4: Nonlinear Relationships and Transformations
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254
Chapter 5 Summarizing Bivariate Data
The scatterplot of these data is reproduced here as Figure 5.26. Because this plot
shows a marked curved pattern, it is clear that no straight line can do a reasonable job
of describing the relationship between x and y. However, the relationship can be described by a curve, and in this case the curved pattern in the scatterplot looks like a
parabola (the graph of a quadratic function). This suggests trying to ﬁnd a quadratic
function of the form
y^ 5 a 1 b1x 1 b2x 2
that would reasonably describe the relationship. That is, the values of the coefﬁcients
a, b1, and b2 in this function must be selected to obtain a good ﬁt to the data.
Average finish time
300
275
250
225
200
10
FIGURE 5.26
20
30
Scatterplot for the marathon data.
40
Age
50
60
70
What are the best choices for the values of a, b1, and b2? In ﬁtting a line to data, we
used the principle of least squares to guide our choice of slope and intercept. Least
squares can be used to ﬁt a quadratic function as well. The deviations, y 2 y^ , are still
represented by vertical distances in the scatterplot, but now they are vertical distances
from the points to a parabola (the graph of a quadratic function) rather than to a line,
as shown in Figure 5.27. We then choose values for the coefﬁcients in the quadratic
function so that the sum of squared deviations is as small as possible.
y
Deviation
FIGURE 5.27
Deviation for a quadratic function.
x
x1
For a quadratic regression, the least squares estimates of a, b1, and b2 are those values
that minimize the sum of squared deviationsg 1 y 2 y^ 2 2 where y^ 5 a 1 b1 x 1 b2 x 2.
For quadratic regression, a measure that is useful for assessing ﬁt is
R2 5 1 2
SSResid
SSTo
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5.4
255
Nonlinear Relationships and Transformations
where SSResid 5 g 1y 2 y^ 2 2. The measure R 2 is deﬁned in a way similar to r 2 for
simple linear regression and is interpreted in a similar fashion. The notation r 2 is used
only with linear regression to emphasize the relationship between r 2 and the correlation coefﬁcient, r, in the linear case.
The general expressions for computing the least-squares estimates are somewhat
complicated, so we rely on a statistical software package or graphing calculator to do
the computations for us.
E X A M P L E 5 . 1 4 Marathon Data Revisited: Fitting
a Quadratic Model
300
75
275
50
25
Residual
Average finish time
For the marathon data, the scatterplot (see Figure 5.26) showed a marked curved
pattern. If the least-squares line is ﬁt to these data, it is no surprise that the line does
not do a good job of describing the relationship (r 2 5 .001 or .1% and se 5 56.9439),
and the residual plot shows a distinct curved pattern as well (Figure 5.28).
250
225
0
−25
200
−50
10
20
30
FIGURE 5.28
Plots for the marathon data of Example 5.14: (a) least-square regression
line; (b) residual plot.
40
Age
(a)
50
60
70
20
10
30
40
x
(b)
50
60
70
Part of the Minitab output from ﬁtting a quadratic regression function to these
data is as follows:
The regression equation is
y = 462 – 14.2 x + 0.179 x-squared
Predictor
Constant
x
x-squared
S = 18.4813
Coef
462.00
–14.205
0.17888
SE Coef
43.99
2.460
0.03025
R-Sq = 92.1%
T
10.50
–5.78
5.91
P
0.002
0.010
0.010
R-Sq(adj) = 86.9%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
2
3
5
SS
11965.0
1024.7
12989.7
MS
5982.5
341.6
F
17.52
P
0.022
The least-squares coefﬁcients are
a ϭ 462.00
b1 ϭ Ϫ14.205
b2 ϭ 0.17888
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256
Chapter 5 Summarizing Bivariate Data
and the least-squares quadratic is
y^ 5 462.00 2 14.205x 1 0.17888x 2
A plot showing the curve and the corresponding residual plot for the quadratic
regression are given in Figure 5.29. Notice that there is no strong pattern in the residual plot for the quadratic case, as there was in the linear case. For the quadratic
regression, R 2 5 .921 (as opposed to .001 for the least-squares line), which means
that 92.1% of the variability in average marathon ﬁnish time can be explained by an
approximate quadratic relationship between average ﬁnish time and age.
