Odds ratio

The odds ratio , also called relative chance , odds ratio , odds ratio ( OR for short ), or seldom called cross product ratio , is a statistical measure that says something about the strength of a connection between two characteristics. It is thus a measure of association in which two opportunities are compared with one another. The odds ratio is independent of the marginal distribution .

Calculation based on probabilities

Number of people …
	with risk factor (F)	without risk factor (F ^c )	Marginal shots
ill (K)	a	b	a + b
not ill (K ^c )	c	d	c + d
Marginal shots	a + c	b + d	a + b + c + d

The absolute frequencies (a, b, c, d) or the probabilities (P) can be used to calculate the odds ratio. The odds ratio is calculated from the probabilities (P) as follows:

P (K | F) = a / (a + c) = the conditional probability of falling ill, if the relevant risk factor is present.

P (K | F ^c ) = b / (b + d) = the conditional probability of falling ill if the relevant risk factor is not present.

{\ displaystyle R (F: F ^ {c}) = {R (F) \ over R (F ^ {c})} = {{P (K | F) \ over {1-P (K | F) )}} \ over {P (K | F ^ {c}) \ over {1-P (K | F ^ {c})}}} = {{P (K | F) \ cdot (1-P ( K | F ^ {c}))} \ over {P (K | F ^ {c}) \ cdot (1-P (K | F))}}}

with R (F) = Odds (F) and R (F ^ c) = Odds (F ^ c).

interpretation

An odds ratio of

exactly 1 means that there is no difference in the chances
> 1 means that the chances of the first group are greater
<1 means that the chances of the first group are smaller.

application

The odds ratio is often used in epidemiology and medicine to find out how closely a suspected risk factor is related to a particular disease. The advantage of opportunity ratios compared to risk ratio is that it can be used in all study designs, i.e. in case-control studies as well as in cross-sectional and intervention studies .

Typically, people with a potential risk factor for a disease are compared with people without this risk factor with regard to the occurrence of the same disease. The data obtained are shown in a cross table , which also makes it easy to calculate the odds directly:

Number of people …
	with risk factor	without risk factor
ill	a	b
not sick	c	d

The following then applies:

{\ displaystyle \ mathrm {Odds \ Ratio} = {\ frac {a / b} {c / d}} = {\ frac {a \ cdot d} {b \ cdot c}}}

The odds ratio is a measure of how much greater the chance in the group with a risk factor is to get sick (in the sense of a quota) compared to the chance in the group without a risk factor. The odds ratio assumes values between 0 and ∞ . A value of 1 means an equal chance ratio.

An example with fictitious data

Suppose you want to investigate the relationship between the occurrence of heart attacks and smoking. 10,000 patients are observed to determine whether they smoke or not and whether they have ever had a heart attack . The result is the following crosstab:

Number of people …
	they smoke	who don't smoke
having heart attack	130	70
without a heart attack	1870	7930

So out of 2,000 people who smoke, 130 have had a heart attack. It results in the odds ratio

{\ displaystyle \ mathrm {Odds \ Ratio} = {\ frac {130 \ cdot 7930} {70 \ cdot 1870}} \ approx 7 {,} 88}

This means that the chance of having a heart attack is almost eight times higher among smokers than among non-smokers. At this point, however, the mathematical difference between opportunity and risk must be pointed out. To make it easier to interpret, the relative risk (see below) should be given instead of the odds ratio if possible.

Difference to the relative risk

Unlike the relative risk , the odds ratio relates to odds and not to probabilities.

The following example should explain the difference between the odds ratio and relative risk:

depression
gender	Yes	No
Female	40	143
male	10	101

Depression with the categories “yes” and “no” is the risk variable, gender with the categories “female” and “male” is the independent (causative) variable.

The prevalence among women is ${\ displaystyle 40: (40 + 143) = 0 {,} 219 \,}$

The prevalence among men is ${\ displaystyle 10: (10 + 101) = 0 {,} 09 \,}$

The relative risk is the quotient of the prevalences ${\ displaystyle 0 {,} 219: 0 {,} 090 = 2 {,} 43 \,}$

The odds ratio, however, is calculated as follows:

For women, the "quota" is ${\ displaystyle 40: 143 = 1: 3 {,} 58 = 0 {,} 280 \,}$

For men, the "quota" is ${\ displaystyle 10: 101 = 1: 10 {,} 1 = 0 {,} 099 \,}$

The odds ratio is the quotient from the "odds" . ${\ displaystyle 0 {,} 280: 0 {,} 099 = 2 {,} 83 \,}$

Or more simply: . ${\ displaystyle (40 \ cdot 101) :( 143 \ cdot 10) = 2 {,} 83 \,}$

Association measures according to Yule

Other dimensions are Yules Q and Yules Y (1912), which George Udny Yule published around 1900.

The proposed association measure (Yules ) can be represented as a transformation of the odds ratio ( ), through which the odds ratio is normalized to the interval between and , whereby if both variables are statistically independent of one another. ${\ displaystyle Q}$ ${\ displaystyle Q}$ ${\ displaystyle Q = (OR-1) / (OR + 1)}$ ${\ displaystyle -1}$ ${\ displaystyle +1}$ ${\ displaystyle Q = 0}$

Yules Y is calculated as follows:

{\ displaystyle Y = {\ sqrt {OR}} - 1 / {\ sqrt {OR}} + 1}

.

