With a true p of 4.4% (or less), our experimental data is extremely aberrant and unlikely – occurring in less than 1 in 40 collections.

This method yields a driftwood find range of 4.4%-9.8%.

Statistical analysis can be somewhat contentious, but these values are in good agreement with the range determined previously.

How many failures to be sure?

The Excel formula BINOMDIST(0,N,p,TRUE) gives the exact probability of failing to find something in N searches, given a find rate of p. When this probability falls below 2.5%, we can be fairly certain that a find rate of p will return at least one success from N trials. (If this probability falls below 1%, we are 99% certain of at least one success in N trials.)

Hence 100 trials is good enough to strongly reject a 5% find rate (99% certainty), because BINOMDIST(0,100,0.05,TRUE) = 0.0059 < 0.01. Similarly, 200 trials will reasonably reject a 2% find rate (95% certainty) and strongly reject a 2.5% find rate (99%).

I believe that this analysis is duplicated on the Urban Dead wiki, so at some point I'll compare results…

Unsuccessful Searches	Unlikely Find Rate	Almost Certainly Not
N	p (95%)	p (99%)
35	10%	12.5%
75	5%	6%
91	4%	5%
122	3%	4%
183	2%	2.5%
367	1%	1.25%
460	0.8%	1%

Technical notes

The following data used in above examples:

Beaches (May 16, 2006)
Total Searches	Nothing	Piece of Driftwood	Gold Coin	Bottle of Beer	Crab	Bite from Crab
371	322	25	16	2	4	2