Interestingly, the WALD, although based on a mechanistic representation of dispersal by wind (Katul et al. 2005), was in the best-fit group of only 15 of the 55 wind dispersal datasets and by this metric performed better than only the Exponential, Gaussian and Lognormal.


The r 2 values showed generally very good fit with high values (in terms of the median value and number of datasets for which r 2 ≥0.7; Table 1) for most functions except the Gaussian and Lognormal, and the patterns of r 2 values followed those of the AICc values.


Fig. 1 shows some example plots of data with the fitted functions.

Tables S4 and S5 present the fitted parameter values for all well-fitting functions for each of the 168 datasets and the supplementary data describing the species and study system.


The Exponential power (ExP; ΔAICc≤4 for 111 cases) and Log-sech (LogS; 119) clearly outperformed the other functions (but note these numbers are ≪168).

The ExP can reduce to an Exponential (b=1) or Gaussian (b=2), and can fit a fat tail (b<1) or a thin tail (b>1).

Of the datasets selected as showing good general fit to the ExP (i.e. r ≥0.7), 117 had a b<1, and 27 had b>1, suggesting that a majority of kernels are fat-tailed.

The LogS is always fat-tailed (power-law tail) and flattens (for b<1) or decreases (b>1) close to 0 distance.

Of the 142 datasets with good general fit to the LogS, 90 had b<1 and 52 b>1.


Although the case study composition of the best-fit group differed among the functions, there was no bias in comparison to the full set of case studies in terms of dispersal mode (χ 2 = 45.9, df = 40, p=0.24), growth form (χ 2 = 21.7, df = 20, p=0.36), plant height (Kruskal Wallis H = 9.6, df = 10, p=0.48) or seed mass (H = 12.5, df = 10, p=0.25).

We also asked whether the study design affected the set of best-fit functions (e.g. was a function allowing a fatter tail less likely to fit kernels measured over shorter distances?), but the maximum distance over which dispersal was measured did not differ among the best-fit groups (H = 15.4, df = 10, p=0.12).


The fitting of the ExP and LogS functions across grouped datasets provided generalised dispersal kernels for a number of well-studied plant growth form/dispersal mode combinations (Table 3, Fig. 2).

Across the 10 combinations both the ExP and LogS gave similar and generally good fits, with r 2 ranging from 0.39-0.97 (median 0.73, r 2 <0.7 for 3 combinations) for the former and 0.32-0.97 (median 0.78, r 2 <0.7 for 3) for the latter; although the patterns of r 2 values across the form/mode combinations differed between ExP and LogS (note that it is not appropriate to compare the fits of the ExP and LogS using AIC as they were fitted to different sets of case studies).

Similar results were seen for the sub-divisions according to seed mass or plant height classes, with r 2 <0.7 for 12 of 38 groups across the ExP and LogS functions.


Median dispersal distances of the fitted functions were in the order Trees (ExP mean of the medians = 20.9 m) > Shrubs (3.05 m) > Herbs and Graminoids (0.38 m).

Vertebrates (excluding rodents; ExP mean of the medians = 22.5 m) dispersed seeds further than Rodents (5.45 m), which dispersed seeds similar distances to Wind for dispersules with an appendage (8.75 m), while dispersal by Ants (0.87 m), Ballistic (0.45 m) and Wind for dispersules with no appendage (0.20 m) resulted in the shortest median distances.

The same patterns were found for the 95 th percentile and for the LogS function (Table 3).

While these means are somewhat confounded as growth form and dispersal mode were not found in all possible combinations, more specific comparisons show the same patterns.

For Trees, dispersal distances are of the order Vertebrate > Wind (with appendage) > Rodent.

For Shrubs, Vertebrate > Ant

. However, for Herbs, Wind-dispersal of dispersules with appendages did not result in longer dispersal distances than by Wind without appendages, Ant or Ballistic.


Taller Herbs, Graminoids or Trees had greater median and 95 th percentile distances than shorter plants within the same growth form/dispersal mode combination, and these differences were substantial (Table 3; Fig 2c,e,j; Fig S1c,e,j) and consistent between the ExP and LogS fits.


Seed mass did not show a consistent pattern of effects on animal-dispersed kernels.

For the ExP fits, lighter seeds were dispersed further for Ant-dispersed Herbs (Fig. 2a), Vertebrate-dispersed Shrubs (Fig. 2g) and Rodent-dispersed Trees (Fig. 2h), but the pattern was reversed for Vertebrate-dispersed Trees (Fig. 2i) and there was little difference between seed mass classes for Ant-dispersed Shrubs (Fig. 2f).

Furthermore, the patterns changed to some degree when using the LogS function in that this indicated heavier seeds dispersed further in Ant-dispersed Herbs (Fig. S1a) and Ant-dispersed Shrubs (Fig. S1f) and no pattern for Rodent-dispersed Trees (Fig. S1h).


This indicates a strong pattern for plant height effects on wind dispersal, but an inconsistent pattern for seed mass effects on animal dispersal.


The two functions ExP and LogS described similar dispersal kernels for each combination of datasets, as described above.

However, these functions have different intrinsic shapes (Nathan et al. 2012) and were fitted to slightly different datasets.

The modelled median and 95 th percentile dispersal distances therefore differed between the ExP and LogS fitted to each combination of datasets (Table 1).

This difference for the median (in terms of the absolute % difference of the LogS value from the ExP value) ranged from 3% to 476%, with a median of 48%.

However, one function did not give a consistently higher or lower median value than the other.


In this paper we present standardised dispersal kernels for case studies representing a range of plant types across a wide geographic range, as well as generalised kernels for well-studied plant growth form/dispersal mode combinations.

While it would be preferable to determine dispersal kernels directly in the system(s) one is studying, dispersal is notoriously difficult and resource-consuming to measure (Bullock, Shea & Skarpaas 2006).

It is indicative of this difficulty that in contrast to our collation of 168 dispersal datasets for 144 species, the COMPADRE database of plant demographic matrices currently includes 637 species and 6242 matrices (http://www.compadre-db.org/Data/Compadre; accessed 02/06/16) and has many more to be added.

Potential uses of our synthesis of empirical dispersal data include: 1) Choosing appropriate dispersal functions in generic modelling studies; 2) Selecting informative dispersal kernels for one's empirical study system; 3) Using representative dispersal kernels in cross-taxon comparative studies.

Below we expand on each of these uses and discuss points that arise, including those relating to data quality.


Many functions are used to describe empirical dispersal kernels and the choice for a particular study often seems to arise from personal preference or experience.

The Gaussian and Exponential have statistically mechanistic basis, in that they describe a movement process (Petrovskii & Morozov 2009).

Some other functions have a similar philosophy.

For example, the 2Dt is a continuous mixture of Gaussian kernels with variance parameters distributed as the inverse of a Gamma distribution (Clark et al. 1999).

However, the functions used are often chosen as 2-parameter equations allowing high leptokurtosis and long (sometimes fat) tails, with little consideration of the underlying mechanism.

This suggests that no one function will be the best as they are generally simple (to allow fitting) statistical descriptions of a complex process.


However, few studies compare the fit of multiple functions to dispersal data.

Martinez and Gonzalez-Taboada (2009) fitted Exponential, 2Dt, Log-normal and Weibull functions (as well as a Weibull-Exponential mixture) to a number of vertebrate and wind-dispersed trees and shrubs in a forest system and found different functions performed better for different species, with no relation between dispersal mode and best-fit function.

Clark et al. ( 2005) fitted Exponential, Gaussian, Inverse Power (i.e. a Power Law function) and 2Dt functions to dispersal data for different trees in a forest plot.

They found Gaussian and 2Dt functions best fitted wind- dispersed seeds, while the Inverse Power best fitted the bird- and monkey-dispersed seeds.

Greene et al. ( 2004), again considering trees from a variety of studies, found the Lognormal performed better than the 2Dt or Weibull.


For this study, we opted for an objective selection of dispersal functions, using 11 listed in a review by Nathan et al. ( 2012).

This approach gave insights into the performance of different functions when fitted to a wide range of dispersal kernels.

The Log-sech (LogS) and Exponential Power (ExP) showed the best fits to the datasets.

It is difficult to pinpoint why these two functions perform best, but this finding suggests these might be useful general functions to use in models.

However, these functions have quite different properties and histories of use in dispersal studies.


The LogS has been used very rarely, just in one study of dispersal of birds (Van Houtan et al. 2010).

The LogS becomes the Cauchy for b=1 (Nathan et al. 2012), which is occasionally used in seed dispersal studies (Seri, Maruvka & Shnerb 2012; Munoz et al. 2013).

The LogS also has unattractive properties, such as no mean value for b≥1, nor does it have moments.


The ExP has been widely used on a variety of taxa (Nathan et al. 2012) and on many seed dispersal data since Clark et al. ( 1998).

It has useful properties such as a mean value (eqn 2), has moments and it reduces to the Exponential or Gaussian for certain values of b.


An interesting finding was that of the 144 datasets showing a good fit of the ExP (i.e. r 2 >0.7), 117 had b<1, which indicates a fat tail.

Similarly nine of the 10 ExP functions estimated for the major growth form/dispersal mode groupings had b<1, as did 17 of the 19 plant height/seed mass subdivisions of these groupings.

As an illustration of the influence of the b parameter, 57 of the 168 datasets had both the ExP and the Exponential in the best-fit group and had b<1 for the ExP (10 had b>1).

The median value of the ExP b in these 57 datasets was 0.445 and the 95 th percentile (as a measure of the tail) of the fitted function was a median of 234% more than that of the fitted Exponential.

Fat-tailed kernels are a popular concept in dispersal ecology (Klein, Lavigne & Gouyon 2006; Nathan et al. 2008), but this property of the ExP allows an explicit test whether empirical kernels are indeed fat-tailed.


