Fitting Curves to Data 1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 5: Fitting Curves to Data Terry Dielman Applied Regression

Fitting Curves to DataFitting Curves to Data 11Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Chapter 5:Chapter 5:

Fitting Curves to DataFitting Curves to DataTerry DielmanTerry Dielman

Applied Regression Analysis:Applied Regression Analysis:A Second Course in Business and A Second Course in Business and Economic Statistics, fourth editionEconomic Statistics, fourth edition


5.1 Introduction5.1 Introduction

In Chapter 4 , the model was presented as:In Chapter 4 , the model was presented as:

where we assumed linear relationships where we assumed linear relationships between between y y and theand the x x variables.variables.

In this chapter we find that this may not be In this chapter we find that this may not be true and consider curvilinear relationships true and consider curvilinear relationships between the variables.between the variables.

ikikiii exxxy 22110


ModelingModeling In general, we regress Y on some In general, we regress Y on some

X which is not a linear function.X which is not a linear function.

Common functions are XCommon functions are X22 , 1/X , 1/X or log(X) or log(X)

In economics, sometimes regress In economics, sometimes regress log(y) on log(x) log(y) on log(x)


5.2 Fitting Curvilinear Relationships5.2 Fitting Curvilinear Relationships

Polynomial Regression – a common Polynomial Regression – a common correction for nonlinearity is to add correction for nonlinearity is to add powers of the explanatory variablepowers of the explanatory variable

In practice a second-order model is In practice a second-order model is often sufficient to describe the often sufficient to describe the relationshiprelationship

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Example 5.1: TelemarketingExample 5.1: Telemarketing

n = 20 telemarketing employeesn = 20 telemarketing employees

Y = average calls per day over 20 Y = average calls per day over 20 workdaysworkdays

X = Months on the jobX = Months on the job

Data set TELEMARKET5Data set TELEMARKET5


Plot of Calls versus MonthsPlot of Calls versus Months

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MONTHS

CA

LL

S

There is an increase in calls with experience, but the rate of increase slows over time.


Fit of a First-Order ModelFit of a First-Order Model

For comparison purposes, we first fit For comparison purposes, we first fit the linear equation and obtained:the linear equation and obtained:

CALLS = 13.6708 + .7435 MONTHSCALLS = 13.6708 + .7435 MONTHS

This equation, which has an RThis equation, which has an R22 of of 87.4%, implies that each month of 87.4%, implies that each month of experience leads to .7435 more calls experience leads to .7435 more calls per day.per day.


Fitting a Second-Order ModelFitting a Second-Order Model

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S = 1.00325 R-Sq = 96.2 % R-Sq(adj) = 95.8 %

- 0.0401182 MONTHS**2

CALLS = -0.140471 + 2.31020 MONTHS

Regression Plot


Regression OutputRegression OutputRegression Analysis: CALLS versus MONTHS, MonthSQ

The regression equation isCALLS = - 0.14 + 2.31 MONTHS - 0.0401 MonthSQ

Predictor Coef SE Coef T PConstant -0.140 2.323 -0.06 0.952MONTHS 2.3102 0.2501 9.24 0.000MonthSQ -0.040118 0.006333 -6.33 0.000

S = 1.003 R-Sq = 96.2% R-Sq(adj) = 95.8%

Analysis of Variance

Source DF SS MS F PRegression 2 437.84 218.92 217.50 0.000Residual Error 17 17.11 1.01Total 19 454.95


Hypothesis Test on Hypothesis Test on 22HH00: : 2 2 = 0 = 0 (Use the linear equation)(Use the linear equation)

HHaa: : 2 2 ≠ 0 ≠ 0 (Quadratic has improved fit)(Quadratic has improved fit)

Test as usual with t = bTest as usual with t = b22/SE(b/SE(b22))

Here t = -.0402/.00633 = -6.33 is Here t = -.0402/.00633 = -6.33 is significant with p-value = .000significant with p-value = .000

Not surprising since RNot surprising since R22 increased 9% increased 9%


Hypothesis Tests "Top Down"Hypothesis Tests "Top Down"

The usual practice is to keep lower-The usual practice is to keep lower-order terms when a high-order term order terms when a high-order term is significant.is significant.

