Epidemiologic Perspectives & Innovations BioMed Central Methodology

Numerous studies have reported inverse associations between birth weight and a range of diseases in later life. These have led to the development of the foetal origins of adult disease hypothesis. However, many such studies have only been able to demonstrate a statistically significant association between birth weight and disease in later life by adjusting for current size. This has been interpreted as evidence that the impact of low birth weight on subsequent disease is somehow dependent on subsequent weight gain, and has led to a broadening of the hypothesis into the developmental origins of health and disease. Unfortunately, much of the epidemiological evidence used for both of these interpretations is prone to a statistical artefact known as the 'reversal paradox'. The aim of this paper is to illustrate why, using vector geometry.


Background
Numerous studies have reported inverse associations between birth weight and a range of diseases in later life. These have led to the development of the 'foetal origins of adult disease hypothesis'. However, many such studies have only been able to demonstrate a statistically significant association between birth weight and disease in later life by adjusting for current size. This has been interpreted as evidence that the impact of low birth weight on subsequent disease is somehow dependent on subsequent weight gain, and has led to a broadening of the hypothesis into the 'developmental origins of health and disease'. Unfortunately, much of the epidemiological evidence used for both of these interpretations is prone to a statistical artefact known as the 'reversal paradox'. The aim of this paper is to illustrate why, using vector geometry.

Materials and methods
This paper introduces the key concepts of vector geometry as applied to multiple regression analysis. This approach is then used to illustrate the similar statistical problems encountered when adjusting for current size or growth when exploring the association between birth weight and disease in later life.

Results
Geometrically, the three covariates -birth size, growth, and current size -span only 2-dimensional space. Regressing disease in later life (i.e. the outcome variable) on any two of these covariates equates to projecting the disease variable onto the plane spanned by the three covariate vectors. The three possible regression modelswhere any two covariates are considered -are therefore equivalent and yield exactly the same model fit (R 2 ).

Conclusion
Vector geometry illustrates why it is impossible to differentiate between the effects of growth from the effects of current size in studies exploring the relationship between size at birth and subsequent disease. For similar reasons, it is impossible to differentiate between the effects of growth and the effects of birth weight. Assessing the 'independent' impact of growth on later disease by adjusting for either birth weight or current size is therefore illusory.