Because of these properties, the ExP might be more generally useful than the LogS.


Our suggestion that no one function would be expected to fit all datasets well is supported by the fact that no function was always in the best-fit group for each dataset.

However, there was strong differentiation among the functions in how well they were represented in the best-fit groups.


Certain functions are very popular in the wider dispersal literature, such as: the Lognormal (Greene et al. 2004), which performed very badly here; or the 2Dt (Jones & Muller-Landau 2008) which performed fairly well. It is particularly interesting that the WALD performed poorly for wind-dispersed datasets.


The WALD is an analytical formulation of a wind-dispersal model (Katul et al. 2005), and has been used widely as a mechanistic model to generate dispersal kernels (Skarpaas & Shea 2007; Bullock et al. 2012).

While its poor performance when used in this paper as a fitted function does not negate such uses, it does suggest that more validation is needed.


The WALD has a density mode at a distance >0 and the fact that many of the 55 wind-dispersed datasets showed monotonically declining density with distance may explain its poor performance.


It would be useful to revisit the WALD theory to analyse why it fails, such as the simplifying assumptions used upon the Langevin and the Fokker-Planck equations (Katul et al. 2005).


Our analysis suggests the Gaussian is a very poor representation of the dispersal process.

This point has long been raised in the dispersal literature (Wallace 1966), but our analysis provides an objective affirmation.

While its mathematical properties make the Gaussian popular in mathematical representations of dispersal, the low kurtosis and thin tail can lead to inaccurate predictions about dispersal distances and spatial dynamics (Clark et al. 1998; Klein, Lavigne & Gouyon 2006).


The second one-parameter function, the Exponential, has had a similarly negative press to the Gaussian (Bullock & Clarke 2000; Nathan et al. 2012), but it performed much better than the Gaussian. While it did not perform as well as almost all the 2- parameter functions (but better than the Lognormal - Table 1), our analyses suggest the Exponential has some credence as a simple function to use both as a comparator for more complex functions when fitting data, and as a straightforward function in mathematical modelling studies.


It might be argued that the good fit in many cases may reflect data quality or inadequate sampling at long distances, in that the tail of the kernel may not be well described.

This point is countered by our analysis showing that the distance over which dispersal was measured in a study did not differ among the best-fit groups for the set of functions.

This does raise the issue that while the tail of the dispersal kernel is often discussed in the literature and has a clear meaning in mathematical formulations of the kernel (Klein, Lavigne & Gouyon 2006), it is not clear what exactly the tail is in empirical measures of dispersal, or what a sufficient measurement of this tail looks like.

Portnoy & Willson (1993) defined the tail as "the set of distance categories beyond the last clear mode of the data".

By this definition, visual examination of each dataset suggested each represented a kernel tail to some degree (e.g. Fig. 1).


As dispersal is intrinsic to plant life histories, governing local and regional population dynamics, genetic structuring, evolutionary processes and community dynamics to name a few, knowledge of dispersal kernels should be a key aim in many studies.

The relative scarcity of such information is therefore limiting in ecology.

The individual case studies and the generic dispersal kernels presented in this paper are therefore a resource to help address this limitation.

A researcher might choose one or more case studies that match to some extent the characteristics of their study system and/or a generalised dispersal kernel which does the same.

While we provide generalised kernels for 10 growth form/dispersal mode combinations, there are a small number of case studies representing 13 other combinations.

The potential uses of such kernels are multiple.

If one has the luxury of choice among multiple case studies, selection could follow alternative criteria, such as fine-scale measures of short-distance dispersal when studying competition or the Janzen- Connell effect vs less detail, but measures over long distances, to assess ability to spread or persist in fragmented landscapes.


Furthermore, the ExP and LogS functions gave somewhat different generalised kernels, as illustrated by variation in the predicted medians and 95 th percentiles.

This partially reflects the intrinsic differences in the functions - the LogS has a power-law tail and a weak effect of distance close to the source, whereas the ExP represents a smoother decline with distance (Nathan et al. 2012).

But the differences also reflect uncertainty in our analyses due to variation among the datasets within each group in terms of the ecology of each system and the data-gathering approach.

Our parametrisation of both functions will allow researchers to use them in combination and represent this uncertainty in the kernel.


Each individual case study, and thus each generalised kernel, relates to a single dispersal mode.

Multiple dispersal agents may be involved in the dispersal of seeds from a plant or of an individual seed.

Thus the 'total dispersal kernel' (Nathan et al. 2008) of a plant or population may require combining kernels for multiple dispersal modes.

To do so one should convolve (Neubert & Parker 2004) the relevant dispersal distance probability density functions (pdf) - i.e. the dispersal location pdf/2πd.


Surprisingly, only three of the 107 studies considered dispersal by multiple modes: ballistic dispersal followed by ant dispersal (Beaumont, Mackay & Whalen 2009); and dispersal by vehicles vs that by wind (Arnold 1981; Bullock et al. 2003).

The same approach could be used if, say, different animal species disperse the seed and one has individual kernels for each animal vector (Lehouck et al. 2009).


Empirical dispersal kernels are useful for multi-species studies.

These might involve modelling how fast species may spread, or potential distributions, under a changing climate.

Current approaches use a small number of empirical datasets (Bullock 2012) or simple, pre-determined dispersal functions, such as the Exponential (Engler & Guisan 2009; Bocedi et al. 2014).

A broader range of empirically-determined kernels, applicable to a wide range of species, should allow more realistic and representative forecasts.


Another use would be to represent dispersal in comparative studies.

While plant demographic data has been used in comparative life history analyses (Salguero-Gomez et al. 2016), dispersal information has not been included.

Other comparative analyses use traits related to dispersal ability, such as plant height, seed mass and dispersal mode (Baeten et al. 2015).

The use of information on the kernel itself should help better integrate a more rounded understanding of dispersal into such studies.


This potential raises the issue of the how dispersal kernels vary according to plant and vector characteristics.

It should be noted that analysis of the differences among classifications are not the main focus of this paper, and the classifications are used more to give ecologists well-defined groups from which to draw dispersal kernels relevant to their needs.

We fitted generalised ExP and LogS functions for growth form and dispersal syndrome combinations and were able to sub-divide these further according to plant height (wind dispersal) or seed mass (animal dispersal) classes.


These functions were fitted to groupings of datasets each of which encompassed a wide range of systems, a diversity of data-gathering methods and variety in the plants and dispersal vectors studied.

For example, of the 30 papers (59 datasets) assessing non-rodent vertebrate dispersal, the majority studied dispersal by small to medium sized birds (15 papers, some papers studied more than one group), such as thrushes, tanagers or mockingbirds.

Larger birds such as toucans or cassowaries were covered in six papers, primates such as tamarins or spider monkeys in four and medium-sized omnivorous mammals such as possums and martens in three.


Bears, deer, fruit bats, iguanas and a large freshwater fish were studied in one paper each, and two papers studied a broad, unspecified group of vertebrate dispersers.

Thus, there is inherent variation among the datasets within each group.

Despite this, the functions fitted to each group had generally high r 2 values, being >0.7 in the great majority of cases and being significant at p<0.05 in all.


Thus we have confidence in the value of these generalised functions for wider use.

The collecting of further dispersal datasets would allow further subdivision of datasets according to key variables such as vertebrate type, local wind speed, size of wind- dispersal appendage, or vegetation type.


Because the datasets varied in terms of distance classes, maximum distance, seed densities and sampling methods, fitting functions across all the data with a and b varying according to group would not have been sensible or feasible.

However, comparisons of the fitted curves and the dispersal quantiles allowed tentative conclusions about differences among these groups.

Our data suggested average dispersal distances vary among growth forms in the order Trees > Shrubs > Herbs, Graminoids, and among dispersal modes in the order Vertebrates (excluding rodents) > Rodents, Wind (with appendage) > Wind (without appendage), Ant, Ballistic.

Taller plants within groups disperse seeds further by wind, but seed mass has an inconsistent relationship with dispersal distance (see also Thomson et al. ( 2011)).

Variation in whether or not lighter-seeded species disperse further than heavier-seeded species may arise if larger seeds are dispersed by larger, more mobile animals (Nathan et al. 2008) and also where a varying number of seeds are dispersed together in a fruit (Jordano 2000).


Our findings build on the analysis by Willson (1993), who carried out log-linear regression analysis of collated dispersal kernels (i.e. fitting an Exponential function).

Interestingly, she concluded wind- dispersed herbs with seed appendages dispersed further than herbs using ballistic or ant dispersal, and wind-dispersed trees and shrubs dispersed further than those dispersed by vertebrates.

These findings contradict ours and probably reflect the current availability of more data (60 papers compared to our 107) as well as the more sophisticated analytical methods that are now available.

However, this does demonstrate the need to continue gathering dispersal data directly as it is likely conclusions will develop as more case studies become available.


Any statistical synthesis of multiple datasets is bound by the number and quality of the studies used.

The 168 datasets represent a great effort by the researchers involved and we hope this synthesis does them justice by using these studies to provide general information for the wider use of dispersal kernels in ecological research.

However, the findings of this paper, especially the generalised dispersal kernels, are likely to be much improved upon if further dispersal data are gathered.

The fitted kernels provided here can also be used to inform direct seed dispersal studies.

Skarpaas, Shea and Bullock (2005) detailed a Monte Carlo approach to designing seed dispersal studies efficiently using information on the likely dispersal kernel, which could be provided by case studies and generalised kernels given in this paper.


While the relatively small number of studies is generally limiting, we can identify three particular areas of research that require focussed activity.

One is to use methods that allow better characterisation of the tail, such as molecular markers or tracking animal dispersers (Bullock, Shea & Skarpaas 2006).


