R help - Nov 2006 - Confidence interval for relative risk

The concrete problem is that I am refereeing
a paper where a confidence interval is
presented for the risk ratio and I do not find
it credible. I show below my attempts to
do this in R. The example is slightly changed
from the authors'.

I can obtain a confidence interval for
the odds ratio from fisher.test of
course

=== fisher.test example ==
 > outcome <- matrix(c(500, 0, 500, 8), ncol = 2, byrow = TRUE)
 > fisher.test(outcome)

         Fisher's Exact Test for Count Data

data:  outcome
p-value = 0.00761
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
  1.694792      Inf
sample estimates:
odds ratio
        Inf

=== end example ==
but in epidemiology authors often
prefer to present risk ratios.

Using the facility on CRAN to search
the site I find packages epitools and Epi
which both offer confidence intervals
for the risk ratio

=== Epi example ==
 > library(Epi)
 > twoby2(outcome[c(2,1),c(2,1)])
2 by 2 table analysis:
------------------------------------------------------
Outcome   : Col 1
Comparing : Row 1 vs. Row 2

       Col 1 Col 2    P(Col 1) 95% conf. interval
Row 1     8   500      0.0157    0.0079   0.0312
Row 2     0   500      0.0000    0.0000      NaN

                                    95% conf. interval
              Relative Risk:    Inf       NaN      Inf
          Sample Odds Ratio:    Inf       NaN      Inf
Conditional MLE Odds Ratio:    Inf    1.6948      Inf
     Probability difference: 0.0157    0.0027   0.0337

              Exact P-value: 0.0076
         Asymptotic P-value: NaN
------------------------------------------------------

=== end example ==
So Epi gives me a lower limit of NaN but the same confidence
interval and p-value as fisher.test

=== epitools example ==
 > library(epitools)
 > riskratio(outcome)
$data
           Outcome
Predictor  Disease1 Disease2 Total
   Exposed1      500        0   500
   Exposed2      500        8   508
   Total        1000        8  1008

$measure
           risk ratio with 95% C.I.
Predictor  estimate lower upper
   Exposed1        1    NA    NA
   Exposed2      Inf   NaN   Inf

$p.value
           two-sided
Predictor  midp.exact fisher.exact  chi.square
   Exposed1         NA           NA          NA
   Exposed2 0.00404821  0.007610478 0.004843385

$correction
[1] FALSE

attr(,"method")
[1] "Unconditional MLE & normal approximation (Wald) CI"
Warning message:
Chi-squared approximation may be incorrect in: chisq.test(xx, correct
correction)

=== end example ==
And epitools also gives a lower limit
of NaN.

=== end all examples ==
I would prefer not to have to tell the authors of the
paper I am refereeing that
I think they are wrong unless I can help them with what they
should have done.

Is there another package I should have tried?

Is there some other way of doing this?

Am I doing something fundamentally wrong-headed?



Michael Dewey
http://www.aghmed.fsnet.co.uk

Hello,

A common way to calculate the CI for a relative risk is the following. Given a
2x2 table of the form:

a b
c d

the log of the risk ratio is given by:

lrr = log[ a/(a+b) / ( c/(c+d) ) ]

which is asymptotically normal with variance:

vlrr = 1/a - 1/(a+b) + 1/c - 1/(c+d).

So an approximate 95% CI for the risk ratio is given by:

exp[ lrr - 1.96*sqrt(vlrr) ], exp[ lrr + 1.96*sqrt(vlrr) ].

A common convention is to add 1/2 to each cell when there are zeros.

So, for the table:

       Col 1 Col 2
Row 1     8   500 
Row 2     0   500

lrr = log[ 8.5/509 / ( 0.5/501 ) ] = 2.817
vllr = 1/8.5 - 1/509 + 1/0.5 - 1/501 = 2.1137

exp[ 2.817-1.96*sqrt(2.1137) ] = .97
exp[ 2.817+1.96*sqrt(2.1137) ] = 289.04

Maybe that is what the authors did.

Best,

-- 
Wolfgang Viechtbauer 
?Department of Methodology and Statistics 
?University of Maastricht, The Netherlands 
?http://www.wvbauer.com/ 

