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Graphical interpretations of data East is east...

R. Allan Reese looks at two graphs showing a statistical model for the number of hawks counted in autumn at a migration hotspot, related to local wind direction. Look for yourself and decide which way the wind is blowing

In his classic Visual Display of Quantitative Information, Edward Tufte nominates a candidate for “the worst graphic ever to make its way into print”, based on its low data density and gratuitous use of colour.1 I offer in competition Figure 1 on the basis that, while it represents reasonable data and appears within a peer-reviewed paper, I consider it impossible to understand without the text explanation.2

Reproduced here are two panels from an overall figure relating daily bird counts to individual predictors. Actual daily counts are not shown, just the marginal fitted lines from a generalised linear mixed model. You can see that numbers of birds are higher with certain wind speeds and directions, but which directions are they? Have a go at answering before reading on.

The main problem arises from the titles of the x-axes, naming wind components. An east– west wind would normally blow from the east, implying that negative speeds in the first panel are westerly winds, so the axis appears drawn with the usual compass orientation (west on the left). As such, it seems to show that higher numbers of birds are seen with easterly winds, while applying the same logic to the second panel suggests that higher numbers are seen with southerly winds (negative northerlies).

If that is your interpretation, you would be wrong on both counts. The text in the paper is very lucid: “counts were expected to be highest on days with locally north and west winds … [and the] species models for the five species of interest generally agreed with the global model. … Birds arriving [from the north] in the area of the Strait during adverse crossing conditions (easterly winds) tend to wait for favourable conditions (westerly winds) to cross to Africa. … Honey Buzzard was the only species that migrated in higher numbers with southerly winds.”

The qualifying “generally” in that quote refers to another problem with these graphs: the y-axis title, “Count per species per day”. Five species of raptor migrated during distinct but overlapping windows each autumn. The response variable appears to be the totalled daily counts for these species, divided by 5, regardless of how many species were seen that day. Moreover, the counts were totalled from

two viewing sites that differed dramatically in the proportions by species.

How could the text and graphs be so contradictory? And why are the graphs so cryptic to discern the simple compass directions? Usually in these columns I offer my own interpretation and presentation of a data set. Unfortunately at present the authors prefer not to share the data in case they carry out more analyses themselves. But the lead author has written to me to say: “I don’t think I would create them [the graphs] any different. I may put a text label highlighting each side (east on the left and west on the right), instead of just saying ‘East–West’ (implied) in the axis label. There is no way that I am going to create a graph with negative values on the right hand side. I suspect many more people would take issue with that approach.”

To me, this is the epitome of Mathsworld – the mindset that numbers must be manipulated as ciphers to get a “right answer”

Allan Reese is an independent statistical consultant and member of the Significance editorial board.

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FIGURE 1 Migration counts at the Strait of Gibraltar per sampled species per day (as predicted by variables within a model), plotted against wind speed and direction. Reproduced from Miller et al.2 with permission of Wiley.

SIGNIFICANCE42 February 2019 © 2019 The Royal Statistical Society

a dozen equally spaced round values”, it would be more informative to choose either the boundaries of the conventional wind (Beaufort) scale or the deciles of numbers of birds counted. The latter would, however, be impractical if the species were shown separately. The speeds may have been supplied as metres per second (m/s), but are these units that readers will relate to? I do not know (10 m/s = 22.4 miles per hour = Beaufort 5–6).

The lumping of species arose from the peer-review process, I am told. The author writes: “I originally analyzed and reported each species separately. The feedback from the initial review emphasized looking at it from a global model perspective, which I did. I have never been a fan of these global random- effects models, but haven’t figured out a great way to perform a multi-species analysis, present it clearly and concisely, pulling out the globally interesting factors.”

This is unfortunate, especially as the graphs show only univariate model fits. They lack visual information on the amount or pattern of supporting data. It would be easy to show lines for the separate species as well as the composite. As noted in the earlier quote, the relationship for Honey Buzzard is different, and plotting daily counts against wind speed for each species might reveal the dominant wind directions and other interactions.

Why do scientific papers include graphs? They may be justified as showing a pattern more immediately than a verbal explanation.

In the graphs reproduced here, we can see the rate of drop-off with wind speed and that there is not a simple westerly/easterly cut-off, but unfortunately the actual wind direction cannot be reconstructed from the separated components. It is likely that the wind speeds of zero on each axis do not refer to still atmospheres.

If the raw wind speeds were supplied they could be displayed as a polar graph (radar chart). We could, for example, group observations into, say, 16 sectors for analysis and plot the relative numbers in each direction for various wind speeds. I would rather plot such polygons (contours) than try a fake three- dimensional surface, but how informative they can be cannot be clear until the plots are made. The winds probably also show seasonal variation (summer into autumn) or associations with other variables such as temperature and rainfall. There is a lot more analysis to perform on this interesting data set which I hope will one day be made public. n

References 1. Tufte, E. R. (1983) The Visual Display of Quantitative Information. Cheshire, CT: Graphics Press.

2. Miller, R. A., Onrubia, A., Martín, B., Kaltenecker, G. S., Carlisle, J. D., Bechard, M. J. and Ferrer, M. (2016)

Local and regional weather patterns influencing

post-breeding migration counts of soaring birds at the

Strait of Gibraltar, Spain, Ibis, 158(1), 106–115. 3. Reese, R. A. (2018) Graphical interpretations of data: Truth and persuasion. Significance, 15(1), 42–43.

rather than used in a semantic context. Wind direction is, of course, a circular variable but is here decomposed into two components at right angles. The positive “horizontal” component is therefore a wind blowing west to east, but its velocity as an increasing “westerly” plots where the compass points to east. Increasing “easterly” winds plot as negative values increasing to the left. An easterly wind is a negative westerly in Mathsworld, but not in any real-world weather discussion.

An immediate fix would be to title the axis explicitly, with “Westerly winds” under the positive section of the axis, and “Easterly winds” under the negative values. This explains the relationship with numbers but highlights the perversity of plotting westerly winds to the “east” as usually drawn.

A better solution is to reverse the sign and plot westerly wind strength going left, and easterly to the right. The sign is arbitrary. Better still to drop the minus signs altogether: label in both directions with positive numbers as the wind speed. Negative values are necessary computationally to position points on the graph, but for the human–computer interface we want to communicate “simply and neatly”.3 The graphs would also be improved with a vertical reference line at zero to show the dichotomy. That would also clarify that the wind speeds cover equal ranges from west to east but stronger winds from the south than the north.

Which wind speeds should be labelled? While it is easy to use the default “about half

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