The variety of methods used to gather data is likely to lead to uncertainty.

While we could not analyse this formally due to the fact that methods used are strongly linked to the characteristics of the system studied, analysis of biases due to methods would be useful.

We did find that estimates of tree seed dispersal by vertebrates were affected by the methods used.

A number of studies combined measures of seed retention, such as gut passage time, with data on movement by the animal vector.

Dispersal distances estimated by this method were in general further than dispersal measured by other methods, such as following vertebrate vectors, seed trapping or using molecular markers.

It is unclear however, whether this represents a bias in the data as methods are usually chosen to match the researcher's understanding of the system, such as knowledge that seeds are being carried a long way by vertebrate dispersers (see Cortes & Uriarte 2013).

While biases according to method are possible, they do not undermine the aim of this paper, which is to synthesise existing information of empirical dispersal kernels.


The second research area concerns our finding that dispersal of trees and by vertebrates are favourite study systems.

Studies on other dispersal modes and growth forms would provide much needed data for relatively understudied dispersal systems; two in particular are dispersal of seeds by humans or by water.


Finally, little is known about how dispersal varies among habitats.

In Fig. S2 we examine the seven species in our analysis for which kernels were measured in different habitat types (usually in different studies, but through the same dispersal mode).

The study methods and habitat contrasts were too inconsistent to allow patterns to be discerned.

It is likely that habitat type and structure will affect the dispersal process (Westcott et al. 2005; Trakhtenbrot et al. 2014) and so more formal contrasts of kernels among habitat types would allow characterisation of intraspecific variation in dispersal.


Furthermore, synthesis of such data would be aided by more complete presentation of the gathered data in papers.

Many papers we used presented data in graphs or other summary forms (e.g. summing across replicates).

Analyses would have more power if data are published in their raw form, and we would encourage researchers to do so.


In their recent meta-analysis of seed dispersal data, Bullock et al. ( 2017) observed that the inverse Gaussian probability density function (pdf) does not fit real dispersal kernel data (empirical seed shadows) for wind-dispersed species very well, despite it being related to the WALD model for dispersal by wind (Katul et al. 2005) [1][2].

Because of this they suggest that the theory behind the WALD model needs to be revisited.

We explain why, regardless of whether or not the WALD itself is a good model, this deduction is not logical. But first it is necessary to review some important definitions, since inconsistency in nomenclature has, unfortunately, resulted in some confusion.

The WALD model (Katul et al. 2005) was derived as a simplification of a highly complex mechanistic model of movement of unattached propagules from a point source by wind [2].

Katul et al. ( 2005) showed that it was reasonable at predicting frequency distributions of controlled-release dispersal distance data, using parameter estimates based only on plant traits and extant weather conditions.

Since their model has the same functional form as the inverse Gaussian distribution (often also called the Wald, although strictly the latter is only a special case of the inverse Gaussian: Seshadri 1993, 1999) they named their model the Wald Analytical Long-Distance (WALD) model [2][3][4].

The inverse Gaussian (or the Wald) and the WALD are not synonymous even though the derivation of both involves the first passage time problem for Brownian motion with drift (Schrödinger 1915) [5].

The WALD is only “analytical” in its original context of the movement of particles under specified wind conditions.

An inverse Gaussian can be fit to natural seed shadow data, without wind dispersal being implied at all, as just another phenomenological model that may happen to describe the data.

Using the original formulation of the WALD and assuming that it is reasonable to expect a real seed shadow to be approximated by propagules behaving as they do from experimental releases, it is then possible to obtain “quasi-mechanistic” (Jongejans et al. 2008) estimates of the WALD parameters [6].


We consider that there are two main reasons why it is illogical to argue that the WALD should be reconsidered simply because an inverse Gaussian does not fit seed shadow data for wind dispersed propagules.

The standard so-called “mechanistic” pdfs used to describe dispersal are derived from quite restricted assumptions about the motion of populations of objects: all individuals start from a single point source, travel under narrowly-defined environmental conditions and are identical in the properties that determine their movement.

This is the case for the various pdfs derived from Brownian motion, random walks and from wind dispersal.

But we know that these assumptions are inappropriate for the population of seeds dispersing from a plant over its lifetime, which is what we require for making inferences about spatial population dynamics.

We should distinguish between (i) models merely intended to summarise positions of naturally dispersed seeds from (ii) those intended to improve our understanding of movement processes – which we may test using artificial releases.

One model may not be sufficient for both purposes.

Even though we may, for simplicity, base simulations of a mechanistic movement model on mean parameter values justified by arguments that levels of variation are low or of little consequence (e.g. Nathan et al. 2002), this does not necessarily mean that we expect these assumptions to be valid for every real seed in a real location under a real set of environmental conditions [7].

Every model is simplistic to some extent, and this is certainly the case for current dispersal pdfs.

A mechanistic model of post-release movement, such as the WALD, describes just one component of a system that may be highly complex (Cousens et al. 2008) [8].

We give two illustrations.

Even if our mechanistic model is perfectly fitted to the particular aspect of the dispersal process that the model is based on, the final dispersed position of a seed may be the sum of a number of sequential random displacement processes.

For example, a fruit may burst explosively, producing randomly distributed seed landing points in still air conditions.

Any wind present at the time of liberation, strong winds subsequent to landing depending where the seed falls, and rain can all provide additional displacement components.

The pdfs of sums of several independent random variables do not typically belong to the same family as any of the constituent random variables.

This issue has been extensively explored in the probability literature (Feller 1971, Gnedenko & Kolmogorov 1968, Hughes 1995, Zolotarev 1986) [9][10][11][12]. If we consider a class of random variables for which the densities of each differ only in terms of a centering parameter and a horizontal scale parameter, the class is called “stable” if the sum of two independent random variables from the class also belongs to the class.

Although the classes of stable heavy-tailed random variables with infinite variance have also been definitively determined, it has been established that the normal (or Gaussian) pdf is the unique stable class with finite variance.

Adding two independent normal variables produces a normal variable (its mean being the sum of the means of the constituents, and its variance the sum of the variances of the constituents).

In contrast, for example, it is easily demonstrated by integration that the sum of two independent random variables with identical one-dimensional Laplace densities (k/2)exp(–k|x|) produces a random variable with density (k/4)(1+k|x|)exp(–k|x|), which looks very different to a Laplace density.


Instead of seed displacements reflecting a number of concatenated individual displacement events for each seed, there may be displacements by multiple vectors, leading to a superposition of dispersals by each vector.

Significant dispersal by a second vector not well modelled by the preferred mechanistic model for the first vector also compromises the ability of the mechanistic model to describe the full dispersal process.

Clark et al. ( 1998) showed that the combination of two exponentially-dispersed sub-populations, representing long- and short-distance dispersal units, produced spatial population dynamics resembling those of a fat-tailed distribution rather than an exponential [13].

In short, then, the value of a mechanistic model in fitting field data rests not only on the aptness of the model for the dispersal process that it captures, but on the absence of significant contribution from other sequential and/or concurrent dispersal mechanisms.

So it is hardly surprising that a real seed shadow may be totally different in shape from a highly simplistic model of one component (e.g. Neubert & Parker 2004) [14].

It is worth noting that the WALD “mechanistically based” density fitted Aleppo Pine dispersal data much better when it was combined with other functions describing the local environment (Schurr et al. 2008) [15].

Moreover, we do not expect some of the modelled mechanisms to be appropriate in a particular circumstance, so in itself the fact that a pdf is “mechanistically-based” does not necessarily give it some sort of higher status than a purely empirical pdf.

For example, we do not expect most animals dispersing seeds to follow a random walk or wind-dispersed seeds to be under Brownian motion.

Ergo, it does not add any additional validity to (say) a multi-generational spatial population dynamics model or to a simulation model used to determine optimal sampling strategies (Skarpaas & Shea 2007), for the dispersal function to be a “mechanistic” pdf [16].


To address the lack of information about the shape and extent of real dispersal kernels, Bullock et al. ( 2017) synthesized empirical information on seed dispersal distances.

Testing the fit of a variety of probability density functions, they found no function was the best-fitting for all datasets but some outperformed others.

Cousens, Hughes and Mesgaran (2017) focus on their specific finding of the generally poor fit of the WALD function to wind dispersal data and use this to argue that mechanistically derived functions would not be expected to fit data particularly well.

We agree in part with this argument and discuss the issues that may lead to poor fit, including the simplifying assumptions of the WALD and the complexity of the dispersal process.

We explain the fundamental linkage between the mechanistic form of the WALD and the derived function used for fitting to data.

We demonstrate however, that the logic that a mechanistically based function could fit to data is valid, under the hypothesis that it encompasses the key processes determining the dispersal kernel.

This argument is supported by the facts that: a) our analyses and others have shown the WALD performs well in a number of cases; and b) the WALD is the best fitting function for an example in which we simulate dispersal data using a realistic representation of variability in the wind dispersal process.


While there are reasons that mechanistically derived functions may not fit well to empirical data, they do in some empirical and simulated cases and this suggests they can capture the dispersal behaviour of real systems.

Mechanistic functions should be explored along with other more general functions when describing empirical data to investigate their simplifying assumptions and to add to our arsenal of functions for analysing dispersal data.

Analyses using these functions are critical if we are to move from simply describing the system in which the data were gathered to gaining more general insights into dispersal and predicting its consequences.


Our synthesis of plant dispersal studies (Bullock et al. 2017) analysed the rich empirical information on seed dispersal distances from studies on a wide variety of plants across many ecosystems worldwide.

A major aspect of our study was the fitting of a number of widely used probability density functions to these data sets, and a comparison of their performance.

We found that many of these straightforward functions described the empirical data well, but the performance of alternative functions varied.

