Two Props

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StatsToDo: Compare Two Proportions in Unpaired Groups

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Comparing two proportions is a common approach to many aspects of clinical research, in comparing efficacies and adverse outcomes under different circumstances, in quality control, and as a basis for meta-ana;ysis to establich evidence for practice.

This page provides the 6 most commonly used methods of comparing the proportion in two groups.

Please note: The data of examples presented in this page are all artificially created to demonstrate the procedures. They do not represent real clinical scenarios.

Chi Square Test for 2x2 contigency tables

This tests the probability that the number of positive and negative cases in two groups are from a similar population, a goodness of fit test. Until tests using confidence interval became popular in the 1990s, this was the principle test used to compare two proportions.

For Chi Square to be used, the total number in the study should exceed 30 cases, and each cell should have at least 5 cases. Short of these numbers, the assumptions of Chi Squares distribution cannot be assured and there is a possibility of misinterpretation.

An example: We wish to study the difference in the preferences for specialty training between the male and female interns in a hospital. We found a group of 16 male interns, 10 of whom chose surgical specialties. We also found a group of 21 female interns, 5 of them chose surgical specialties. The Chi Squares is 4.15 and α=0.04. From this we can conclude that male and female interns differ significantly (at the p<0.05 decision level), in choosing surgical specialties for training.

Fisher's Exact probability

This estimates the probability of observing the difference in proportions in two groups, departing from the null hypothesis that the two groups are the same. The calculation provided estimates the probability for the two tail test.

As the probability is estimated using permutation, no assumption of the underlying binomial distribution is made, and the power of the model is 100%. Because of this, the test remains valid even when the sample size is small. Fisher's test is therefore often used when the sample or cell size are insufficient for the chi square test.

The test uses Factorial numbers repeatedly, so that computing time increases exponentially with the number of cases. Therefore this test is only used when the conditions for a Chi Square Test cannot be satisfied. With the high speed and large memory capacity of modern computers however, Fisher's Test can still be easily computed for sample size as large as 1000

An example

We repeat the study in the Chi Square Panel, but use Fisher's Exact Probability instead.

We wish to study the difference in the preferences for specialty training between the male and female interns in a hospital. We found a group of 16 male interns, 10 of whom chose surgical specialties. We also found a group of 21 female interns, 5 of them chose surgical specialties. Fisher's Exact Probability = 0.02, and we can conclude that male and female interns differ significantly (at the p<0.05 decision level), in choosing surgical specialties for training.

Risk Difference

This is often used to evaluate the results of a controlled trial, where risk is the proportion of the group with positive outcomes, so that risk in group 1 risk1=Pos1/(Pos1+Neg1), and risk in group 2 r2=Pos2/(Pos2+Neg2). Risk difference is then rd=r1-r2.

A particular useful notation is the Number Needed To Treat (NNT), where NNT = 1 / RD rounded up to the next integer. NNT represents the number of cases that need to be treated differently to affect the change of outcome in a single case. NNT therefore provides a nuanced indicator of how clinically relevant the risk difference is

There are two methods to estimate the confidence interval of the risk difference, based on the assumption of distribution of the risk difference.

The Standard Method assumes that risk difference is a normally distributed variable, and its Standard Error can be estimated. The confidence interval is therefore rd - z(SE) to rd + z(SE), where z is the normal deviate, and depends on the confidence the user required (usually 95% confidence, so that z=1.65 for 1 rail amd 1.96 for 2 tail). This is commonly used, particularly to produce effect size and Standard Error to be combined with other studies in a meta-analysis.

The problem with the Standard method is that the assumption of normal distribution is only true when the sample size is large and the risk is near the 0.5 value. As the risk value approaches 0 or 1, the distribution becomes increasingly asymmetrical, with a narrower tail towards the extreme values (0 or 1) and longer tail towards the center (0.5). This error is accentuated if the sample size is small.

The Exact method, based on the correct assumption that risk difference is binomially distributed, is therefore more accurate. This produces confidence intervals that are asymmetric, and without the Standard Error. Although these results are more accurate, they cannot be easily used in meta-analysis

An example :

We wish to study whether preoperative antibiotics reduces postoperative infections in a randomised controlled trial. Group 1 are 11 patients that received antibiotics, and 4 of them had postoperative infections. Groups 2 are 15 patients that received no antibiotics and 12 of them had postoperative infections.

