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Confidence interval of a quotient by Fieller's method (for unpaired data)

fiellervarname[ifexp] [inrange],by(grouping_varname)[level(real)reverse]

fiellerinumerator_mean numerator_SD numerator_n denominator_meandenominator_SD denominator_n[,Level(real)]

by...:may be used withfieller; see help by.

Description

fiellercalculates the confidence interval for a quotient of two independent samples of normally distributed data invarname. The samples are identified by the grouping variable in theby()option.

fielleriis the immediate form of the command. Means, standard deviations andns are given sequentially, with the numerator's statistics first.

Options

by()is required forfieller. It identifies the grouping variable, which may be either numeric or string. The smallest value defines the group that will go into the numerator and the next-smallest value is the group in the denominator.

reverse(forfieller) allows the user to invert the quotient should theby()variable be coded such that the group intended to be in the denominator is the lesser value.

levelallows the user to choose the level of the confidence interval. It defaults toc(level). See help creturn.

RemarksThe confidence interval is calculated as described in Motulsky (1995). The method is based upon an eponymous theorem by Edgar C. Fieller.

The method is intended for normally distributed data, but variances need not be identical between the two groups.

fiellerandfielleriare for uncorrelated data (independent groups). Confidence intervals for quotients of paired data should be calculable by the official Stata commandpkequivwith thefielleroption, if the dataset can be arranged to meet the requirements of the command. See help pkequiv.The confidence interval relies upon the solution to a quadratic equation, which doesn't necessarily have both roots real; therefore, the confidence interval might not always be calculable. This will happen when the quotient is not statistically signficsignficantly different from zero at the specified level of Type I error rate—the mean of the denominator should be "large" in comparison to its standard deviation.

Reference

Harvey Motulsky,

Intuitive BiostatisticsOxford: Oxford University Press, 1995. pp. 285–86.

Examples

. sysuse bplong

. // A recommended preliminary: -regress- and plot residuals with, forexample, -pnorm-

. bysort when: fieller bp, by(sex) level(90)

. fieller bp if when == 2, by(sex)

. // An alternative approach (for large samples) using the delta method;less favorably considered

. tabulate sex, generate(Sex)

. regress bp Sex1 Sex2 if when == 2, noconstant

. nlcom _b[Sex1] / _b[Sex2]

. // Another alternative, using generalized linear modeling (again, forlarge samples)

. glm bp Sex1 if when == 2, family(gaussian) link(log) eform nolog

. fielleri 278 5.5 4 254 2.88 4, level(99)

AuthorE-mail Joseph Coveney if you observe any problems.

Also seeManual:

[R] pkequiv