No single function fitted all data sets well, but certain functions – the exponential power and log-sech – were the best performing on average.

We then fitted these two functions to data sets that represented combinations of dispersal mode and plant growth form.

These functions fit the combined data sets well despite variation among studies in empirical methods, local conditions, vegetation type and the exact dispersal process.

The analysis of alternative dispersal kernels and presentation of generalized kernels for growth form/dispersal mode groups provides a rich resource for ecologists, and we described ways by which this improved information might enrich spatial ecology.

We take this opportunity to correct typographical errors in our paper in the equations for: the 2Dt, which should be 𝑏−1/𝜋𝑎2 (1 + −𝑏/𝑑2/𝑎2); the gamma, which should be 1/2𝜋𝑎2Γ(𝑏) 𝑏−2/𝑑/𝑎 exp (−𝑑/𝑎); and the Weibull, which should be 𝑏/2𝜋𝑎2 𝑑𝑏−2exp (−𝑑𝑏/𝑎𝑏).


Among other probability density functions, our analysis included the WALD, which is based on a mechanistic description of seed dispersal by wind (Katul et al. 2005).

We noted the fact that when fitted to datasets describing seed dispersal by wind, the WALD performed poorly compared with many other functions, in that it was among the best-fitting functions in relatively few cases.

Cousens, Hughes and Mesgaran (2017) focus on this specific aspect of our paper and suggest one would not expect the WALD to fit empirical data particularly well. They give two closely interlinked reasons for this, which can be summarised as an argument that the simplifications of the WALD make it unlikely to fit the complexity of real data.

The WALD is based on an assumption of a single seed release height and unvarying environmental conditions (including wind speed) during the dispersal period.

We do not disagree with this argument in general – in fact we made a similar argument in our paper.

But, we show below how Cousens, Hughes and Mesgaran (2017) over-simplify the issues and therefore unnecessarily downplay the utility of mechanistically based functions in describing empirical data.


The WALD function is based upon simplifications to an idealised three-dimensional Lagrangian stochastic dispersal model for the trajectories of air particles having no mass in turbulent flows, where the drift and diffusion terms are determined by assuming a high Reynolds number and well-mixed conditions, modelled by a generalised Fokker-Planck equation (Thomson 1987; Katul et al. 2005).

The final function form for the WALD is derived to make further simplifying assumptions, which we discuss below.

These simplifications result in an inverse Gaussian distribution, which is considerably more useful to ecologists than some cumbersome stochastic differential equation which retains the full complexity of dispersal by wind.

The equation given in our paper (see also Nathan et al. ( 2012)) is the re-parameterised WALD suitable for fitting to dispersal data by finding solutions for the parameters a and b, whereby the probability density of seeds at distance d =  √𝑏/√8𝜋3𝑑5 exp (−𝑏(𝑑−𝑎)2/2𝑎2𝑑), which in this form is the 2-dimensional dispersal location kernel (see Bullock et al. 2017).

This is derived by Katul et al. ( 2005)) from the mechanistic model, which allows calculation of a dispersal kernel from measures of plants and the environment.

Specifically, 𝑎 = 𝐻𝑈̅/𝐹 and 𝑏 = 2/ (𝐻/𝜎̅), where H is the seed release height, F is the seed terminal velocity, 𝑈̅ is the mean wind speed at the height of seed release and 𝜎̅ is a turbulent flow parameter (Katul et al. 2005; Bullock et al. 2012).

Since the WALD is mechanistically derived and parameterised by plant traits and environmental variables, one may use these readily available data to predict dispersal and spread without a priori obtained dispersal data (Bullock et al. 2012; Hemrová et al. 2017).

This means the fitted function is fundamentally linked to the theory of the mechanistic model.


As is clear in our paper, we agree that there are good reasons why the WALD may not fit empirical data well, but these are several.

Cousens, Hughes and Mesgaran (2017) give one suggestion.

In our paper we suggested two additional and equally valid mechanisms by which the model might not fit to empirical data.

Considering the underlying theory, simplifying assumptions in the WALD include (Katul et al. 2005): flow is vertically homogeneous; seed terminal velocity is achieved instantly after seed release; the seed settling time is assumed to be much longer than the vertical velocity integral timescale; and the simplifications to the Thomson (1987) model, including Gaussian fluctuations and the use of Kolmogorov scaling within the inertial subrange to arrive at the diffusion coefficient.

Many of the coefficients in the WALD are averaged, which allows the full equations to be simplified from the underlying equations.

One such coefficient is the mean wind speed, given by 𝑈̅ in Katul et al. ( 2005).

There is variation over a season in the wind speed a falling seed might experience, as Cousens, Hughes and Mesgaran (2017) state, but it will also vary over the time that the seed takes to fall and hence 𝑈̅ could be modelled as a function of time, which would result in an intractable non-closed-form equation.

Another example is the seed release height, as this will vary naturally. As Cousens, Hughes and Mesgaran (2017) suggest, one could sum up all the possible release heights of individual seeds and the corresponding WALDs to get a new kernel, which is hard to work with. Or one might convolve some distribution of seed release heights (e.g. a Gaussian) with a WALD.

This might fit better, but there are now extra parameters and one could go on like this with a large number of possible combinations.


Despite these issues, there are good reasons for fitting the WALD to data.

The logic that a mechanistically based function might fit well to data is valid, as it is hoped that it encompasses the relevant processes determining the dispersal kernel and so captures the dispersal kernel.

Thus, clear hypotheses are set up about the determinants of the realised kernel.

Indeed, the WALD has been used by others when fitting functions to data.

In the original paper proposing the WALD, Katul et al. ( 2005) proposed and implemented fitting it to measured dispersal kernels, while also introducing the assumptions in doing so which are being discussed here.

The WALD has been tested in some studies in which alternative functions are compared.

Studying a wind-dispersed tree, Norghauer, Nock and Grogan (2011) found the WALD and Weibull functions gave comparable and better fits to dispersal data than the lognormal.

Lara-Romero et al. ( 2014) fitted functions to seedling data for two herbs using inverse modelling and found the WALD was at least as good a fit to the data as the 2Dt, exponential power and lognormal.


It should also be clarified that in our study the WALD was by no means a poor fit to wind dispersal data in all cases, belying the implication that it will never fit empirical data well.

Of the 55 wind dispersal data sets, the WALD was in the best-fit group for 15, and had an r2>0.9 for 25.

One way to examine the ability of the WALD to describe dispersal data from a varying environment is to draw dispersal distances from WALD functions representing variation in parameter values, and then assess how well a single WALD fits these data in turn.

Cousens, Hughes and Mesgaran (2017) do this, but their example is unrealistic.

They make draws from a WALD with the parameters a and b varying independently each time “according to a uniform distribution of several orders of magnitude”.

In reality, these parameters are unlikely to vary either uniformly or over such wide ranges as they are based on plant and environmental variables (e.g. seed release height, wind speed), which are likely to be more tightly distributed and closer to the mean.

Furthermore the parameters are correlated, i.e. both reflect wind conditions and plant height, and so do not vary independently.

In Fig. 1 we develop a more realistic example, using data from Bullock et al. ( 2012) in which we showed that variation in horizontal wind speed over a season follows a Weibull distribution.

Generating a dispersal data set using a WALD sampled over a Weibull distribution of wind speeds, we find that the WALD is a better fit (Fig. 1) than the log-sech or the other functions that we investigated in Bullock et al. ( 2017).

This shows that it is sensible to ask the question whether a WALD fits wind dispersal data.

We would note however, that if multiple parameters of the WALD (e.g. H, F, U) were allowed to vary over realistic distributions and convolved with the WALD, then the resulting distribution might take on a number of forms, and not necessarily the WALD.


In Bullock et al. ( 2017) we showed that it is possible to summarise the complex and variable dispersal process using simple functions over a large number of empirical data sets.

We found none of the functions we used gave best fit overall, suggesting no single function captures the dispersal process intrinsically.

Mechanistically based functions may fail to describe such data for reasons set out by us in the original paper, by Cousens, Hughes and Mesgaran (2017), and expanded upon here.

We advocate however that these functions are explored along with other more general functions when describing empirical data both to assess whether their simplifying assumptions are valid when tested in the real world and to add to our arsenal of possible functions for analysing data.

Parametric summaries of dispersal data are critical if we are to use the past and ongoing work of ecologists in gathering dispersal data for more than simply describing the system in which the data were gathered.

Prediction in ecology aims both to explain systems and to forecast, or anticipate, future changes (Mouquet et al. 2015).

In line with both aims, our paper synthesized dispersal information and provided general dispersal functions.

These are of use to researchers who may either not have the necessary data to model their system or may not be interested in case specific kernels.

These general and better validated kernel functions would be useful, for example, in species distribution modelling (Miller & Holloway 2015), analysing spatial networks (Marleau, Guichard & Loreau 2014) and predicting responses to climate change (Santini et al. 2016).


Survey data were derived from the Study of Assets and Health Dynamics Among the Oldest Old (AHEAD), a US national probability sample of noninstitutionalized persons born in 1923 or earlier (i.e., those aged 70 years or older in 1993) (22).

Subjects were selected by using a multi- stage area probability design and a dual-frame sample of Medicare recipients.

Within sampled households, one age- eligible person was sampled; when that person had a spouse, he or she was also included in the sample irrespective of age.

The overall response rate of 80 percent yielded an inter- viewed sample of 8,222 persons from 6,047 households.


For these analyses, the following were dropped from the sample: 775 age-ineligible spouses to limit the analysis to those aged 70 years or older; 791 proxy interviews, which are inappropriate for measuring the dependent variable; and 532 persons for whom data were missing or invalid, princi- pally US Census tract identifier or the dependent variable, cognitive status.