In In bb00 + b + b11 x + b x + b22 x x2 2 we would retain we would retain the the bb11 term even if it had an term even if it had an insignificant t-ratio, if the insignificant t-ratio, if the bb22 term term was significant.was significant.


Higher and higher?Higher and higher?

To see if the polynomial has even a To see if the polynomial has even a higher order, we fit a cubic equation.higher order, we fit a cubic equation.

The table below shows the second-The table below shows the second-order model was sufficient.order model was sufficient.

ModelModel p for highest p for highest order termorder term

RR22 Adj RAdj R22 SSee

LinearLinear 0.0000.000 87.4%87.4% 86.7%86.7% 1.7871.787

QuadraticQuadratic 0.0000.000 96.2%96.2% 95.8%95.8% 1.0031.003

CubicCubic 0.5090.509 96.3%96.3% 95.7%95.7% 1.0201.020


Centering the XCentering the X

When polynomial regression is used, When polynomial regression is used, multicollinearity often results multicollinearity often results because x and xbecause x and x2 2 are correlated.are correlated.

This can be eliminated by subtracting This can be eliminated by subtracting x-bar (the mean) from each xx-bar (the mean) from each x

2andUse )x(xxx


5.2.2 Reciprocal Transformation of the 5.2.2 Reciprocal Transformation of the xx Variable Variable

Another curvilinear relationship that Another curvilinear relationship that is in common use is:is in common use is:

Here Here yy and and xx are inversely related are inversely related but the relationship is not linear.but the relationship is not linear.

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y

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Example 5.2Example 5.2

We are interested in the relationship We are interested in the relationship between gas mileage and a car's between gas mileage and a car's horsepower.horsepower.

An the next page is a plot of the An the next page is a plot of the highway mpg (HWYMPG) and highway mpg (HWYMPG) and horsepower (HP) for 147 cars listed in horsepower (HP) for 147 cars listed in the October 2002 the October 2002 Road and TrackRoad and Track..


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HP

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PG

Highway MPG versus HorsepowerHighway MPG versus Horsepower


Modeling the RelationshipModeling the Relationship A regression of HWYMPG on HP yieldsA regression of HWYMPG on HP yields

HWYMPG = 38.73 - .0477 HP with RHWYMPG = 38.73 - .0477 HP with R22 = 59.4% = 59.4%

This does not fit too well because as This does not fit too well because as horsepower increases, mileage decreases, horsepower increases, mileage decreases, but the rate of decrease is slower for but the rate of decrease is slower for more-powerful cars.more-powerful cars.

Although other models, including a Although other models, including a quadratic, might work, we regressed quadratic, might work, we regressed HWYMPG on 1/HP.HWYMPG on 1/HP.


Regression ResultsRegression Results

The regression equation isHWYMPG = 13.6 + 2692 HPINV

Predictor Coef SE Coef T PConstant 13.6310 0.6493 20.99 0.000HPINV 2962.4675 111.7526 24.09 0.000

S = 2.93107 R-Sq = 80.0% R-Sq(adj) = 79.9%

Analysis of Variance

Source DF SS MS F PRegression 1 4987.0 4987.0 580.48 0.000Residual Error 145 1245.1 8.6Total 146 6232.7


7006005004003002001000

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HP

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Data and Reciprocal FitData and Reciprocal Fit


5.2.3 Log Transformation of the 5.2.3 Log Transformation of the xx Variable Variable

Yet another curvilinear equation is:Yet another curvilinear equation is:

where ln(where ln(xx) is the natural logarithm ) is the natural logarithm of of xx..

It is assumed that the It is assumed that the xx values are values are positive because ln(0) is undefined.positive because ln(0) is undefined.

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Example 5.4 Fuel ConsumptionExample 5.4 Fuel Consumption

n = 51 (50 states plus Washington, D.C.)n = 51 (50 states plus Washington, D.C.)

