Revisiting the interaction between birth weight and current body size in the foetal origins of adult disease

Abstract

The four models proposed for exploring the foetal origins of adult disease (FOAD) hypothesis use the product term between size at birth and current size to determine the relative importance of pre- and post-natal growth on disease in later life. This is a common approach for testing the interaction between an exposure (in this instance size at birth) and an effect modifier (in this instance current size)—incorporating the product term obtained by multiplying the exposure and effect modifier variables within a statistical regression model. This study examines the mathematical basis for this approach and uses computer simulations to demonstrate two potential statistical flaws that might generate misleading findings. The first of these is that the expected value of the partial regression coefficient for the product term (between exposure and effect modifier) will be zero when the outcome, exposure and effect modifier are all continuously distributed and follow a multivariate normal distribution. This is because testing the product interaction term amounts to testing for multivariate normality among the three variables, irrespective of the pair-wise correlations amongst them. The second flaw is that it is possible to generate a statistically significant interaction between exposure and effect modifier, even when none exists, simply by categorising either or both of these variables. These flaws pose a serious challenge to the four models approach proposed for exploring the FOAD hypothesis. The interaction between exposure and effect modifier variables should be interpreted with caution both here and elsewhere in epidemiological analyses.

Appendix: derivation of the correlations between XZ and, Y, X and Z

Take three multivariate normal random variables X, Y, and Z, where Y is the outcome in a regression model with X and Z and their product interaction term (XZ) as covariates:

$$ Y = B_{0} + B_{1} X + B_{2} Z + B_{3} XZ + \varepsilon $$

(A1)

where B ₀ is the intercept; B ₁ the partial regression coefficient for X; B ₂ the partial regression coefficient for Z; B ₃ the partial regression coefficient for XZ; and ε the residual error. To prove that the partial regression coefficient for product interaction term XZ is zero, under the assumption of multivariate normality and irrespective of pair-wise correlations amongst Y, X and Z, it is sufficient to prove that the standardized partial regression coefficient for XZ (β ₃) is zero. Using matrix algebra for the regression model [32], one can write:

$${\bf{Y}}\, = \,{\bf{AB}}\, \Rightarrow \,{\bf{A}'{Y}}\, = \,{\bf{A}}'{\bf{AB}}\, \Rightarrow \,{\left( {{\bf{A}}'{\bf{A}}} \right)}^{{ - 1}} {\left( {{\bf{A}}'{\bf{Y}}} \right)}\, = \,{\bf{B}}$$

where: Y is a vector (n × 1) of standardized Y values; A is a matrix (n × 3) of standardize X, Z and XZ covariate values; B is a vector (3 × 1) of the standardized partial regression coefficients; and the prime, ′, denotes the matrix transpose. Furthermore, denoting ${\mathbf{M}} = {\left( {{\mathbf{A}}^{{ - {\mathbf{1}}}} {\mathbf{A}}} \right)}$, M is therefore the correlation matrix for the three covariates. Thus,

$$ {\mathbf{B}} = {\left( {\begin{array}{*{20}c} {{\beta _{1} }} \\ {{\beta _{2} }} \\ {{\beta _{3} }} \\ \end{array} } \right)} = {\mathbf{M}}^{{ - {\mathbf{1}}}} {\mathbf{A}}'{\mathbf{Y}} = {\left( {\begin{array}{*{20}c} {1} & {{r_{{X,Z}} }} & {{r_{{X,XZ}} }} \\ {{r_{{X,Z}} }} & {1} & {{r_{{Z,XZ}} }} \\ {{r_{{X,XZ}} }} & {{r_{{Z,XZ}} }} & {1} \\ \end{array} } \right)}^{{ - 1}} {\left( {\begin{array}{*{20}c} {{r_{{Y,X}} }} \\ {{r_{{Y,Z}} }} \\ {{r_{{Y,XZ}} }} \\ \end{array} } \right)}. $$

(A2)

Solving Eq. A2 for the coefficient β ₃, according to standard matrix algebra (27):

$$ \beta _{3} = \frac{1} {{\det {\left( {\mathbf{M}} \right)}}}{\left[ {r_{{Y,X}} {\left( {r_{{X,Z}} r_{{Z,XZ}} - r_{{X,XZ}} } \right)} + r_{{Y,Z}} {\left( {r_{{X,Z}} r_{{X,XZ}} - r_{{Z,XZ}} } \right)} + r_{{Y,XZ}} {\left( {1 - r^{2}_{{X,Z}} } \right)}} \right]} $$

(A3)

where: $\det ({\bf{M}}) = 1 + 2r_{{X,Z}} r_{{X,XZ}} r_{{Z,XZ}} - r^{2}_{{X,Z}} - r^{2}_{{X,XZ}} - r^{2}_{{Z,XZ}}$ is the matrix determinant of M; and r _a,b is the correlation between a and b, substituting X, Y, Z, or XZ for a and b as appropriate.

