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Risk ratios and odds ratios—what are they?
ARTICLE IN PRESS Midwifery (2004) 20, 169–170
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Risk ratios and odds ratiosFwhat are they? Malcolm Campbell, BA (Hons) (Mathematics), MSc (Statistics), PhD (Statistics) (Lecturer in Statistics) School of Nursing, Midwifery & Health Visiting, University of Manchester, Coupland III Building, Oxford Road, Manchester M13 9PL, UK
Downe et al. (2004) use relative risk ratios and odds ratios. What are they?
How do you interpret probabilities, odds, relative risk and relative odds?
A risk ratio (sometimes called relative risk, both abbreviated to RR) and an odds ratio (OR) are ways of expressing a comparison between two proportions. For example, in this paper, there are two randomised groups in the trial, women in the lateral position and women in the supported sitting position in the passive second stage of labour. The authors wanted to compare the proportions of instrumental births, episiotomies and perineal suturing in the two groups to see if there was any effect due to maternal position. A risk ratio gives the relative probability or chance of something happening in one group compared with the other, while an odds ratio gives the relative odds. Do you know what the difference is between probability and odds?
Probability takes values on a scale from 0 to 1, where 0 is impossible, 0.5 is as likely as not, and 1 is certain. The others take values from 0 to positive infinity, where 1 has a special meaning. An odds of 1 means that the event has the same chance of occurring as not occurring (the probability of the event is 0.5), while a risk ratio or odds ratio of 1 means that in the two groups being compared, the event has the same chance of occurring. Staying with our example of the event as ‘having an instrumental delivery’, a risk ratio or odds ratio of 1 for the lateral group compared with the sitting group would have meant that instrumental delivery was equally likely in the two groups (or equivalently the groups would have the same proportion of instrumental deliveries). You could think of risk ratios and odds ratios as measures of clinical effect, a value of 1 indicating no intervention effect or no difference between the groups.
Is probability an uncertainty and are odds what you see at a bookmaker’s? Sort of. Probability or risk can be defined as the number of times some event happens divided by the total number of times it either does or does not happen. Odds are the number of times the event happens divided by the number of times it does not happen. If the event is ‘having an instrumental delivery’, the probability of an instrumental delivery is the number of instrumental deliveries divided by the total number of deliveries, while the odds of an instrumental delivery are the number of instrumental deliveries divided by the number of noninstrumental deliveries. Most people find probability easier to understand but gamblers may think more naturally in odds. E-mail address: [email protected] (M. Campbell).
So what about values in this study then? Have a look at Table 3. In the lateral group, there were 16 instrumental and 33 spontaneous births; in the sitting group, there were 30 and 28, respectively. So the probability or risk of an instrumental birth in the lateral group is 16/49, while that in the sitting group is 30/58. The risk ratio of instrumental birth for the sitting group compared with the lateral group as a reference group is the ratio of the two probabilities, 30/58 divided by 16/49, which is 1.58. This means that women in the sitting group are more than one and a half as likely as those in the lateral group to have an instrumental delivery. The odds of an instrumental birth for the lateral group are 16/33, while those for the sitting
0266-6138/$ - see front matter & 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.midw.2004.03.004
ARTICLE IN PRESS 170
group are 30/28. The odds ratio for the sitting group compared with the lateral group is 30/28 divided by 16/33, or 2.20. This means that for women in the sitting group compared with women in the lateral group, the odds of an instrumental birth are more than twice as high.
Just a minuteFthe paper gives the risk ratio of instrumental birth for the lateral group compared with the sitting group as 0.63. Where did that come from? Yes, it might be a little confusing as they are using the sitting group as the reference group, but it has the same meaning. This risk ratio is the probability of instrumental birth in the lateral group, 16/49, divided by the probability in the sitting group, 30/ 58, which is 0.63 (and this is just 1 over the risk ratio for sitting compared with lateral). Women in the lateral group are just under two-thirds as likely as women in the sitting group to have an instrumental delivery.
I think I can grasp the meaning of a risk ratio. Why do we need odds ratios as well? Although odds ratios may seem harder to interpret, they can be used in more complicated analyses to allow for the behaviour of other variables. As the authors point out, while the position of the women may affect whether a birth is instrumental, so too might factors such as position of the fetal head at the diagnosis of full dilation, the use of oxytocin during labour, or maternal BMI. If these factors differ between the groups, they may influence the observed probabilities, risk ratios and odds ratios
M. Campbell
associated with instrumental birth. So the authors fitted logistic regression models to predict whether a birth was instrumental or spontaneous from such confounding variables together with position in the passive second stage of labour. These models allowed the authors to estimate an odds ratio for position adjusted for the confounding variables. They found that the odds ratio of 2.2 for sitting compared with lateral rose to 2.3 when adjusted for the other variables.
Where can I find out more about risk ratios and odds ratios? Simon’s website (STATSFSTeve’s Attempt to Teach Statistics) shows how they are calculated and interpreted with examples at http://www.childrensmercy.org/stats/definitions/ or.htm (accessed 5 February 2004), while Kirkwood and Sterne (2003, pp. 148–164) give a highly detailed and readable description.
Acknowledgements I would like to thank an anonymous statistical reviewer for helpful comments.
