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A graph is full of answers, and the only work a reader needs to do is to bring the right questions, and know how to interrogate the graph. A good place to start exploring a graph is to apply a few questions:

1 What exactly does the horizontal X axis represent?

2 What exactly does the vertical Y axis represent?

3 When X increases, what happens to Y?

Reading information from graphs becomes easier with practice, and a few points about the graphs in this book might be helpful. Some of the points are obvious, but others will take some thought before insights can be pulled out of graphs.

The graphs in Figures B and C come from a massive dataset for tropical forests around the world. Taylor et al. (2017) compiled information on rates of wood growth, with basic information on each location's annual precipitation and average temperature (this example shows up again in Chapter 2). We know that trees need large amounts of water, because water evaporates (transpires) from leaves as the leaves absorb carbon dioxide from the air. We might have an idea that forests with a higher water supply should grow faster. This idea may or may not be true for forests, so checking the evidence from many studies lets us determine if this idea is worthy of our confidence. Indeed, forests with the highest amounts of precipitation grew more stemwood than those with a moderate supply. A few key features are worth pointing out. The X axis for precipitation spans a sevenfold range, from 1000–6000 mm yr⁻¹. The Y axis for stem growth also spans a large range, but the line in the graph goes only from a bit more than 4 Mg ha⁻¹ yr⁻¹ to something less than 8 Mg ha⁻¹ yr⁻¹. Note that a sixfold change in the X axis (precipitation) gives at most a twofold range in stem growth, so the response is not as dramatic as if a twofold difference in precipitation gave a sixfold change in stem growth. Water matters, but not as much as we might have expected.

FIGURE B Stem growth in tropical forests is higher for sites with higher precipitation (left). The association with temperature is more complex, increasing at low temperatures and declining at the highest temperatures

(Source: from data in Taylor et al. 2017).

FIGURE C The influence of both factors can be examined together by examining the response of growth to temperature for three precipitation groupings (lower left). Putting sites into precipitation groups leaves out some of the information in the full dataset, and a 3D graph makes use of all the information. 3D graphs can work well for illustrating the overall trend surface, but specific details may be easier to read on 2D graphs

(Source: from the database compiled by Taylor et al. 2017).

A second point for the first graph is that the average across all the studies follows a simple trend: a given increase in precipitation gives about the same amount of increase in stem growth, regardless of whether we look at the dry end or the wet end of the spectrum. We might have guessed that a small increase in water for dry sites would have a bigger value for tree growth than the same increase on a site that is already wet, but the available evidence would not support that generalization.

Just because the line in a graph goes up does not mean the trend warrants high confidence. If we chose a set of three numbers at random, the odds are good that the average trend would go up or down, rather than be flat. But if we choose a set with 100 random numbers, the odds are very high indeed that the trend would be flat (as there was no chance of the value of Y being related to X). This third point is illustrated with the shaded band around the line in the first graph, which represents the 95% confidence interval around the line. This means the evidence says the true trend would fall within that band about 95% of the time if the sampling were repeated. If we plotted a line with 240 random points for stem growth for random levels of precipitation, the line would be close to flat. If a flat line was placed into Figure B, it would fall outside the shaded bands of the confidence interval, so high confidence is warranted that sites with high rates of forest growth also tend to have higher precipitation. The odds are less than one in one thousand that growth and precipitation are unrelated (a flat line; this is the “P” value in statistics, which was <0.001 for this trend). This relationship of course does not prove that having more water is the key to producing more growth, but it does show the idea is not counter to the available evidence.

A third point about the first graph is that the dispersion of points around the line is broad indeed. Two tropical forests that have the same amount of precipitation might easily differ by twofold in growth rates. Even if confidence had been warranted in an average effect of precipitation, the average trend would not give a strong prediction for any single observation: half of all observations are always above average, and half below, no matter how much confidence is warranted in the overall trend.

