ODDS, ODDS Ratios and Relative Risk What are ODDS? Odds, like probability, contain the number of times the event occurred in the numerator, BUT the odds denominator contains only the number of times the event did not occur. The difference between odds and probability may be better understood by thinking of the chance of drawing an ace from a deck of 52 cards. The probability of drawing an ace is the number of times an ace will be drawn divided by the total number of cards or 4 of 53 or 1 of 13 (7.69%). The odds, on the other hand, are the number of times an ace will be drawn divided by the number of times it will not be drawn or 4 to 48 or 1:12. The odds in favor of an event or a proposition are expressed as the ratio of a pair of integers or probabilities, such that the first represents the relative likelihood that the event will happen, and the second, the relative likelihood it won't, as in "the odds that a randomly chosen day of the week is a Sunday are one to six," which is sometimes written 1:6, or 1/6. In probability theory and statistics, where the variable "p" is the probability in favor of the event, and the probability against the event is therefore 1-p, "the odds" of the event are the quotient of the two, or . That value may be regarded as the relative likelihood the event will happen, expressed as a fraction (if it is less than 1), or a multiple (if it is equal to or greater than one) of the likelihood that the event will not happen. In the example just given, saying the odds of a Sunday are "one to six" or, less commonly, "one-sixth" means the probability of picking a Sunday randomly is one-sixth the probability of not picking a Sunday. While the mathematical probability of an event has a value in the range from zero to one, "the odds" in favor of that same event lies between zero and infinity. The odds against the event with probability given as "p" are . Consider the Table below: SMOKE? YES NO Grand Total FATHERSMOKE YES 58 212 270 NO 20 152 172 78 364 442 Assume that “SMOKE?” is the dependent variable which we are trying to explain or predict. “FATHERSMOKE” is the independent variable which we are using to explain or predict “SMOKE”. The random probability that someone smokes is 78/442 or 17.65%. The probability that someone smokes, given that their father smokes, is 58/270 or 21.48%. The odds of smoking if your father smokes is 58/212 or .27 to 1. In other words, if your father smokes, the odds that you do not smoke are 3.65:1. Note that when reporting odds, the ratio is implied, but never actually computed in its reduced form. Odds are expressed as “a to b”. PLEASE NOTE THAT ODDS ARE NOT THE SAME AS PROBABILITIES – THE UNITS ARE NOT PERCENTAGES!! What are Odds Ratios? The odds ratio is a way of comparing whether the probability of a certain event is the same for two groups, where the “event” is the dependent variable and the “groups” are the values of the independent variable. In the above example, the event of interest is whether or not someone smokes – the dependent variable. The independent variable of interest is whether or not the father smokes. From the above example, your odds of smoking, if your father smokes is .27:1. Your odds of not smoking, if your father smokes is 3.65:1. Your odds of smoking if your father does not smoke is 20/152 or .13:1. The odds of not smoking if your father does not smoke is 7.6:1. So…the odds ratio of smoking depending on if your father smokes is .27/.13 or 2.08. The interpretation here is that your odds of smoking are twice as high if your father smokes versus if your father does not smoke. An odds ratio of 1 implies that the event is equally likely in both groups. An odds ratio greater than one implies that the event is more likely in the first group (father smokes). An odds ratio less than one implies that the event is less likely in the first group. Typically, the confidence interval for the odds ratio is built to determine if 1 is a possible outcome of the odds ratio – if it is, the likelihood of the event is not significantly different between the two groups. Note that odds ratios are reported as “the odds of x are 1.2 times the odds of y”. So again, there are no percentages here! Odds ratios only