300
10
0
260
Residual
Average finish time
280
240
220
−10
−20
200
−30
180
10
20
30
40
Age
(a)
50
60
10
70
20
30
40
x
(b)
50
60
70
FIGURE 5.29
Quadratic regression of Example 5.13:
(a) scatterplot; (b) residual plot.
Linear and quadratic regression are special cases of polynomial regression. A polynomial regression curve is described by a function of the form
y^ 5 a 1 b1x 1 b2 x 2 1 b3 x 3 1 c1 bk x k
which is called a kth-degree polynomial. The case of k 5 1 results in linear regression
1 y^ 5 a 1 b1x2 and k 5 2 yields a quadratic regression 1 y^ 5 a 1 b1x 1 b2 x 22 . A quadratic curve has only one bend (see Figure 5.30(a) and (b)). A less frequently encountered special case is for k 5 3, where y^ 5 a 1 b1x 1 b2 x 2 1 b3 x 3, which is called a
cubic regression curve. Cubic curves have two bends, as shown in Figure 5.30(c).
y
y
y
FIGURE 5.30
Polynomial regression curves:
(a) quadratic curve with b2 Ͻ 0;
(b) quadratic curve with b2 Ͼ 0;
(c) cubic curve.
x
(a)
x
x
(b)
(c)
E X A M P L E 5 . 1 5 Fish Food
Data set available online
Sea bream are one type of fish that are often raised in large fish farming enterprises.
These fish are usually fed a diet consisting primarily of fish meal. The authors of the
paper “Growth and Economic Profit of Gilthead Sea Bream (Sparus aurata, L.) Fed
Sunflower Meal (Aquaculture [2007]: 528–534) describe a study to investigate
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5.4
Nonlinear Relationships and Transformations
257
whether it would be more profitable to substitute plant protein in the form of sunflower meal for some of the fish meal in the sea bream’s diet.
The accompanying data are consistent with summary quantities given in the
paper for x 5 percent of sunflower meal in the diet and y 5 average weight (in grams)
of fish after 248 days.
Sunflower Meal (%)
Average Fish Weight
0
6
12
18
24
30
36
432
450
455
445
427
422
421
Figure 5.31 shows a scatterplot of these data. The relationship between x and y does
not appear to be linear, so we might try using a quadratic regression to describe the
relationship between sunflower meal content and average fish weight.
Averagee fish weight
460
FIGURE 5.31
Scatterplot of average fish weight versus sunflower meal content for the
data of Example 5.15.
450
440
430
420
0
10
20
Sunflower meal (%)
30
40
Minitab was used to fit a quadratic regression function and to compute the corresponding residuals. The least-squares quadratic regression is
y^ ϭ 439 ϩ 1.22x Ϫ 0.053x 2
A plot of the quadratic regression curve and the corresponding residual plot are
shown in Figure 5.32.
Notice that the residual plot in Figure 5.32(b) shows a curved pattern (cubic)—not
something we like to see in a residual plot. This suggests that we may want to consider
something other than a quadratic curve to describe the relationship between x and y.
Looking again at the scatterplot of Figure 5.31, we see that a cubic function might be
a better choice because there appear to be two “bends” in the curved relationship—one
at around x 5 12 and another at the far right hand side of the scatterplot.
Using the given data, Minitab was used to fit a cubic regression, resulting in the
curve shown in Figure 5.33(a). The cubic regression is then
y^ ϭ 431.5 ϩ 5.39x Ϫ 0.37x 2 ϩ 0.006x 3
The corresponding residual plot, shown in Figure 5.33(b), does not reveal any
troublesome patterns that would suggest a choice other than the cubic regression.
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258
Chapter 5 Summarizing Bivariate Data
Fitted Line Plot
Average fish weight = 439.0 + 1.220 sunflower meal (%)
−0.05324 Sunflower Meal (%)**2
S
9.46736
R-Sq
69.0%
R-Sq(adj) 53.5%
450
10
5
440
Residuals
Averagee fish weight
460
430
0
420
−5
410
−10
0
10
20
Sunflower meal (%)
30
40
0
10
(a)
20
Sunflower meal (%)
30
40
(b)
FIGURE 5.32
Quadratic regression plots for the fish food data
of Example 5.15: (a) least-squares quadratic regression; (b) residual plot for quadratic regression.
