Stratified 2x2 Tables: Miettinen-Nurminen and Friends

A single 2x2 table

For one 2x2 table, RD, RR, and OR give the risk difference, relative risk, and odds ratio with their score confidence intervals:

RD(7, 10, 3, 10) # y1, n1, y2, n2

   p1  p2  RD       SE        lower     upper
1 0.7 0.3 0.4 0.204939 -0.001673089 0.8016731

RR(7, 10, 3, 10)

   p1  p2       RR     SElog     lower    upper
1 0.7 0.3 2.333333 0.5255383 0.8329862 6.536056

OR(7, 10, 3, 10)

      odd1      odd2       OR     SElog     lower    upper
1 2.333333 0.4285714 5.444444 0.9759001 0.8040183 36.86729

The Miettinen-Nurminen (MN) score intervals for a single stratum are RDmn1, RRmn1, and ORmn1:

RDmn1(7, 10, 3, 10)

         p1          p2          RD       lower       upper 
 0.70000000  0.30000000  0.40000000 -0.04910451  0.71457639

RRmn1(7, 10, 3, 10)

       p1        p2        RR     lower     upper 
0.7000000 0.3000000 2.3333333 0.9264279 6.8474376

ORmn1(7, 10, 3, 10)

      odd1       odd2         OR      lower      upper 
 2.3333333  0.4285714  5.4444444  0.8215414 36.0809910

These intervals perform well even in extreme cases such as zero cells:

RDmn1(10, 10, 0, 10)

       p1        p2        RD     lower     upper 
1.0000000 0.0000000 1.0000000 0.6636451 1.0000000

RRmn1(10, 10, 1, 10)

       p1        p2        RR     lower     upper 
 1.000000  0.100000 10.000000  2.474357 55.882702

Stratified analysis by the MN score method

When the data have strata (centers, studies, subgroups), provide a data.frame with columns y1, n1, y2, n2, one row per stratum:

d0 = data.frame(y1=c(25, 23), n1=c(339, 370), y2=c(28, 40), n2=c(335, 364))
RDmn(d0)

$Strata
          p1         p2           RD       lower        upper
1 0.07374631 0.08358209 -0.009835777 -0.05173666  0.031557500
2 0.06216216 0.10989011 -0.047727948 -0.08979481 -0.007312602

$Common
           p1            p2            RD         lower         upper 
 0.0677085653  0.0972940300 -0.0295854647 -0.0589400053 -0.0008563609

RRmn(d0)

$Strata
          p1         p2        RR     lower     upper
1 0.07374631 0.08358209 0.8823220 0.5284979 1.4721821
2 0.06216216 0.10989011 0.5656757 0.3470167 0.9195091

$Common
        p1         p2         RR      lower      upper 
0.06770624 0.09729930 0.69585539 0.48873173 0.98979282

ORmn(d0)

$Strata
        odd1       odd2        OR     lower    upper
1 0.07961783 0.09120521 0.8729527 0.5004020 1.522954
2 0.06628242 0.12345679 0.5368876 0.3160012 0.912411

$Common
      odd1       odd2         OR      lower      upper 
0.07235566 0.10843574 0.67410200 0.45950894 0.98888123

The Strata part shows the per-stratum MN intervals, and the Common part shows the common (pooled) estimate with its stratified score confidence interval.

Since version 1.0.0:

RDmn and RRmn agree with ratesci::scoreci(weighting="MN", skew=FALSE) by Pete Laud to 8 decimal places, including zero-cell strata.
ORmn uses the inverse variance weighted score statistic with the score bias correction following Laud, and agrees with ratesci::scoreci(contrast="OR", stratified=TRUE, skew=FALSE) to at least 7 significant digits.

Inverse variance (meta-analysis) and CMH methods

The *inv functions pool by the inverse variance method and report both the fixed effect and DerSimonian-Laird random effects results with Cochran’s Q test - the standard meta-analysis summary:

d1 = data.frame(y1=c(25, 23), n1=c(339, 370), y2=c(28, 40), n2=c(335, 364))
RRinv(d1)

$RRs
          p1         p2        RR     SElog     lower     upper      pwi     pwsi
1 0.07374631 0.08358209 0.8823220 0.2641578 0.5257455 1.4807394 52.02783 48.29184
2 0.06216216 0.10989011 0.5656757 0.2510544 0.3458355 0.9252634 47.97217 51.70816

$Heterogeneity
         Q      prob
1 1.487959 0.2225334

$tau2
[1] 0.03240235

$Fixed
         RR     lower     upper
1 0.6985385 0.4889789 0.9979083

$Random
         RR     lower    upper
1 0.7011316 0.4536439 1.083637

The Cochran-Mantel-Haenszel common odds ratio:

ORcmh(d1)

$ORs
        odd1       odd2        OR     SElog     lower     upper
1 0.07961783 0.09120521 0.8729527 0.2866293 0.4977502 1.5309816
2 0.06628242 0.12345679 0.5368876 0.2728489 0.3145096 0.9165007

$Common
         OR    SElog     lower     upper
1 0.6740762 0.196439 0.4586694 0.9906452

Example: aspirin and coronary heart disease

aspirinCHD contains six trials of aspirin versus placebo for post-myocardial infarction mortality:

aspirinCHD

   y1   n1  y2   n2
1  49  615  67  624
2  44  758  64  771
3  27  317  32  309
4 102  832 126  850
5  85  810  52  406
6 246 2267 219 2257

ORmn(aspirinCHD)$Common

     odd1      odd2        OR     lower     upper 
0.1128906 0.1246222 0.9030600 0.7969606 1.0232842

ORinv(aspirinCHD)$Common

         OR      SElog     lower    upper
1 0.9034556 0.06395705 0.7970149 1.024111

Choosing a method