Dealing with ‘the spectre of "spurious" correlations': hazards in comparing ratios and other derived variables with a randomization test to determine if a biological interpretation is justified

Abstract

We note the continuing widespread use of regressions of mathematically dependent (derived or confounded) variables [e.g. comparisons of standardized ratios: X/Y versus Z/Y, or the part versus the whole: X versus (X + Y)] in all disciplines of biology and ecology. These may lead to 'spurious' correlations as even random numbers would produce similarly statistically significant results. We developed a randomization test to determine the probability of obtaining the observed correlation coefficient by chance alone. Only if the regression remains statistically significant after the results of the randomization test are taken into account (random coefficient subtracted from the observed coefficient) is any sort of biological interpretation justified. We demonstrate that the often compared expressions, ln[(Y + X)/X] (e.g. relative growth rate) versus lnX (e.g. original mass), are negatively and significantly correlated whatever values of X and Y are used; thus, conclusions from such comparisons that seedlings from smaller seeds grow faster than from larger seeds are spurious. Derived variables are only likely to be meaningfully correlated if X and Y are correlated from the outset and the researcher can then decide if the 'actual' derived relationship is worth reporting.

Autumn migrating passerines at a desert edge: Do birds depart for migration after reaching a threshold fuel load or vary it according to the rate of fuel deposition?

“…Yet, one has to bear in mind that the results of these analyses could produce spurious results (Williams et al, 2022). This is because fuel deposition rate, an independent factor in the analyses, is calculated using both departure fuel load and stopover duration, which are dependent factors in the same statistical models.…”

Contrasting leaf thickness & saturated water content explain diverse structural/physiological properties of arid species

“…1a,b). ( Q + D )/ A is equivalent to ( Qv•z + D v •z ) and Q / A to Qv•z so that their significant relationship with Q v and z is no surprise as they are correlated by definition (Williams et al 2022). Garnier & Laurent (1994) reported a positive slope of 0.5 between ( D + Q )/ A (= 1 + Q v / D v ) and z for grasses while ours was 1.2, no doubt because of the additive effect of the associated increasing Q v with rising succulence.…”

“…Here, we examine, theoretically and empirically, how three standard measures of succulence relate to their possible underlying properties, such as leaf thickness ( z ), and dry matter and saturated water contents on the basis of mass ( D m and Q m ) and volume ( D v and Q v ). We were aware of the problem of comparing confounded variables as they will be correlated by definition (Williams et al 2022), unless they remain constant in a particular study. Thus we note, for example, that the type of relationship of D v with specific leaf area [SLA = 1/( D v •z)], proposed as of potential interest by Wilson et al .…”

High variation in handling times confers 35‐year stability to predator feeding rates despite community change

“…Pearson (1897) referred to the nonzero correlation of ratios involving uncorrelated numerator and correlated denominator variables as being spurious (but see Haig [2003] for a discussion of this term). When inference is being made regarding the relationship of the two numerator variables, the issue is indeed a major problem that has plagued—and continues to plague—diverse scientific disciplines (e.g., Atkinson et al, 2004; Håkanson & Stenström‐Khalili, 2009; Jackson & Somers, 1991; Kenney, 1982; Williams et al, 2022), leading many to infer a relationship between measured variables when in fact none exists. Closely related issues can confound the interpretation of regression analyses (Freckleton, 2011; Spearman, 1904) as, for example, in the debate over density‐dependent population regulation (Wolda et al, 1994).…”