## Odds Ratio and Yule's Q

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### Introduction

The odds ratio is an important option for testing and quantifying the association between two raters making dichotomous ratings. It should probably be used more often with agreement data than it currently is.

The odds ratio can be understood with reference to a 2×2 crossclassification table:

Crossclassification frequencies for binary ratings by two raters
Rater 1 Rater 2
+ -
+ a b a + b
- c d c + d
a + c b + d Total
By definition, the odds ratio, OR, is
```             [a/(a+b)] / [b/(a+b)]
OR  =  -----------------------,    (1)
[c/(c+d)] / [d/(c+d)]
```
but this reduces to
```              a/b
OR  =   -----,                     (2)
c/d
```
or, as OR is usually calculated,
```              ad
OR  =   ----.                      (3)
bc
```
The last equation shows that OR is equal to the simple crossproduct ratio of a 2×2 table.

#### Intuitive explanation

The concept of "odds" is familiar from gambling. For instance, one might say the odds of a particular horse winning a race are "3 to 1"; this means the probability of the horse winning is 3 times the probability of not winning.

In Equation (2), both the numerator and denominator are odds. The numerator, a/b, gives the odds of a positive versus negative rating by Rater 2 given that Rater 1's rating is positive. The denominator, c/d, gives the odds of a positive versus negative rating by Rater 2 given that Rater 1's rating is negative.

OR is the ratio of these two odds--hence its name, the odds ratio. It indicates how much the odds of Rater 2 making a positive rating increase for cases where Rater 1 makes a positive rating.

This alone would make the odds ratio a potentially useful way to assess association between the ratings of two raters. However, it has some other appealing features as well. Note that:

```            a/b       a/c       d/b       d/c       ad
OR =  -----  =  -----  =  -----  =  -----  =  ----.
c/d       b/d       c/a       b/a       bc
```

From this we see that the odds ratio can be interpreted in various ways. Generally, it shows the relative increase in the odds of one rater making a given rating, given that the other rater made the same rating--the value is invariant regardless of whether one is concerned with a positive or negative rating, or which rater is the reference and which the comparison.

The odds ratio can be interpreted as a measure of the magnitude of association between the two raters. The concept of an odds ratio is also familiar from other statistical methods (e.g., logistic regression).

#### Yule's Q

OR can be transformed to a -1 to 1 scale by converting it to Yule's Q (or a slightly different statistic, Yule's Y.)

For example, Yule's Q is

```            OR - 1
Q  =  --------.
OR + 1

```

#### Log-odds ratio

It is often more convenient to work with the log of the odds ratio than with the odds ratio itself. The formula for the standard error of log(OR) is very simple:

slog(OR) = square-root(1/a  +  1/b  +  1/c  +  1/d).
Knowing this standard error, one can easily test the significance of log(OR) and/or construct confidence intervals. The former is accomplished by calculating:
z = log(OR)/slog(OR).
and referring to a table of the cumulative distribution of the standard normal curve to determine the p-value associated with z.

Confidence limits are calculated as:

log(OR) ± zL × slog(OR).
where zL is the z value defining the appropriate confidence limits, e.g., zL = 1.645 or 1.96 for a two-sided 90% or 95% confidence interval, respectively.

Confidence limits for OR may be calculated as:

exp[log(OR) ± zL × slog(OR)].

Alternatives are to estimate confidence intervals by the nonparametric bootstrap (for description, see the Raw agreement indices page) or to construct exact confidence intervals by considering all possible distributions of the cases in a 2×2 table.

Once one has used log OR or OR to assess association between raters, one may then also perform a test of marginal homogeneity, such as the McNemar test.

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### Pros and Cons: the Odds Ratio

#### Pros

 The odds ratio is very easily calculated. Software for its calculation is readily available, e.g., SAS PROC FREQ and SPSS CROSSTABS. It is a natural, intuitively acceptable way to express magnitude of association. The odds ratio is linked to other statistical methods.

#### Cons

 If underlying trait is continuous, the value of OR depends on the level of each rater's threshold for a positive rating. That is not ideal, as it implies the basic association between raters changes if their thresholds change. Under certain distributional assumptions (so-called "constant association" models), this problem can be eliminated, but the assumptions introduce extra complexity. While the odds ratio can be generalized to ordered category data, this again introduces new assumptions and complexity. (See the Loglinear, association, and quasi-symmetry models page).

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### Extensions and alternatives

#### Extensions

• More than two categories. In an N×N table (where N > 2), one might collapse the table into various 2×2 tables and calculate log(OR) or OR for each. That is, for each rating category k = 1, ..., N, one would construct the 2×2 table for the crossclassification of Level k vs. all other levels for Raters 1 and 2, and calculate log OR or OR. This assesses the association between raters with respect to the Level k vs. not-Level k distinction.

This method is probably more appropriate for nominal ratings than for ordered-category ratings. In either case, one might consider instead using Loglinear, association, or quasi-symmetry models.

• Multiple raters. For more than two raters, a possibility is to calculate log(OR) or OR for all pairs of raters. One might then report, say, the average value and range of values across all rater pairs.

#### Alternatives

Given data by two raters, the following alternatives to the odds ratio may be considered.

• In a 2×2 table, there is a close relationship between the odds ratio and loglinear modeling. The latter can be used to assess both association and marginal homogeneity.

• Cook and Farewell (1995) presented a model that considers formal decomposition of a 2×2 table into independent components which reflect (1) the odds ratio and (2) marginal homogeneity.

• The tetrachoric and polychoric correlations are alternatives when one may assume that ratings are based on a latent continuous trait which is normally distributed. With more than two rating categories, extensions of the polychoric correlation are available with more flexible distributional assumptions.

• Association and quasi-symmetry models can be used for N×N tables, where ratings are nominal or ordered-categorical. These methods are related to the odds ratio.

• When there are more than two raters, latent trait and latent class models can be used. A particular type of latent trait model called the Rasch model is related to the odds ratio.

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### References

Either of the books by Agresti are excellent starting points.

Agresti A. Categorical data analysis. New York: Wiley, 1990.

Agresti A. An introduction to categorical data analysis. New York: Wiley, 1996.

Bishop YMM, Fienberg SE, Holland PW. Discrete nultivariate analysis: theory and practice. Cambridge, Massachusetts: MIT Press, 1975

Cook RJ, Farewell VT. Conditional inference for subject-specific and marginal agreement: two families of agreement measures. Canadian Journal of Statistics, 1995, 23, 333-344.

Fleiss JL. Statistical methods for rates and proportions, 2nd Ed. New York: John Wiley, 1981.

Khamis H. Association, measures of. In Armitage P, Colton T (eds.), The Encyclopedia of Biostatistics, Vol. 1, pp. 202-208. New York: Wiley, 1998.

Somes GW, O'Brien, KF. Odds ratio estimators. In Kotz L, Johnson NL (eds.), Encyclopedia of statistical sciences, Vol. 6, pp. 407-410. New York: Wiley, 1988.

Sprott DA, Vogel-Sprott MD. The use of the log-odds ratio to assess the reliability of dichotomous questionnaire data. Applied Psychological Measurement, 1987, 11, 307-316.

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Last updated: 21 August 2006 (added counter)

(c) 2006 John Uebersax PhD    email