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StatsToDo : Odd and Risk Interconversion Explained
 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 StatsToDo : 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 StatsToDo 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))