Fieller's theorem - AbsoluteAstronomy.com

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, Fieller's theorem allows the calculation of a confidence interval

Confidence interval

In statistics, a confidence interval is a particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate. It is an observed interval , in principle different from sample to sample, that frequently includes the parameter of interest, if the...

for the ratio of two means

Arithmetic mean

In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

Approximate confidence interval

Variables a and b may be measured in different units, so there is no way to directly combine the standard error

Standard error

Standard error can refer to:* Standard error , the estimated standard deviation or error of a series of measurements* Standard error stream, one of the standard streams in Unix-like operating systems...

s as they may also be in different units. The most complete discussion of this is given by Fieller (1954).

Fieller showed that if a and b are (possibly correlated

Correlation

In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

) means of two samples with expectations

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

and

, and variances

and

and covariance

, and if

are all known, then a (1 − α) confidence interval (m_L, m_U) for

is given by

where

Here

is an unbiased estimator

Bias of an estimator

In statistics, bias of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called unbiased. Otherwise the estimator is said to be biased.In ordinary English, the term bias is...

based on r degrees of freedom, and

is the

-level deviate from the Student's t-distribution based on r degrees of freedom.

Three features of this formula are important in this context:

a) The expression inside the square root has to be positive, or else the resulting interval will be imaginary.

b) When g is very close to 1, the confidence interval is infinite.

c) When g is greater than 1, the overall divisor outside the square brackets is negative and the confidence interval is exclusive.

Approximate formulae

These equations approximation to the full formula, and are obtained via a Taylor series expansion of a function of two variables and then taking the variance (i.e. a generalisation to two variables of the formula for the approximate standard error for a function of an estimate).

Case 1

Assume that a and b are jointly normally distributed, and that b is not too near zero (i.e. more specifically, that the standard error of b is small compared to b),

From this a 95% confidence interval can be constructed in the usual way (degrees of freedom for t^* is equal to the total number of values in the numerator and denominator minus 2).

This can be expressed in a more useful form for when (as is usually the case) logged data is used, using the following relation for a function of x and y, say ƒ(x, y):

to obtain either,

Case 2

Assume that a and b are jointly normally distributed, and that b is near zero (i.e. SE(b) is not small compared to b).

First, calculate the intermediate quantity:

You cannot calculate the confidence interval of the quotient if

, as the CI for the denominator μ_b will include zero.

However if

then we can obtain

Other

One problem is that, when g is not small, CIs can blow up when using Fieller's theorem. Andy Grieve has provided a Bayesian solution where the CIs are still sensible, albeit wide. Bootstrapping

Bootstrapping

Bootstrapping or booting refers to a group of metaphors that share a common meaning: a self-sustaining process that proceeds without external help....

provides another alternative that that does not require the assumption of normality.

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