Risk of infection in the group receiving antibiotic is r1 = 4/11 = 0.31 (31%), and risk in the group that did not receive antibiotic is r2 = 12/15 = 0.8 (80%). Because the sample size is small, the exact method is used. The risk difference is 0.31 - 0.8 = -0.49 (-49%), with the 95% confidence interval -0.71 to -0.12. In other words, those receiving antibiotics were 12% to 70% less likely to have postoperative infections than those not receiving antibiotics.

If the traditional method was used, the 95% confidence interval would be -0.81 to -0.17 (17% to 81%)

Number needed to treat is based on the risk difference, which is the same in both methods. NNT = 1 / RD = 1 / 0.49 = 2.03, rounded upwards to 3. In other words, for the reduction of 1 case of infection, three more patients will need to receive pre-operative antibiotics.

Risk Ratio (also known as Relative Risks)

Although risk difference is a useful research model, it assumes that the sample size in the two groups are roughly the same. Large discrepancies in sample size will therefore introduce bias to the results. The model is therefore less useful in epidemiological studies, where the sample size cannot be easily equalized. For example, to study death rate comparing smokers with non-smokers, the number of smokers are very much less than non-smokers.

Risk Ratio is therefore introduced, initially to cope with the circumstances around epidemiological studies. The model is so effective that it is now increasing adapted for general comparison of 2 proportions.

Risk ratio, as all other ratios, is log normally distibuted. This means the log(risk ratio) is normally distributed, and used for all calculations. The results are then transformed back to non-logarithmic value by the exponential function.

An example

We wish to study whether men are more likely to have car accidents than women. We collected a database of drivers, and found that out of 500 men 23 had an accident in the past year, while out of 530 women 10 had an accident during the same period.

Risk of car accident amongst men is r1 = 23/500 = 0.046 (4.6%), risk of accident in women is r2 = 10/530 = 0.019 (1.9%). Risk Ratio is rr = 0.046/0.019 = 2.44, with 95% CI 1.17 to 5.07. In other words, male drivers have 1.17 to 5.07 times the risk of having a car accident compared with female drivers.

Odds Ratio

While risk is the proportion of positive cases in the sample (risk = P / (P+N)), odd is the ratio of the positive and negative cases in the sample (odd = P / N). Althogh risks and odds are both conceptul representations of proportion, and risk ratio and odds ratio both represents differences in proportions in 2 groups, they are numerically different, and have different properties.

Gamblers commonly use odds to represent the ratio of winning against losing, while clinicians often prefer risks as it represents predictions or expected outcomes
Risk is unidirectional, in that the group (treatment or characteristic) precurs the outcome, while odds represents the relationship between before and after, without a requirement of directions. Because of this, odds ratio can be used more flesibly. Examples are the relationship between social class and educational achievement, when it is not clear which is the cause and which is the effect. Another example is in retrospective paired controlled studies, where cases are grouped according to outcome, and the test is whether the precursor variable is different
Odd, because it is mathematically simpler, can be more flexiblly used, particularly in multivariate statistical models.

Statisticians therefore often prefer using odds and odds ratio for their calculations. In any case odd and risk can be easily interconverted as odd = risk / (1 - risk) and risk = odd / (1 + odd)

An Example We wish to study the relationship between the deflexion of the fetal head and occipital posterior position in early labour. We do not know, and are unconcerned, which comes first, but merely want to know if there is an association.

We found 20 babies with occipital posterior position in early labour, 12 of whom has a deflexed head (Pos1=12) and 8 with a flexed head (Neg1=8). Odd1 is therefore o1 = 12 / 8 = 1.5

We also found 35 babies with occipital anterior position, 10 of which had deflexed head (Pos2=10) and 25 with a flexed head (Neg2=25). The odd is o2 = 10 /25 = 0.4

The Odds ration is or = 1.5 / 0.4 = 3.75, the 95% confidence interval 1.2 to 11.9 (2 tail). As the confidence interval does not include the null value of 1, we can conclude that occipital posterior and deflexion of the fetal head in ealy labour are related.