Background
Numerous studies over the past two decades have found inverse associations between birth weight and a range of chronic diseases -associations which gave rise to the 'foetal origins of adult disease hypothesis'. This argues that under-nutrition or growth retardation in utero can have adverse long-term effects on the development of vital organ systems, thereby increasing the risk of a range of metabolic and related disorders such as: hypertension [1]; diabetes [2]; arteriosclerosis [3]; and obesity [4]. However, many such studies have only been able to demonstrate a statistically significant association between birth weight and disease in later life by adjusting for current size [5]. This has been interpreted as evidence that the impact of low birth weight on subsequent disease is somehow dependent on subsequent weight gain, and has led to a broadening of the hypothesis into the 'developmental origins of health and disease' (DOHaD) [6].
Two mechanisms have been postulated to explain the impact of current size on the relation between birth weight and disease in later life. On the one hand, some researchers argue that current size helps to distinguish between those individuals who are genetically small and essentially healthy at birth (i.e. those who remain relatively small in later life), and those who are small at birth as a result of intrauterine growth retardation (i.e. those who subsequently attain a normal or above normal body size, given better conditions for postnatal growth) [7]. On the other hand, other researchers argue that intrauterine conditions leading to growth retardation and low birth weight can elicit permanent yet adaptive physiological responses that are intended to prepare the foetus for a postnatal environment in which nutritional resources are scarce and growth is compromised [8]. In this second mechanism, low birth weight babies who subsequently experience better than expected conditions for postnatal growth are thought to be ill-adapted to cope with normal or excessive nutrition and, as a result, have an increased risk of metabolic and related disorders [8]. Both mechanisms appear plausible, and it is feasible that both might operate at the same time, although the first focuses on developmental damage to organ systems as a result of intrauterine growth retardation, while the latter suggests that its physiological effects are only maladaptive in postnatal environments where growth is no longer compromised.
To help establish the relative importance of pre-and postnatal events on disease in later life, Lucas et al. [9] proposed that four analytical models should be used to establish the role of size at birth, current size and the interaction between the two. However, some researchers have recently questioned the validity of this approach, arguing that it might be inappropriate to adjust for current body size [5], and that testing the interaction between size at birth and current size is equivalent to testing the multivariate normality of birth size, current size and the disease outcome [10]. Our previous studies have confirmed that such adjustments can create a statistical artefact known as the 'reversal paradox' -perhaps better known as 'Simpson's paradox' in the analysis of categorical data [11]. Some researchers might assume that focussing on postnatal weight gain gets around this problem, particularly in studies of children where higher than average postnatal growth amongst low birth weight infants is often interpreted as 'catch-up growth' -a pattern of compensatory growth exhibited by those who have experienced growth retardation in utero but are subsequently able to recover what is presumed to be their 'intended' growth trajectory.
In fact, much of the epidemiological evidence used to support a focus on weight gain rather than current weight to explore the 'DOHaD' is based on similarly questionable statistical models -the only difference being in their interpretation. For example, for studies examining systolic blood pressure as the health outcome of interest, most only find a statistically significant inverse relationship with birth weight after adjustment for current weight or body mass index [5]. When there is no adjustment for one or more measures of current body size, the relation between birth weight and blood pressure is substantially reduced and is often not statistically significant [12,13]. For those researchers interested in growth rather than attained size, the statistical effect of adjusting for current weight seems to indicate that there is an interaction between birth weight and current body weight, and that it is more likely to be postnatal growth than size at birth that is relevant to health in later life. This is because the relation between birth weight and blood pressure is substantially weaker without adjustment for current weight [12,13]. In practice this proves to be simply an alternative interpretation of the same, ambiguous statistical relationship -an issue the present study sets out to address using vector geometry to illustrate how focusing on current size or growth, and their associated interpretations, are equally problematic. This is because both scenarios use similar statistical models which are prone to the same statistical artefact, even though they arrive at very different conclusions. Moreover, the present study aims to show that although growth appears to have a larger impact, this cannot be statistically differentiated from that of current size.
To this end, we begin with a concise introduction to vector geometry and use this to illustrate the multiple regression analyses commonly used to explore the foetal origins of adult disease hypothesis. We then demonstrate that the common practice of regressing disease outcomes on birth size and current size does not address the question of whether growth has a greater impact than birth size or current size. For this illustration we use adult systolic blood pressure (BP) as the outcome, with birth weight (BW), and current weight (CW) as potential covariates. A fourth covariate, weight gain (WG), is defined as the difference between current weight and birth weight (CW -BW) and for simplicity, all four variables are treated as continuous. For those interested in a fuller explanation of the basic geometric tools involved, these have been summarised in the Appendix.

Representation of variables as vectors
Vector geometry is a very useful tool for providing nonstatisticians with an intuitive understanding of statistical theory, such as correlation and regression [14,15]. We use vector geometry to illustrate 'simple' (one covariate) and 'multiple' (two or more covariates) regression analyses. To do this, we switch from the more familiar domain of 'variable space' to the less familiar domain of 'subject space'. In variable space, two variables are represented within a plane by a scatter plot, whereas in subject space the same two variables are represented within a plane by two scaled vectors with lengths equal to the standard deviation (SD) of their corresponding variables. The number of dimensions needed to represent variables in subject space is no greater than the number of variables. Although it is impossible to visualize more than three dimensions, only two dimensions are needed to illustrate correlation and simple regression, and only three dimensions are required to illustrate multiple regression.

Correlation and simple regression
When variables are represented as scaled vectors, the correlation between the original variables equates to the cosine of the angle between their corresponding vectors. Furthermore, the simple regression coefficient of one variable (the dependent variable) regressed on the other variable (the covariate) is equivalent to the orthogonal projection of the first vector on the second, i.e. a line perpendicular to the second vector is drawn from the end of the first vector, and the intersection of the line with the second vector determines the length and direction of the projection of the first vector onto the second vector. For instance, for two variables X and Y represented by vectors x and y, their correlation coefficient (ρ xy ) is given by cos(θ xy ) where θ xy is the angle between x and y -see Figure  1. The simple regression coefficient of the variable X(b X ), when Y is regressed on X, is the length of the perpendicular projection of y on x divided by the length of x (denoted ||x||), i.e. b X = (||y||/||x||)cos(θ xy ) -see Figure 1.

