Introduction

In binomial statistics, the relationship between the number of cases with a positive characteristics (NPos) and that without the positive characteristics (NNeg) can be expressed in two ways

• Proportions, probability, and risk are mathematically the same. In most clinical studies, and in this page, the term risk is used
  risk = NPos / (NPos + NNeg)
• Odd is a term originated from gambling, but it has desirable statistical properties, so are also frequently used
  odd = NPos / NNeg

With increasing emphasis on evidence based practices, there is a need to consolidate evidence from diverse publications, so there is a need to inter-convert risks, odds, risk differences, risk ratios, and odds ratio from one format to another.

This page provides the algorithms for such conversions

Definitions

Numbers

• NPos is the number of cases with a positive characteristics observed in a population, sample, or group
• NNeg is the number of cases without a positive characteristics observed in a population, sample, or group
• Risk in a population, sample, or group is risk = NPos / (NPos + NNeg)
• Odd in a population, sample, or group is odd = NPos / NNeg
• Risk can be converted to odd odd = risk / (1 - risk)
• Odd can be converted to risk risk = odd / (1 + odd)
• e.g. A new drug was given to 20 patients with an illness, and 10 improved.
  • The risk of improvement = 10 / (10+10) = 0.5
  • The odd of improvement = 10 / 10 = 1
  • Converting risk to odd odd = risk / (1 - risk) = 0.5 / (1 - 0.5) = 0.5 / 0.5 = 1
  • Converting odd to risk risk = odd / (1 + odd) = 1 / (1 + 1) = 1 / 2 = 0.5

Two Group Comparisons

Comprison of binomial outcomes between two groups are common, particularly when collating Evidence for medical management. Usually there is a passive group, which represents the baseline or population reference, and the indexed group, which represents the group of interest or that receiving a new intervention. On this page, in order to avoid confusion, the indexed group is named group 1, and the baseline or passive group group 2

For example: In a controlled trial, 20 patients were given a new drug, and 10 improved, while 30 patients were given a placebo, and 6 improved.

• Group 1 (indexed group): Npos = 10, NNeg = 10, risk1 = 10/20 = 0.5, odd1 = 10/10 = 1.0
• Group 2 (passive group): NPos = 6, NNeg = 24, risk2 = 6/30 = 0.2, odd2 = 6/24 = 0.25

Three comparisons between the 2 groups can be made. Using the same example,

• Risk Difference (RD) = risk 1 - risk2 = 0.5-0.2 = 0.3.

  From this, NNT (Numbers Needed to Treat), the number of cases needed to be given the intervention to change the outcome by 1 case, is ceiling(1 / Risk Difference) = ceiling(1/0.3) = ceiling(3.33) = 4

• Risk Ratio (RR), sometimes also called Relative Risks = risk1 / risk2 = 0.5 / 0.2 = 2.5
• Odds Ratio (OR) = odd1 / odd2 = 1.0 / 0.25 = 4

Interconversions between Risk Difference, Risk Ratio, and Odds Ratio requires firstly the decomposition of the index, then recomposition to the new index. To do so, the value of Patient Expected Event Rate (PEER) is required. PEER is essentially the background risk, which is risk of the passive or reference group (risk2 in this page).

The indecis can be decomposed as follows, using the same example

• For Risk Difference :
  risk2 = PEER = 0.2
  risk1 = Risk Difference + risk2 = 0.3 + 0.2 = 0.5
• For Risk Ratio:
  risk2 = PEER = 0.2
  risk1 = Risk Ratio x risk2 = 2.5 x 0.2 = 0.5
• risks can be converted to odds by odd = risk / (1 - risk)
  odd1 = risk1 / (1 - risk1) = 0.5 / (1-0.5) = 0.5 / 0.5 = 1
  odd2 = risk2 / (1 - risk2 = 0.2 / (1-0.2) = 0.2 / 0.8 = 0.25
• For Odds Ratio :
  risk2(PEER) = 0.2
  odd2 = risk2 / (1 - risk2) = 0.2 / (1-0.2) = 0.2 / 0.8 = 0.25
  odd1 = Odds Ratio x odd2 = 4 x 0.25 = 1
  risk1 = odd1 / (1 + odd1) = 1 / (1 + 1) = 1 / 2 = 0.5

References

Journal Evidence-Based Medicine 1997;2:103-4.

