statistical analysis Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers propertie ...

, the rule of three states that if a certain event did not occur in a sample with subjects, the interval from 0 to 3/ is a 95%

confidence interval In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as ...

for the rate of occurrences in the

population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using ...

. When is greater than 30, this is a good approximation of results from more sensitive tests. For example, a pain-relief drug is tested on 1500 human subjects, and no

adverse event An adverse event (AE) is any untoward medical occurrence in a patient or clinical investigation subject administered a pharmaceutical product and which does not necessarily have a causal relationship with this treatment. An adverse event can ther ...

is recorded. From the rule of three, it can be concluded with 95% confidence that fewer than 1 person in 500 (or 3/1500) will experience an adverse event. By symmetry, for only successes, the 95% confidence interval is . The rule is useful in the interpretation of

clinical trial Clinical trials are prospective biomedical or behavioral research studies on human subject research, human participants designed to answer specific questions about biomedical or behavioral interventions, including new treatments (such as novel v ...

s generally, particularly in phase II and phase III where often there are limitations in duration or

statistical power In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...

. The rule of three applies well beyond medical research, to any trial done times. If 300 parachutes are randomly tested and all open successfully, then it is concluded with 95% confidence that fewer than 1 in 100 parachutes with the same characteristics (3/300) will fail.

Derivation

A 95%

is sought for the probability ''p'' of an event occurring for any randomly selected single individual in a population, given that it has not been observed to occur in ''n''

Bernoulli trial In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is ...

s. Denoting the number of events by ''X'', we therefore wish to find the values of the parameter ''p'' of a

binomial distribution In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no qu ...

that give Pr(''X'' = 0) ≤ 0.05. The rule can then be derived"Professor Mean" (2010
"Confidence interval with zero events"
The Children's Mercy Hospital. Retrieved 2013-01-01. either from the Poisson approximation to the binomial distribution, or from the formula (1−''p'')^''n'' for the probability of zero events in the binomial distribution. In the latter case, the edge of the confidence interval is given by Pr(''X'' = 0) = 0.05 and hence (1−''p'')^''n'' = .05 so ''n'' ln(1–''p'') = ln .05 ≈ −2.996. Rounding the latter to −3 and using the approximation, for ''p'' close to 0, that ln(1−''p'') ≈ −''p'' (Taylor's formula), we obtain the interval's boundary 3/''n''. By a similar argument, the numerator values of 3.51, 4.61, and 5.3 may be used for the 97%, 99%, and 99.5% confidence intervals, respectively, and in general the upper end of the confidence interval can be given as

\frac

, where

1-\alpha

is the desired confidence level.

Extension

The

Vysochanskij–Petunin inequality In probability theory, the Vysochanskij– Petunin inequality gives a lower bound for the probability that a random variable with finite variance lies within a certain number of standard deviations of the variable's mean, or equivalently an upper ...

shows that the rule of three holds for

unimodal In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal pr ...

distributions with finite

variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...

beyond just the binomial distribution, and gives a way to change the factor 3 if a different confidence is desired.

Chebyshev's inequality In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from th ...

removes the assumption of unimodality at the price of a higher multiplier (about 4.5 for 95% confidence).

Cantelli's inequality In probability theory, Cantelli's inequality (also called the Chebyshev-Cantelli inequality and the one-sided Chebyshev inequality) is an improved version of Chebyshev's inequality for one-sided tail bounds. The inequality states that, for \lambda > ...

is the one-tailed version of Chebyshev's inequality.

Notes

References

* * * Ziliak, S. T.; D. N. McCloskey (2008). ''The cult of statistical significance: How the standard error costs us jobs, justice, and lives''. University of Michigan Press. {{ISBN, 0472050079 Clinical trials Statistical approximations Medical statistics Nursing research

Derivation

Extension

See also

Notes

References