Fitted Line Plot
Average fish weight = 431.5 + 5.387 sunflower meal (%)
−0.3657 sunflower meal (%)**2 + 0.005787 sunflower meal (%)**3
S
2.64725
R-Sq
98.2%
R-Sq(adj) 96.4%
450
2
1
Residual
Averagee fish weight
460
440
0
−1
430
−2
−3
420
0
10
20
Sunflower meal (%)
30
40
(a)
0
10
20
Sunflower meal (%)
30
40
(b)
FIGURE 5.33
Cubic regression plots for the fish
food data of Example 5.15:
(a) least-squares cubic regression;
(b) residual plot for cubic regression.
Based on analysis of these data, we might recommend using sunflower meal for
about 12% of the diet. Sunflower meal is less costly than fish meal, but using more
than about 12% sunflower meal is associated with a decrease in the average fish
weight. It is not clear what happens to average fish weight when sunflower meal is
used for more than 36% of the diet, the largest x value in the data set.
Transformations
An alternative to ﬁnding a curve to ﬁt the data is to ﬁnd a way to transform the
x values and/or y values so that a scatterplot of the transformed data has a linear appearance. A transformation (sometimes called a reexpression) involves using a simple funcCopyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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5.4
Nonlinear Relationships and Transformations
259
tion of a variable in place of the variable itself. For example, instead of trying to describe
the relationship between x and y, it might be easier to describe the relationship between
!x and y or between x and log( y). And, if we can describe the relationship between,
say, !x and y, we will still be able to predict the value of y for a given x value. Common
transformations involve taking square roots, logarithms, or reciprocals.
E X A M P L E 5 . 1 6 River Water Velocity and Distance from Shore
As fans of white-water rafting know, a river ﬂows more slowly close to its banks
(because of friction between the river bank and the water). To study the nature of
the relationship between water velocity and the distance from the shore, data were
gathered on velocity (in centimeters per second) of a river at different distances (in
meters) from the bank. Suppose that the resulting data were as follows:
Distance
.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
Velocity 22.00 23.18 25.48 25.25 27.15 27.83 28.49 28.18 28.50 28.63
A graph of the data exhibits a curved pattern, as seen in both the scatterplot and the
residual plot from a linear ﬁt (see Figures 5.34(a) and 5.34(b)).
Velocity
FIGURE 5.34
Plots for the data of Example 5.16:
(a) scatterplot of the river data;
(b) residual plot from linear ﬁt.
Residual
1
30
29
28
27
26
25
24
23
22
0
–1
0 1 2 3 4 5 6 7 8 9 10
Distance
(a)
0
1
2
3
4 5 6
Distance
(b)
7
8
9 10
Let’s try transforming the x values by replacing each x value by its square root.
We deﬁne
xr 5 !x
The resulting transformed data are given in Table 5.2.
T AB LE 5 .2 Original and Transformed Data of Example 5.16
Original Data
Data set available online
Transformed Data
x
y
xЈ
y
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
22.00
23.18
25.48
25.25
27.15
27.83
28.49
28.18
28.50
28.63
0.7071
1.2247
1.5811
1.8708
2.1213
2.3452
2.5495
2.7386
2.9155
3.0822
22.00
23.18
25.48
25.25
27.15
27.83
28.49
28.18
28.50
28.63
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260
Chapter 5
Summarizing Bivariate Data
Figure 5.35(a) shows a scatterplot of y versus xЈ (or equivalently y versus !x).
The pattern of points in this plot looks linear, and so we can ﬁt a least-squares line
using the transformed data. The Minitab output from this regression appears
below.
Regression Analysis
The regression equation is
Velocity = 20.1 + 3.01 sqrt distance
Predictor
Constant
Sqrt dis
Coef
20.1102
3.0085
S = 0.6292
StDev
0.6097
0.2726
R-Sq = 93.8%
T
32.99
11.03
P
0.000
0.000
R-Sq(adj) = 93.1%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
8
9
SS
48.209
3.168
51.376
MS
48.209
0.396
F
121.76
P
0.000
The residual plot in Figure 5.35(b) shows no indication of a pattern. The resulting regression equation is
y^ 5 20.1 1 3.01xr
An equivalent equation is
y^ 5 20.1 1 3.01!x
The values of r 2 and se (see the Minitab output) indicate that a line is a reasonable
way to describe the relationship between y and xЈ. To predict velocity of the river at
a distance of 9 meters from shore, we ﬁrst compute xr 5 !x 5 !9 5 3 and then
use the sample regression line to obtain a prediction of y:
y^ 5 20.1 1 3.01xr 5 20.1 1 13.012 132 5 29.13
Velocity
Residual
29
28
27
26
25
24
FIGURE 5.35
Plots for the transformed data of Example 5.16: (a) scatterplot of y versus
xЈ; (b) residual plot resulting from a
linear ﬁt to the transformed data.