Under the assumption of multivariate normality, it can be shown that [33, 34]:

$$ \begin{aligned}{} & \sigma _{{X,XZ}} = E{\left( X \right)}\sigma _{{X,Z}} + E{\left( Z \right)}\sigma ^{2}_{X} \\ & \sigma _{{Y,XZ}} = E{\left( X \right)}\sigma _{{Y,Z}} + E{\left( Z \right)}\sigma _{{Y,X}} \\ & \sigma _{{Z,XZ}} = E{\left( X \right)}\sigma ^{2}_{Z} + E{\left( Z \right)}\sigma _{{X,Z}} \\ \end{aligned} $$

(A4)

where: σ _a,b is the covariance of a and b; E(a) the expected value of a; and $\sigma ^{2}_{a}$ the variance of a, again substituting X, Y, Z, or XZ for a and b as appropriate.

Now, denoting $V_{X} = {\sigma _{X} } \mathord{\left/ {\vphantom {{\sigma _{X} } {E(X)}}} \right. \kern-\nulldelimiterspace} {E(X)}$ and $V_{Z} = {\sigma _{Z} } \mathord{\left/ {\vphantom {{\sigma _{Z} } {E(Z)}}} \right. \kern-\nulldelimiterspace} {E(Z)}$, where E(X) ≠ 0 and E(Z) ≠ 0 (which can be guaranteed by adding an arbitrary constant to X or Z), writing $D = {\sqrt {V^{2}_{X} + V^{2}_{Z} + 2r_{{X,Z}} V_{X} V_{Z} + V^{2}_{X} V^{2}_{Z} (1 + r^{2}_{{Y,X}} )} }$, and utilising Eq. A4, the correlations between X and XZ (r _X,XZ ), Y and XZ (r _Y,XZ ), and Z and XZ (r _Z,XZ ) can be written:

$$ \begin{aligned}{} & r_{{X,XZ}} = {{\left( {V_{X} + r_{{X,Z}} V_{Z} } \right)}} \mathord{\left/ {\vphantom {{{\left( {V_{X} + r_{{X,Z}} V_{Z} } \right)}} D}} \right. \kern-\nulldelimiterspace} D \\ & r_{{Y,XZ}} = {{\left( {r_{{Y,X}} V_{X} + r_{{Y,Z}} V_{Z} } \right)}} \mathord{\left/ {\vphantom {{{\left( {r_{{Y,X}} V_{X} + r_{{Y,Z}} V_{Z} } \right)}} D}} \right. \kern-\nulldelimiterspace} D, \\ & r_{{Z,XZ}} = {{\left( {r_{{X,Z}} V_{X} + V_{Z} } \right)}} \mathord{\left/ {\vphantom {{{\left( {r_{{X,Z}} V_{X} + V_{Z} } \right)}} D}} \right. \kern-\nulldelimiterspace} D \\ \end{aligned} $$

(A5)

and Eq. A3 then becomes:

$$ \beta _{3} = \frac{1} {{D*\det {\left( {\mathbf{M}} \right)}}}{\left\{ \begin{aligned}{} & r_{{Y,X}} {\left[ {r_{{X,Z}} {\left( {r_{{X,Z}} V_{X} + V_{Z} } \right)} - {\left( {V_{X} + r_{{X,Z}} V_{Z} } \right)}} \right]} + \\ & r_{{Y,Z}} {\left[ {r_{{X,Z}} {\left( {V_{X} + r_{{X,Z}} V_{Z} } \right)} - {\left( {r_{{X,Z}} V_{X} + V_{Z} } \right)}} \right]} + \\ & {\left( {1 - r^{2}_{{X,Z}} } \right)}{\left( {r_{{Y,X}} V_{X} + r_{{Y,Z}} V_{Z} } \right)} \\ \end{aligned} \right\}} = \frac{{{\left\{ 0 \right\}}}} {{D*\det {\left( {\mathbf{M}} \right)}}} = 0. $$

Testing the product term is thus to test the linearity and multivariate normality of the three continuous variables [17, 18]. The interaction effect is scale-dependent since transformation of the original variables can attenuate the interaction coefficient or produce a non-zero effect that is otherwise zero for the original variables.