> -----Original Message-----
> From: r-help-bounces at stat.math.ethz.ch [mailto:r-help-
> bounces at stat.math.ethz.ch] On Behalf Of Michael Dewey
> Sent: Friday, November 10, 2006 15:43
> To: r-help at stat.math.ethz.ch
> Subject: [R] Confidence interval for relative risk
> 
> The concrete problem is that I am refereeing
> a paper where a confidence interval is
> presented for the risk ratio and I do not find
> it credible. I show below my attempts to
> do this in R. The example is slightly changed
> from the authors'.
> 
> I can obtain a confidence interval for
> the odds ratio from fisher.test of
> course
> 
> === fisher.test example ==> 
>  > outcome <- matrix(c(500, 0, 500, 8), ncol = 2, byrow = TRUE)
>  > fisher.test(outcome)
> 
>          Fisher's Exact Test for Count Data
> 
> data:  outcome
> p-value = 0.00761
> alternative hypothesis: true odds ratio is not equal to 1
> 95 percent confidence interval:
>   1.694792      Inf
> sample estimates:
> odds ratio
>         Inf
> 
> === end example ==> 
> but in epidemiology authors often
> prefer to present risk ratios.
> 
> Using the facility on CRAN to search
> the site I find packages epitools and Epi
> which both offer confidence intervals
> for the risk ratio
> 
> === Epi example ==> 
>  > library(Epi)
>  > twoby2(outcome[c(2,1),c(2,1)])
> 2 by 2 table analysis:
> ------------------------------------------------------
> Outcome   : Col 1
> Comparing : Row 1 vs. Row 2
> 
>        Col 1 Col 2    P(Col 1) 95% conf. interval
> Row 1     8   500      0.0157    0.0079   0.0312
> Row 2     0   500      0.0000    0.0000      NaN
> 
>                                     95% conf. interval
>               Relative Risk:    Inf       NaN      Inf
>           Sample Odds Ratio:    Inf       NaN      Inf
> Conditional MLE Odds Ratio:    Inf    1.6948      Inf
>      Probability difference: 0.0157    0.0027   0.0337
> 
>               Exact P-value: 0.0076
>          Asymptotic P-value: NaN
> ------------------------------------------------------
> 
> === end example ==> 
> So Epi gives me a lower limit of NaN but the same confidence
> interval and p-value as fisher.test
> 
> === epitools example ==> 
>  > library(epitools)
>  > riskratio(outcome)
> $data
>            Outcome
> Predictor  Disease1 Disease2 Total
>    Exposed1      500        0   500
>    Exposed2      500        8   508
>    Total        1000        8  1008
> 
> $measure
>            risk ratio with 95% C.I.
> Predictor  estimate lower upper
>    Exposed1        1    NA    NA
>    Exposed2      Inf   NaN   Inf
> 
> $p.value
>            two-sided
> Predictor  midp.exact fisher.exact  chi.square
>    Exposed1         NA           NA          NA
>    Exposed2 0.00404821  0.007610478 0.004843385
> 
> $correction
> [1] FALSE
> 
> attr(,"method")
> [1] "Unconditional MLE & normal approximation (Wald) CI"
> Warning message:
> Chi-squared approximation may be incorrect in: chisq.test(xx, correct >
correction)
> 
> === end example ==> 
> And epitools also gives a lower limit
> of NaN.
> 
> === end all examples ==> 
> I would prefer not to have to tell the authors of the
> paper I am refereeing that
> I think they are wrong unless I can help them with what they
> should have done.
> 
> Is there another package I should have tried?
> 
> Is there some other way of doing this?
> 
> Am I doing something fundamentally wrong-headed?
> 
> 
> 
> Michael Dewey
> http://www.aghmed.fsnet.co.uk

Michael Dewey <info at aghmed.fsnet.co.uk> wrote
>>> Subject: [R] Confidence interval for relative risk
>>>
>>> The concrete problem is that I am refereeing
>>> a paper where a confidence interval is
>>> presented for the risk ratio and I do not find
>>> it credible. I show below my attempts to
>>> do this in R. The example is slightly changed
>>> from the authors'.
>>>
>>> I can obtain a confidence interval for
>>> the odds ratio from fisher.test of
>>> course
>>>
>>> === fisher.test example ==>>>
>>> > outcome <- matrix(c(500, 0, 500, 8), ncol = 2, byrow =
TRUE)
>>> > fisher.test(outcome)
>>>
>>>          Fisher's Exact Test for Count Data
>>>
>>> data:  outcome
>>> p-value = 0.00761
>>> alternative hypothesis: true odds ratio is not equal to 1
>>> 95 percent confidence interval:
>>>   1.694792      Inf
>>> sample estimates:
>>> odds ratio
>>>         Inf
>>>
>>> === end example ==>>>
Since

    RR   = p1 +        1               p1=a/(a+b), p2=c/(c+d)
                 --------------
                   p2         1
                 -----   +   ---
                  1-p2        OR

you can backsolve for the RR CI.
> x <- function(p1, p2, oddsr) p1 + 1/(p2/(1-p2) + 1/oddsr)
> 1/x(8/508, 0, 1/1.694792)
[1] 1.650734

Another approach is the Cornfield method -- which is a profile likelihood based
CI
using the Gibbs Chi-square (LRTS), but IIRC originally used the Pearson
chi-square.
I don't think there are R implementations.

David Duffy.

