Growth models and the expected distribution of fluctuating asymmetry

Abstract

Multiplicative error accounts for much of the size‐scaling and leptokurtosis in fluctuating asymmetry. It arises when growth involves the addition of tissue to that which is already present. Such errors are lognormally distributed. The distribution of the difference between two lognormal variates is leptokurtic. If those two variates are correlated, then the asymmetry variance will scale with size. Inert tissues typically exhibit additive error and have a gamma distribution. Although their asymmetry variance does not exhibit size‐scaling, the distribution of the difference between two gamma variates is nevertheless leptokurtic. Measurement error is also additive, but has a normal distribution. Thus, the measurement of fluctuating asymmetry may involve the mixing of additive and multiplicative error. When errors are multiplicative, we recommend computing log E(l) − log E(r), the difference between the logarithms of the expected values of left and right sides, even when size‐scaling is not obvious. If l and r are lognormally distributed, and measurement error is nil, the resulting distribution will be normal, and multiplicative error will not confound size‐related changes in asymmetry. When errors are additive, such a transformation to remove size‐scaling is unnecessary. Nevertheless, the distribution of l − r may still be leptokurtic. © 2003 The Linnean Society of London, Biological Journal of the Linnean Society, 2003, 80, 57–65.

Associations between Floral Asymmetry and Individual Genetic Variability Differ among Three Prickly Pear (Opuntia echios) Populations

“…This assumption has been questioned [34,35], but was tested explicitly in our dataset [23]. By comparing alternative models, we were able to show that, at least for the flower petals of our prickly pear cacti, a normal distribution of developmental errors fitted best [23], (see also [36] for other examples in petals).…”

A Longitudinal Study of Changes in Fluctuating Asymmetry with Age in Jamaican Youth

“…Why these studies of human FA give seemingly contradictory patterns is unclear, but it is probably a combination of many factors such as population differences, socioeconomic differences, different kinds of traits, changes in confounding variables with age, and methodological differences [3,20,27]. It is possible that some authors have transformed data inappropriately, which can give the appearance of a decrease in FA with body size [12,13]. Although not statistically significant, variation in measurement error with trait size and across years suggests that measurement error may have been higher for the youngest children in our sample (see Materials and Methods), but such a trend cannot explain the consistent patterns of change in FA that we observed.…”

“…Most previous studies of developmental changes in FA have been conducted on traits that are molted and re-grown, such as feathers [5,[8][9][10] or components of an exoskeleton [4,11]. For traits like bones and stems that instead grow by adding to what is already present, developmental errors may accumulate because asymmetry at one point in time directly affects the degree of asymmetry at time t + 1 [3,12].…”

Fluctuating Asymmetry in Two Common Freshwater Fishes as a Biological Indicator of Urbanization and Environmental Stress within the Middle Chattahoochee Watershed

“…Using this total area, we calculated percent total area represented by each code. Because values of 15% imperviousness and greater have been shown to cause water quality impairment [47], we summed the total area percentages for all codes (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) representing imperviousness above 15%. Similarly, we used an impervious surface raster (file name: Ga. Land Use Trends Impervious Surface Cover 2008 from www.georgiaspatial.org) to quantify percent impervious surface and used the North American Equal Area Conic projection for accuracy.…”

“…Unlike directional asymmetry and antisymmetry (see [6]), FA results in random deviations from bilateral symmetry that yield symmetrical frequency distributions [4,7,8] and is usually not considered to have a genetic component (but see [9]). Therefore, FA can be used to indirectly quantify developmental instability of an organism due to environmental influences.…”