NegRisk Conversion in Binary Data Models

• NegRisk Conversion is the transformation of conditional relative-risk parameters into marginal ones using log-mean regression coefficients and mediator distributions.
• The framework employs a recursive regression graph model to explicitly decompose the treatment effect and mediator influence on binary outcomes.
• It utilizes a weighted-average structure to isolate direct and indirect effects, applicable in both single and multiple mediator settings.

Searching arXiv for the specified paper and closely related work on log-mean / relative-risk regression. Search query: arXiv (Lupparelli, 2018) Lupparelli relative risk log-mean regression NegRisk Conversion denotes the conversion from conditional relative-risk parameters to marginal relative-risk parameters within a recursive regression graph model for binary data, using log-mean regression coefficients parameterized as logarithms of relative risks. In the formulation associated with Lupparelli’s recursive-graph framework, the central object is a decomposition of a marginal log-relative risk into a direct conditional log-relative risk plus a deviation term that depends on the mediator distribution and, in the single-mediator case, on mediator–treatment interaction terms. The underlying motivation is the discrete-data analogue of the classical problem of effect distortion under marginalization, familiar from linear Gaussian settings and often associated with Cochran, but here developed for binary mediators and binary outcomes, including multivariate product outcomes (Lupparelli, 2018).

1. Formal setting and graph structure

The general setup uses three blocks of binary variables. $X$ is a single binary treatment or explanatory variable with levels $0/1$. $Z_U=\{Z_u:u\in U\}$ is a collection of binary intermediate variables. $Y_V=\{Y_v:v\in V\}$ is a collection of binary outcome variables. For any $D\subseteq V$, the product outcome is defined as

$Y^D=\prod_{v\in D}Y_v,$

so that $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$. Similarly, for any $E\subseteq U$,

$Z^E=\prod_{u\in E}Z_u.$

The recursive regression graph is a directed acyclic specification in which $X$ has directed edges to each $0/1$0 and to each $0/1$1, each $0/1$2 has a directed edge to each $0/1$3, and among the final block $0/1$4 there may be bi-directed edges to allow dependence of residuals. Equivalently, the joint distribution factorizes as

$0/1$5

This graph-theoretic factorization is the structural basis for the conversion formulas. It separates the modeling of mediator responses to treatment from the modeling of outcomes given treatment and mediators, which in turn permits an explicit comparison between conditional and marginalized relative-risk parameters (Lupparelli, 2018).

2. Log-mean regression parameterization

For the single-mediator case $0/1$6, the framework is expressed through three regressions.

First, the regression of $0/1$7 on $0/1$8 is

$0/1$9

with

$Z_U=\{Z_u:u\in U\}$0

Hence

$Z_U=\{Z_u:u\in U\}$1

Second, the regression of $Z_U=\{Z_u:u\in U\}$2 on $Z_U=\{Z_u:u\in U\}$3 is

$Z_U=\{Z_u:u\in U\}$4

with

$Z_U=\{Z_u:u\in U\}$5

By definition,

$Z_U=\{Z_u:u\in U\}$6

Third, the marginal regression of $Z_U=\{Z_u:u\in U\}$7 on $Z_U=\{Z_u:u\in U\}$8 alone is

$Z_U=\{Z_u:u\in U\}$9

with

$Y_V=\{Y_v:v\in V\}$0

so that

$Y_V=\{Y_v:v\in V\}$1

The key feature of this parameterization is that all regression coefficients are directly interpretable on the relative-risk scale rather than the odds-ratio scale. This makes the subsequent conversion formula a statement about how marginal relative risks are induced by conditional relative risks together with the mediator distribution (Lupparelli, 2018).

3. Conditional-to-marginal conversion for one mediator

For a single mediator, the univariate Relative-Risk formula states that the marginal log-relative risk of $Y_V=\{Y_v:v\in V\}$2 on $Y_V=\{Y_v:v\in V\}$3 can be written as

$Y_V=\{Y_v:v\in V\}$4

where the deviation term $Y_V=\{Y_v:v\in V\}$5 is

$Y_V=\{Y_v:v\in V\}$6

with

$Y_V=\{Y_v:v\in V\}$7

An equivalent expression on the relative-risk scale is

$Y_V=\{Y_v:v\in V\}$8

In this decomposition, the total marginal effect of $Y_V=\{Y_v:v\in V\}$9 on $D\subseteq V$0 is the direct conditional effect $D\subseteq V$1 multiplied by a weighted-average factor capturing distortion through $D\subseteq V$2. The formal role of $D\subseteq V$3 is therefore to quantify the discrepancy between the conditional and marginalized parameterizations after summing over the mediator. This suggests that “conversion” is not a mere algebraic relabeling of coefficients, but a structurally constrained marginalization result in which treatment-induced shifts in mediator prevalence and mediator-associated outcome risk jointly determine the marginal effect (Lupparelli, 2018).