Multiple regression
Regressing variable Y on the two variables X and Z is equivalent, within vector geometry, to finding the orthogonal projection of the vector y onto the plane spanned by the vectors x and z, then using the parallelogram rule to find the contributing proportions of x and z that yield the projected vector y p . For instance, if we denote the regression equation for these variables as: Y = b X X+b Z Z, where b X and b Z are partial regression coefficients, then using vector geometry: y p = b X x+b Z z, where b X and b Z are the proportions (i.e. the projection weights) of the vectors x and z that make up y p -see Figure 2.
Within vector geometry, the P-value for partial regression coefficients obtained when controlling for other covariates is derived from the projection of vectors for the outcome and each covariate onto the subspace perpendicular to all other covariates. For instance, when regressing Y on both X and Z, the P-value for the partial regression coefficient for X is derived from the projection of x and y onto the subspace perpendicular to z. Since the entire model space is only three dimensions (spanned by x, y, and z), the subspace perpendicular to z is a plane, denoted V ⊥z -(a) The correlation between variables Y and X (ρ XY ) is the cosine of θ xy , the angle between vectors x and y; the projec-tion of y on x (denoted y p ) has the length ||y||·cos(θ xy ) Figure 1 (a) The correlation between variables Y and X (ρ XY ) is the cosine of θ xy , the angle between vectors x and y; the projection of y on x (denoted y p ) has the length ||y||·cos(θ xy ). Vector y p lies in the same direction as vector x and may therefore be expressed as a multiple of x: y p = b X x, where b X = (||y||/||x||)cos(θ xy ) -the simple regression coefficient for X when Y is regressed on X. (b) If θ xy = 90° (i.e. π/2 radians), then x and y are orthogonal (denoted x ⊥ y), the correlation between X and Y is zero: ρ XY = cos(90°) = cos(π/2) = 0 and the regression coefficient for Y regressed on X is also zero.  Figure 2. The P-value derived from the F ratio test for the partial regression coefficient b X is given as [15]: where y x⊥z is the projection of y ⊥z on x ⊥z (which is equivalent to the projection of y on x ⊥z ), and n is the sample size [16]. The value of the F test with 1 and n-3 degrees of freedom is equivalent to that of the t-test with n-3 degrees of freedom. A detailed explanation can be found in Wickens' excellent book [15].

A geometrical illustration of adjustment for current weight in DOHaD
When examining the relationship between birth weight and disease in later life, most studies have found that the correlation and (simple) regression coefficients between the two are close to zero or slightly negative [5]. However, taking hypertension as an example, when blood pressure is simultaneously regressed on birth weight and current weight, the adjustment for current weight tends to reduce or reverse any positive association between birth weight and blood pressure, and accentuate any existing negative association between the two [17,18]. This is an effect known as the 'reversal paradox' [11].
To illustrate this geometrically, blood pressure, birth weight and current weight can be represented as vectors, bp, bw and cw respectively, where the correlation between blood pressure and birth weight is nearly zero (corr(BP, BW) ≈ 0) -hence bp and bw are almost orthogonal. Assuming that bp and bw are orthogonal, the projection of bp(bp p ) on the plane spanned by bw and cw is also orthogonal to bw -see Figure 3. Within the multiple regression model BP = b BW BW + b CW CW, the partial regression coefficient for birth weight (b BW ) can be derived using the parallelogram rule by projecting bp p onto the vector bw The projection of y(y p ) onto the plane spanned by x and z comprises the appropriate proportions of the vectors x and z where the proportion of vector x is b X and the proportion of z is b Z . These proportions are derived by means of the parallelogram rule: y p is projected onto x parallel to the direction of z to obtain the proportion b X of x. Likewise, y p is projected onto z parallel to the direction of x to obtain the proportion b Z of z. The P-value for the partial regression coefficient of X(b X ), when Y is regressed on X whilst also adjusting for Z, is derived within vector geometry from the projection of y and x onto the subspace perpendicular to z (V ⊥z ). parallel to the direction of cw -see Figure 3. Consequently, b BW is not zero, but negative, due to the positive angles between bp and cw and between bw and cw. In terms of variables, b BW is negative due to the positive correlations between blood pressure (BP) and birth weight (BW), and between current weight (CW) and birth weight (BW).
In general, the smaller the angle between bp and cw, and the smaller the angle between bw and cw, the greater the length of the projection of bp p on bw along the vector cw.
In other words, the greater the positive correlation between blood pressure and current weight, and the greater the positive correlation between birth weight and current weight, the greater the absolute value of b BW . Moreover, while the partial regression coefficient for birth weight (b BW ) is not zero when blood pressure is simultaneously regressed on birth weight and current weight, birth weight nevertheless contributes nothing to 'explaining' the variance in blood pressure. This is because birth weight and blood pressure are uncorrelated -i.e. they are orthogonal in vector space.