make sense if you understand the odds. What is Relative Risk? A more direct measure comparing the probabilities in two groups is the relative risk, which is also known as the risk ratio. The relative risk is simply the ratio of the two conditional probabilities. From the previous example, if the father smokes, the relative risk of smoking is the probability of smoking if the father smokes divided by the probability of smoking if the father does not smoke. Mathematically, this is (58/270)/(20/172) or 21.48%/11.63% or 1.8474. In other words, if your father smokes, you are 1.85 times more likely to smoke than if your father did not smoke. Like the odds ratio, a relative risk equal to 1 implies that the event is equally probable in both groups. A relative risk greater than 1 implies that the event is more likely in the first group. A relative risk less than 1 implies that the event is less likely in the first group. The SAS Code to generate the 2x2 contingency table, the odds ratios, the relative risk is: Proc freq data=smoke; tables Fathersmoke*smoke/chisq relrisk riskdiff; Run; The output will look like this: Table of FATHERSMOKE by smoke FATHERSMOKE(FATHERSMOKE) smoke(SMOKE?) Frequency Percent Row Pct Col Pct 1 1 2 Total 2 Total 58 212 270 13.12 47.96 61.09 21.48 78.52 74.36 58.24 20 152 172 4.52 34.39 38.91 11.63 88.37 25.64 41.76 78 364 442 17.65 82.35 100.00 Frequency Missing = 4 DF Value Prob The Chi Square statistic here Chi-Square 1 7.0195 0.0081 would indicate that there is a Likelihood Ratio Chi- 1 7.3503 0.0067 1 6.3579 0.0117 1 7.0036 0.0081 Statistic Square Continuity Adj. ChiSquare Mantel-Haenszel ChiSquare Phi Coefficient 0.1260 Contingency 0.1250 Coefficient Cramer's V 0.1260 significant relationship between smoking and whether or not your father smokes. What this does not tell us, is whether you are more or less likely to smoke if your father smokes. Column 1 Risk Estimates (Asymptotic) 95% (Exact) 95% Confidence Limits Confidence Limits Risk ASE Row 1 0.2148 0.0250 0.1658 0.2638 0.1673 0.2687 Row 2 0.1163 0.0244 0.0684 0.1642 0.0725 0.1739 Total 0.1765 0.0181 0.1409 0.2120 0.1421 0.2153 Difference 0.0985 0.0350 0.0300 0.1671 Difference is (Row 1 - Row 2) The Column 1 Risk Estimates tell us that, of those whose smoke, 21.48% have fathers who smoke and 11.63% do not smoke. The difference is 9.85%, where the 95% Confidence Interval ranges from 3% to 16.71%. Since 0 is not embedded in this interval, the difference is statistically significant. Column 2 Risk Estimates (Asymptotic) 95% (Exact) 95% Confidence Limits Confidence Limits Risk ASE Row 1 0.7852 0.0250 0.7362 0.8342 0.7313 0.8327 Row 2 0.8837 0.0244 0.8358 0.9316 0.8261 0.9275 Total 0.8235 0.0181 0.7880 0.8591 0.7847 0.8579 -0.0985 0.0350 -0.1671 -0.0300 Difference Difference is (Row 1 - Row 2) The Column 2 Risk Estimates tell us that, of those whose do not smoke, 78.52% have fathers who smoke and 88.37% have fathers who do not smoke. The difference is 9.85%, where the 95% Confidence Interval ranges from 16.71% to 3%. Since 0 is not embedded in this interval, the difference is statistically significant. Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits Case-Control (Odds Ratio) 2.0792 1.2005 3.6012 Cohort (Col1 Risk) 1.8474 1.1536 2.9585 Cohort (Col2 Risk) 0.8885 0.8180 0.9651 The Estimates of the Relative Risk tell us that the odds ratio for smoking is 2.0792 – where your odds of smoking are two times greater if your father smokes. Note that this calculation was generated as – odds of smoking if father smokes/odds of smoking if father does not smoke. Mathematically this is expressed as (58/212)/(20/152). Since 1 is not embedded in the Confidence Interval – which ranges from 1.2005 to 3.6012 – we would conclude that the odds of smoking are statistically higher if the father smokes. The relative risk of smoking if the father smokes, is 1.8474. This calculation was generated as the probability of smoking if father smokes/probability of smoking if father does not smoke. Since 1 is not embedded in the Confidence Interval – which ranges from 1.1536 to 2.9585 – we would conclude that the relative risk of smoking statistically significant, when the father smokes versus when he does not smoke.