Richard Guo’s contribution to the Discussion of ‘Parameterizing and simulating from causal models’ by Evans and Didelez

I congratulate Evans and Didelez warmly on this innovative and inspiring paper. Here, I offer my interpretation of the approach in terms of two factorizations and a density ratio.

Any distribution over $Z,X,Y$ can be factorized in two ways:

$$(P_{ZX},P_{Y\mid Z,X}){\displaystyle \underset{C}{\overset{C^{-1}}{\leftrightarrows }}}{P}_{ZXY}{\displaystyle \underset{A}{\overset{A^{-1}}{\rightleftarrows }}}(P_{ZX},P_{Y\mid X},{\varphi }_{ZY\mid X}).$$

(1)

Factorization $C^{-1}$ is the usual one; it is factorization $A^{-1}$ that is the focus of the two authors. More concretely, suppose ${\varphi }_{ZY\mid X}(z,y\mid x)$ is a conditional copula density, i.e. for every value of x, $(z,y)\mapsto {\varphi }_{ZY\mid X}(z,y\mid x)$ is a density function over $[0,1{]}^2$ with uniform margins. To obtain $A^{-1}(P_{ZXY})$⁠, take $P_{ZX}$ and $P_{Y\mid X}$ to be the corresponding component and let ${\varphi }_{ZY\mid X=x}$ be the density function of $(F(Z\mid X=x),F(Y\mid X=x))$ for every x in the support of X. Conversely, to compose $P_{XYZ}$ from the three pieces, we have

$$\begin{array}{rl}{p}_{ZXY}(z,x,y)& =A(p_{ZX},p_{Y\mid X},{\varphi }_{ZY\mid X})\\ & =p_{ZX}(z,x){\displaystyle \underset{=p_{Y\mid ZX}(y\mid z,x)}{{\displaystyle \underbrace{p_{Y\mid X}(y\mid x){\varphi }_{ZY\mid X}(F(z\mid x),F(y\mid x)\mid x)}}}},\end{array}$$

(2)

where $F(z\mid x)$ and $F(y\mid x)$ are defined by $P_{ZX}$ and $P_{Y\mid X}$⁠, respectively. Note that although $P_{ZX}$ appear on both sides of equation (1), the relation between $P_{Y\mid Z,X}$ and the pair $(P_{Y\mid X},{\varphi }_{ZY\mid X})$ is not a separate bijection because $F(z\mid x)$ is needed to map one to the other. In other words, the map between $P_{Y\mid Z,X}$ and $(P_{Y\mid X},{\varphi }_{ZY\mid X})$ itself depends on $P_{ZX}$⁠.

Suppose $P_{ZXY}^{*}$ is a related distribution, of which the margin $P_{Y\mid X}^{*}$ is our model of interest. We require that $P_{ZXY}^{*}$ is related to $P_{ZXY}$ through a density ratio r, given by

$$\frac{p^{*}(z,x,y)}{p(z,x,y)}=r(z,x;p),$$

such that (a) r does not depend on y, (b) $r>0$P-almost everywhere, and (c) r can be identified from P. Then, by integrating out y on both sides of $p^{*}(z,x,y)=r(z,x;p)p(z,x,y)$⁠, we have

$$r(z,x;p)=\frac{p^{*}(z,x)}{p(z,x)},p^{*}(y\mid z,x)=p(y\mid z,x).$$

(3)

That $p^{*}(z,x,y)$ being ‘cognate’ with respect to $p(z,x,y)$ amounts to choosing

$$r(z,x;p)=\frac{w(z\mid x)}{p(z\mid x)}$$

for some weight $w(z\mid x)$⁠, such as $w(z\mid x)=p(z)$ for estimating the average treatment effect and $w(z\mid x)=p(z\mid x=1)$ for estimating the effect of the treatment on the treated.

With $r(x,y;p)$ chosen and fixed, the frugal parametrization is to represent p (and hence $p^{*}$⁠) through $p_{ZX}$⁠, $p_{Y\mid X}^{*}$⁠, and ${\varphi }_{ZY\mid X}^{*}$⁠, i.e. the following three pieces in box:

$$\begin{array}{rl}(P_{ZX},P_{Y\mid Z,X}){\displaystyle \underset{C}{\overset{C^{-1}}{\leftrightarrows }}}& P_{ZXY}{\displaystyle \underset{A}{\overset{A^{-1}}{\rightleftarrows }}}(\boxed{{\displaystyle P_{ZX}}},P_{Y\mid X},{\varphi }_{ZY\mid X})\\ (P_{ZX}^{*},P_{Y\mid Z,X}^{*}){\displaystyle \underset{C}{\overset{C^{-1}}{\leftrightarrows }}}& P_{ZXY}^{*}{\displaystyle \underset{A}{\overset{A^{-1}}{\rightleftarrows }}}(P_{ZX}^{*},\boxed{{\displaystyle P_{Y\mid X}^{*}}},\boxed{{\displaystyle {\varphi }_{ZY\mid X}^{*}}}).\end{array}$$