References Downe, S., Gerret, D., Renfrew, M., 2004. A prospective trial on the effect of position in the passive second stage of labour on birth outcome in nulliparous women using epidural analgesia. Midwifery 20 (2), 157–168. Kirkwood, B.R., Sterne, J.A.C., 2003. Essential Medical Statistics 2nd Edition. Blackwell Science, Oxford.
www.elsevier.com/locate/midw
Risk ratios and odds ratiosFwhat are they? Malcolm Campbell, BA (Hons) (Mathematics), MSc (Statistics), PhD (Statistics) (Lecturer in Statistics) School of Nursing, Midwifery & Health Visiting, University of Manchester, Coupland III Building, Oxford Road, Manchester M13 9PL, UK
Downe et al. (2004) use relative risk ratios and odds ratios. What are they?
How do you interpret probabilities, odds, relative risk and relative odds?
A risk ratio (sometimes called relative risk, both abbreviated to RR) and an odds ratio (OR) are ways of expressing a comparison between two proportions. For example, in this paper, there are two randomised groups in the trial, women in the lateral position and women in the supported sitting position in the passive second stage of labour. The authors wanted to compare the proportions of instrumental births, episiotomies and perineal suturing in the two groups to see if there was any effect due to maternal position. A risk ratio gives the relative probability or chance of something happening in one group compared with the other, while an odds ratio gives the relative odds. Do you know what the difference is between probability and odds?
Probability takes values on a scale from 0 to 1, where 0 is impossible, 0.5 is as likely as not, and 1 is certain. The others take values from 0 to positive infinity, where 1 has a special meaning. An odds of 1 means that the event has the same chance of occurring as not occurring (the probability of the event is 0.5), while a risk ratio or odds ratio of 1 means that in the two groups being compared, the event has the same chance of occurring. Staying with our example of the event as ‘having an instrumental delivery’, a risk ratio or odds ratio of 1 for the lateral group compared with the sitting group would have meant that instrumental delivery was equally likely in the two groups (or equivalently the groups would have the same proportion of instrumental deliveries). You could think of risk ratios and odds ratios as measures of clinical effect, a value of 1 indicating no intervention effect or no difference between the groups.
Is probability an uncertainty and are odds what you see at a bookmaker’s? Sort of. Probability or risk can be defined as the number of times some event happens divided by the total number of times it either does or does not happen. Odds are the number of times the event happens divided by the number of times it does not happen. If the event is ‘having an instrumental delivery’, the probability of an instrumental delivery is the number of instrumental deliveries divided by the total number of deliveries, while the odds of an instrumental delivery are the number of instrumental deliveries divided by the number of noninstrumental deliveries. Most people find probability easier to understand but gamblers may think more naturally in odds. E-mail address: [email protected] (M. Campbell).
So what about values in this study then? Have a look at Table 3. In the lateral group, there were 16 instrumental and 33 spontaneous births; in the sitting group, there were 30 and 28, respectively. So the probability or risk of an instrumental birth in the lateral group is 16/49, while that in the sitting group is 30/58. The risk ratio of instrumental birth for the sitting group compared with the lateral group as a reference group is the ratio of the two probabilities, 30/58 divided by 16/49, which is 1.58. This means that women in the sitting group are more than one and a half as likely as those in the lateral group to have an instrumental delivery. The odds of an instrumental birth for the lateral group are 16/33, while those for the sitting
0266-6138/$ - see front matter & 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.midw.2004.03.004
ARTICLE IN PRESS 170
group are 30/28. The odds ratio for the sitting group compared with the lateral group is 30/28 divided by 16/33, or 2.20. This means that for women in the sitting group compared with women in the lateral group, the odds of an instrumental birth are more than twice as high.
Just a minuteFthe paper gives the risk ratio of instrumental birth for the lateral group compared with the sitting group as 0.63. Where did that come from? Yes, it might be a little confusing as they are using the sitting group as the reference group, but it has the same meaning. This risk ratio is the probability of instrumental birth in the lateral group, 16/49, divided by the probability in the sitting group, 30/ 58, which is 0.63 (and this is just 1 over the risk ratio for sitting compared with lateral). Women in the lateral group are just under two-thirds as likely as women in the sitting group to have an instrumental delivery.
I think I can grasp the meaning of a risk ratio. Why do we need odds ratios as well? Although odds ratios may seem harder to interpret, they can be used in more complicated analyses to allow for the behaviour of other variables. As the authors point out, while the position of the women may affect whether a birth is instrumental, so too might factors such as position of the fetal head at the diagnosis of full dilation, the use of oxytocin during labour, or maternal BMI. If these factors differ between the groups, they may influence the observed probabilities, risk ratios and odds ratios
M. Campbell
associated with instrumental birth. So the authors fitted logistic regression models to predict whether a birth was instrumental or spontaneous from such confounding variables together with position in the passive second stage of labour. These models allowed the authors to estimate an odds ratio for position adjusted for the confounding variables. They found that the odds ratio of 2.2 for sitting compared with lateral rose to 2.3 when adjusted for the other variables.
Where can I find out more about risk ratios and odds ratios? Simon’s website (STATSFSTeve’s Attempt to Teach Statistics) shows how they are calculated and interpreted with examples at http://www.childrensmercy.org/stats/definitions/ or.htm (accessed 5 February 2004), while Kirkwood and Sterne (2003, pp. 148–164) give a highly detailed and readable description.
Acknowledgements I would like to thank an anonymous statistical reviewer for helpful comments.
References Downe, S., Gerret, D., Renfrew, M., 2004. A prospective trial on the effect of position in the passive second stage of labour on birth outcome in nulliparous women using epidural analgesia. Midwifery 20 (2), 157–168. Kirkwood, B.R., Sterne, J.A.C., 2003. Essential Medical Statistics 2nd Edition. Blackwell Science, Oxford.