StatTools : Odd and Risk Interconversion Explained

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Related Links:
Odd and Risk Interconversion Program Page
Unpaired Proportions Explained Page

Explanation References
Risk and proportions are common parameters in statistics, and the most common methods of comparing two groups of binary measurements are Risk Differences, Risk Ratios, or Odds Ratios.

On many occasions, there is a need to convert risks and odds, sometimes as intermediary steps during Bayesian Probability analysis, in prediction mathematics, and when carrying out meta-analysis harmonising results produced using risks or odds in different calculations.

This page supports the programs provided by the Odd and Risk Interconversion Program Page , and provides the definitions and formulas for the calculations.

Definitions

Group 1 and Group 2 are terms designating where the data come from. Conventionally, and as a default in StatTools :

  • Group 1 is usually the group of interest, the index group, the group receiving the intervention, the new treatment, the new drug.
  • Group 2 is usually the background group, the control group, representing the population or expected measurements

Numbers Positive (NPos) is the number of subjects which are positive in the attribute of interest

Numbers Negative (NNeg) is the number of subjects which are negative in the attribute of interest

Risk is the proportion of numbers positive Risk = NPos / (NPos + NNeg)

  • Risk1 is risk calculated from data of group 1
  • Risk2 is risk calculated from data of group 2
  • StatTools follows common practice, designates group 2 is the population of background group, the Patient Expected Event Rate (PEER) is the same as Risk 2
  • Risk Difference is the difference between the two groups. Risk Difference = Risk 1 - Risk 2
  • The number needed to treat (NNT) is the expected number of cases that needs to be treated with the intervention to result in a single case with a different outcome. NNT = ceiling(1 / absolute(Risk Difference))
  • Risk Ratio is the ratio of the risks from the two groups. Risk Ratio = Risk 1 / Risk 2

Odd is the ratio of numbers positive and negative Odd = NPos / NNeg

  • Odd1 is odd calculated from data of group 1
  • Odd2 is odd calculated from data of group 2
  • Odds Ratio is the ratio of the odds from the two groups. Odds Ratio = Odd1 / Odd2

Calculations Used in conversion

Odd to Risk

  • Odds = NPos / NNeg; therefore
  • NPos = NNeg x Odds
  • Risk = NPos / (NPos + NNeg)
    = NNeg x Odds / (NNeg x Odds + NNeg)
    = NNeg x Odds / (NNeg x (Odds + 1))
    = Odds / (1 + Odds)
  • Risk = Odd / (1 + Odd)

Risk to Odd

  • Risk = Odds / (1 + Odds); therefore
  • Odds = Risk (1 + Odds) = Risk + Risk x Odds
  • Odds - Risk x Odds = Risk
  • Odds(1 - Risk) = Risk
  • Odds = Risk / ( 1 - Risk)

Deconstruct Risk Difference and PEER to Risk1 and Risk2

  • Risk2 = PEER
  • Risk1 = Risk Difference + Risk2

Deconstruct Risk Ratio and PEER to Risk1 and Risk2

  • Risk2 = PEER
  • Risk1 = Risk Ratio x Risk2

Deconstruct Odds Ratio and PEER to Risk1 and Risk2

  • Risk2 = PEER
  • Odd2 = Risk2 / (1 - Risk2)
  • Odd1 = Odds Ratio x Odd2
  • Risk1 = Odd1 / (1 + Odd1)

Reconstruct all Effects from Risk1 and Risk2

  • Risk Difference = Risk1 - Risk2
  • Number Needed to Treat = ceiling(1 / absolute(Risk Difference))
  • Risk Ratio = Risk1 / Risk2
  • Odds Ratio = Odd1 / Odd2 = (Risk1/(1 - Risk1)) / (risk2 / (1-Risk2))