cci 6 2 0 7, exact. Cornfield Estimates vs add Haldane-Anscombe adjustment - Who knew an OR would be so interesting?

[Brian Brummer]

Hi All,

I conducted a small CC study for a nosocomial food-borne outbreak. The results are:

cci 6 2 0 7, exact

                                                         Proportion
                 |   Exposed   Unexposed  |      Total      exposed
-----------------+------------------------+------------------------
           Cases |         6           2  |          8       0.7500
        Controls |         0           7  |          7       0.0000
-----------------+------------------------+------------------------
           Total |         6           9  |         15       0.4000
                 |                        |
                 |      Point estimate    |    [95% conf. interval]
                 |------------------------+------------------------
      Odds ratio |                .       |    3.436748           . (Cornfield)
 Attr. frac. ex. |                1       |    .7090272           . (Cornfield)
 Attr. frac. pop |              .75       |
                 +-------------------------------------------------
                                  1-sided Fisher's exact P = 0.0056
                                  2-sided Fisher's exact P = 0.0070

This gives an undefined upper 95% ci and lower 85% ci of 3.4. This is fine but left me curious since I am used to the "traditional way" of calculating the 95% Ci using:

95% CI = e^{ln(OR) +-1.96√(1/a + 1/b + 1/c + 1/d)}

However, both the upper and lower 95% Ci are undefined since the OR is undefined.

While I am okay with Stata output and generally understand the interpretation, I was curious at how it was able to define the lower Ci so i ventured down the epitab to find the cornfield method and popped it into the wolfram app and calculated the lower Ci to be 1.8 (link at the end of post).

This made me uneasy since stating "the cornfield" method might not be reproducible for many...atleast, it was not for me...

So I wanted to find a way of calculating 95% Ci using the "traditional method" and found that the Haldane-Anscombe adjustment of adding fixed effects (usually 0.5) deals with this robustly and crucially for me, allows the OR to be calculated in the traditional way. Stata's immediate calculator rejects non-integer values and so comparing the Cornfield vs Haldane-Ansocme adjustment become a bit more laborious than expected. i could add a fixed effect of 1 but this may cause a larger bias towards the null (already a criticism of the adjustment.

cci 6.5 2.5 0.5 7.5, exact 
'6.5' found where integer expected

For interest sake, this results:

			Total
Exposed	Unexposed
Cases	6.5	2.5	9
Control	0.5	7.5	8
Total	7	10	17
OR	Point estimate	95% Ci lower	95% Ci upper
39.00	1.57	969.24
p- value	0.0254

Anyway, my supervisors don't seem to have a clear idea on how they would deal with this so I decided to invite the forum's inputs.

So, what would you do?

Cheers
Brian

PS advice like "you should have sample exposed controls" is not particularly helpful at this point.

Links to wolfram app
Wolfram calculation of lower 95% Ci using Cornfield method

https://www.wolframalpha.com/input?i...5D%5D+%3D0.025

*this calculation does not give the same estimate as stata. This is still under investigation.
Wolfram calculation of upper 95% Ci using Cornfield method

https://www.wolframalpha.com/input?i...%5D+%3D1-0.025

Wolfram calculation of lower 95% Ci using traditional method

https://www.wolframalpha.com/input?i...%5C%2841%29%5D
Wolfram calculation of upper 95% Ci using Cornfield method

https://www.wolframalpha.com/input?i...%5C%2841%29%5D

[Leonardo Guizzetti]

The thing that bothers me with these ad hoc corrections in a sparse sample like this is that they end up having a huge influence over the estimated OR and its precision. Ultimately you know the OR is undefined and its precision by approximate methods approaches infinity.

You could try to use penalized logistic regression (or a Bayesian analog) which has the benefit of being able to converge when separation exists, as it does here. I would -search firthlogit- for one implementation. You will still probably get a huge OR, but it will be estimable in theory.

[Joseph Coveney]

With sparse data you get an imprecise estimate. I'm not sure that that makes the estimate any more interesting or worthy of agonizing over. Using the official Stata exlogistic and two user-written commands (see output below), the estimates (tabulated below) are all within about what you'd expect of each other.

