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ips6e_supp_material - SUPPLEMENTAL MATERIAL: TRANSFORMING...

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SUPPLEMENTAL MATERIAL: TRANSFORMING DATA, DECISION ANALYSIS Introduction This brief supplement presents material useful to some readers of IPS that was re- moved from the main text for reasons of length. It has two sections: 1. A section on transforming relationships. This is intended to follow Section 2.6 of IPS, which now ends Chapter 2. 2. A supplement to Section 4.5, brieFy discussing decision analysis. The natural place for this material is immediately following the discussion of tree diagrams, IPS page 301. 1
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Supplemental Section 2.6 Transforming Relationships Let’s start by revisiting Example 2.18 on page 120 of IPS. How is the weight of an animal’s brain related to the weight of its body? Figure 2.31 (which repeats Figure 2.17) is a scatterplot of brain weight against body weight for 96 species of mammals. 1 The line is the least-squares regression line for predicting brain weight from body weight. The outliers are interesting. We might say that dolphins and humans are smart, hippos are dumb, and African elephants are just big. That’s because dolphins and humans have larger brains than their body weights suggest, hippos have smaller brains, and the elephant is much heavier than any other mammal in both body and brain. EXAMPLE 2.37 The plot in Figure 2.31 is not very satisfactory. Most mammals are so small relative to elephants and hippos that their points overlap to form a blob in the lower-left corner of the plot. The correlation between brain weight and body weight is r =0 . 86, but this is misleading. If we remove the elephant, the correlation for the other 95 species is r = 0 . 50. Figure 2.32 is a scatterplot of the data with the four outliers removed to allow a closer look at the other 92 observations. We can now see that the relationship is not linear. It bends to the right as body weight increases. Biologists know that data on sizes often behave better if we take logarithms 2
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1000 500 1500 2000 2500 0 2800 2600 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 Brain weight, grams Body weight, kilograms Hippo Elephant 3000 4500 4000 3500 Dolphin Human FIGURE 2.3 1 Scatterplo t o f b rain w eight a gainst bo dy weight for 96 species of mammals.
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0 200 400 600 0 300 200 100 400 500 Brain weight, grams Body weight, kilograms FIGURE 2.3 2 Scatterplot of brain weight against body weight for mammals, with outliers removed. 0 1 2 3 –1 2 1 0 3 Logarithm of brain weight Logarithm of body weight FIGURE 2.3 3 Scatterplot of the logarithm of brain weight against the logarithm of bodyweight for 96 species of mammals.
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before doing more analysis. Figure 2.33 plots the logarithm of brain weight against the logarithm of body weight for all 96 species. The effect is almost magical. There are no longer any extreme outliers or very in±uential observations. The pattern is very linear, with correlation r =0 . 96. The vertical spread about the least-squares line is similar everywhere, so that predictions of brain weight from body weight will be about equally precise for any body weight (in the log scale).
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ips6e_supp_material - SUPPLEMENTAL MATERIAL: TRANSFORMING...

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