The problem of missing cognitive data is discussed in more depth in the Results section.


To eliminate the household level of clustering, we randomly sampled one person per household, which eliminated 1,009 persons.

Consistent with most neighborhood theories that assume an urban setting (23, 24), we limited the sample to persons living in US Census tracts that are at least 75 percent urban.

Census tracts in urban areas encompass a smaller physical area than those in rural settings.

Additionally, the concept of neighborhood carries a different substantive meaning in many rural areas, where residences may be very sparsely dis- tributed.


The final analytic sample size was 3,442 persons.

Weights (provided in the AHEAD data) were used to adjust for variation in the probabilities of selection, including the oversampling of African Americans, Hispanics, and resi- dents of Florida.

Weights were further adjusted for our an- alytic selection of one person per household.

Consequently, the analytic sample was nationally representative of persons aged 70 years or older living in the community in urban areas in 1993.


AHEAD utilized a multidimensional measure of cogni- tion based on the Telephone Interview for Cognitive Status (25).

Most AHEAD studies have used a total score rather than establishing a cutoff point to designate cognitive im- pairment.

We followed suit, given that our goal was not to estimate the prevalence of cognitive impairment but rather to explore multiple levels of influence on overall cognitive performance.

The reliability and validity of the AHEAD cog- nitive measure has been established previously (26-28).


First, memory was assessed with immediate and delayed word recall tasks.

The interviewer read a list of 10 nouns and asked the respondent to recall as many as possible.

After 5 minutes of asking other cognitive questions, the re- spondent was again asked to recall the nouns.

Scores represent the number of words correctly recalled.


Second, working memory was assessed by using the Serial 7's subtraction test.

Respondents were asked to subtract 7 from 100 and to continue subtracting 7 from each subsequent number for a total of five trials.

Scores represent the number of correct subtractions.


Third, mental status was measured by tasks assessing knowledge, language, and orientation: 1) counting backward for 10 continuous numbers beginning with 20, as quickly as possible; 2) reporting the correct date; 3) naming objects; and 4) naming the president and vice-president.

Scores represent the number of correct responses.


Cognitive function is the sum of component scores (i.e., all correct responses), ranging from 0 to 35.


Individual-level risk factors encompassed several domains.

Educational attainment was assessed as the highest grade of school or year of college completed.

Other socioeconomic- related measures included household wealth and household income, both logged.

Standard sociodemographic informa- tion included gender, age, marital status (married, widowed, separated/divorced, never married), and ethnicity (non- Hispanic White, African American, Hispanic, other).

In the health domain, activities of daily living represent a count (0-6) of the respondent's self-reported difficulties with personal-care tasks.

Depressive symptoms were measured with a count of eight items from the Center for Epidemio- logic Studies Depression Scale (29), scored yes (1) or no (0) for being experienced during ''much of the time in the past week'' (a ¼ 0.78).

Eight physician-identified major health conditions included heart problems, stroke, psychiatric prob- lems, high blood pressure, diabetes, cancer, lung disease, and arthritis.


Neighborhood-level constructs were operationalized with 1990 US Census tract data (STF3A data), which were linked with geocodes to the individual-level data.

US Census tracts are designed to be relatively homogeneous units with re- spect to population characteristics, economic status, and living conditions at the time they are established.

Ideally, they contain between 1,000 and 8,000 people, with an opti- mum size of 4,000, although there is wide variation because of population shifts between the decennial US Census data collections.

Geographic size varies, depending on popula- tion density.

US Census tracts are particularly suitable rep- resentations of neighborhoods in urban areas because of the dense concentration of persons.

In this paper, we treat ''neighborhood'' and ''Census tract'' as equivalent terms.


The 3,442 participants in our analytic sample resided in 1,217 US Census tracts, with the number of participants per tract ranging from 1 to 31.

These tracts consist of 5,344 residents on average (median, 4,678), ranging from 342 to 71,872.


Neighborhood education was assessed as the proportion of residents aged 25 years or older without a high school de- gree.

Our analysis of education also controlled for the com- peting effect of median household income (logged) at the US Census tract level.

Among the potential neighborhood- level economic controls, median household income was chosen because it had the lowest zero-order correlation with proportion of residents without a high school degree (r ¼ -0.699), reducing potential multicollinearity.


For all analyses, normalized grand sample weights were applied so that findings could be generalized to the urban population of older US adults.

Descriptive statistics were calculated with the SVY procedure in the Stata software package, which takes into account the complex sample de- sign in estimating standard errors (30).

Hierarchical linear models were estimated with HLM software, version 6.01 (31).

Contextual-level variables were grand mean centered to facilitate interpretation.

Tests of statistical significance for random-effects models were one sided (31).


The gross variance in cognition associated with neighbor- hood context was first estimated with a null, or unconditional, model that contained only a random intercept.

Second, the impact of individual-level sociodemographic and health risk factors was tested.

Third, the main effect of neighborhood- level educational attainment on the average level of cogni- tion was assessed.

Fourth, median household income at the neighborhood level was controlled.

Fifth, the cross-level interaction between individual-level educational attainment and neighborhood-level educational attainment was assessed.

Deviance statistics assessed model fit, and sequential differ- ences between models were considered with the likelihood ratio test.


Individual-level characteristics (weighted) of the sample are shown in table 1.

Females outnumbered males, the av- erage respondent was in his or her late seventies, most were non-Hispanic White, nearly half were widowed, average education approximated high school graduation, and both income and wealth were variable.

In terms of health, levels of assistance with activities of daily living requirements and number of depressive symptoms were relatively low, and most participants had not experienced serious health prob- lems, although nearly half reported high blood pressure and a substantial minority had arthritis or heart problems.

Par- ticipants responded correctly to nearly 60 percent of the cognitive items, on average.


At the neighborhood level, wide variation in US Census tract characteristics was evident.

The average US Census tract consisted of one quarter of residents aged 25 years or older without a high school degree, but some areas had a high density of such persons (maximum, 86.32 percent) whereas others virtually lacked such persons (minimum, 0.94 percent).

On average, median household income was slightly higher than the $30,056 reported in the 1990 US Census for the population as a whole (32).

Table 2 presents the hierarchical linear models of cogni- tion (level 1 is the individual, level 2 is the US Census tract).

As shown at the bottom of the table, models 2, 3, 4, and 6 represent significant improvements in fit over previous mod- els (i.e., the deviance statistic significantly decreases), but model 5 does not.

In addition, statistically significant ran- dom variation (s) in cognition across US Census tracts was sustained with each sequential model.


As shown in model 1 (the null model), there was signif- icant variation in cognition across the US Census tracts (s ¼ 6.22, p < 0.001).

The intraclass correlation (q; between- group variation/[between þ within-group] variation) was sizable (0.20) and indicated that 20 percent of the variation in cognition in this sample could be attributed to living in different US Census tracts.

Model 2 added individual-level sociodemographic variables, which accounted for 22 per- cent of the level 1 within-group variance.

Consistent with Downloaded from https://academic.oup.com/aje/article/163/12/1071/97210 by Institut Armand-Frappier user on 16 May 2024 previous research (20), low educational attainment, being older, being an ethnic minority, and living in a household with low income were significant risk factors for poor cognitive function.

Model 3 added individual-level health variables, which accounted for an additional 2 percent of the level 1 variance in cognition.

As shown, high levels of diffi- culties with activities of daily living, high levels of depres- sive symptomatology, and having suffered a stroke were negatively associated with cognition, whereas cancer and ar- thritis were positively associated with cognition.

Additionally, female gender was positively and significantly associated with cognition, although it was of borderline significance in the previous model.

The coefficients for the remaining individual- level sociodemographic variables were virtually unchanged by controlling for health status.


Model 4 introduced level 2 neighborhood educational at- tainment.

Living in a low-education area was associated with low cognitive function, net of individual-level educa- tion and covariates.

Neighborhood-level educational level significantly improved the fit of the model, but, in compar- ison to model 3, it did not affect the amount of intercept variance at level 2 accounted for by the model.

The coef- ficients for the individual-level variables were largely un- altered, with the exception of that for Hispanic ethnicity, which decreased noticeably, meaning that neighborhood ed- ucation accounted for some of the effect on cognition of being Hispanic.

As shown in model 5, inclusion of median household income did not significantly improve the fit of the model, and its coefficient was not significant.

The main ef- fect of neighborhood education remained significant, and it actually increased by about 68 percent, indicating that its effect may have been suppressed by not controlling for me- dian household income.

Median household income ac- counted for nearly negligible additional intercept variance at level 2, and the coefficient for Hispanic ethnicity became nonsignificant


The interaction between individual- and neighborhood- level education was examined next.

First, we tested whether the slope for individual-level education (from model 5) varied significantly across neighborhoods.

This variation was nearly significant (s 11 ¼ 0.03, v 2 (df ¼ 640) ¼ 699.31, p ¼ 0.06; based on 643 US Census tracts with sufficient variation).

We then modeled the random neighborhood variability in the level 1 education slope as a function of neighborhood-level education (refer to model 6), finding a significant cross-level interaction.

The between-group variation increased because the education variable was now conditional.


Figure 1 illustrates the conditional impact of level 1 ed- ucation on the association between neighborhood education and cognition among these older adults.

For neighborhood education, two values-the minimum and maximum-were substituted into the regression equation.

For individual-level education, three values (the mean plus and minus one stan- dard deviation) were substituted.

Mean values were used for significant continuous variables, and 0 was the value for other variables.


As shown in figure 1, as the proportion of neighborhood residents without a high school degree in- creased, cognition decreased.

However, this decrease was not commensurate across the population: The decline was steepest among those with the least education.

Cognitive differentials between persons with low educational attainment and high educational attainment were larger in low-education neighborhoods than in high-education neighborhoods.