References

Altman D (1994) Practical Statistics for medical Research. Chapman Hall, London. ISBN 0 412 276205 (First Ed. 1991)

Chi square : p.252
Fisher's Exact probability : p.233

Altman GD, Machin D, Bryant TN, Gardner MJ (2000) Statistics with confidence (2nd Ed). BMJ Books ISBN 0 7279 1375 1

Odds ratio : p.60-62
Relative risks : p.57-59
Risk difference : p.233

Data input : table with 4 columns :
  - Col 1 = number of Positives in Grp 1
  - Col 2 = number of Positives in Grp 2
  - Col 3 = number of Negatives in Grp 1
  - Col 4 = number of Negatives in Grp 2
Additional 5th column
  - for Risk Difference, Risk Ratio and Odds Ratio
  - column for % confidence of Confidence Interval

Chi Square :
Fisher's Exact Probability :
Risk Difference (Standard) :
Risk Difference (Exact) :
Risk Ratio (Relative Risks) :
Odds Ratio :

Chi Square Fisher P Risk Difference Risk Ratio Odds Ratio F G H

Chi Square Test for 2x2 contingency table

# Program 1 Chi Sq 

dat = ("
P1   P2  N1  N2
 12    5   4  16
150  100 210 100
 55   34  12  20
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data 
# vectors for result
XSq <- vector()
P <- vector()
# Calculate
for(i in 1 :nrow(df))
{
  mx <- matrix(c(df$P1[i],df$P2[i],df$N1[i],df$N2[i]),nrow=2,ncol=2)
  test <- chisq.test(mx)
  XSq <- append(XSq, test$statistic)
  P <- append(P, test$p.value)
}
# Add results to data frame for display
df$XSq <- XSq
df$P <- P
df # input data + chiSq + p

The results are

> df # input data + chiSq + p
   P1  P2  N1  N2      XSq           P
1  12   5   4  16 7.631331 0.005736296
2 150 100 210 100 3.283567 0.069976688
3  55  34  12  20 4.682993 0.030462625

Fisher's Exact Probability for 2x2 contingency table

# Program 2 Fisher's Exact Probability 
# data input
dat = ("
P1   P2  N1  N2
12    5   4  16
 3    5   4   1
14   12  11  13
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data 
# vector for results
P <- vector()
# Calculate
for(i in 1 :nrow(df))
{
  mx <- matrix(c(df$P1[i],df$P2[i],df$N1[i],df$N2[i]),nrow=2,ncol=2)
  test <- fisher.test(mx)
  P <- append(P, test$p.value)
}
# combine result in data frame for display
df$P <- P
df # input data + Fisher's Exact probability

The results are

> df # input data + Fisher's Exact probability
  P1 P2 N1 N2           P
1 12  5  4 16 0.002979686
2  3  5  4  1 0.265734266
3 14 12 11 13 0.777511985

This subpanel provides R algorithms for risk differences, divided into 3 sections

Section 1 Calculates risks in groiup1 and 2 (R1, R2), the risk difference (RD), and the Numbers needed to treat (NNT). These parameters are common to both Standard and Exact algorithms
Section 2 takes the dataframe created in section 1, calculates the Standard Error (SE) to the risk difference, and the confidence intervals using the standard model, assuming that risk difference is approximately normally distributed
Section 3 takes the dataframe created in section 1, and calculates the exact confidence interva;, based on the assumtion that risk difference is binomially distributed

Section 1: Input data and the basic risk parameters

# Risk Difference : Section 1. Initial calculations
# data input
dat = ("
P1   P2  N1  N2  Pc
 12   5   4  16  90
150	100	210	100  99
 55  34  12  20  95
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data 
# 3a: calculate risk difference
df$R1 <- df$P1 / (df$P1 + df$N1)  # Risk 1
df$R2 <- df$P2 / (df$P2 + df$N2)  # Risk 2
df$RD <- df$R1 - df$R2            # Risk Difference
df$NNT = ceiling(1 / df$RD)
df # show initial risk difference

The initial data frame produced are as follows.

P1, N1, P2, N2 are number of positives and negatives in groups 1 and 2
Pc = oercent of confidence in confidence interval. usuallu 90, 95, or 99
R1 and R2 are risk of being positive in groups 1 and 2
RD = R1 - R2 = risk difference
NNT = Number needed to treat = cwiling (1 / RD)

> df # show initial risk difference
   P1  P2  N1  N2 Pc        R1        R2          RD NNT
1  12   5   4  16 90 0.7500000 0.2380952  0.51190476   2
2 150 100 210 100 99 0.4166667 0.5000000 -0.08333333 -12
3  55  34  12  20 95 0.8208955 0.6296296  0.19126589   6

Section 2. Calculation of Standard Error and confidence interval using the standard model, assuming risk difference to be approximately normally distributed.