4. Multivariate extension and product outcomes

The same logic extends from a single outcome to any product outcome $D\subseteq V$4 with $D\subseteq V$5. For the single-mediator multivariate case, the formula becomes

$D\subseteq V$6

with

$D\subseteq V$7

Accordingly, the same form of decomposition holds for every joint event represented by $D\subseteq V$8. Because $D\subseteq V$9 corresponds to simultaneous occurrence of all outcomes indexed by $Y^D=\prod_{v\in D}Y_v,$0, the conversion formula applies not only to marginal effects on individual binary outcomes but also to relative risks for multivariate response events. This is consistent with the recursive graph specification, where the final outcome block may include bi-directed edges permitting residual dependence among the $Y^D=\prod_{v\in D}Y_v,$1’s (Lupparelli, 2018).

A plausible implication is that the framework is particularly suited to settings in which scientific interest centers on co-occurrence patterns rather than isolated endpoints. In that sense, the conversion mechanism generalizes beyond scalar mediation-style interpretations and becomes a device for studying treatment effects on joint binary profiles under explicit marginalization over intermediate variables.

5. Multiple mediators and weighted-average structure

When $Y^D=\prod_{v\in D}Y_v,$2, the framework allows a vector $Y^D=\prod_{v\in D}Y_v,$3 of mediators, while dropping any $Y^D=\prod_{v\in D}Y_v,$4-by-$Y^D=\prod_{v\in D}Y_v,$5 interaction terms for interpretability. The joint regression of a product outcome $Y^D=\prod_{v\in D}Y_v,$6 on $Y^D=\prod_{v\in D}Y_v,$7 is

$Y^D=\prod_{v\in D}Y_v,$8

For mediator products $Y^D=\prod_{v\in D}Y_v,$9, $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$0, the regressions are

$P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$1

The resulting marginal formula is

$P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$2

where

$P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$3

Equivalently,

$P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$4

where

$P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$5

This weighted-average representation clarifies how the marginal relative risk is assembled from mediator-pattern-specific contributions. The conversion depends on all mediator configurations $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$6, weighted by their probabilities under each treatment level. The absence of $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$7 interactions in this multiple-mediator version is presented as an interpretability restriction; it yields a decomposition in which the direct term $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$8 is separated from a mediator-aggregation term built from the $P(Y^D=1)=P(Y_v=1\text{ for all }v\in D)$9 coefficients and the distribution of $E\subseteq U$0 conditional on $E\subseteq U$1 (Lupparelli, 2018).

6. Special cases and interpretive consequences

Several simplifications identify when conditional and marginal relative risks coincide or when the marginal effect is purely indirect in the sense of the stated formulas.

If $E\subseteq U$2, that is, $E\subseteq U$3, then $E\subseteq U$4 and therefore

$E\subseteq U$5

If $E\subseteq U$6, that is, $E\subseteq U$7, then

$E\subseteq U$8

which reduces to a pure weighted-averaged indirect effect. The Gaussian path-analysis analogue is described as

$E\subseteq U$9

whereas in the present binary relative-risk framework the indirect component emerges as the ratio of two weighted averages.

If $Z^E=\prod_{u\in E}Z_u.$0, that is, $Z^E=\prod_{u\in E}Z_u.$1, and one also assumes no $Z^E=\prod_{u\in E}Z_u.$2 interaction, $Z^E=\prod_{u\in E}Z_u.$3, then again $Z^E=\prod_{u\in E}Z_u.$4 and

$Z^E=\prod_{u\in E}Z_u.$5

These special cases delimit the sense in which conversion is needed. When treatment does not alter mediator prevalence, when outcomes are conditionally independent of mediators, or when specific interaction structures vanish, conditional and marginal relative risks coincide. Otherwise, marginalization introduces a nontrivial distortion term. A common misconception would be to treat conditional and marginal relative risks as interchangeable in binary recursive systems; the formulas show that their equality is a special-case property, not a generic feature (Lupparelli, 2018).

7. Estimation workflow and relation to the motivating application

The practical implementation guidelines are explicit. One fits the two, or multivariate, log-mean regression models: $Z^E=\prod_{u\in E}Z_u.$6 via a log-$Z^E=\prod_{u\in E}Z_u.$7 linear predictor, and $Z^E=\prod_{u\in E}Z_u.$8 via separate univariate regressions. Estimation may use standard maximum-likelihood or the algorithms for log-mean-linear models associated with Lupparelli and Roverato (2017). Interaction terms $Z^E=\prod_{u\in E}Z_u.$9 are then examined and may be dropped if non-significant for parsimony. From the fitted model one extracts $X$0, the mediator effects $X$1, and the parameters $X$2 for all $X$3 in order to compute $X$4. The deviation term $X$5 is then obtained by substitution into the relevant formula, and the marginal log-relative risk follows as

$X$6

with exponentiation yielding the relative-risk scale (Lupparelli, 2018).

The motivating context described for the broader framework is the analysis of morphine data to assess the effect of preoperative oral morphine administration on postoperative pain relief. More generally, the original problem is framed as one arising in medical and social science settings where discrete variables make the relationship between marginal and conditional effects challenging. The framework therefore occupies a specific methodological niche: it provides a recursive-graph, log-mean regression treatment of binary data in which the conversion from conditional to marginal relative risks is explicit, multivariate, and structurally tied to mediator distributions under treatment.