To further investigate contingencies between individual- and neighborhood-level education, we estimated model 3 from table 2 separately for tertiles of neighborhood education: high (1-16 percent without a high school degree), medium (17-29 percent without a high school degree), and low (30- 86 percent without a high school degree).

Individual-level educational attainment significantly affected cognition within each neighborhood type, but the coefficient was significantly smaller for those living in high-education areas (b ¼ 0.31;


Our findings provide compelling evidence that cognitive functioning in late life does not manifest in isolation from social context, and this study is the first of its kind known to identify particular aspects of neighborhoods that are associated with cognition among older adults.

First, we found that cog- nition varied significantly across urban US Census tracts.

Second, cognition was associated with educational profiles of the neighborhood, in that older adults living in concen- trated low-education areas were worse off cognitively, on average, than those living in areas where most residents had attained at least a high school education.

Importantly, the main effect of contextual-level education occurred net of personal characteristics of the individual, most notably his or her own educational attainment level, meaning that the contextual effect was not merely the manifestation of com- positional differences across neighborhoods.

Third, the cog- nitive impact of neighborhood education was sustained when controlling for median household income at the contextual level, demonstrating that education is uniquely important.

Fourth, the cognitive impact of one's own educational attain- ment was contingent on neighborhood education, with low educational attainment being somewhat more consequential in low-education areas than in high-education areas.

Thus, cognition in late life is a product of the personal charac- teristics of the individual in interaction with his or her environment.


High school graduation was the modal level of educa- tional attainment for this sample of older persons, but, for the most part, these older persons live surrounded by younger cohorts.

For these subsequent cohorts, educational attain- ment has risen considerably generation by generation.

Thus, it could reasonably be expected that older persons might be surrounded, on average, by younger neighbors with higher levels of educational attainment.

Since education is associ- ated with cognition, this general environmental advantage may offset individual deficits in education.

However, low educational attainment of one's neighbors, as evidenced by low rates of high school graduation, is not normative for the general population but evidence of the type of limited achievement that often goes with socioeconomic disadvan- tage.

In this situation, older persons whose cognition is compromised by their own limited education may find little compensation in their surroundings.


Applying a taxonomy developed by Catalano and Pickett (33), we suggest that the specific etiologic connection be- tween neighborhood education and cognition in late life may be generated in three ways.

First, persons living in low- education areas may be disproportionately exposed to chronic, stressful life conditions that create multiple hazards such as reduced employment opportunities, marginal occupations, and truncated incomes, which lead to a concomitant dearth of safe physical resources (e.g., well-lit walkways, bicycle paths, parks, gyms) and social resources (e.g., well-kept shopping areas, social clubs, neighborhood organizations), thereby hindering engagement in both physical and social activities, which have been linked to better cognitive func- tioning (11, 34, 35).

Second, there is an unequal distribution of cognitively stimulating coping resources in low-education neighborhoods (e.g., medical screening facilities, health clinics, bookstores, libraries), which have been linked to better cognitive function (36-39), and are scarce or absent because of the lack of an educated consumer demand or monetary tax base for such resources.

Third, low-education neighborhoods may be characterized by a high tolerance for illness, which stems from untreated chronic conditions (e.g., diabetes), elevated exposure to hazards, and limited coping strategies, generating widespread cognitive deficits, which are viewed as normative or socially acceptable, es- pecially for older persons who may already have other forms of impairment such as those that limit their mobility (40, 41).


We focused on education as the single most relevant neighborhood socioeconomic characteristic in terms of cog- nition among these older adults because education is con- ceptualized as being the catalyst for various other forms of neighborhood disadvantage, as described above.

These other neighborhood-level characteristics (e.g., proportion of Downloaded from https://academic.oup.com/aje/article/163/12/1071/97210 by Institut Armand-Frappier user on 16 May 2024 residents living below the poverty level, proportion of households receiving public assistance, proportion of un- employed adults) are highly correlated with neighborhood education (average r ¼ 0.757) and each other (average r ¼ 0.802) and, in additional analysis, were found to form a sin- gle principal component.

We replicated the neighborhood main-effects model by using this principal component in- stead of neighborhood education, and we found that the principal component approached significance (b ¼ ÿ0.24; standard error, 0.13; p ¼ 0.066).

However, we prefer the results with neighborhood education because they suggest two types of influence on older persons' cognition: 1) a gen- eral impact of socioeconomic disadvantage (shared in com- mon with the other socioeconomic indicators) and 2) a specific impact of educational disadvantage.


The finding that cognitive deficits associated with being Hispanic became nonsignificant once the neighborhood characteristics were controlled deserves mention.

It appears that neighborhood context may be of special cognitive im- portance for these persons and that contextual-level education and income may account for the observed individual-level association between Hispanic background and cognition.

This finding should be interpreted with caution, however, because of the small proportion of Hispanics in the sample (4 percent).

Ethnic variation in cognition was not the focus of our analysis, and ethnicity was included primarily as a control variable.

Targeted studies are needed to fully inves- tigate how neighborhood socioeconomic characteristics may be uniquely consequential to older Hispanics, especially given previous findings of neighborhood effects among older Mexican Americans (2).


This study has several strengths.

First, the data were de- rived from a large, national probability sample of older adults, enhancing external validity and generalizability of the findings.

Second, the measure of cognition was multi- faceted, and it tapped into both memory capabilities and the mental status of participants.

Third, linkage of the individual- level data with US Census tract data provided a unique opportunity to assess the impact of neighborhood-level ed- ucation on cognition.

Fourth, controlling for a comprehen- sive list of individual-level risk factors enabled us to largely eliminate their potentially confounding impact on the results.


There are also limitations to acknowledge.

First, as dis- cussed in other analyses of the AHEAD data (27), results are somewhat biased toward a cognitively well-functioning population because we excluded institutionalized persons, proxy-assisted interviews, and persons for whom a large portion of cognitive data was missing.

Our analysis of the impact of missing cognitive data on our findings supports this possibility, in that those missing cognitive data do differ in several ways from participants in our analytic sample.

Importantly, however, missing cognitive data is not a system- atic function of neighborhood clustering, and we are confident in the basic nature of the relation between neighborhood edu- cation and cognition.

Second, selection effects (endogeneity) related to unique characteristics of persons who reside in certain residential areas may have affected the findings, and social selection remains an alternative explanation for our results.

Third, the study was cross-sectional; thus, causal pathways cannot be definitively determined.


Overall, however, by showing that cognition varied across urban neighborhoods and that the neighborhood-level edu- cation accounted for some of this variation, controlling for neighborhood-level income, individual-level education, and other risk factors, we conclude that social context is indeed consequential to cognitive function among older adults.

Fur- thermore, it is apparent that education-related neighbor- hood risk factors for poor cognition may be offset by one's own educational attainment, particularly in disadvantaged neighborhoods.

This finding suggests that context-specific health assessments may facilitate the early identification and treatment of cognitive health problems, and targeting older persons with limited education in disadvantaged neigh- borhoods may prove especially beneficial.

In addition, pro- moting educational attainment in the general population in disadvantaged neighborhoods may prove cognitively benefi- cial to its aging residents because it may lead to meliorations in stressful life conditions and coping deficiencies in the neighborhood.


During the past decade, there has been a dramatic increase in the use of multilevel modeling in epidemiologic analysis.

There were 10 times as many papers in Index Medicus identified by the terms 'multilevel' and 'epidemiology' in 2005 than there were in 1995.

In recent years, the American Journal of Epidemiology has published papers that use multilevel analyses to consider the associations between group-level characteristics and a range of health indicators [1-7].

Accompanying these content-specific contributions have been several discussions of the strengths and limitations of multilevel modeling techniques [8, 9] and the rationale for applying these methods to public health [10, 11].

In this issue of the Journal, Wight et al. [ 12] examine whether neighborhood context influences the cognitive function of older adults.

This paper contributes to the literature that considers the role of group-level education as a determinant of individual health [13, 14].

The authors show that older adults living in areas characterized by low aggregate education have lower cognitive function than do older adults living in areas characterized by high aggregate education, independent of the participants’ own education.

In many ways, this analysis is representative of the growing multilevel analysis literature.

This study makes use of a large, nationally representative cross-sectional sample in the United States.

Participants were geocoded to their census tract of residence, and US Census data were used to characterize census tracts (equated here with 'neighborhoods') and to test for an association between neighborhood educational level and individual-level cognition.

However, this paper is especially successful on several fronts and provides us with an opportunity to reflect on directions that might fruitfully move the field forward.

It is the purpose of this commentary to offer thoughts about 1) specificity of association, 2) multilevel causal pathways, and 3) heterogeneity of associations, motivated by the work of Wight et al. and building on the extant literature on multilevel epidemiologic analyses.


One key challenge facing studies of the macrolevel context and health is the relative paucity of theoretical development compared with the rapid progress of methodological development [9, 15, 16].

This limitation places epidemiology in sharp contrast with other disciplines, such as sociology, where substantial theoretical work has developed strong frameworks that provide the basis for posing and testing hypotheses on the relations between macrolevel exposures and individual outcomes [17-19].

The principal danger of having underdeveloped theoretical frameworks is that multilevel analyses run the risk of becoming another form of 'black box epidemiology' [20], where the associations between 'exposures' and 'outcomes' are explored, and statistically significant associations are reported, without a theoretical basis for why such associations should exist and why specific factors would cause particular diseases.

Although atheoretical epidemiologic explorations can reveal associations that then generate mechanistic hypotheses that can guide future research [21], they risk unearthing statistically significant relations that represent only the play of chance.

Multilevel analysis in all aspects of health would do well to focus on evaluating specific theoretically driven hypotheses, transcending black box approaches in order to fruitfully identify macrolevel domains that may influence particular health outcomes.

