{smcl}
{* 2010-09-30}{...}
{cmd:help rdci} and {cmd:help rdcii} {right:Version 1.2 2010-09-30}

{hline}

{title:Title}

{p2colset 5 13 13 2}{...}
{p2col:{hi:rdci} {hline 2}}Improved asymptotic confidence interval for the difference between two 
independent proportions{p_end}
{p2col:{hi:rdcii}}{p_end}
{p2colreset}{...}


{title:Syntax}

{p 8 17 2}
{cmdab:rdci}
{it:var_case var_exposed}
{ifin}
{weight}
[{cmd:,} {it:options}]

{p 8 17 2}
{cmdab:rdcii}
{it:#a #b #c #d}
[{cmd:,} {it:options}]

{synoptset 20 tabbed}{...}
{synopthdr}
{synoptline}
{syntab:Main}
{synopt:{opt l:evel(#)}}set confidence level; default is prevailing setting (see {help creturn}){p_end}
{synopt:{opt z:sot}}allow t-over-two to vary as {it:z}{c 178}-over-two (Agresti-Caffo confidence interval){p_end}
{synopt:{opt c:c}}apply continuity correction (Wallenstein confidence interval){p_end}
{synopt:{opt nobrute:force}}turn off brute-force bracketing for root finding (Miettinen-Nurminen confidence interval){p_end}
{synopt:{opt tol:erance(#)}}relative tolerance for root-finding convergence (Miettinen-Nurminen confidence interval){p_end}
{synopt:{opt ltol:erance(#)}}absolute tolerance for root-finding convergence (Miettinen-Nurminen confidence interval){p_end}
{synopt:{opt v:erbose}}displays iteration log for root-finding (Miettinen and Nurminen confidence interval){p_end}
{synopt:{opt init:ial(numlist)}}initial values for root-finding (Miettinen-Nurminen confidence interval){p_end}
{synoptline}
{p2colreset}{...}
{p 4 6 2}
{cmd:by} may be used with {cmd:rdci}; see {helpb by}.{p_end}
{p 4 6 2}
{cmd:fweight}s are allowed with {cmd:rdci}; see {help weight}.


{title:Description}

{pstd}
{cmd:rdci} calculates confidence intervals for the difference between two independent 
proportions by four methods that are considered to be improvements over the conventional Wald 
method.  {cmd:rdcii} is the immediate form of the command.


{title:Options}

{dlgtab:Main}

{phang}
{cmd:level} set confidence level; it must lie between 0.1 and 99.9%, inclusive; default is the Stata 
{help level} setting.

{phang}
{cmd:zsot} adds {it:z}{c 178}-over-two successes and failures instead of a fixed value of two each 
for the Agresti-Caffo method

{phang}
{cmd:cc} applies Yates's continuity correction in Wallenstein's method; this mimics a so-called exact 
confidence interval

{phang}
{cmd:nobruteforce} forgoes brute-force incrementing through parameter space to set brackets on 
occasions when initial brackets fail to yield roots with Ridders's method

{phang}
{cmd:tolerance} sets the relative error tolerance for root finding for Ridders's method; default is 1e-6.

{phang}
{cmd:ltolerance} sets the absolute error tolerance for Ridders's method; not invoked by default.

{phang}
{cmd:verbose} displays iteration history for root finding; helpful in diagnosis of 
problematic cases.

{phang}
{cmd:initial} allows manually setting brackets for root finding.


{title:Remarks}

{pstd}
{cmd:rdci} calculates confidence intervals for differences between two independent proportions 
by four methods:  Miettinen-Nurminen, Agresti-Caffo, Newcombe Method 10 (so-called score method or 
hybrid score method), and Wallenstein.  These risk difference confidence intervals are considered 
to have better statistical properties (i.e., are stabler, more accurate{c 150}have better "coverage") 
than the conventional Wald confidence interval, especially for small- to moderate-sized samples.

{pstd}
As with {cmd:cs} (see {help epitab}), nonzero nonmissing values in the {it:cases} variable 
for {cmd:rdci} are cases, and nonzero nonmissing values in the {it:exposure} variable indicate 
exposure.  The sequence of {it:#a}, {it:#b}, {it:#c} and {it:#d} in {cmd:rdcii} is identical to that 
for the analogous {cmd:csi} (see {help epitab}), as well.