Morris JA, Gardner MJ (1988) Calculating confidence intervals for relative risks (Odds ratios) and standardised ratio and rates. BMJ 296 7th May. 1313-1316

Sistrom CL, Garvan CW (2004) Proportions, Odds, and Risk. Radiology 230:12-19

Schechtman E (2002) Odds Ratio, Relative Risk, Absolute Risk Reduction, and the Number Needed to Treat - Which of These Should We Use? Value in Health 5:5:431-436

Input Data

Data Input for Interconversion Between Risks and Odds:    - A single column of data to be converted.       -Each row the data to be converted Data Input for Interconversion Between Risk Difference, Risk Ratio, and Odds Ratio    - Two columns separated by white spaces       -Each row a set of data to be converted       -Column 1 is the value to be converted, Risk Difference, Risk Ratio, or Odds Ratio       -Column 2 is the Patient Expected Event Rate (PEER).       -As the common practice is to designate Group 1 as the group of interest or intervention,         and group 2 as the control, background, or population value group,         PEER is the same as risk or proportion of positives in group 2.

Convert Risks to Odds :
Convert Odds to Risks :
Convert from Risk Difference & PEER :
Convert from Risk Ratio & PEER :
Convert from Odds Ratio & PEER :

The five programs for Odds/Risks conversion in R codes are as follows

Program 1: Conversion of risks to odds

myDat = ("
Risks
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
df$Odds <- df$Risks / (1 - df$Risks)
df      # display risk to odds results

The results are

> df      # display risk to odds results
  Risks      Odds
1   0.1 0.1111111
2   0.2 0.2500000
3   0.3 0.4285714
4   0.4 0.6666667
5   0.5 1.0000000
6   0.6 1.5000000
7   0.7 2.3333333
8   0.8 4.0000000
9   0.9 9.0000000

Program 2: Conversion of Odds to Risks

myDat = ("
Odds
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
df$Risks <- df$Odds / (1 + df$Odds)
df      # display risk to odds results

The results are

> df      # display risk to odds results
  Odds     Risks
1  0.5 0.3333333
2  1.0 0.5000000
3  1.5 0.6000000
4  2.0 0.6666667
5  2.5 0.7142857
6  3.0 0.7500000
7  3.5 0.7777778
8  4.0 0.8000000
9  4.5 0.8181818

Program 3: Convert from Risk Difference (RD) and Patient Expected Event Rate (PEER)

myDat = ("
RD    PEER
-0.1   0.2
0.2    0.2
-0.3   0.5
-0.4   0.5
0.5    0.1
0.6    0.2
0.7    0.1
0.8    0.1
0.9    0.05
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
risk2 <- df$PEER
risk1 <- df$RD + df$PEER
nnt <- ceiling(1 / abs(df$RD))
riskRatio <- risk1 / risk2
odd1 <- risk1 / (1 - risk1)
odd2 <- risk2 / (1 - risk2)
oddsRatio <- odd1 / odd2
df$Risk1 <- risk1
df$Risk2 <- risk2
df$NNT <- nnt
df$RR <- riskRatio
df$Odd1 <- odd1
df$Odd2 <- odd2
df$OR <- oddsRatio
df  # display results of conversion from Risk Difference

The results are as follows

> df  # display results of conversion from Risk Difference
    RD PEER Risk1 Risk2 NNT   RR       Odd1       Odd2          OR
1 -0.1 0.20  0.10  0.20  10  0.5  0.1111111 0.25000000   0.4444444
2  0.2 0.20  0.40  0.20   5  2.0  0.6666667 0.25000000   2.6666667
3 -0.3 0.50  0.20  0.50   4  0.4  0.2500000 1.00000000   0.2500000
4 -0.4 0.50  0.10  0.50   3  0.2  0.1111111 1.00000000   0.1111111
5  0.5 0.10  0.60  0.10   2  6.0  1.5000000 0.11111111  13.5000000
6  0.6 0.20  0.80  0.20   2  4.0  4.0000000 0.25000000  16.0000000
7  0.7 0.10  0.80  0.10   2  8.0  4.0000000 0.11111111  36.0000000
8  0.8 0.10  0.90  0.10   2  9.0  9.0000000 0.11111111  81.0000000
9  0.9 0.05  0.95  0.05   2 19.0 19.0000000 0.05263158 361.0000000