-- 
| David Duffy (MBBS PhD)                                         ,-_|\
| email: davidD at qimr.edu.au  ph: INT+61+7+3362-0217 fax: -0101  /     *
| Epidemiology Unit, Queensland Institute of Medical Research   \_,-._/
| 300 Herston Rd, Brisbane, Queensland 4029, Australia  GPG 4D0B994A v

At 14:43 10/11/2006, Michael Dewey wrote:

After considerable help from list members and some digging of my own
I have prepared a summary of the findings which I have posted (see
link below). Broadly there were four suggestions
1 - Wald-type intervals,
2 - transforming the odds ratio confidence interval,
3 - method based on score test,
4 - method based on likelihood.

Method 3 was communicated to me off-list
  ==========================I haven't followed all of the thread either but
has someone already
pointed out to you confidence intervals that use the score method? 
For example Agresti (Categorical Data Analysis 2nd edition, p. 77-78) 
note that 'although computationally more complex, these methods often 
perform better'. However, they also note that 'currently they are not 
available in standard software'.

But then again, R is not standard software: the code (riskscoreci) 
can be found from here: 
http://www.stat.ufl.edu/~aa/cda/R/two_sample/R2/index.html

best regards,
Jukka Jokinen

===============================================and so I reproduce it here.
Almost a candidate for the fortunes package there.
The other three can be found from the archive
under the same subject although not all in the same thread.

Methods 3 and 4 seem to have more going for them as far
as I can judge.

Thanks to David Duffy, Spencer Graves,
Jukka Jokinen, Terry Therneau, and Wolfgan Viechtbauer.
The views and calculations presented here and in the summary
are my own and
any errors are my responsibility not theirs.

The summary document is available from here

http://www.zen103156.zen.co.uk/rr.pdf


Original post follows.

===================================>The concrete problem is that I am
refereeing
>a paper where a confidence interval is
>presented for the risk ratio and I do not find
>it credible. I show below my attempts to
>do this in R. The example is slightly changed
>from the authors'.
>
>I can obtain a confidence interval for
>the odds ratio from fisher.test of
>course
>
>=== fisher.test example ==>
> > outcome <- matrix(c(500, 0, 500, 8), ncol = 2, byrow = TRUE)
> > fisher.test(outcome)
>
>         Fisher's Exact Test for Count Data
>
>data:  outcome
>p-value = 0.00761
>alternative hypothesis: true odds ratio is not equal to 1
>95 percent confidence interval:
>  1.694792      Inf
>sample estimates:
>odds ratio
>        Inf
>
>=== end example ==>
>but in epidemiology authors often
>prefer to present risk ratios.
>
>Using the facility on CRAN to search
>the site I find packages epitools and Epi
>which both offer confidence intervals
>for the risk ratio
>
>=== Epi example ==>
> > library(Epi)
> > twoby2(outcome[c(2,1),c(2,1)])
>2 by 2 table analysis:
>------------------------------------------------------
>Outcome   : Col 1
>Comparing : Row 1 vs. Row 2
>
>       Col 1 Col 2    P(Col 1) 95% conf. interval
>Row 1     8   500      0.0157    0.0079   0.0312
>Row 2     0   500      0.0000    0.0000      NaN
>
>                                    95% conf. interval
>              Relative Risk:    Inf       NaN      Inf
>          Sample Odds Ratio:    Inf       NaN      Inf
>Conditional MLE Odds Ratio:    Inf    1.6948      Inf
>     Probability difference: 0.0157    0.0027   0.0337
>
>              Exact P-value: 0.0076
>         Asymptotic P-value: NaN
>------------------------------------------------------
>
>=== end example ==>
>So Epi gives me a lower limit of NaN but the same confidence
>interval and p-value as fisher.test
>
>=== epitools example ==>
> > library(epitools)
> > riskratio(outcome)
>$data
>           Outcome
>Predictor  Disease1 Disease2 Total
>   Exposed1      500        0   500
>   Exposed2      500        8   508
>   Total        1000        8  1008
>
>$measure
>           risk ratio with 95% C.I.
>Predictor  estimate lower upper
>   Exposed1        1    NA    NA
>   Exposed2      Inf   NaN   Inf
>
>$p.value
>           two-sided
>Predictor  midp.exact fisher.exact  chi.square
>   Exposed1         NA           NA          NA
>   Exposed2 0.00404821  0.007610478 0.004843385
>
>$correction
>[1] FALSE
>
>attr(,"method")
>[1] "Unconditional MLE & normal approximation (Wald) CI"
>Warning message:
>Chi-squared approximation may be incorrect in: chisq.test(xx, correct
>correction)
>
>=== end example ==>
>And epitools also gives a lower limit
>of NaN.
>
>=== end all examples ==>
>I would prefer not to have to tell the authors of the
>paper I am refereeing that
>I think they are wrong unless I can help them with what they
>should have done.
>
>Is there another package I should have tried?
>
>Is there some other way of doing this?
>
>Am I doing something fundamentally wrong-headed?
>
>
>
Michael Dewey
http://www.aghmed.fsnet.co.uk