# Calculate SE and CI (standard)
dfS <- df  # copy df from section 1
dfS # optional display of input data
# vectors for results
SE <- vector()      # standard error
LL1 <- vector()     # lower limit 1 tail
UL1 <- vector()     # upper limit q tail
LL2 <- vector()     # lower limit 2 tail
UL2 <- vector()     # upper limit 2 tail
# calculation for each row of data
for(i in 1 : nrow(dfS))
{
  r1 = as.numeric(dfS$R1[i])
  dfS$R1[i] = sprintf(r1,fmt="%#.4f")
  r2 = as.numeric(dfS$R2[i])
  dfS$R2[i] = sprintf(r2,fmt="%#.4f")
  p1 = dfS$P1[i]
  p2 = dfS$P2[i]
  n1 = dfS$N1[i]
  n2 = dfS$N2[i]
  pc = dfS$Pc[i]
  rd = as.numeric(dfS$RD[i])
  dfS$RD[i] = sprintf(rd,fmt="%#.4f")
  se = sqrt((r1 * (1.0 - r1) / (p1 + n1)) + (r2 * (1.0 - r2) / (p2 + n2)))
  SE <- append(SE,sprintf(se,fmt="%#.4f"))
  prob = (1 - pc / 100) / 1 
  z = qnorm(1 - prob)
  LL1 <- append(LL1, sprintf(rd - z * se,fmt="%#.4f"))
  UL1 <- append(UL1, sprintf(rd + z * se,fmt="%#.4f"))
  prob = (1 - pc / 100) / 2 
  z = qnorm(1 - prob)
  LL2 <- append(LL2, sprintf(rd - z * se,fmt="%#.4f"))
  UL2 <- append(UL2, sprintf(rd + z * se,fmt="%#.4f"))
}
dfS$SE <- SE
dfS$LL1 <- LL1
dfS$UL1 <- UL1
dfS$LL2 <- LL2
dfS$UL2 <- UL2
dfS # input data and results

The results for section 2 are as follows

SE = Standard Error of risk difference
LL1 and UL1 = lower and upper limits of confidence interval, 1 tail
LL2 and UL2 = lower and upper limits of confidence interval, 2 tail

> dfS # input data and results
   P1  P2  N1  N2 Pc     R1     R2      RD NNT     SE     LL1    UL1     LL2    UL2
1  12   5   4  16 90 0.7500 0.2381  0.5119   2 0.1427  0.3291 0.6948  0.2772 0.7466
2 150 100 210 100 99 0.4167 0.5000 -0.0833 -12 0.0439 -0.1854 0.0187 -0.1964 0.0297
3  55  34  12  20 95 0.8209 0.6296  0.1913   6 0.0807  0.0585 0.3240  0.0331 0.3494

Section 3. confidence interval (exact model), assuming risk difference to have the binomial distribution

# Calculate CI (Exact)
#   Subroutine to estimate confidence interval
ExactLimits <- function(np,ng,z)  # numbers of positive and negatives, z
{
  n = np + ng
  q = ng / n
  z2 = z * z
  A = 2 * np + z2;
  B = z * sqrt(z2 + 4.0 * np * q)
  C = 2 * (n + z2)
  lower = (A - B) / C
  upper = (A + B) / C
  return (c(lower, upper))
}

# Main program
dfE <- df  # copy df from section 1
dfE # optional display of input data
LL1 <- vector()
UL1 <- vector()
LL2 <- vector()
UL2 <- vector()
for(i in 1 : nrow(dfE))
{
  r1 = as.numeric(dfE$R1[i])
  dfE$R1[i] = sprintf(r1,fmt="%#.4f")
  r2 = as.numeric(dfE$R2[i])
  dfE$R2[i] = sprintf(r2,fmt="%#.4f")
  p1 = dfE$P1[i]
  p2 = dfE$P2[i]
  n1 = dfE$N1[i]
  n2 = dfE$N2[i]
  pc = dfE$Pc[i]
  rd = as.numeric(dfE$RD[i])
  dfE$RD[i] = sprintf(rd,fmt="%#.4f")
  prob = (1 - pc / 100) / 1 
  z = qnorm(1 - prob)
  ar = ExactLimits(p1,n1,z)
  lower1 = ar[1]
  upper1 = ar[2];
  ar = ExactLimits(p2,n2,z)
  lower2 = ar[1];
  upper2 = ar[2];
  lower = rd - sqrt((r1 - lower1)^2 + (upper2 - r2)^2)
  upper = rd + sqrt((r2 - lower2)^2 + (upper1 - r1)^2)
  LL1 <- append(LL1, sprintf(lower,fmt="%#.4f"))
  UL1 <- append(UL1, sprintf(upper,fmt="%#.4f"))
  prob = (1 - pc / 100) / 2 
  z = qnorm(1 - prob)
  ar = ExactLimits(p1,n1,z)
  lower1 = ar[1]
  upper1 = ar[2];
  ar = ExactLimits(p2,n2,z)
  lower2 = ar[1];
  upper2 = ar[2];
  lower = rd - sqrt((r1 - lower1)^2 + (upper2 - r2)^2)
  upper = rd + sqrt((r2 - lower2)^2 + (upper1 - r1)^2)
  LL2 <- append(LL2, sprintf(lower,fmt="%#.4f"))
  UL2 <- append(UL2, sprintf(upper,fmt="%#.4f"))
}
dfE$LL1 <- LL1
dfE$UL1 <- UL1
dfE$LL2 <- LL2
dfE$UL2 <- UL2
dfE # result output