Method	OR	LB	UB
exlogistic	20	2	—
firthlogit	40	3	6000
penlogit1	40	3	2000
penlogit2	30	2	5000

LB, UB = 95% confidence bounds; penlogit1 = normal(0, 10) prior; penlogit2 = log-F(0, 1, 1, 1) prior

.ÿ
.ÿversionÿ17.0

.ÿ
.ÿclearÿ*

.ÿ
.ÿinputÿbyte(outÿgrpÿcount)

ÿÿÿÿÿÿÿÿÿÿoutÿÿÿÿÿÿÿgrpÿÿÿÿÿcount
ÿÿ1.ÿ0ÿ0ÿ7
ÿÿ2.ÿ1ÿ0ÿ2
ÿÿ3.ÿ0ÿ1ÿ0
ÿÿ4.ÿ1ÿ1ÿ6
ÿÿ5.ÿend

.ÿ
.ÿccÿoutÿgrpÿ[fweight=count],ÿexact

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿgrpÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿProportion
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿExposedÿÿÿUnexposedÿÿ|ÿÿÿÿÿÿTotalÿÿÿÿÿÿexposed
-----------------+------------------------+------------------------
ÿÿÿÿÿÿÿÿÿÿÿCasesÿ|ÿÿÿÿÿÿÿÿÿ6ÿÿÿÿÿÿÿÿÿÿÿ2ÿÿ|ÿÿÿÿÿÿÿÿÿÿ8ÿÿÿÿÿÿÿ0.7500
ÿÿÿÿÿÿÿÿControlsÿ|ÿÿÿÿÿÿÿÿÿ0ÿÿÿÿÿÿÿÿÿÿÿ7ÿÿ|ÿÿÿÿÿÿÿÿÿÿ7ÿÿÿÿÿÿÿ0.0000
-----------------+------------------------+------------------------
ÿÿÿÿÿÿÿÿÿÿÿTotalÿ|ÿÿÿÿÿÿÿÿÿ6ÿÿÿÿÿÿÿÿÿÿÿ9ÿÿ|ÿÿÿÿÿÿÿÿÿ15ÿÿÿÿÿÿÿ0.4000
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|ÿÿÿÿÿÿPointÿestimateÿÿÿÿ|ÿÿÿÿ[95%ÿconf.ÿinterval]
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ|------------------------+------------------------
ÿÿÿÿÿÿOddsÿratioÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.ÿÿÿÿÿÿÿ|ÿÿÿÿ3.436748ÿÿÿÿÿÿÿÿÿÿÿ.ÿ(Cornfield)
ÿAttr.ÿfrac.ÿex.ÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1ÿÿÿÿÿÿÿ|ÿÿÿÿ.7090272ÿÿÿÿÿÿÿÿÿÿÿ.ÿ(Cornfield)
ÿAttr.ÿfrac.ÿpopÿ|ÿÿÿÿÿÿÿÿÿÿÿÿÿÿ.75ÿÿÿÿÿÿÿ|
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ+-------------------------------------------------
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ1-sidedÿFisher'sÿexactÿPÿ=ÿ0.0056
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ2-sidedÿFisher'sÿexactÿPÿ=ÿ0.0070

ÿNote:ÿExactÿconfidenceÿlevelsÿnotÿpossibleÿwithÿzeroÿcountÿcells.

.ÿ
.ÿexlogisticÿoutÿgrpÿ[fweight=count],ÿnolog
note:ÿCMLEÿestimateÿforÿgrpÿisÿ+inf;ÿcomputingÿMUE.

ExactÿlogisticÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿÿÿÿÿÿÿÿÿ15
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿModelÿscoreÿÿÿ=ÿÿÿ8.166667
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿPrÿ>=ÿscoreÿÿÿ=ÿÿÿÿÿ0.0070
---------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿOddsÿratioÿÿÿÿÿÿÿSuff.ÿÿ2*Pr(Suff.)ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+-------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿgrpÿ|ÿÿÿ17.27956*ÿÿÿÿÿÿÿÿÿÿ6ÿÿÿÿÿÿ0.0112ÿÿÿÿÿÿ1.812866ÿÿÿÿÿÿÿ+inf
---------------------------------------------------------------------------
(*)ÿmedianÿunbiasedÿestimatesÿ(MUE)

.ÿ
.ÿquietlyÿfirthlogitÿoutÿi.grpÿ[fweight=count],ÿnolog

.ÿppl
95%ÿCIÿLBÿ=ÿ1.0679888ÿORÿ=ÿ2.9095219
95%ÿCIÿUBÿ=ÿ8.6730116ÿORÿ=ÿ5843.0697

.ÿquietlyÿestimatesÿrestoreÿUnconstrained

.ÿfirthlogitÿ,ÿor

ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNumberÿofÿobsÿ=ÿÿÿÿÿ15
ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿWaldÿchi2(1)ÿÿ=ÿÿÿ4.99
Penalizedÿlogÿlikelihoodÿ=ÿ-7.0780083ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿProbÿ>ÿchi2ÿÿÿ=ÿ0.0254

