Conditional Odds-Ratio Inference

Why Condition on the Margins?

For a two-by-two table, the odds ratio measures the association between a binary group indicator and a binary outcome. Conditional inference fixes the row and column totals and treats one cell count as random. The nuisance probabilities disappear from its distribution, leaving a model indexed by the odds ratio alone. This gives a direct route to conditional tests and confidence limits.

Write the table as follows, with \(X\) denoting the upper-left cell count:

Here \(N\) is the total sample size, \(r\) is the first row total, and \(c\) is the first column total. For probabilities \(p_1\) and \(p_2\) of Outcome 1 in Groups 1 and 2, respectively, the population odds ratio is

The table’s sample cross-product odds ratio is \(X(N-r-c+X)/\{(r-X)(c-X)\}\), when the denominator is nonzero. Conditional maximum likelihood estimation, introduced below, is a different calculation in general.

The Conditional Noncentral Hypergeometric Distribution

Given the margins, the possible values of \(X\) form the support \(S=\{m,m+1,\ldots,M\}\), where \(m=\max(0,r+c-N)\) and \(M=\min(r,c)\). Under independent binomial sampling within the two groups, the probability of observing \(X=x\), before conditioning, contains the factors

For fixed margins, the terms depending on \(x\) reduce, up to a factor constant in \(x\), to \(\binom{r}{x}\binom{N-r}{c-x}\theta^x\). Thus the conditional distribution is

This is the noncentral hypergeometric distribution. At \(\theta=1\), it is the ordinary hypergeometric distribution used in Fisher’s exact test. When \(S\) has at least two values, the conditional likelihood for an observed value \(a\) is proportional to \(w_a\theta^a/Z(\theta)\).

Conditional Maximum Likelihood Estimation

It is convenient to use the log-odds parameter \(\eta=\log\theta\). For observed \(X=a\), the conditional log-likelihood, up to an additive constant independent of \(\theta\), is \(a\eta-\log Z(e^\eta)\). Its derivative is \(a-E_\theta[X]\); differentiating the conditional mean with respect to \(\eta\) gives \(\operatorname{Var}_\theta(X)\).

Proof. Writing the finite sum \(Z(e^\eta)=\sum_{x\in S}w_xe^{\eta x}\), differentiation gives

If \(S\) contains at least two values, every support point has positive probability for finite \(\theta>0\), so this variance is strictly positive. As \(\theta\) tends to zero, the term \(w_m\theta^m\) dominates the finite sum; dividing numerator and denominator by \(\theta^m\) shows that the distribution concentrates at \(m\). Dividing by \(\theta^M\) instead shows concentration at \(M\) as \(\theta\) tends to infinity. The mean is therefore continuous, strictly increasing, and has limits \(m\) and \(M\). For an interior observation, the equation \(E_\theta[X]=a\) has exactly one solution. The log-likelihood derivative \(a-E_\theta[X]\) is positive below that solution and negative above it, proving it is the unique maximizer. For \(a=m\), the derivative is negative for every finite \(\theta\); for \(a=M\), it is positive. These give the stated boundary suprema. If \(S=\{m\}\), then \(Z(\theta)=w_m\theta^m\), so \(\theta^m/Z(\theta)=1/w_m\); the likelihood is constant. \(\square\)

Exact Conditional Tests

To test \(H_0:\theta=\theta_0\) against \(H_1:\theta>\theta_0\), the conditional exact p-value is \(P_{\theta_0}(X\ge a)\). For \(H_1:\theta<\theta_0\), it is \(P_{\theta_0}(X\le a)\). Each probability sums the relevant conditional mass function over the observed tail. A two-sided exact test requires a specified convention: for example, the equal-tailed convention doubles the smaller one-sided tail probability, capped at one. Discreteness means that exact tests can be conservative; their null rejection probability is no greater than the stated level.

For the \(N=4,r=2,c=2\) table, the weights are \(1,4,1\) on \(S=\{0,1,2\}\). If \(a=2\), then under \(H_0:\theta=1\), \(P(X=2)=1/6\). The one-sided p-value against \(\theta>1\) is \(P(X\ge2)=1/6\). The equal-tailed two-sided convention gives \(\min(1,2\min\{P(X\le2),P(X\ge2)\})=1/3\). This illustrates why a two-sided p-value should be reported with its convention.

Exact Confidence Limits by Inversion

A central equal-tailed \(100(1-\alpha)\%\) conditional confidence interval is obtained by inverting the two one-sided tests. For a non-singleton support and observed value \(a\), its lower limit \(L\) solves \(P_L(X\ge a)=\alpha/2\), and its upper limit \(U\) solves \(P_U(X\le a)=\alpha/2\). These tail probabilities are finite sums of the noncentral hypergeometric probabilities. If an observed value is an endpoint of the support, the corresponding limit is a boundary value: \(L=0\) when \(a=m\), and \(U=\infty\) when \(a=M\). These conventions describe limits in the extended positive parameter space; the odds ratio itself is positive and finite.

The tail probabilities move in the required directions as \(\theta\) changes. To see this, write \(\eta=\log\theta\). For any function \(g\) on the finite support, differentiation of the weighted sum gives

Taking \(g(X)=\mathbf{1}\{X\ge k\}\) makes this covariance nonnegative: if \(X'\) is an independent copy of \(X\), twice the covariance equals \(E[(g(X)-g(X'))(X-X')]\), whose integrand is nonnegative because \(g\) is nondecreasing. Thus upper-tail probabilities are nondecreasing in \(\theta\), and lower-tail probabilities are nonincreasing. This establishes the monotonicity needed for test inversion. The resulting interval has coverage at least \(1-\alpha\): each excluded tail event has probability at most \(\alpha/2\) under the true parameter, and the union bound gives total noncoverage at most \(\alpha\). Discreteness can make coverage greater than the nominal level.

Degenerate Margins and Common Pitfalls

• Treating the sample cross-product ratio as the conditional maximum likelihood estimate. They are generally different; calculate the conditional likelihood or solve \(E_{\hat\theta_c}[X]=a\) for an interior observation.
• Applying endpoint rules without checking the support size. First check whether \(m<M\). If \(m=M\), the conditional likelihood is constant, not maximized uniquely at either boundary.
• Reporting an exact two-sided p-value without its definition. Different two-sided conventions can produce different values. State the tail or ordering convention used.
• Assuming an exact interval must have finite endpoints. An endpoint observation can yield a zero or infinite limit; that is a consequence of the conditional support, not a calculation failure.