The results are as follows. Please note that the confidence interval is not symmetrical on both side of the risk difference, as binimial distribution has a narrower distribution towards the extremes (0 or 1) than towrds the center (0.5)

> dfE # result output 
   P1  P2  N1  N2 Pc     R1     R2      RD NNT     LL1    UL1     LL2    UL2
1  12   5   4  16 90 0.7500 0.2381  0.5119   2  0.3043 0.6594  0.2413 0.6887
2 150 100 210 100 99 0.4167 0.5000 -0.0833 -12 -0.1835 0.0183 -0.1939 0.0291
3  55  34  12  20 95 0.8209 0.6296  0.1913   6  0.0578 0.3202  0.0324 0.3436

# Risk Ratio
# Data input
dat = ("
P1   P2  N1  N2  Pc
 12   5   4  16  90
150 100	210 100  99
 55  34  12  20  95
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data 
# vectors for calculations
R1 <- vector()
R2 <- vector()
RR <- vector()
LRR <- vector()
SE <- vector()
LL1 <- vector()
UL1 <- vector()
LL2 <- vector()
UL2 <- vector()
# Calculations
for(i in 1:nrow(df))
{
  p1 = df$P1[i]                  # number in grp 1 pos
  p2 = df$P2[i]                  # number in grp 2 pos
  n1 = df$N1[i]                  # number in grp 1 neg
  n2 = df$N2[i]                  # number in grp 2 neg
  pc = df$Pc[i]                  # % confidence
  r1 = p1 / (p1 + n1)            # Risk Grp 1
  R1 <- append(R1,sprintf(r1,fmt="%#.4f"))
  r2 = p2 / (p2 + n2)            # Risk Grp 2
  R2 <- append(R2,sprintf(r2,fmt="%#.4f"))
  rr = r1 / r2                   # risk ratio
  RR <- append(RR,sprintf(rr,fmt="%#.4f"))
  lrr = log(rr)                  # log(risk ratio)
  LRR <- append(LRR,sprintf(lrr,fmt="%#.4f"))
  se = sqrt(1.0 / p1 - 1.0 /(p1 + n1) + 1.0 / p2 - 1.0 / (p2 + n2)) # Standard Error (lrr)
  SE <- append(SE,sprintf(se,fmt="%#.4f"))
  prob = (1 - pc / 100) / 1 
  z = qnorm(1 - prob)
  lower = exp(lrr - z * se)       # lower margin of confidence interval 1 tail
  upper = exp(lrr + z * se)       # upper margin of confidence interval 1 tail
  LL1 <- append(LL1,sprintf(lower,fmt="%#.4f"))
  UL1 <- append(UL1,sprintf(upper,fmt="%#.4f"))
  prob = (1 - pc / 100) / 2 
  z = qnorm(1 - prob)
  lower = exp(lrr - z * se)       # lower margin of confidence interval 2 tail
  upper = exp(lrr + z * se)       # upper margin of confidence interval 2 tail
  LL2 <- append(LL2,sprintf(lower,fmt="%#.4f"))
  UL2 <- append(UL2,sprintf(upper,fmt="%#.4f"))
}
# combine results into data frame for display
df$R1 <- R1
df$R2 <- R2
df$LRR <- LRR
df$SE <- SE
df$RR <- RR
df$LL1 <- LL1
df$UL1 <- UL1
df$LL2 <- LL2
df$UL2 <- UL2
df # show data frame with resulkts