Web links

Individual evidence

^ Ludwig Fahrmeir , Rita artist, Iris Pigeot , and Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 114.
^ Heinz Holling, Bernhard Schmitz: Handbook Statistics, Methods and Evaluation . Hogrefe Verlag, 2010, ISBN 978-3-8409-1848-3 , pp. 295 ( limited preview in Google Book search).
↑ Chapter Cross product ratio (odds ratio) in the glossary for data collection and statistical analysis (accessed on January 6, 2008)
^ G. Arminger, Clifford C. Clogg, ME Sobel: Handbook of Statistical Modeling for the Social and Behavioral Sciences . Springer Science & Business Media, 2013, ISBN 978-1-4899-1292-3 , pp. 260 ( limited preview in Google Book Search).
↑ Martin Gross: Classes, Shifts, Mobility: An Introduction . Springer-Verlag, 2014, ISBN 978-3-531-19943-6 , pp. 137 ( limited preview in Google Book search).
^ Anthony J. Viera: Odds ratios and risk ratios: what's the difference and why does it matter? In: Southern Medical Journal . tape 101 , no. 7 , July 2008, ISSN 1541-8243 , p. 730-734 , doi : 10.1097 / SMJ.0b013e31817a7ee4 , PMID 18580722 .
^ Achim Bühl, Peter Zöfel: SPSS 12 . Pearson studies, Munich 2005
↑ Joachim Hartung, Bärbel Elpelt, Karl-Heinz Klösener: Statistics: Teaching and manual applied statistics; with numerous, fully calculated examples . Oldenbourg Verlag, 2005, ISBN 978-3-486-57890-4 , pp. 444 ( limited preview in Google Book search).
↑ Elmar Klemm: Introduction to Statistics: For the Social Sciences . Springer-Verlag, 2013, ISBN 978-3-322-83376-1 , pp. 276 ( limited preview in Google Book search).
^ Stephan Hagemann: Measures for association analysis in data mining: foundation, analysis and test . Diplomica Verlag, 2008, ISBN 978-3-8366-5718-1 , p. 25 ( limited preview in Google Book search).
↑ Elmar Klemm: Introduction to Statistics: For the Social Sciences . Springer-Verlag, 2013, ISBN 978-3-322-83376-1 , pp. 277 ( limited preview in Google Book search).
^ Franz Petermann, Michael Eid : Handbuch der Psychologische Diagnostik . Hogrefe Verlag, 2006, ISBN 978-3-8409-1911-4 , pp. 372 ( limited preview in Google Book search).

[1] Ludwig Fahrmeir , Rita artist, Iris Pigeot , and Gerhard Tutz : Statistics. The way to data analysis. 8., revised. and additional edition. Springer Spectrum, Berlin / Heidelberg 2016, ISBN 978-3-662-50371-3 , p. 114.

[2] Heinz Holling, Bernhard Schmitz: Handbook Statistics, Methods and Evaluation . Hogrefe Verlag, 2010, ISBN 978-3-8409-1848-3 , pp. 295 ( limited preview in Google Book search).

[3] Chapter Cross product ratio (odds ratio) in the glossary for data collection and statistical analysis (accessed on January 6, 2008)

[4] G. Arminger, Clifford C. Clogg, ME Sobel: Handbook of Statistical Modeling for the Social and Behavioral Sciences . Springer Science & Business Media, 2013, ISBN 978-1-4899-1292-3 , pp. 260 ( limited preview in Google Book Search).

[5] Martin Gross: Classes, Shifts, Mobility: An Introduction . Springer-Verlag, 2014, ISBN 978-3-531-19943-6 , pp. 137 ( limited preview in Google Book search).

[6] Anthony J. Viera: Odds ratios and risk ratios: what's the difference and why does it matter? In: Southern Medical Journal . tape 101 , no. 7 , July 2008, ISSN 1541-8243 , p. 730-734 , doi : 10.1097 / SMJ.0b013e31817a7ee4 , PMID 18580722 .

[7] Achim Bühl, Peter Zöfel: SPSS 12 . Pearson studies, Munich 2005

[8] Joachim Hartung, Bärbel Elpelt, Karl-Heinz Klösener: Statistics: Teaching and manual applied statistics; with numerous, fully calculated examples . Oldenbourg Verlag, 2005, ISBN 978-3-486-57890-4 , pp. 444 ( limited preview in Google Book search).

[9] Elmar Klemm: Introduction to Statistics: For the Social Sciences . Springer-Verlag, 2013, ISBN 978-3-322-83376-1 , pp. 276 ( limited preview in Google Book search).

[10] Stephan Hagemann: Measures for association analysis in data mining: foundation, analysis and test . Diplomica Verlag, 2008, ISBN 978-3-8366-5718-1 , p. 25 ( limited preview in Google Book search).

[11] Elmar Klemm: Introduction to Statistics: For the Social Sciences . Springer-Verlag, 2013, ISBN 978-3-322-83376-1 , pp. 277 ( limited preview in Google Book search).

[12] Franz Petermann, Michael Eid : Handbuch der Psychologische Diagnostik . Hogrefe Verlag, 2006, ISBN 978-3-8409-1911-4 , pp. 372 ( limited preview in Google Book search).