Where no such theoretical or mechanistic work has been conducted for a particular macrolevel determinant, it would be productive for the field if we focused our energy on developing it.


Wight et al. [ 12] present an analysis predicated on a specific neighborhood characteristic (aggregate area-level education) with the theoretical potential to predict a specific outcome (cognitive function).

The authors build their hypothesis on two observations: that educational attainment at the individual level predicts a range of other factors that may be salutary, or deleterious, for health; and that high levels of education are cognitively protective for individuals [22-25].

Therefore, the authors admirably bring together the evidence from previous work, present plausible theoretical reasons as to why the macrolevel determinant of interest may matter for this particular outcome, and use multilevel analysis to assess their hypothesis.

Wight et al. [ 12] also show results from an analysis in which they calculated a composite measure of neighborhood 'disadvantage' and were able to replicate the central neighborhood main effects that are the focus of this paper.

The authors note that they prefer treating neighborhood education as the exposure because the analysis illustrates a specific impact of educational disadvantage.

We suggest that the choice of neighborhood educational attainment as the central independent variable in this analysis is not only preferable but also the only meaningful choice based on the articulated mechanistic pathways, and it is a core strength of this analysis.

However, this discussion also highlights the challenges of discerning the association of one potential macrolevel exposure with an outcome when it is highly collinear with other macrolevel exposures.

There may be causal relations among macrolevel characteristics that need to be better understood before we can untangle their separate and combined influences on health.

Multilevel analyses can move further from atheoretical approaches by explicitly discussing the pathways that may link macrolevel exposures with individual outcomes and, more ambitiously, through attempts at testing these various pathways.

The paper by Wight et al. [ 12] is perhaps more successful than most papers on the first of these counts but, as is often the case in multilevel analyses, is limited on the second.


Wight et al. suggest three plausible mechanisms for explaining the observed association between low neighborhood educational attainment and poor individual cognitive function.

These mechanisms include potential decreased individual activity due to neighborhood stressors, more limited availability of cognitively stimulating material resources such as libraries, and a greater social tolerance for illness.

However, similar to many multilevel analyses, this paper does not explore which of the potential mechanisms might actually explain the observed main-effects association.

In many ways, this shortcoming lays bare an essential tension in epidemiologic analysis more broadly, and one that is particularly germane to multilevel analyses.

The predominant regression modeling techniques are designed to control for potential confounding.

Regression approaches are not designed to consider mediation and are even more ill-suited to differentiating between confounding and mediation.

This issue clearly is relevant to all epidemiologic analysis.

However, it is particularly challenging in the multilevel context, where exposures may be quite distal from the outcomes.

The pressure to demonstrate multilevel associations in models with extensive covariate control is substantial and, to some extent, understandable.

The relative newness of examining distal exposures places an extra onus on investigators to demonstrate that their work is not simply a reflection of unmeasured confounding.

Therefore, it is commonplace in papers that present multilevel analyses to adjust for the individual-level variables that may either confound, but could also plausibly mediate, the observed association between the macrolevel determinant and individual health.

Insofar as macrolevel factors influence individual-level risk factors as well as individual-level outcomes, adjusting for individual-level factors will ultimately render all multilevel associations statistically insignificant [15, 26, 27].

The decision about which variables to include in multilevel models then is a particular challenge faced by analyses interested in explicating the pathways that link macrolevel exposures with individual outcomes.

The analysis by Wight et al. [ 12] is commendable for its judicious use of individual-level controls to reassure us that the findings are not simply a reflection of confounding by well-recognized risks for poor cognitive function without overadjustment for covariates that likely mediate the association of interest.

However, in the interest of testing hypothesized pathways between macrolevel exposures and individual outcomes, methodological development that provides tools to isolate the role of specific mediators of multilevel associations would go a long way toward furthering the contribution of multilevel analyses.


One additional note about this issue is in order.

Suppose that this analysis had included adjustment for limited physical activity, a variable that can plausibly be considered a confounder of the relation of interest but could also plausibly be considered a mediator (as suggested by the authors [12]).

Suppose further that the final model then did not show a statistically significant association between neighborhood education and individual cognitive function.

To what extent would such a model suggest that neighborhood educational attainment is not an 'important' contextual determinant of health?

This question gets at the heart of causal determination and the extent to which, in epidemiology, we are interested in causes distal from the individual.

If a compositional characteristic of a neighborhood—the general level of education—causes less individual activity, and if that lower level of individual activity causes residents in that neighborhood to have lower cognitive function, are we interested only in limited physical activity as a cause, or are we also interested in what might be considered 'causes of causes' [28]? Ultimately, this is not an empiric question as much as a reflection of the values of the epidemiologic research community that influence our analyses and the relevance of our work to the broader public health community.

To some extent, it is self-evident that features of social context influence health.

We epidemiologists must then choose whether we want to contribute to the body of empiric evidence that demonstrates what aspects of social context may matter for what.

We suggest that epidemiology, as the science of public health, might do well to consider causes at multiple levels regardless of their proximity to disease outcomes, providing a broader understanding of the processes that produce health and disease in populations and a wider array of potential points of intervention in the disease process.


Much of our energy in epidemiologic analysis is focused on optimizing our explanatory models, explaining to the extent possible the risk of individual disease as a function of individual 'independent' covariates (exposures), and treating residual variation as random error.

However, as we broaden the epidemiologic lens of inquiry to include determinants at different levels of aggregation, we are beginning to understand that much of the residual variability from individual-level models is not random variability but rather a reflection of a web of interrelations among covariates at multiple levels of influence [10, 29, 30].

Multilevel modeling expands on the individual-level regression approach by including explanatory variables at different levels of aggregation and provides one opportunity to better explore a more comprehensive range of factors that might explain variability in individual health and disease.

Fundamentally, however, multilevel approaches remain concerned with the same goal of optimizing model-based explanation of the likelihood of individual disease.

Regression model-based approaches explicitly endeavor to explain dependent variables (health outcomes) as a function of a fixed set of variables in the hope of providing generalizable explanations.

Relations between covariates at multiple levels of aggregation in nature are unlikely to be unidirectional or simple.

Several authors have called for the broader use of more flexible analytic tools that can take into account mediation, multidirectional causation, and nonlinear relations in epidemiologic research [31-35].

Unfortunately, at this time, most of these methods remain largely inaccessible because of both limitations of most readily available data and challenges in implementing more sophisticated analytic strategies.


Wight et al. [ 12] demonstrate an interaction between neighborhood education and individual education in explaining individual cognitive function.

They surmise that this modification of the neighborhood education–individual cognition effect by individual education might reflect the greater import of social context for those who are already vulnerable, in this case, persons with low levels of education.

The authors suggest that this might guide context-specific health interventions informed by this work.

While this approach is indeed instructive, this demonstration of heterogeneity of association by Wight et al. only scratches the surface of the potential dynamic processes that shape individual cognition in older adults.

Insights from some epidemiologic disciplines such as infectious disease epidemiology [36-39] and, to some extent, the emerging epidemiologic study of frailty among the elderly [40] might lend themselves to broader adoption of epidemiologic methods that account for the complexity of nature across multiple levels.

Broad acceptance of multilevel techniques in the epidemiologic literature presents epidemiologists with a way to consider exposures that extend beyond typical individual exposures or risk behaviors and that may encompass more distal variables that are causes of these more proximal causes of disease.

However, introduction of these methods has also brought challenges for epidemiology.

Methodologically, multilevel analytic techniques force us to consider the analytic difficulties of differentiating between confounders and mediators and of assessing indirect, reciprocal relations as part of a more comprehensive causal framework.

Conceptually, these techniques force us to rethink, yet again, what we might mean by 'cause' and to reconsider the uneasy alliance between the most widely used sufficient-cause frameworks in epidemiology that focus on proximate exposures [41] and the notion of causes of causes.

Through clear exploration of specific associations, and potential heterogeneity within these associations, papers such as the one in this issue of the Journal by Wight et al. [ 12] are a small step in the right direction.

Future epidemiologic theoretical and methodological work will be needed to take bigger steps to satisfactorily address the challenges faced in multilevel analyses.


Galea and Ahern [1] have provided an insightful commentary on our article [2], raising several noteworthy observations pertaining to the current state of and future directions for multilevel epidemiologic research.

We concur that theory-driven multilevel studies are needed, along with explication of the mechanisms that link levels of influence on health, particularly since these mechanisms may be more readily amenable to intervention than structural determinants of health.

For example, in our analyses of the association between neighborhood education and late-life cognition [2], we recommend promoting educational attainment in the general population in disadvantaged neighborhoods because it may prove cognitively beneficial to aging residents of these neighborhoods.

An argument could be made instead for promoting alteration to the hypothesized mechanisms by which we believe educational attainment would benefit aging residents, such as engaging in physical and social activities or increasing access to cognitively stimulating material resources, because these mechanisms are less wieldy from a policy standpoint and are more proximal to the target population.

However, without longitudinal data to test the impact of these mechanisms, this possibility is purely speculative.

By specifying causal mechanisms, theory-based investigations are well poised to distinguish mediators from confounders or analytic noise.


As Galea and Ahern [1] note, a drawback to implementing theory-based multilevel investigations concerns the current lack of appropriate data.

With growing evidence for apparent area-level effects, there will hopefully be greater impetus to develop data sets that specify the potential mechanisms linking levels of influence and that account for the inherent nesting of individuals within contexts.

To date, many multilevel studies have been undertaken with data not originally designed to assess multiple levels of causality (as was the case with data from the Study of Assets and Health Dynamics Among the Oldest Old (AHEAD) analyzed in our study [2]), thereby hindering examinations of heterogeneous associations or limiting the set of variables available for inclusion in multivariate models.