The results are as follows

R1 and R2 = risk in grp 1 and 2
RR = Risk Ratio (Relative Risks) = R1 / R2
LRR and SE = log(Risk Ratio) and its Standard Error
LL1 and UL1 = lower and upper border of confidence interval, 1 tail
LL2 and UL2 = lower and upper border of confidence interval, 2 tail

> df # show data frame with resulkts
   P1  P2  N1  N2 Pc     R1     R2     LRR     SE     RR    LL1    UL1    LL2    UL2
1  12   5   4  16 90 0.7500 0.2381  1.1474 0.4162 3.1500 1.8479 5.3697 1.5886 6.2462
2 150 100 210 100 99 0.4167 0.5000 -0.1823 0.0943 0.8333 0.6692 1.0377 0.6537 1.0624
3  55  34  12  20 95 0.8209 0.6296  0.2653 0.1190 1.3038 1.0721 1.5855 1.0326 1.6461

# Odds Ratio
# Data input
dat = ("
P1   P2  N1  N2  Pc
 12   5   4  16  90
150 100 210 100  99
 55  34  12  20  95
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data 
# vectors for calculations
O1 <- vector()
O2 <- vector()
OR <- vector()
LOR <- vector()
SE <- vector()
LL1 <- vector()
UL1 <- vector()
LL2 <- vector()
UL2 <- vector()
# Calculations
for(i in 1:nrow(df))
{
  p1 = df$P1[i]                    # number of Pos in grp 1
  p2 = df$P2[i]                    # number of Pos in grp 2
  n1 = df$N1[i]                    # number of Neg in grp 1
  n2 = df$N2[i]                    # number of Neg in grp 2
  pc = df$Pc[i]                    # % in confidence
  o1 = p1 / n1                     # odd grp 1
  O1 <- append(O1,sprintf(o1,fmt="%#.4f"))
  o2 = p2 / n2                     # odd grp 2
  O2 <- append(O2,sprintf(o2,fmt="%#.4f"))
  or = o1 / o2                     # odds Ratio
  OR <- append(OR,sprintf(or,fmt="%#.4f"))
  lor = log(or)                    # log(odds ratio)
  LOR <- append(LOR,sprintf(lor,fmt="%#.4f"))
  se = sqrt(1.0 / p1 + 1.0 / p2 + 1.0 / n1 + 1.0 / n2)  # Standard Error of log(or)
  SE <- append(SE,sprintf(se,fmt="%#.4f"))
  prob = (1 - pc / 100) / 1 
  z = qnorm(1 - prob)
  lower = exp(lor - z * se)       # lower limit of confidence interval 1 tail
  upper = exp(lor + z * se)       # upper limit of confidence interval 1 tail
  LL1 <- append(LL1,sprintf(lower,fmt="%#.4f"))  
  UL1 <- append(UL1,sprintf(upper,fmt="%#.4f"))
  prob = (1 - pc / 100) / 2 
  z = qnorm(1 - prob)
  lower = exp(lor - z * se)       # lower limit of confidence interval 2 tail
  upper = exp(lor + z * se)       # upper limit of confidence interval 2 tail
  LL2 <- append(LL2,sprintf(lower,fmt="%#.4f"))
  UL2 <- append(UL2,sprintf(upper,fmt="%#.4f"))
}
df$O1 <- O1
df$O2 <- O2
df$LOR <- LOR
df$SE <- SE
df$OR <- OR
df$LL1 <- LL1
df$UL1 <- UL1
df$LL2 <- LL2
df$UL2 <- UL2
df # show data frame with resulkts

The results are as follows

O1 and O2 = Odd in grp 1 and 2
OR = Odds Ratio = O1 / O2
LOR and SE = log(Odds Ratio) and its Standard Error
LL1 and UL1 = lower and upper border of confidence interval, 1 tail
LL2 and UL2 = lower and upper border of confidence interval, 2 tail

> df # show data frame with resulkts
   P1  P2  N1  N2 Pc     O1     O2     LOR     SE     OR    LL1     UL1    LL2     UL2
1  12   5   4  16 90 3.0000 0.3125  2.2618 0.7719 9.6000 3.5699 25.8160 2.6969 34.1728
2 150 100 210 100 99 0.7143 1.0000 -0.3365 0.1773 0.7143 0.4729  1.0789 0.4524  1.1277
3  55  34  12  20 95 4.5833 1.7000  0.9918 0.4254 2.6961 1.3393  5.4273 1.1713  6.2058

Contents of F:15

Contents of G:16

Contents of H:17

Contents of D:3

Contents of E:4

Contents of F:5