This dispersion of points around the average trend is the fourth important story in the first graph. In statistics, the dispersion is the variance of the sample, and a number is often applied to trends in graphs that characterizes this dispersion around the average. The correlation coefficient (r) describes how tightly the data clump near the average trend, and the square of the correlation coefficient (r²) tells the proportion of all the variability in Y values that relates to the level of the X values. In the first graph, the r² for stem growth in relation to precipitation is 0.04 (4%). We can be strongly confident that stem growth on average increases with precipitation, but that knowledge accounts for only a very small part of the full distribution of growth rates of tropical forests. The idea seems likely to be true, but it gives very little power for precise predictions.

The second graph in Figure B shows the growth rates of the same forests, but in relation to the average annual temperature. The confidence band is a bit tighter in this case, and the dispersion of points around the trend is not as large. The probability that random noise would account for the pattern is quite small (less than one in a thousand), so high confidence is warranted in the association between stem growth and temperature. The average trend with temperature accounts for about 23% of all the variation in growth rates among sites (r² = 0.23). Does higher temperature directly cause higher growth rates of forests? Possibly, but the association between two things does not mean that one causes the other. It's possible that soil nutrient supply is the real driver of growth, and soils in warmer parts of the Tropics have higher nutrient supplies. Confidence in whether one thing actually drives another depends on further evidence (and often direct experimentation).

The growth of a forest with a given temperature could depend on water supply. The range of sites could be divided into three groups: sites with less than 2000 mm yr⁻¹, 2000–4000 mm yr⁻¹, and more than 4000 mm yr⁻¹ (Figure C). The trends between temperature and stem growth are similar across these three groups at temperatures below 23 °C, but at higher temperatures growth seemed to decline more on drier sites than on wetter sites. This breakdown of the temperature relationship into three precipitation groups increases that amount of variation accounted for in growth to 31%, and very high confidence is warranted that predictions of temperature responses of growth differ among the precipitation groups.

Separating the sites into precipitation groups actually throws away some information that might be useful. For example, a site with 1950 mm yr⁻¹ precipitation would be tallied in the driest group, and one with 2050 mm yr⁻¹ would be separated into the medium group. Yet these two sites would be more similar to each other than the 2050 mm yr⁻¹ site would be to another in the medium group with 3950 mm yr⁻¹ site. Another version of the analysis could be done with all the data from each site allowed to influence the trend, and then a full three‐dimensional pattern can be developed. The second graph in Figure B has two horizontal axes. The temperature axis increases to the right, and “backward” into the 3D space. The precipitation axis goes the other way, increasing to the left and also going backward into the space. This graph shows how any given level of temperature, and any level of precipitation, connect to give an estimate of the expected rate of stem growth. Keeping all the information on precipitation included (rather than lumping into three groups) increases the variation accounted for to 34%. A key difference is that this full‐information analysis shows that growth continues to increase at high temperatures if the precipitation is high, but levels off (with no decline) on drier sites. This might seem like a small improvement in the pattern, but the improvement does warrant very high confidence.

It can be challenging to read the values for stem growth on the 3D graph, compared with straightforward 2D graphs. The grid lines give some help for visualizing how the overall trend changes, and the use of colors helps peg a value to any given point on the surface. Overall, 3D graphs can be very useful for illustrating overall trends, but 2D graphs might be more useful when the precise values of variables need to be identified.

Why do temperature and precipitation relate to only about one‐third of all the variation in stem growth among tropical forests? Two points are important. This analysis used only annual averages, and two sites with similar annual average might differ in important seasonal ways. A given amount of rain spread evenly across 12 months might have very different effects on growth than if all the rain fell during a 4‐month rainy season (with no rain for 8 months). The second point is that stem growth depends on a wide range of ecological factors, including soil nutrient supplies, and the genotypes of trees present. Attempts to explain forest growth often go beyond the ability of graphs to capture the relationship, using simulation models and other tools that have a chance to capture variations in growth patterns that go beyond two or three dimensions (Chapter 7).