A geometrical illustration of adjustment for weight gain in DOHaD
Since weight gain (WG) can be defined as the change in body weight from birth to the current time (i.e. current weight, CW, less birth weight, BW), all three variables are mathematically related such that each can be derived from the other two. Within vector geometry, this mathematical relationship means that the three vectors representing the three variables (bw, cw and wg) span only two dimensions (i.e. a plane). In statistical terminology, the three variables are collinear, and consequently only two (not all three) can be entered simultaneously as covariates within multiple regression analyses.
From a geometrical perspective, the equivalent to regressing blood pressure simultaneously on all three covariates would be to project the vector for blood pressure (bp) onto the plane spanned by the three vectors representing birth weight (bw), current weight (cw), and weight gain (wg). However, it is impossible to assess the length of the projection bp p to determine partial regression coefficients using the parallelogram rule, because the direction of this projection onto any one of the three covariate vectors (bw, cw, or wg) is now parallel to the direction of the plane spanned by the other two vectors. This dilemma results from these three covariates being multicollinear, and it can only be avoided by discarding one of the three vectors involved -equivalent to removing the corresponding variable from the regression model. Indeed, partial regression coefficients may only be determined for just two of the three covariates, since the space spanned by all three variables is only two-dimensional. Moreover, no matter which two covariates are chosen, the subspace upon which the outcome bp is projected remains the same: it is the plane V w , spanned by bw, cw and wg. For this reason, in all three of the possible regression models -where BP is regressed on: (i) BW and CW; (ii) WG and BW; or (iii) CW and GW -the bp p projections are identical, as are the R 2 values -see Figure 4.
To illustrate this situation, compare the following two models: where: in Model 1, blood pressure (BP) is regressed on birth weight (BW) and current weight (CW), with b 11 , b 12 , b 13 , and ε 1 being the model intercept, partial regression coefficients for birth weight and current weight, and the residual error, respectively; in Model 2, current weight is replaced by weight gain (WG) and the regression coefficients and residual error are now b 21 , b 22 , b 23 and ε 2 , respectively. Despite these differences, for the reasons mentioned earlier, these two models have the same degree of fit (R 2 ), and the residuals of both are identical (ε 1 = ε 2 ).
Using the parallelogram rule to derive partial regression coefficients for each model, consider the line L bw1 , which runs parallel to bw from the tip of bp p to intersect cw and wg -see Figure 4. The partial regression coefficient b 13 for CW in Model 1 is the length of cw intersected by L bw1 , i.e. the length of the vector OC divided by the length of cw. Similarly, the partial regression coefficient b 23 for WG in Model 2 is the length of the vector OG divided by the length of wg. Since cw = bw + wg, the line L bw2 is parallel to L bw1 and, by elementary trigonometry, the ratio of the lengths of OC and cw is identical to the ratio of the lengths of OG and wg -see Figure 4. Therefore, although birth weight has different partial regression coefficients in each model (b 12 ≠ b 22 ), the partial regression coefficients for current weight in Model 1 and weight gain in Model 2 are identical (b 13 = b 23 ).
When using vector geometry to determine the P-value for the partial regression coefficient of current weight in Model 1 or weight gain in Model 2, whilst adjusting for birth weight, it is necessary to identify the corresponding vector subspace perpendicular to bw. This is the same vector subspace for each model and is a plane, denoted V ⊥bw -see Figure 4. Thus, the partial regression coefficient Pvalue for current weight in Model 1 is derived by projecting bp and cw onto V ⊥bw . Similarly, the partial regression coefficient P-value for weight gain in Model 2 is derived by projecting bp and wgonto V ⊥bw . Since cw = bw + wg, the projection of cw or wg onto V ⊥bw is identical, albeit in the reverse direction to bw -see Figure 4. Consequently, regressing blood pressure on either current weight or weight gain, whilst also adjusting for birth weight, yields identical partial regression coefficient P-values for current weight in Model 1 and weight gain in Model 2.
In general, when birth weight is a covariate in multiple regression together with current weight, or any variable that is a linear combination of birth weight and current weight (such as weight gain), the partial regression coefficients for either of these will be identical in magnitude, as will their respective P-values -even though the direction of the coefficients will depend on the nature of the linear relationship concerned.
Finally, we now consider a third model, where blood pressure (BP) is simultaneously regressed on weight gain (WG) and current weight (CW): Model 1 includes birth weight (bw) and current weight (cw) as covariates; Model 2 includes birth weight (bw) and weight gain (wg) as covariates where b 31 , b 32 , b 33 and ε 3 are the intercept, the partial regression coefficients for weight gain and current weight, and the residual error, respectively. We know that Models 1 to 3 have identical R 2 values and identical residuals (i.e. ε 1 = ε 2 = ε 3 ). Furthermore, it can be shown that the partial regression coefficients for birth weight in Model 1 and weight gain in Model 3 are identical (b 32 = -b 12 ) with identical P-values. Thus, when current weight is a covariate in multiple regression analyses together with either birth weight or weight gain, the absolute partial regression coefficients for birth weight and weight gain are identical, as are their P-values, model fit and, hence, the proportion of variance explained.

