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Determining the Odds Ratio and Confidence Interval for Binary Outcomes

David Zurakowski, PhD and James Di Canzio, MS

Departments of Orthopaedic Surgery and Biostatistics, The Children's Hospital


Nearly half of all the scientific manuscripts that are published in leading medical journals have statistical errors or omissions, often serious enough to call the authors' conclusions into question. Although there have been many textbooks written on the subject of statistics, they are usually focused on formulas and derivations requiring a statistical background. Recently, the American College of Physicians published guidelines (1) on presenting statistical data in an effort to improve both clarity and credibility in medical research. The authors of these guidelines begin their book by describing some critical differences between clinical and statistical significance. We are reminded of several important distinctions:

Statistics Medicine For statistical methods to be useful they must adhere to certain standards for reporting results and interpreting data. The book published by the American College of Physicians provides an excellent guide to statistical terminology and a checklist for investigators who are interested in reporting clin-ical trials. Numerous statistical techniques are covered, including descriptive statistics for categorical and continuous data, P-values, confidence intervals, multiple comparisons, association and correlation, linear and logistic regression, analysis of variance (ANOVA), decision making analysis, sensi-tivity, specificity, receiver operating characteristic (ROC) curves, meta-analysis, economic evaluations, cost-outcome ratios, clinical practice guidelines, and survival analysis.
Significance reflects influence chance has on the outcome Significance reflects biological value of the outcome
Derived from groups of individuals Practiced on specific individuals
Conclusions require adequate data Decisions often made with insufficient data
Answers are probabilistic Requires committed decisions in individual cases
Always requires measurement Sometimes requires intuition

Dr. Zurakowski is Principal Biostatistician for the Department of Orthopaedic Surgery at The Children's Hospital

James Di Canzio
is Biostatistician in the Clinical Core Program Office at The Children's Hospital

Please address correspondence to:
David Zurakowski, PhD
Department of Orthopaedic Surgery
The Children's Hospital
300 Longwood Ave.
Boston, MA 02115
Zurakowski@a1.tch.harvard.edu

One statistical technique commonly used in orthopaedic clinical research is logistic regression analysis. During the 1940's at the Mayo Clinic in Rochester, MN, Berkson found a logistic relationship between dosage of a drug and death of animals. In 1953, he introduced the logistic equation into bioassay. (2) Analysis of data from the Framingham Heart Study (3) in the 1960's brought into prominence the development of the most popular statistical method in modern epidemiology, the logistic regression model.

The logistic regression model is currently used in orthopaedic research when the outcome of interest is a binary ("yes/no") event. In fact, when analyzing the effect of a predictor variable (e.g., age, gender, anatomical location, stage of disease, extent of tear) on mortality or need for surgery or reoperation, the logistic equation is an appropriate tool. In practice, the outcome variable has the values 0 (no event) or 1 (event), and the explanatory or predictor variables are potential risk factors. Logit units are related to probability (P) by the logistic relation, z = ln[P/(1-P)], where ln denotes the natural logarithm and z is a function of the ratio of P, the probability of the event, and 1-P, the complementary probability that the event does not occur. Orthopaedic researchers often wish to analyze the relationship between a predictor variable and an outcome variable, each of which has two possibilities. Logistic regression can be performed to evaluate several predictor variables simultaneously to determine which variables are independently associated with the outcome.

Suppose, for example, that history of fever is examined to determine whether it differentiates between septic arthritis and transient synovitis among children who present with an acutely irritable hip. Data from a recent study by Kocher et al. (4) can be summarized as follows:

For those patients with a history of fever, the odds of hav-ing septic arthritis are 67/7 = 9.57, whereas the odds for those with no history of fever are 15/79 = 0.19, The odds ratio (OR) is the ratio of these two odds: 9.57/0.19 = 50.4. This means that patients with a history of fever are approximately 50 times more likely to have septic arthritis than those without a history of fever. The odds ratio is sometimes called the cross-product ratio because it can be calculated by multiplying the counts in the diagonal cells and dividing as follows:
  Septic Arthritis Transient Synovitis Total Patients
History of Fever 67 (a) 7 (b) 74
No History of Fever 15 (c) 79 (d) 94
Total Patients 82 86 168
OR = ad/bc = (67 x 79) / (7 x 15) = 5293/105 = 50.4

The odds ratio is only a point estimate. The precision of this estimate can be described with a confidence interval (CI), which describes the statistical significance of the association between two variables within a specific range. The width of the CI reflects the amount of variability inherent in the OR. There is a tradeoff between precision and confidence. Wider confidence intervals provide great certainty but are less precise. Narrower intervals are more specific but less certain that the truth is within the confidence interval. The most common CI in medicine is 95%.

Several methods are commonly used to construct confi-dence intervals around the odds ratio. A simple method for constructing confidence intervals (5) , based on a Taylor series expansion, can be expressed as follows: CI = (ad/bc) exp( z sqrt (1/a + 1/b + 1/c + 1/d)), where z is the value of the standard normal distribution with the specific level of confidence, and exp is the base of the natural logarithm. The 95% CI would be derived as follows:

95% CI = (ad/bc) exp( 1.96 sqrt (1/a + 1/b + 1/c = 1/d))
= [(67 x 79) / (7 x 15)] exp ( 1.96 sqrt (1/67 + 1/7 + 1/15 + 1/79))
= (50.4) exp( 0.95)
Lower limit = (50.4) exp(-0.95) = (50.4) (0.39) = 19.7
Upper limit = (50.4) exp(+0.95) = (50.4) (2.58) = 130.0

Therefore, among children presenting with an acutely irritable hip, those with a history of fever are on average 50 times more likely to have septic arthritis compared to those with no history of fever. Based on a 95% CI, the lower limit of the OR is approximately 20 and the upper limit is 130. These limits provide the ballpark for the OR. Since the 95% CI does not include 1.0, the results are significant at the 0.05 level (i.e., P <0.05).

Odds ratios and confidence intervals can be quickly and easily obtained by surgeons with a small hand calculator. These tools are useful in clinical and laboratory research, and can also be used to determine confidence intervals (if not provided) while reading the literature. Proper understanding and utilization of the fundamental statistical principles and calculations presented here will allow for more reliable analysis, interpretation, and communication of clinical information by all physicians.


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References
1. Lang TA, Secic M. How to report statistics in medicine. Philadelphia, PA: American College of Physicians, 1997.
2. Berkson J. A statistically precise and relatively simple method of estimating bioassay with quantal response, based on the logistic function. J Am Stat Assoc 1953; 48:565-599.
3. Truett J, Cornfield J, Kannel W. A multivariate analysis of the risk of coronary heart disease in Framingham. J Chronic Dis 1967; 20:511-524.
4. Kocher MS, Zurakowski D, Kasser JR. Differentiating between septic arthritis and transient synovitis of the hip in children: an evidence-based clinical prediction algorithm. J Bone Joint Surg 1999; 81-A:1662-1670.
5. Hennekens CH, Buring JE. Epidemiology in medicine. Boston: Little, Brown, 1987:252-258.

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