# It is not just weight that affects the fuel

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It is not just weight that affects the fuel consumption; other characteristics may have an effect. Engine sizes may be different and the shapes may not be the same. One could make the model more complicated by incorporating these other factors into it. Although this would reduce the variability in fuel consumption, one should not make the function so complicated that it passes through every single point. Such an approach would ignore the natural variability in measure- ments and attach too much importance to random variation. Henri Poincare, in The Foundations of Science [Science Press, New York, 1913 (reprinted 1929), p. 169] expresses this very well when he writes, Pass to an example of a more scienti fi c character. I wish to determine an experimental law. This law, when I know it, can be represented by a curve. I make a certain number of isolated observations; each of these will be represented by a point. When I have obtained these different points, I draw a curve between them, striving to pass as near to them as possible and yet preserve for my curve a regular form, without angular points, or in fl ections too accentuated, or brusque variation of the radius of curvature. This curve will represent for me the probable law, and I assume not only that it will tell me the values of the function intermediate
Abraham Abraham ˙ C01 November 8, 2004 0:33 1.5 Data Plots and Empirical Modeling 19 4 3 (a) 2 7 6 5 4 3 2 Weight GP100M 100 200 (b) 300 2 3 4 5 6 7 Displacement GP100M FIGURE 1.4 (a) Pairwise scatter plot of y (gallons/100 miles) against weight. (b) Pairwise scatter plot of y (gallons/100 miles) against displacement. (c) Pairwise scatter plot of y (gallons/100 miles) against number of cylinders. (d) Three- dimensional plot of y (gallons/100 miles) against weight and displacement between those which have been observed, but also that it will give me the observed values themselves more exactly than direct observation. This is why I make it pass near the points, and not through the points themselves. Here, we have described a two-dimensional representation of fuel consump- tion y and weight x 1 . Similar scatter plots can be carried out for fuel consumption ( y ) and displacement x 2 and also fuel consumption ( y ) and number of cylinders x 3 . The graphs shown in Figures 1.4b and 1.4c indicate linear relationships, even though the strengths of these relationships differ. We notice from Figure 1.4 that each pairwise scatter plot exhibits linear- ity. Is this enough evidence to conclude that the model with the three explana- tory variables should be linear also? The answer to this question is no in general. Although the linear model, y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 3 + ε , may pro- vide a good starting point, the model may miss more complicated associations.
Abraham Abraham ˙ C01 November 8, 2004 0:33 20 Introduction to Regression Models 4 5 6 (c) 7 8 2 3 4 5 6 7 Number of cylinders GP100M 300 2 2.5 Displacement 200 3.5 4.5 5.5 3 6.5 GP100M 4 (d) 100 Weight FIGURE 1.4 (Continued) Two-dimensional displays are unable to capture the joint relationships among the response and more than one explanatory variable. In order to bring out the