Contingency Tables, Odds Ratios, and Relative Risks | Binomial Tests and 2×2 Tables

A contingency table (crosstab) organizes two or more categorical variables at the same time. In particular, the 2 × 2 table is the basic form for quantifying the direction and magnitude of an association through odds ratios (ORs) and relative risks (RRs) as well as tests of independence.

Structure of 2×2 contingency table#

	With result Y=1	No result Y=0	total
Exposure X=1	A	B	A+B
No exposure X=0	C	D	C+D
total	A+C	B+D	N

A to D are cell frequencies that satisfy the simultaneous condition, and the sum of rows and columns is the marginal frequency. Although it is optional to place one in a row and one in a column, the numerator/denominator and interpretation of the effect size depend on the arrangement, so the definition should be clearly stated.

Joint Probability, Conditional Probability, and Independence#

equal conditional probabilities

P(Y=1\mid X=1)=P(Y=1\mid X=0)=P_Y

Definition of independence

P(X=i,Y=j)=P(X=i)P(Y=j)

Study Design Determines Interpretation#

design	Sampling criteria	Quantities that are easy to estimate directly
Cohort/randomized trial	Tracking from exposure/intervention group	Risk, RR, risk difference, OR
case control study	Extract based on presence/absence of results	OR (Generally, risk and RR cannot be directly estimated)
cross-sectional study	Measure exposure and outcomes at a single point in time	Prevalence ratio/OR

Even with the same four numbers, the meaning of the denominator changes depending on who is extracted from which population. Check the research design before looking at the calculation formula.

Odds and the Odds Ratio#

odds of probability p

\operatorname{odds}=\frac{p}{1-p}

Odds ratio between exposed and unexposed groups

OR=\frac{A/B}{C/D}=\frac{AD}{BC}

OR=1 indicates no association, OR>1 indicates higher odds of outcome in the exposed group, and OR<1 indicates lower odds. For example, OR=2 is not twice the probability, but twice the odds.

Estimate the OR on a logarithmic scale#

log odds ratio

\log(OR)=\log A+\log D-\log B-\log C

approximate variance

\operatorname{Var}[\log(OR)]\approx\frac1A+\frac1B+\frac1C+\frac1D

standard error

SE[\log(OR)]=\sqrt{\frac1A+\frac1B+\frac1C+\frac1D}

test statistic

Z=\frac{\log(OR)}{SE[\log(OR)]}

CI_{1-\alpha}(OR)=\exp\left\{\log(OR)\pm z_{1-\alpha/2}SE[\log(OR)]\right\}

Risk and Relative Risk#

Risk of exposure group

R_1=\frac{A}{A+B}

Risk for unexposed group

R_0=\frac{C}{C+D}

relative risk

RR=\frac{R_1}{R_0}=\frac{A/(A+B)}{C/(C+D)}

RR=1 is equal risk, RR=2 is twice the risk of the exposed group, and RR=0.5 is half the risk. It is naturally interpreted in cohort studies and trials where the follow-up period and denominator can be defined.

Estimate RR on a logarithmic scale#

Approximate variance of log relative risk

\operatorname{Var}[\log(RR)]\approx\left(\frac1A-\frac1{A+B}\right)+\left(\frac1C-\frac1{C+D}\right)

standard error

SE[\log(RR)]=\sqrt{\left(\frac1A-\frac1{A+B}\right)+\left(\frac1C-\frac1{C+D}\right)}

test statistic

Z=\frac{\log(RR)}{SE[\log(RR)]}

CI_{1-\alpha}(RR)=\exp\left\{\log(RR)\pm z_{1-\alpha/2}SE[\log(RR)]\right\}

Difference Between OR and RR#

item	OR	RR
amount to compare	odds	risk (probability)
unrelated value	1	1
case control study	Estimable	Usually not directly estimated
rare results	Approximate to RR	direct
If the result is not rare	It tends to appear larger than RR in the direction away from 1.	Clinical interpretation is relatively intuitive

p\approx0\quad\Longrightarrow\quad\frac{p}{1-p}\approx p\quad\Longrightarrow\quad OR\approx RR

Why the Log-Scale Variance Formulas Work: The Delta Method#

The variance of a nonlinear function is found by a first-order approximation near the estimated value. This is the basis of the log standard error of OR/RR.

\operatorname{Var}[f(X)]\approx[f'(\mu)]^2\operatorname{Var}(X)

Since the derivative of log(x) is 1/x, the logarithmic variance of the counts is approximately 1/x. When we add and subtract independent logarithmic cell frequencies, the variances add up and we get 1/A+1/B+1/C+1/D of log(OR). In RR, since the numerator and denominator in the same row are shared, covariance is taken into consideration, and −1/(A+B) and −1/(C+D) appear.

Zero Cells and Small Samples#

If any cell is 0, the normal OR or log standard error will be infinite or undefined. Since mechanical 0.5 addition can change the estimation target, we will consider the exact method, conditional estimation, penalized logistic regression, etc. depending on the purpose, and specify the method to be adopted.

Reporting in Practice#

Specify row/column definitions, target population, and research design
Check the four cells and marginal frequencies, and indicate how to handle missing values.
Select OR, RR, and risk difference if necessary to suit your design and purpose.
Report point estimates and 95% confidence intervals
Explain not only the p-value of the independence test, but also the direction, magnitude, and clinical significance of the effect.
Describes limitations of small samples, zero cells, confounding, and selection bias.