Difficulties in summing log-normal distributions for abundance and potential solutions

The log-normal distribution, often used to model animal abundance and its uncertainty, is central to ecological modeling and conservation but its statistical properties are less intuitive than those of the normal distribution. The right skew of the log-normal distribution can be considerable for highly uncertain estimates and the median is often chosen as a point estimate. However, the use of the median can become complicated when summing across populations since the median of the sum of log-normal distributions is not the sum of the constituent medians. Such estimates become sensitive to the spatial or taxonomic scale over which abundance is being summarized and the naive estimate (the median of the distribution representing the sum across populations) can become grossly inflated. Here we review the statistical issues involved and some alternative formulations that might be considered by ecologists interested in modeling abundance. Using a recent estimate of global avian abundance as a case study (Callaghan et al. 2021), we investigate the properties of several alternative methods of summing across species’ abundance, including the sorted summing used in the original study (Callaghan et al. 2021) and the use of shifted log-normal distributions, truncated normal distributions, and rectified normal distributions. The appropriate method of summing across distributions was intimately tied to the use of the mean or median as the measure of central tendency used as the point estimate. Use of the shifted log-normal distribution, however, generated scale-consistent estimates for global abundance across a spectrum of contexts. Our paper highlights how seemingly inconsequential decisions regarding the estimation of abundance yield radically different estimates of global abundance and its uncertainty, with conservation consequences that are underappreciated and require careful consideration

Difficulties in summing log-normal distributions for abundance and potential solutions.

The log-normal distribution, often used to model animal abundance and its uncertainty, is central to ecological modeling and conservation but its statistical properties are less intuitive than those of the normal distribution. The right skew of the log-normal distribution can be considerable for highly uncertain estimates and the median is often chosen as a point estimate. However, the use of the median can become complicated when summing across populations since the median of the sum of log-normal distributions is not the sum of the constituent medians. Such estimates become sensitive to the spatial or taxonomic scale over which abundance is being summarized and the naive estimate (the median of the distribution representing the sum across populations) can become grossly inflated. Here we review the statistical issues involved and some alternative formulations that might be considered by ecologists interested in modeling abundance. Using a recent estimate of global avian abundance as a case study (Callaghan et al. 2021), we investigate the properties of several alternative methods of summing across species' abundance, including the sorted summing used in the original study (Callaghan et al. 2021) and the use of shifted log-normal distributions, truncated normal distributions, and rectified normal distributions. The appropriate method of summing across distributions was intimately tied to the use of the mean or median as the measure of central tendency used as the point estimate. Use of the shifted log-normal distribution, however, generated scale-consistent estimates for global abundance across a spectrum of contexts. Our paper highlights how seemingly inconsequential decisions regarding the estimation of abundance yield radically different estimates of global abundance and its uncertainty, with conservation consequences that are underappreciated and require careful consideration

Parameter values used in the simulation study and global bird abundance study.

Log-normal parameter values μ and σ used in both the simulation study and the re-analysis of Callaghan et al.’s global bird abundance data. (PDF)</p

Zika virus dynamics: When does sexual transmission matter?

The Zika virus (ZIKV) has captured worldwide attention with the ongoing epidemic in South America and its link to severe birth defects, most notably microcephaly. ZIKV is spread to humans through a combination of vector and sexual transmission, but the relative contribution of these transmission routes to the overall epidemic remains largely unknown. Furthermore, a disparity in the reported number of infections between males and females has been observed. We develop a mathematical model that describes the transmission dynamics of ZIKV to determine the processes driving the observed epidemic patterns. Our model reveals a 4.8% contribution of sexual transmission to the basic reproductive number, R0. This contribution is too minor to independently sustain an outbreak but suggests that vector transmission is the main driver of the ongoing epidemic. We also find a minor, yet statistically significant, difference in the mean number of cases in males and females, both at the peak of the epidemic and at equilibrium. While this suggests an intrinsic disparity between males and females, the differences do not account for the vastly greater number of reported cases for females, indicative of a large reporting bias. In addition, we identify conditions under which sexual transmission may play a key role in sparking an epidemic, including temperate areas where ZIKV mosquito vectors are less prevalent