For example, the power to detect cross-level interactions is limited when there are too few cases per macro-level unit to provide sufficient variability in variables of interest.

Although there are approaches for solving the problem of "data sparseness" (e.g., cluster analysis, sample stratification), the problem would best be addressed by designing projects, a priori, to contain ample data points per contextual unit.

Similarly, testing causal mechanisms between levels can occur only if these pathways are operationalized as variables.

Analysis of data sets specifically developed to test such pathways would represent a sizable improvement over secondary analysis of data sets developed for other purposes, which leave the investigator with the option of making use of only those constructs that happen to be available.


By deriving theory-driven research models and applying them to appropriate data sets, future multilevel epidemiologic studies stand to make substantial contributions to our understanding of not only why but how contexts matter to health.


In 1977, Professors Paul Horowicz and Paul De Weer (my Washington University colleague), the founding editor and associate editor, respectively, of AJP Cell Physiology, invited me to write a review article for their new journal.

Paul De Weer, an Na ϩ pump expert, and I had discussed my ideas about the role of Na ϩ pumps and a postulated Na ϩ pump inhibitor/ natriuretic hormone in the pathogenesis of hypertension.

The present article is a personal account of the origin of those ideas and the developments in the field since my hypothesis was published in the journal's first volume (29); Table 1 provides a timeline.


Two key early influences were my experiences treating heart failure (HF) patients with cardiotonic steroids (CTSs, e.g., digoxin) as a medical student and intern, and my 18 months in Daniel Tosteson's laboratory studying the Na ϩ pump, where I routinely used inhibition by the CTS ouabain (pronounced, "wä=ba n") (255) to measure pump activity.

In 1958 I first met Tosteson who told me that "the Na-K-ATPase (NKA, or Na ϩ pump) has just been discovered (264).

You'd better hurry up and work on it before everything about the pump is found out and there's nothing left to do"--so I went to work in his laboratory on a stipend from an NIH training grant.

I'm still working on the pump 60 years later (43)!


My research and clinical experience with ouabain led me to ponder the question of how a pump inhibitor could enhance cardiac contraction (the positive inotropic effect) and thereby benefit patients with HF.

The story of how I fortuitously found the answer by discovering the Na/Ca exchanger (NCX) (14), the "missing link," while studying squid axon Na ϩ pumps as an NIH Special Fellow in Cambridge and Plymouth, England, is described elsewhere (28, 35).

When we discovered the NCX, our 1973 paper (245) we postulated a "privileged communication" between the extracellular fluid and the sarcoplasmic reticulum (SR), involving the PM NCX (and, presumably, PM Na ϩ pumps and the tiny intervening space between the PM and SR).

A similar idea, the ASM "superficial buffer-barrier", was later proposed by Van Breemen (285).

These views were bolstered by evidence for local Na ϩ gradients in heart (298) and ASM (235) and by discoveries about the subcellular localization of Na ϩ pumps and NCX in PM microdomains (see The Plot Thickens-A Lot, below).

This concept differs from the more widely distributed "fuzzy space" (165) that has recently been questioned (90, 187).


␣2 NKA ouabain binding sites modulate basal BP* Table 3 regulation and, thus, a link to arterial contraction, TPR, and BP.


While scouring the literature for relevant clinical data, I found a number of reports that raised the possibility that a natriuretic hormone, or "third factor," might be involved (37, 45, 54).

Some studies even suggested that this factor was an "active sodium transport" (i.e., Na ϩ pump) inhibitor (72, 80) [the senior author, Neal Bricker, had taught me renal patho- physiology and clinical nephrology when I was a medical student!].


I leaped at the idea and wove a seemingly logical (but rather naïve) story about how salt retention, by an un- known mechanism, triggers the secretion of an endogenous Na ϩ pump inhibitor (i.e., a ouabain-like compound or "OLC").

I was unaware that Szent-Gyorgi had postulated an endogenous CTS 24 years earlier (274), and that Dahl had already shown that a "humoral agent" was instrumental in "salt" hypertension in rats (50).


The OLC would be expected to: 1) inhibit Na ϩ pumps in the kidneys, thereby promoting natriuresis to com- pensate for the salt retention; and 2) inhibit Na ϩ pumps in arterial myocytes, thereby promoting Ca 2ϩ entry via NCX [see Fig. 1, taken from the original article (29)].


The myocyte Ca 2ϩ retention should, of course, increase arterial tone, TPR and BP.


In fact Mason and Braunwald had shown that local ouabain infusion increases forearm vascular resistance in humans (198).


Numerous novel findings have demonstrated that the model in Fig. 1 is overly simplistic.


Nevertheless, the concepts of functional Na ϩ pump-NCX coupling and an OLC with vasotonic (and cardiotonic) actions have now been verified, and many of the details have been filled in.


My 1977 paper (29) was published while I was abroad.

Upon my return, my laboratory colleagues (at Washington University Medical School in St. Louis, MO) all stopped what they were doing to watch me enter my office at the back of the laboratory.

They wanted to see my astonishment when I opened the door and found my desk buried under a mound of several thousand reprint request cards (the article has been cited Ͼ1,700 times).

Interest in those ideas was soon reflected in de Wardener and MacGregor's review, "Dahl's hypothesis that a saluretic substance (plays a) role in essential hyperten- sion" (55), and in Kaplan's reference to my "biochemical hypothesis" (147).


While my AJP Cell article was in press, Haddy and Over- beck (113) also proposed that an OLC is a critical player in the pathogenesis of salt-sensitive hypertension.

They, however, suggested that the critical effects of pump inhibition were reduced electrogenic Na ϩ pump current and consequent myo- cyte depolarization (112), but this mechanism cannot work in the steady state.

Partial pump block will cause a transient decline in Na ϩ transport (and pump current), but when [Na ϩ ] CYT rises, pumped Na ϩ efflux must recover and must equal Na ϩ influx in the steady state to avoid continued net gain of Na ϩ (24).

In other words, chronically, partially inhibited pumps operate with a higher [Na ϩ ] CYT but unchanged pump current.


These proposals about the role of an OLC in hypertension spurred interest in purifying and identifying this compound.

One consequence was that a postdoctoral applicant, John Hamlyn, arrived at my office in 1981, shortly after I moved to the University of Maryland School of Medicine in Baltimore. He told me he wanted to purify this new hormone. I thought to myself, "Sure, you and who else?" Nevertheless, he had some good ideas so I agreed to take him on and let him try, perhaps swayed a bit by the fact that he grew up in the '60s in Plymouth, UK, the site of our NCX discovery.

Little did I- or any of us-realize how terribly difficult this quest for the holy spirit would be.


Most opportunely, shortly after John's arrival I was intro- duced to Avinoam Kowarski and Bruce Hamilton, two clinical endocrinology colleagues who were studying hypertension and had collected plasmas from a number of patients and controls.

They agreed to collaborate and provided us with de-identified plasmas, and John devised a direct but complex biochemical assay to measure Na ϩ -K ϩ -ATPase (NKA) inhibitory activity in the plasmas.

On several occasions I hovered over John's shoulder as he performed the assay, and I marveled at his skill; it left me with no doubt about the quality of his data.

My mentors, Howard Schneiderman, Dan Tosteson, David Gold- man and Alan Hodgkin (26) had imbued me with the need to


During the next decade, Upjohn funded and participated in our research as follows: I was tasked with procuring, literally, tons of plasma.

A good source was blood banks at our univer- sity and other nearby large medical institutions that performed plasmapheresis.

John worked out the initial steps of extraction and purification on large, normal phase chromatographic col- umns, but the latter proved unsatisfactory.

Accordingly, sub- sequent work was done on smaller, reverse-phase HPLC (high- pressure liquid chromatography) columns that gave far supe- rior resolution and reliability.

At every step, the eluates were carefully screened for NKA inhibitory activity, and the most promising, high-activity samples were saved for further anal- ysis.

Once false-positive had been excluded and the location of relevant biological activities was understood, the isolation steps worked out in Baltimore were refined and scaled up for batch processing at Upjohn.


In 1987 I attended a meeting in Switzerland and stopped off to visit Reuter in Bern.

Harald asked me if I had seen the latest issue of Biochemical and Biophysical Research Communica- tions.

I hadn't, so he escorted me to the library, retrieved the issue from the rack, and opened it to an article by Tamura and Inagami (278).

I skimmed through the report in which they claimed to have purified an endogenous NKA inhibitor of molecular mass 336 Da, with "striking similarity toѧ ouabain," and was alarmed.

I immediately phoned Hamlyn in Baltimore.

He hadn't seen the article either, so I started to read it to him.

Before I'd gotten very far, he interrupted me: "They missed it!" John, who has an exceptional understanding of steroid chem- istry, recognized that they, too, had seen what appeared to be the same bioactive polar material eluting from their columns, but that they had been led astray by a co-eluting contaminant.

When I returned to Baltimore, I sought John out.

He wasn't in his office, but an 8½ ϫ 11 inch photocopy of an Inagami portrait was hanging from each of his bookcases-and from each of the eye-level shelves in his laboratory.

As John ex- plained, "I want to know who my competition is."


In fact, there was no shortage of competition.

In 1988, a prominent Tokyo group, Goto and Yamada, described a bio- active material in dog plasma that "closely resembles ouabain" (103).

Their extracts, while impure, almost certainly contained OLC, but they, too, were thrown off the scent by a contaminant that had an estimated mass of only 343 Da; ouabain's mass is 585 Da.

The following year, another very active group, Hallaq and Haupert in Boston, described the ouabain-like positive inotropic effects of a partially purified material from bovine hypothalamus (116).

Clearly, the key to winning the "race" was to put sufficient pure material in the hands of an analytical chemist who made a living by identifying small molecules.