Translating Latent Correlation Into Response Correlation

From a Gaussian Field to Binary Responses

A spatial or temporal random field describes dependence on a latent scale. For a binary outcome, however, the observed data are thresholded: each response records only whether an underlying value lies above or below a cutoff. Thresholding changes the strength of association. In particular, the correlation between two latent values is generally not the correlation between the two observed binary responses.

The probit link makes this translation especially explicit. With a Gaussian latent field and independent standard normal observation errors, each binary response can be represented as a thresholded normal variable. The pairwise response correlation then follows from a bivariate normal probability, but it depends on both the latent correlation and the two marginal success probabilities.

These assumptions include conditional independence of the responses given the field: once \(U_i\) and \(U_j\) are known, the remaining errors are independent. Any marginal dependence between the responses comes from their shared or correlated field values. The unit residual variance is the standard probit convention; it fixes the latent scale.

The Response-Correlation Formula

Set \(Z_k=(U_k+\epsilon_k)/s_k\). Then \(Z_i,Z_j\) are standard normal with correlation \(r_{ij}\), and \(Y_k=1\{Z_k>-a_k\}\). Let \(\Phi\) denote the standard normal cumulative distribution function and \(\Phi_2(x,y;r)\) the cumulative distribution function of a standard bivariate normal pair with correlation \(r\), evaluated at \((x,y)\).

For the joint probability, the event \(Z_i>-a_i,Z_j>-a_j\) is equivalent, after changing both signs, to \(-Z_i<a_i,-Z_j<a_j\). The pair \((-Z_i,-Z_j)\) is still standard bivariate normal with correlation \(r_{ij}\), giving the stated \(\Phi_2\) probability. The response covariance and correlation are therefore

The denominator is the product of the marginal standard deviations of the two Bernoulli responses. The formula applies when both marginal probabilities are strictly between zero and one; at a degenerate margin, response correlation is undefined. Notice that even if \(r_{ij}\) is held fixed, changing either \(a_i\) or \(a_j\) generally changes the response correlation.

Why Prevalence Changes the Translation

When both thresholds are zero, each response has success probability one half. In that special case, the bivariate normal quadrant identity gives

One justification is to write the joint success probability as \(Q(r)=\Phi_2(0,0;r)\). The derivative of the bivariate normal probability with respect to its correlation at these zero cutoffs is the bivariate normal density at the origin, \(Q'(r)=1/(2\pi\sqrt{1-r^2})\). Since \(Q(0)=1/4\), integrating from \(0\) to \(r\) gives \(Q(r)=1/4+\arcsin(r)/(2\pi)\). Subtracting the product of the margins, \(1/4\), and dividing by the response standard-deviation product, \(1/4\), yields the arcsine formula.

Away from zero thresholds, the general bivariate normal CDF formula is needed. This is not a minor adjustment: rare outcomes have small Bernoulli variances, and their response correlations can differ markedly from the latent correlation.

Why the Logistic Case Is Different

For a logistic mixed model, write \(q_k(U_k)=\operatorname{expit}(\eta_k+U_k)\), where \(\operatorname{expit}(x)=1/(1+\exp(-x))\). If the responses are conditionally independent given the random effects, then

Consequently, \(\operatorname{Cov}(Y_i,Y_j)=\operatorname{Cov}\{q_i(U_i),q_j(U_j)\}\). Unlike the probit case, a normal random-effects distribution passed through the expit function does not produce a bivariate normal threshold probability with a simple closed-form correlation map. Both the marginal probabilities and the joint probability generally require integration over the random-effects distribution, often by numerical quadrature or simulation.

This difference concerns marginalization, not conditional dependence: under either model, conditional independence can coexist with marginal association. Nor does it mean logistic models cannot produce response correlations. They can; the correlation is obtained from the nonlinear expectations above, rather than from the probit bivariate normal CDF identity.

Assumptions, Checks, and Limitations

The probit translation relies on a correctly specified bivariate Gaussian latent field, independent standard normal observation errors, and the stated threshold rule. If the field is not Gaussian, the bivariate normal CDF formula need not hold even when its variances and covariance are correctly specified. If observation errors are correlated, their covariance must also enter the liability covariance; omitting it gives the wrong \(r_{ij}\). The linear predictors and field distribution must likewise correspond to the population or conditioning scheme for which the marginal correlation is being reported.

A useful check is to compute the marginal probabilities from the same model used for the joint probability, and then verify that the resulting joint probability is compatible with those margins. For example, it must satisfy \(\max\{0,p_i+p_j-1\}\leq P(Y_i=1,Y_j=1)\leq\min\{p_i,p_j\}\). Numerical integration should also be checked for stability when probabilities are near zero or one. The response-correlation denominator becomes small in those cases, so rounding error in the joint probability can have a substantial effect on the calculated correlation.

A common mistake is to report the correlation of the spatial field as the correlation of the binary outcomes. The correction is to first include independent probit-error variance to obtain the standardized liability correlation, then use the thresholds and bivariate normal probability to calculate response covariance and correlation. A second mistake is to apply the probit arcsine formula at nonzero thresholds; that shortcut is valid only when both marginal success probabilities are one half.

The probit formula is a pairwise translation under a specified latent model, not a universal conversion rule for all binary data. For logistic models, direct marginal integration is the appropriate alternative. In either case, observed binary responses contain less information about dependence than the latent Gaussian values: thresholding discards magnitude, and estimated response correlations can be imprecise when outcomes are sparse. Pairwise response correlations also do not replace specifying the joint random-effects model when the goal is to describe dependence among many locations or times.