23
22
1
2
Distance
(a)
3
.8
.6
.4
.2
0
−.2
−.4
−.6
−.8
1
2
Distance
(b)
3
In Example 5.16, transforming the x values using the square root function
worked well. In general, how can we choose a transformation that will result in a
linear pattern? Table 5.3 gives some guidance and summarizes some of the properties
of the most commonly used transformations.
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5.4
Nonlinear Relationships and Transformations
261
TA BLE 5. 3 Commonly Used Transformations
Transformation
Mathematical
Description
No transformation
y^ 5 a 1 bx
Square root of x
y^ 5 a 1 b!x
Log of x *
y^ 5 a 1 b log10 1x2
or
y^ 5 a 1 b ln 1x2
1
y^ 5 a 1 ba b
x
Reciprocal of x
Log of y *
(Exponential
growth or decay)
Try This Transformation When
log10 1 y^ 2 5 a 1 bx
or
ln 1 ^y2 5 a 1 bx
The change in y is constant as x changes. A 1-unit increase in x is associated
with, on average, an increase of b in the value of y.
The change in y is not constant. A 1-unit increase in x is associated with
smaller increases or decreases in y for larger x values.
The change in y is not constant. A 1-unit increase in x is associated
with smaller increases or decreases in the value of y for larger x values.
The change in y is not constant. A 1-unit increase in x is associated with
smaller increases or decreases in the value of y for larger x values. In addition, y has a limiting value of a as x increases.
The change in y is not constant. A 1-unit increase in x is associated with
larger increases or decreases in the value of y for larger x values.
* The values of a and b in the regression equation will depend on whether log10 or ln is used, but the y^ ’s and r 2 values will be identical.
E X A M P L E 5 . 1 7 Loons on Acidic Lakes
A study of factors that affect the survival of loon chicks is described in the paper
“Does Prey Biomass or Mercury Exposure Affect Loon Chick Survival in Wisconsin?” (The Journal of Wildlife Management [2005]: 57–67). In this study, a relationship between the pH of lake water and blood mercury level in loon chicks was
observed. The researchers thought that this might be because the pH of the lake water
might be related to the type of fish that the loons ate. The accompanying data (read
from a graph in the paper and shown in Table 5.4) is x 5 lake pH and y 5 blood
T A B LE 5 .4 Data and Transformed Data from Example 5.17
Data set available online
Lake pH (x)
Blood Mercury Level (y)
Log(y)
5.28
5.69
5.56
5.51
4.90
5.02
5.02
5.04
5.30
5.33
5.64
5.83
5.83
6.17
6.22
6.15
1.10
0.76
0.74
0.60
0.48
0.43
0.29
0.09
0.10
0.20
0.28
0.17
0.18
0.55
0.43
0.40
0.0414
Ϫ0.1192
Ϫ0.1308
Ϫ0.2218
Ϫ0.3188
Ϫ0.3665
Ϫ0.5376
Ϫ1.0458
Ϫ1.0000
Ϫ0.6990
Ϫ0.5528
Ϫ0.7696
Ϫ0.7447
Ϫ0.2596
Ϫ0.3665
Ϫ0.3979
(continued)
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262
Chapter 5 Summarizing Bivariate Data
T A B L E 5 .4 Data and Transformed Data from Example 5.17—cont'd
Lake pH (x)
Blood Mercury Level (y)
Log(y)
6.05
6.04
6.24
6.30
6.80
6.58
6.65
7.06
6.99
6.97
7.03
7.20
7.89
7.93
7.99
7.99
8.30
8.42
8.42
8.95
9.49
0.33
0.26
0.18
0.16
0.45
0.30
0.28
0.22
0.21
0.13
0.12
0.15
0.11
0.11
0.09
0.06
0.09
0.09
0.04
0.12
0.14
Ϫ0.4815
Ϫ0.5850
Ϫ0.7447
Ϫ0.7959
Ϫ0.3468
Ϫ0.5229
Ϫ0.5528
Ϫ0.6576
Ϫ0.6778
Ϫ0.8861
Ϫ0.9208
Ϫ0.8239
Ϫ0.9586
Ϫ0.9586
Ϫ1.0458
Ϫ1.2218
Ϫ1.0458
Ϫ1.0458
Ϫ1.3979
Ϫ0.9208
Ϫ0.8539
mercury level (mg/g) for 37 loon chicks from different lakes in Wisconsin. A scatterplot is shown in Figure 5.36(a).
