**Introduction**

**Calculation of the Odds Ratio**

_{1}” represents the odds of the event of interest for Group 1, and “PG

_{2}” represents the odds of the event of interest for Group 2.

**Standard New**

**Treatment**

**Treatment**

**Event Happens ** a b

**Event does**c d

**not happen**

**Significance Tests for the Odds Ratio**

**Fisher’s Exact**

*p*” is the Fisher’s Exact Probability, “a, b, c, d” represent the counts in the cells, and “n” represents the total sum of the values in all four cells.

**Chi-Square**

^{2}) assumes that the numbers in the cells represent counts and not proportions or averages, and it assumes that the value of the expecteds is 5 or greater in 80% or more of the cells. The value of the probability must be evaluated through a table of Fisher’s Exact Probability values for one degree of freedom to obtain the significance value for the test. Most statistical computer programs such as Stata and SPSS will calculate the Fisher’s Exact and Chi-Square values and provide the significance value of the result. The Chi-Square formula is:

**Likelihood Ratio Chi-Square**

*represents expected values, and “ln” indicates the log to be taken.*

_{i}**Standard Error and Confidence Intervals for the Odds Ratio**

**Examples of Uses of the Odds Ratio**

**Determination of results of a drug study**

*Staphylococcus aureus*(SA). Although the mortality rate for this disease ranges from 25% to 47% (6), let us assume that in the population of interest, White males aged 30 to 60 the mortality rate is 38% with the standard antibiotic treatment of penicillin, methycillin, vancomycin and other antibiotics. However, a new drug has been developed that attacks the bacteria’s ability to protect itself from the human immune system rather than interfering with cell wall development. The question is this: What are the odds of dying with the new drug as opposed to the standard antibiotic therapy protocol? The odds ratio is a way of comparing whether the odds of a certain outcome is the same for two different groups (9).

**(a/b)/(c/d)**= (152/17)/

(262/103) = 8.94/2.41 =

**3.71**

**(a**×

**d)/(b**×

**c)**(this is called the

*cross-product*). The result is the same:

(17 × 248) = (15656/4216) =

**3.71**.

*Table 1. Results from fictional SA endocarditis treatment study*

*more likely*to experience the event (death) than the second group. An OR of less than 1 means that the first group was

*less*likely to experience the event. However, an OR value below 1.00 is not directly interpretable. The degree to which the first group is less likely to experience the event is not the OR result. It is important to put the group expected to have higher odds of the event in the first column. It is not valid to try to determine how much less the first group’s odds of the event was than the second group’s. When the odds of the first group experiencing the event is less than the odds of the second group, one must reverse the two columns so that the second group becomes the first and the first group becomes the second. Then it will be possible to interpret the difference because that reversal will calculate how many more times the second group experienced the event than the first. If we reverse the columns in the example above, the odds ratio is: (5/22)/(45/28) = (0.2273/1.607) = 0.14 and as can be seen, that does not tell us that the new drug group died 0.14 times less than the standard treatment group. In fact, this arrangement produces a result that can only be interpreted as “the odds of the first group experiencing the event is less than the odds of the second group experiencing the event”. The degree to which the first group’s odds are lower than that of the second group is not known.

**Odds ratio in epidemiology studies**

*post hoc*if different groups had different outcomes on a particular measure. For example, Friese

*et al.*(10) conducted a study to find out if there were different probabilities for having a larger number of surgeries for breast cancer for women whose initial diagnostic procedures included a needle biopsy versus for women who did not have an initial breast biopsy. Through use of the odds ratio, they discovered that use of the needle biopsy was associated with a reduced probability of multiple surgeries. The odds ratio table for this study would have the following structure (Table 2):

*Table 2. Table format for epidemiology study*

*et al.*obtained an OR of 0.35 and concluded that use of the needle biopsy as an initial diagnostic test reduced the probability of multiple surgeries by 0.35% for women with breast cancer. (Note: This table should have been changed because an OR value of 0.35 cannot be directly interpreted. All that can be said is that the women who had an initial needle biopsy had fewer surgeries than women who did not have the biopsy.)