Discussion
Lucas et al. [9] have previously discussed the algebraic relationship of regression coefficients amongst the three multivariable models presented above. However, in this article, we used vector geometry to demonstrate why these three models are effectively equivalent. Not only do the partial regression coefficients exhibit algebraic relationships, but the coefficient P-values and the variances explained are identical. The crucial issue, therefore, remains the interpretation of these models. For instance, when adjusting for birth weight it is impossible to differentiate between the effects of current weight or weight gain on blood pressure, since either covariate gives rise to identical coefficient P-values and an equivalent proportion of outcome variance explained. Conversely, when adjusting for current weight, the impact of weight gain on blood pressure is identical to that of birth weight, albeit in the opposite direction. For these reasons, the apparent finding that weight gain has an 'independent' statistical relationship with blood pressure may not reflect any genuine aetiological relationship.
From a clinical viewpoint, higher weight gain is equivalent to higher current weight if one adjusts for birth weight (i.e. holds birth weight constant). Under these circumstances, arguing that weight gain is related to blood pressure is equivalent to arguing that current weight is related to blood pressure, which we know to be true. Furthermore, while adjusting for current weight tends to create a stronger inverse relationship between birth weight and blood pressure, it will also strengthen the positive relationship between blood pressure and current weight. It is therefore unclear whether it is current weight or weight gain that contributes to elevated blood pressure, or both. Indeed, the stronger relationship between current weight and blood pressure after adjusting for birth weight might be interpreted as either: (i) that the impact of weight is cumulative and linear; or (ii) that heavier people also have, on average, larger birth weights. In the latter scenario, adjusting for birth weight would be interpreted as removing its 'protective' effect on blood pressure, thereby increasing the strength of its relationship with current weight. An alternative interpretation of the same regression model would be that, holding current weight constant, those with greater weight gain must have a lower birth weight, and hence the greater the weight gain the higher the blood pressure.

Conclusion
As we have seen using vector geometry, weight gain can be a proxy for either current weight (by adjusting for birth weight) or birth weight (by adjusting for current weight).
(a) The two-dimensional plot of three subjects with measure-ments of two variables: systolic blood pressure [SBP] and body height in variable space; (b) the three-dimensional plot of the same data in subject space Figure 6 (a) The two-dimensional plot of three subjects with measurements of two variables: systolic blood pressure [SBP] and body height in variable space; (b) the three-dimensional plot of the same data in subject space. Consequently, the role of weight gain in many of the regression models commonly adopted to compare the pre-and post-natal developmental origins of health and disease is essentially ambiguous.

Competing interests
The author(s) declare that they have no competing interests.
The projection of bp on V w and V ⊥cw is vector bp p and bp ⊥cw respectively Figure 5 The projection of bp on V w and V ⊥cw is vector bp p and bp ⊥cw respectively. Since cw = bw + wg, the projections of bw and wg on V ⊥cw (bw ⊥cw and wg ⊥cw respectively) will be in opposite directions (though parallel). Therefore, if the angle between bw ⊥cw and bp ⊥cw is φ, the angle between wg ⊥cw and bp ⊥cw will be (π -φ). From elementary trigonometry: cos(φ) = -cos(π -φ). Hence, in Model 3, after adjustment for current weight (CW), the P-value for weight gain (WG) is identical to that for birth weight (BW) in Model 1. From the two lines, L cw1 and L cw2 running parallel to cw, it is apparent that the absolute values of the two partial regression coefficients for birth weight (BW) and weight gain (WG) are identical.