{pstd}
The command calls {cmd:ridder} during the estimation of Miettinen-Nurminen confidence interval.  
{cmd:ridder} is a user-written command that will need to be installed for {cmd:rdci} to estimate 
Miettinen-Nurminen confidence intervals.  Because the command calls {cmd:ridder}, any global 
macro {it:S_1} that the user has defined will be overwritten.

{pstd}
As with any root-finding algorithm, on occasion, {cmd:ridder} can fail.  The affected 
Miettinen-Nurminen confidence limit will be set to missing when this occurs.  The {opt verbose} option 
will allow the user to see the iteration log of {cmd:ridder} and diagnose whether a missing confidence 
limit results from failure of convergence or of bracketing.  Adjusting the tolerance ({opt tolerance}, 
{opt ltolerance} or both) if the former and manually setting the brackets {opt initial(numlist)} 
if the latter often allows for successful root finding in such cases.  One or two numbers are allowed 
in {opt initial(numlist)}.  If one number is entered, the root-finding interval will be (-1, {it:#}] 
in the search for the lower confidence limit and [{it:#}, +1) for the upper confidence limit.  If 
two numbers are entered, then the second number will form this latter lower bracket in the search for 
the upper confidence limit.  

{pstd}
If root finding fails during a first attempt, {cmd:rdci} detects this and implements an 
exhuastive scan of the parameter space on the affected side of the point estimate.  This may be turned 
off with the {opt nobruteforce} option should the the user wish to do so, for example, when setting 
the bracket points manually or to limit output when the user invokes the {opt verbose} option.  The 
exhaustive scan, however, ought to reduce the need for user intervention.

{pstd}
Note that the tolerance settings are for the difference between the objective function's result and the 
target chi-square statistic for the confidence level; it is not the tolerance for the values of the 
confidence limits.  (Run the ancillary do-file for examples illustrating the meaning of 
this, and for suggested settings of {opt ltolerance} to assure reasonably good tolerance for the 
confidence limits.)

{pstd}
When the expectation of successes or failures is fewer than two in either exposure group, Wallenstein's 
method might lead to a confidence limit that implies that an individual proportion lies 
outside of the parameter space for proportions.  {cmd:rdci} tests for this and applies adjustments 
recommended by Wallenstein.  The Wallenstein method performs relatively poorly in general when the 
observed successes (failures) in both exposure groups are zero, especially when the two sample sizes are 
the same.  (Run the ancillary do-file to see examples of this behavior.)

{pstd}
When the so-called t-over-two parameter is fixed at two (the default, as suggested by Agresti and Caffo), 
the Agresti-Caffo method might not be accurate with very small sample sizes when the {opt level} is 
set at unusually very low levels (<10%).  There will not be a problem when more conventional values 
for {opt level} are used (>80%).  Allowing the t-over-two parameter to vary with level (which is done 
by invoking the {opt zsot} option) might also aid accuracy.

{pstd}
The subroutine for the Wallenstein method is recursive, as is the main program (calling itself 
indirectly via {cmd:ridder}), but the recursion depth is only one, and so there is scant chance 
of exhausting memory.


{title:Examples}

{phang}{cmd:. rdcii 7 12 9 2, level(90)}

{phang}{cmd:. rdci case exp [freq=pop]}


{title:References}

{pstd}
A. Agresti and B. Caffo, Simple and effective confidence intervals for proportions and 
differences of proportions result from adding two successes and two 
failures. {it:The American Statistician} {bf:54}:280{c -}88, 2000.

{pstd}
O. Miettinen and M. Nurminen, Comparative analysis of two rates. {it:Statistics in Medicine} 
{bf:4}:213{c -}26, 1985.

{pstd}
R. G. Newcombe, Interval estimation for the difference between independent proportions: 
comparison of eleven methods. {it:Statistics in Medicine} {bf:17}:873{c -}90, 1998.

{pstd}
S. Wallenstein, A non-iterative accurate asymptotic confidence interval for the difference 
between two proportions. {it:Statistics in Medicine} {bf:16}:1329{c -}36, 1997.


{title:Acknowledgements}

{pstd}
Code for calculating Wilson intervals of the individual proportions (used in calculation of 
Newcombe's Method 10 confidence intervals) is adapted from {cmd:ciwi} by Nicholas J. Cox.


{title:Author}

{pstd}
Joseph Coveney jcoveney@bigplanet.com


{title:Also see}

{psee}
Manual:  {bf:[ST] epitab}

{psee}
Online:  {helpb epitab}, {helpb ci}, {helpb ridder} (if installed)