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿOddsÿratioÿÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿ1.grpÿ|ÿÿÿÿÿÿÿÿÿ39ÿÿÿ63.93122ÿÿÿÿÿ2.23ÿÿÿ0.025ÿÿÿÿÿ1.569359ÿÿÿÿ969.1855
ÿÿÿÿÿÿÿ_consÿ|ÿÿÿ.3333333ÿÿÿ.2434323ÿÿÿÿ-1.50ÿÿÿ0.132ÿÿÿÿÿ.0796616ÿÿÿÿ1.394789
------------------------------------------------------------------------------
Note:ÿ_consÿestimatesÿbaselineÿodds.

.ÿ
.ÿpenlogitÿoutÿgrpÿ[fweight=count],ÿorÿnprior(grpÿ0ÿ10)ÿppl(grp)


PenalizedÿlogisticÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNo.ÿofÿobsÿÿ=ÿÿÿÿÿÿÿÿÿ15

Normalÿpriorÿforÿgrp:ÿexactÿpriorÿmedianÿORÿ(95%ÿPL):ÿ1.00ÿ(0.00,ÿ492.71)
Dataÿapprox.ÿequivalentÿtoÿprior:ÿcases=0.53ÿÿnoncases=0.53ÿÿexp(offset)=1

Penalizedÿlogÿlikelihoodÿ=ÿ-9.867014

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿOddsÿRatioÿÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿgrpÿ|ÿÿÿÿ42.0245ÿÿÿ67.13519ÿÿÿÿÿ2.34ÿÿÿ0.019ÿÿÿÿÿ1.835214ÿÿÿÿ962.3176
ÿÿÿÿÿÿÿ_consÿ|ÿÿÿ.3582299ÿÿÿ.2651434ÿÿÿÿ-1.39ÿÿÿ0.165ÿÿÿÿÿ.0839742ÿÿÿÿ1.528191
------------------------------------------------------------------------------
--------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿ[95%ÿPLÿConf.ÿInterval]
-------------+------------------------
ÿÿÿÿÿÿÿÿÿgrpÿ|ÿÿÿÿ3.115304ÿÿÿÿ2433.693
--------------------------------------

.ÿ
.ÿpenlogitÿoutÿgrpÿ[fweight=count],ÿorÿlfprior(grpÿ0ÿ1ÿ1ÿ1)ÿppl(grp)


PenalizedÿlogisticÿregressionÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿNo.ÿofÿobsÿÿ=ÿÿÿÿÿÿÿÿÿ15

Log-Fÿpriorÿforÿgrp:ÿexactÿpriorÿmedianÿORÿ(95%ÿPL):ÿ1.00ÿ(0.00,ÿ647.79)
Dataÿapprox.ÿequivalentÿtoÿprior:ÿcases=0.50ÿÿnoncases=0.50ÿÿexp(offset)=1

Penalizedÿlogÿlikelihoodÿ=ÿ-6.8299418

------------------------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿOddsÿRatioÿÿÿStd.ÿerr.ÿÿÿÿÿÿzÿÿÿÿP>|z|ÿÿÿÿÿ[95%ÿconf.ÿinterval]
-------------+----------------------------------------------------------------
ÿÿÿÿÿÿÿÿÿgrpÿ|ÿÿÿ31.20028ÿÿÿ50.72349ÿÿÿÿÿ2.12ÿÿÿ0.034ÿÿÿÿÿ1.289222ÿÿÿÿ755.0737
ÿÿÿÿÿÿÿ_consÿ|ÿÿÿ.3780314ÿÿÿ.2802326ÿÿÿÿ-1.31ÿÿÿ0.189ÿÿÿÿÿ.0884173ÿÿÿÿ1.616288
------------------------------------------------------------------------------
--------------------------------------
ÿÿÿÿÿÿÿÿÿoutÿ|ÿ[95%ÿPLÿConf.ÿInterval]
-------------+------------------------
ÿÿÿÿÿÿÿÿÿgrpÿ|ÿÿÿÿ2.461649ÿÿÿÿ4629.794
--------------------------------------

.ÿ
.ÿexit

endÿofÿdo-file


.

Ad hoc ppl computes likelihood-profile 95% confidence bounds for firthlogit; the Wald CIs shown in the regression table should be ignored. ppl and the command that it calls are attached.

search penlogit

for that user-written command.