Explore Paradox stock photos. Download royalty-free images, illustrations, vectors, clip art, and video for your creative projects on Adobe Stock. Immune against the transportation paradox. An immune cost matrix satisfies z(C,a,b) ≤ z(C,ˆa,bˆ) for all supply vectors a and ˆa with a ≤ a and for all corresponding demand vectors b and bˆ with b ≤ ˆb. So regardless of the choice of the supply and demand vectors, the transportation paradox does not arise when the cost matrix C is.
BackgroundNumerous 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. BackgroundNumerous 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 ; diabetes ; arteriosclerosis ; and obesity. 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' (DOHaD).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). 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. 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.
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 post-natal events on disease in later life, Lucas et al. 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 , 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.
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. Representation of variables as vectorsVector geometry is a very useful tool for providing non-statisticians with an intuitive understanding of statistical theory, such as correlation and regression ,.
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 regressionWhen 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. 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. (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.
Multiple regressionRegressing 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. 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).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 – see Figure. A geometrical illustration of adjustment for current weight in DOHaDWhen 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.
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 ,. This is an effect known as the 'reversal paradox'.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. 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 parallel to the direction of cw – see Figure. 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). A geometrical illustration of multiple regression for blood pressure ( BP, represented by vector bp), regressed simultaneously on birth weight ( BW, represented by vector bw) and current weight ( CW, represented by vector cw).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 DOHaDSince 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. 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). Pro evolution soccer 2012 pc download torrent. 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. Model 1 includes birth weight ( bw) and current weight ( cw) as covariates; Model 2 includes birth weight ( bw) and weight gain ( wg) as covariates. In determining the partial regression coefficients, blood pressure ( bp) is projected ( bp p) onto the plane ( V w) spanned by bw, cw and wg. The point O is the origin of the vectors bp, bp p, bw, cw and wg; C and G are the intersections of the line L bw1 (running from the end of bp p parallel to bw) with vectors cw and wg. The line L bw2, crossing the tips of cw and wg, runs parallel to L bw1.
DiscussionLucas et al. 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. Basic geometric toolsThe most common form of geometry in clinical research occurs in what is termed variable space, illustrated for instance by a scatter plot. In the scatter plot of two variables, say X and Y, each with n independent observations ( X 1.
X n) and ( Y 1. Y n), there will be n points in 2-dimensional space (i.e. The axes represent variables X and Y, and the points are the observations made on each subject. In place of using variables as axes, the same data may be displayed in what is termed 'subject space', using subjects as the axes (of which there would now be n) and the variables X and Y become two points (in n-dimensional space). By connecting the origin with each point, X and Y become vectors in n-dimensional space, with coordinates ( X 1. X n) and ( Y 1.
Y n) respectively.Figure and illustrate the difference between variable and subject space using a numerical example. Suppose the body height and systolic blood pressure of three subjects A, B and C are measured.
In variable space, the data are displayed as three points representing the three subjects in a two-dimensional scatter plot (Figure ). In contrast, in subject space, the data are displayed as two points representing the two variables in a three-dimensional scatter plot (Figure ). (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.Although it is impossible to visualize n-dimensional space, we only need two dimensions (i.e.
A plane) to visualize the relative relationship between the two vectors representing X and Y. We effectively 'drop' the original axes, retaining only the relative relationship between the vectors representing the variables. In general, the number of dimensions needed to represent variables in subject space is no greater than the number of variables. Whilst it remains impossible to visualize four or more dimensions, using this condensed form of vector space, only two dimensions are required to illustrate the principles of simple regression, and only three dimensions are required to illustrate the principles of multiple regression. It is therefore useful to represent the original variable, e.g.
X, as scaled vector, x, where each original data point, X i, is transformed to x i such that the length of the vector ( x ) is equal to the standard deviation (SD) of the original variable. This is achieved using the following formula:x i = X i − ( ∑ i = 1 n X i n ) n − 1. The projection of bp on V wand 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. Huxley RR, Shiell AW, Law CM. The role of size at birthand postnatal catch-up growth in determining systolic blood pressure: a systematic of the literature.
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$begingroup$ I guess the first remark solves my issue; i.e., a 'vector of vectors' is a slippery term that does not behave like a regular vector, even though it is used fairly often to explain matrix multiplication (among other topics like orthogonal projection). The second remark is probably a misunderstanding of what I wrote, or perhaps of my use of transposes for $vec x$ as to not clutter the page with columns: when dotting $vec a$ and $vec b$ as $vec acdot vec b$, both vectors ought to be of identical dimensions, but of course, representing it as a matrix-vector product transposes $vec a$. $endgroup$–Nov 18 '18 at 22:06.
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