The likelihood can be obtained through

$$\begin{array}{rl}p(z,x,y)=\frac{p^{*}(z,x,y)}{r(z,x;p)}& =\frac{A(p^{*}(z,x),p^{*}(y\mid x),{\varphi }^{*}(z,y\mid x))}{r(z,x;p)}\\ & =\frac{A(p(z,x)r(z,x;p),p^{*}(y\mid x),{\varphi }^{*}(z,y\mid x))}{r(z,x;p)},\end{array}$$

where the second line uses equation (3). When ${\varphi }^{*}(z,y\mid x)$ is a conditional copula density, using equation (2), it follows that

$$p(z,x,y)=p(z,x){\displaystyle \underset{=p(y\mid z,x)}{{\displaystyle \underbrace{p^{*}(y\mid x){\varphi }^{*}(F^{*}(z\mid x),F^{*}(y\mid x)\mid x)}}}}.$$

(4)

Indeed, by uniform margins of the copula, one can check that

$$\int p^{*}(y\mid x){\varphi }^{*}(F^{*}(z\mid x),F^{*}(y\mid x)\mid x)\mathrm{d}y=\int \mathrm{d}{F}^{*}(y\mid x){\varphi }^{*}(F^{*}(z\mid x),F^{*}(y\mid x)\mid x)=1.$$

Further, in equation (4), the arguments of ${\varphi }^{*}$ depend on $F^{*}(y\mid x)$ and $F^{*}(z\mid x)$⁠: the former is derived from $p^{*}(y\mid x)$ and the latter is the conditional distribution function pertaining to $p^{*}(z,x)=p(z,x)r(z,x;p)$⁠, given by

$$F^{*}(z\mid x)=\frac{{\int }_{-\mathrm{\infty }}^{z}p(z^{\prime },x)r(z^{\prime },x;p)\mathrm{d}{z}^{\prime }}{{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}p(z^{\prime },x)r(z^{\prime },x;p)\mathrm{d}{z}^{\prime }}.$$

(5)

Hence, equation (4) provides an explicit expression for $p(z,x,y)$ in terms of $p_{ZX}$⁠, $p_{Y\mid X}^{*}$⁠, and ${\varphi }_{ZY\mid X}^{*}$⁠, which depends on the pre-specified density ratio $r(x,y;p)$ through equation (5). Multiplying equation (4) by the density ratio simply yields the expression for $p^{*}(z,x,y)$⁠.

Sequentially randomized trial

 

Example 1

For Figure 1, with $r(a,l,b;p)=$$p(b)/p(b\mid a,l)$⁠, we can parametrize P and $P^{*}$ in terms of the three pieces in box below:

$$\begin{array}{rl}P(A,L,B,Y)& {\displaystyle \underset{A}{\overset{A^{-1}}{\rightleftarrows }}}(\boxed{{\displaystyle P_{ALB}}},P_{Y\mid AB},{\varphi }_{YL\mid AB})\\ P^{*}(A,L,B,Y)& {\displaystyle \underset{A}{\overset{A^{-1}}{\rightleftarrows }}}(P_{ALB}^{*},\boxed{{\displaystyle P_{Y\mid AB}^{*}}},\boxed{{\displaystyle {\varphi }_{YL\mid AB}^{*}}}).\end{array}$$

Figure 1.

(a) $P(A,L,B,Y)$ and (b) $P^{*}(A,L,B,Y)$⁠.

Partially marginal model

 

Example 2

Suppose we have an observational study with baseline covariates $Z=(Z_1,Z_2)$⁠, treatment X and outcome Y. Imagine that we want to study how $Z_1$ modifies the effect of X on Y. Hence, we want to choose $P^{*}(Z,X,Y)$ such that $P^{*}(Y\mid X,Z_1)$ aligns with our intended marginal model $P(Y\mid Z_1,\text{do}(X))$⁠. In the meantime, we need to use both $Z_1$ and $Z_2$ to control for the confounding between X and Y. This is called a ‘partially’ marginal model because the marginal model is conditional on a partial collection of baseline covariates. To facilitate this analysis, we can choose density ratio

$$r(z_1,z_2,x;p)=p(x)/p(x\mid z_1,z_2)$$

and parametrize p (and hence $p^{*}$⁠) in terms of

$$p(z_1,z_2,x),p^{*}(y\mid z_1,x),{\varphi }_{Z_2,Y\mid Z_1,X}^{*}.$$

Author notes

Conflict of interests: None declared.