The pattern in this scatterplot is typical of exponential decay, with the change in
y as x increases much smaller for large x values than for small x values. You can see
that a change of 1 in pH is associated with a much larger change in blood mercury
level in the part of the plot where the x values are small than in the part of the plot
where the x values are large. Table 5.3 suggests transforming the y values (blood mercury level in this example) by taking their logarithms.
Two standard logarithmic functions are commonly used for such transformations—
the common logarithm (log base 10, denoted by log or log10) and the natural logarithm
(log base e, denoted ln). Either the common or the natural logarithm can be used; the
only difference in the resulting scatterplots is the scale of the transformed y variable. This
can be seen in Figures 5.36(b) and 5.36(c). These two scatterplots show the same pattern,
and it looks like a line would be appropriate to describe this relationship.
Table 5.4 displays the original data along with the transformed y values using
yЈ 5 log(y). The following Minitab output shows the result of fitting the least-squares
line to the transformed data:
Regression Analysis: Log(y) versus Lake pH
The regression equation is
Log(y) ϭ 0.458 Ϫ 0.172 Lake pH
Predictor
Coef
SE Coef
T
P
Constant
0.4582
0.2404
1.91
0.065
Lake Ph
Ϫ0.17183
0.03589
Ϫ4.79
0.000
S ϭ 0.263032
R-Sq ϭ 39.6%
R-Sq(adj) ϭ 37.8%
(continued)
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5.4
Nonlinear Relationships and Transformations
263
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
35
36
SS
1.5856
2.4215
4.0071
MS
1.5856
0.0692
F
22.92
P
0.000
1.2
Blood mercury level (y)
1.0
0.8
0.6
0.4
0.2
0.0
5
6
7
8
Lake pH (x)
9
10
(a)
0.0
−0.2
Log (y)
−0.4
−0.6
−0.8
−1.0
−1.2
−1.4
5
6
7
8
Lake pH (x)
9
10
9
10
(b)
0.0
−0.5
ln y
−1.0
−1.5
−2.0
−2.5
−3.0
FIGURE 5.36
Plots for the data of Example 5.17:
(a) scatterplot of the loon data;
(b) scatterplot of the transformed
data with yЈ ϭ log(y); (c) scatterplot
of transformed data with yЈ ϭ ln(y).
−3.5
5
6
7
8
Lake pH (x)
(c)
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264
Chapter 5
Summarizing Bivariate Data
The resulting regression equation is
yЈ ϭ 0.458 Ϫ 0.172x
or, equivalently
log(y) ϭ 0.458 Ϫ 0.172x
Fitting a Curve Using Transformations The objective of a regression analysis is
usually to describe the approximate relationship between x and y with an equation of
the form y ϭ some function of x.
If we have transformed only x, ﬁtting a least-squares line to the transformed data
results in an equation of the desired form, for example,
y^ 5 5 1 3xr 5 5 1 3 !x where xr 5 !x
or
1
1
y^ 5 4 1 .2xr 5 4 1 .2 where xr 5
x
x
These functions specify lines when graphed using y and xЈ, and they specify curves
when graphed using y and x, as illustrated in Figure 5.37 for the square root
transformation.
160
140
120
y
100
80
60
40
20
0
0
10
20
30
Transformed x = square root x
40
50
(a)
160
140
120
y
100
80
60
40
20
0
FIGURE 5.37
(a) A plot of y^ 5 5 1 3xr where
xr 5 !x; (b) a plot of
y^ 5 5 1 3!x.
0
500
1000
1500
2000
2500
x
(b)
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