Program 4: Convert from Risk Ratio (Relative Risks, RR) and Patient Expected Event Rate (PEER)

myDat = ("
RR    PEER
0.5   0.2
2.0   0.2
0.4   0.5
0.2   0.5
6.0   0.1
4.0   0.2
8.0   0.1
9.0   0.1
19.0  0.05
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
risk2 <- df$PEER
risk1 <- df$RR * df$PEER
riskDiff <- risk1 - risk2
nnt <- ceiling(1 / abs(riskDiff))
odd1 <- risk1 / (1 - risk1)
odd2 <- risk2 / (1 - risk2)
oddsRatio <- odd1 / odd2
df$Risk1 <- risk1
df$Risk2 <- risk2
df$RD <- riskDiff
df$NNT <- nnt
df$Odd1 <- odd1
df$Odd2 <- odd2
df$OR <- oddsRatio
df # Display results from Risk Ratio conversions

The results are

> df # Display results from Risk Ratio conversions
    RR PEER Risk1 Risk2   RD NNT       Odd1       Odd2          OR
1  0.5 0.20  0.10  0.20 -0.1  10  0.1111111 0.25000000   0.4444444
2  2.0 0.20  0.40  0.20  0.2   5  0.6666667 0.25000000   2.6666667
3  0.4 0.50  0.20  0.50 -0.3   4  0.2500000 1.00000000   0.2500000
4  0.2 0.50  0.10  0.50 -0.4   3  0.1111111 1.00000000   0.1111111
5  6.0 0.10  0.60  0.10  0.5   2  1.5000000 0.11111111  13.5000000
6  4.0 0.20  0.80  0.20  0.6   2  4.0000000 0.25000000  16.0000000
7  8.0 0.10  0.80  0.10  0.7   2  4.0000000 0.11111111  36.0000000
8  9.0 0.10  0.90  0.10  0.8   2  9.0000000 0.11111111  81.0000000
9 19.0 0.05  0.95  0.05  0.9   2 19.0000000 0.05263158 361.0000000

Program 5: Convert from Odds Ratio (OR) and Patient Expected Event Rate (PEER)

myDat = ("
OR    PEER
0.44	0.2
2.67	0.2
0.25	0.5
0.11	0.5
13.5	0.1
16	0.2
36	0.1
81	0.1
361	0.05
         ") 
df <- read.table(textConnection(myDat),header=TRUE)  # conversion to data frame    
risk2 <- df$PEER
odd2 <- risk2 / (1 - risk2)
odd1 <- df$OR * odd2;
risk1 <- odd1 / (1 + odd1)
riskDiff <- risk1 - risk2
riskRatio <- risk1 / risk2
nnt <- ceiling(1 / abs(riskDiff))
df$Odd1 <- odd1
df$Odd2 <- odd2
df$Risk1 <- risk1
df$Risk2 <- risk2
df$RD <- riskDiff
df$NNT <- nnt
df$RR <- riskRatio
df # Display results of conversion from Odds Ratio

The results are

> df # Display results of conversion from Odds Ratio
      OR PEER    Odd1       Odd2     Risk1 Risk2         RD NNT         RR
1   0.44 0.20  0.1100 0.25000000 0.0990991  0.20 -0.1009009  10  0.4954955
2   2.67 0.20  0.6675 0.25000000 0.4002999  0.20  0.2002999   5  2.0014993
3   0.25 0.50  0.2500 1.00000000 0.2000000  0.50 -0.3000000   4  0.4000000
4   0.11 0.50  0.1100 1.00000000 0.0990991  0.50 -0.4009009   3  0.1981982
5  13.50 0.10  1.5000 0.11111111 0.6000000  0.10  0.5000000   2  6.0000000
6  16.00 0.20  4.0000 0.25000000 0.8000000  0.20  0.6000000   2  4.0000000
7  36.00 0.10  4.0000 0.11111111 0.8000000  0.10  0.7000000   2  8.0000000
8  81.00 0.10  9.0000 0.11111111 0.9000000  0.10  0.8000000   2  9.0000000
9 361.00 0.05 19.0000 0.05263158 0.9500000  0.05  0.9000000   2 19.0000000