*! version 1.0 29apr2016

Title

choi_lr_test -- Choi et al.'s likelihood ratio test for 2x2 tables (choi_lr_test and choi_lr_testi)

Syntax

choi_lr_test var_case var_exposed [if] [in]
[weight]
[, choi_lr_test_options]

choi_lr_testi #a #b #c #d [, choi_lr_testi_options]

choi_lr_test_options

Options
cornfield use Cornfield approximation to calculate confidence intervals (CIs) of the odds ratio

woolf use Woolf approximation to calculate standard error (SE) and CI of the odds ratio

exact calculate Fisher's exact p

level(#) set confidence level; default is level(95)

k(#k) calculate the 1/#k likelihood support interval (LSI) for the odds ratio

var_case equals 1 or 0 for cases or controls, respectively.

var_exposed equals 1 or 0 for subjects who are, or are not, exposed.

fweights are allowed; see weight.

choi_lr_testi_options

Options
cornfield use Cornfield approximation to calculate CI of the odds ratio

woolf use Woolf approximation to calculate SE and CI of the odds ratio

exact calculate Fisher's exact p

level(#) set confidence level; default is level(95)

k(#k) calculate the 1/#k likelihood support interval for the odds ratio

#a equals the number of exposed cases.

#b equals the number of unexposed cases.

#c equals the number of exposed controls.

#d equals the number of unexposed controls.

Description

choi_lr_test is used for 2x2 tables when the parameter of interest
is the odds ratio. It is similar to the cc and cci commands
but also calculates the maximum likelihood estimate (MLE) of the odds
ratio conditioned on the total number of exposed subjects, Choi et al.'s
likelihood ratio test of the null hypothesis that the odds ratio = 1, and the
1/k likelihood support interval (LSI) for this odds ratio. By default,
it calculates the 1/6.8 LSI which equals the 95% CI for normally
distributed random variables. See Choi et al. (2015)
for additional details and Blume (2002) for an introduction
to likelihood methods for measuring statistical evidence.

choi_lr_test and choi_lr_testi use the cc and cci programs,
respectively, to calculate point estimates and CIs for the
unconditioned odds ratio, along with attributable or prevented fractions for the
exposed and total population. choi_lr_testi is the immediate form of
choi_lr_test; see immed. Also see [R] logistic for
related commands.

This program may not be used for analyzing stratified 2x2 tables.

A Shiny app (Choi 2016) and an R package (Choi 2011) are also available for performing these
calculations.

Options for choi_lr_test

Options for choi_lr_testi

Options

cornfield requests that the
Cornfield (1956) approximation be used to calculate the
CI of the odds ratio. (This approximation does not use the
continuity correction use by Cornfield in his 1956 paper.) By default,
choi_lr_test reports an exact interval.

woolf requests that the
Woolf (1955) approximation, also known as the Taylor
expansion, be used for calculating the SE and CI
for the odds ratio.

exact requests that Fisher's exact
p be calculated rather than the chi-squared and its significance level. We
(Dupont & Plummer) recommend using Choi's likelihood ratio chi-squared statistic
whenever samples are small. We also recommend using the 1/6.8 LSI instead of the 95%
CI for the odds ratio for small samples. When the least-frequent cell contains
1,000 cases or more, there will be no appreciable difference between the
significance levels of any of the tests calculated by this program.
The exact significance level will take
considerably longer to calculate. exact
does not affect whether exact CIs are calculated. Commands
always calculate exact CIs where they can, unless cornfield or
woolf is specified.

level(#) specifies the confidence level, as a
percentage, for CIs. The default is level(95) or as
set by set level.

k(#k) specifies a positive number used to calculate the 1/#k LSI for
the odds ratio. The 1/6.8259358 LSI is always calculated. (This LSI is typically
refered to by us and others as the 1/6.8 LSI.) It provides the same level of support
for the true location of the odds ratio as a 95% CI.

Examples

Setup

. webuse ccxmpl

List the data

. list

Calculate odds ratio, etc.

. choi_lr_test case exposed [fw=pop]

Immediate form of above command

. choi_lr_testi 4 386 4 1250

Same as above, but calculate Fisher's exact p rather than the
chi-squared

. choi_lr_testi 4 386 4 1250, exact

Stored results

choi_lr_test and choi_lr_testi store the following in r():

Scalars defined in the choi_lr_test and choi_lr_testi programs

r(or_cond) MLE of odds ratio conditioned on the total number of exposed subjects

r(clr) Conditional likelihood ratio for the null hypothesis that OR = 1

r(chi2_clr) Choi's LR chi-squared statistic. This is equation (9) of Choi et al. (2015)

r(p_choi) p-value from Choi's likelihood ratio statistic

r(lr6pt8lsi_lb) Lower bound of the 1/6.8259358 LSI for the odds ratio

r(lr6pt8lsi_ub) Upper bound of the 1/6.8259358 LSI for the odds ratio

r(lrklsi_lb) Lower bound of the 1/k LSI for the odds ratio

r(lrklsi_ub) Upper bound of the 1/l LSI for the odds ratio

Scalars defined in the cc and cci programs

r(p) two-sided p-value from the 2x2 chi^2 statistic without continuity correction

r(p1_exact) chi-squared or one-sided exact significance

r(p_exact) two-sided exact significance

r(or) Unconditioned MLE of the odds ratio

r(lb_or) lower bound of CI for or

r(ub_or) upper bound of CI for or

r(afe) attributable (prev.) fraction among exposed

r(lb_afe) lower bound of CI for afe

r(ub_afe) upper bound of CI for afe

r(afp) attributable fraction for the population

r(chi2) 2x2 chi^2 statistic without continuity correction

References

Blume JD. 2002. Likelihood methods for measuring statistical evidence.
Stat Med 21:2563-99.

Choi et al. 2015. Elucidating the Foundations of Statistical Inference with 2 x 2
Tables. PLoS ONE 10(4): e0121263. doi:10.1371/journal.pone.0121263

Choi, L. 2011. ProfileLikelihood: profile likelihood for a parameter in commonly
used statistical models; 2011. R package version 1.1.
https://cran.r-project.org/web/packages/ProfileLikelihood/index.html

Choi, L. 2016. Likelihood ratio statistics for 2 x 2 tables. http://statcomp2.vanderbilt.edu:37212/dalep/ProfileLikelihood/

Cornfield, J. 1956. A statistical problem arising from retrospective studies. In Vol. 4 of Proceedings of the Third Berkeley Symposium, ed. J. Neyman, 135-148. Berkeley, CA: University of California Press.

Woolf, B. 1955. On estimating the relation between blood group disease. Annals of Human Genetics 19: 251-253. Reprinted in Evolution of Epidemiologic Ideas: Annotated Readings on Concepts and Methods, ed. S. Greenland, pp.
108-110. Newton Lower Falls, MA: Epidemiology Resources.