FUELCON = fuel consumption per capitaFUELCON = fuel consumption per capita

POP = state populationPOP = state population

AREA = area of state in square milesAREA = area of state in square miles

POPDENS = population densityPOPDENS = population density

Data Set FUELCON5Data Set FUELCON5


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DENSITY

FU

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NPlot of Fuelcon versus DensityPlot of Fuelcon versus Density

r = -.454


Effect of the TransformationEffect of the Transformation

The graph has one point (D.C.) on The graph has one point (D.C.) on the right with all others clumped to the right with all others clumped to the left.the left.

It is hard to see what type of It is hard to see what type of relationship there is until some relationship there is until some adjustments are made.adjustments are made.

Here take the natural log of density Here take the natural log of density to "pull" the extreme point back in.to "pull" the extreme point back in.


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LogDensity

FU

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NConsumption versus LogdensityConsumption versus Logdensity

r = -.527


Linear and Log RegressionsLinear and Log Regressions

The regression equation isFUELCON = 495 - 0.025 DENSITY

Predictor Coef SE Coef T PConstant 465.628 9.481 52.28 0.000DENSITY -0.025 0.007 -3.56 0.001

S = 65.1675 R-Sq = 20.6% R-Sq(adj) = 19.0%

The regression equation isFUELCON = 597 – 24.5 LOGDENS

Predictor Coef SE Coef T PConstant 597.19 29.96 22.15 0.000LOGDENS -24.53 5.65 -4.34 0.000

S = 62.1561 R-Sq = 27.8% R-Sq(adj) = 26.3%


5.2.4 Log Transformations of Both the 5.2.4 Log Transformations of Both the yy and and xx Variables Variables

Here the natural log of Here the natural log of yy is the dependent is the dependent variable and the natural log of variable and the natural log of xx is the is the independent variable:independent variable:

Comparing results with other models may Comparing results with other models may be difficult since we are not modeling be difficult since we are not modeling yy itself. itself.

Economists sometimes use this to Economists sometimes use this to estimate price elasticity (estimate price elasticity (y y is demand andis demand and x x price; price; bb11 is estimated elasticity). is estimated elasticity).

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Example 5.4 Imports and GDPExample 5.4 Imports and GDP

The gross domestic product (GDP) and The gross domestic product (GDP) and dollar amount of total imports dollar amount of total imports (IMPORTS) for 25 countries was (IMPORTS) for 25 countries was obtained from the World Fact Book.obtained from the World Fact Book.

For both variables, low values clump For both variables, low values clump together and higher values spread together and higher values spread out, suggesting log transformations out, suggesting log transformations for both.for both.


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Scatterplot of Imports vs GDPScatterplot of Imports vs GDP


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LogGDP

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Scatterplot of LogImp vs LogGDPScatterplot of LogImp vs LogGDP


Two Regression ModelsTwo Regression Models

Regression Analysis: IMPORTS versus GDP

Predictor Coef SE Coef T PConstant 22.32 19.24 1.16 0.258GDP 0.105671 0.008452 12.50 0.000

S = 87.00 R-Sq = 87.2% R-Sq(adj) = 86.6%

Regression Analysis: LogImp versus LogGDP

Predictor Coef SE Coef T PConstant -1.1275 0.4346 -2.59 0.016LogGDP 0.86703 0.07877 11.01 0.000

S = 0.9142 R-Sq = 84.0% R-Sq(adj) = 83.4%

Not directly comparable


The RThe R22 Compare Different Things Compare Different Things

The 87.2 % RThe 87.2 % R22 for the no-log model is for the no-log model is the percentage of variation in the percentage of variation in ImportsImports explained. explained.

The 84.0% for the second model is The 84.0% for the second model is the percentage of variation in the percentage of variation in ln(Imports)ln(Imports) explained. explained.

If you converted the fitted values of If you converted the fitted values of the second model back to Imports the second model back to Imports you might find the log model better.you might find the log model better.


What Transformation to UseWhat Transformation to Use

It is probably best to try several.It is probably best to try several. A quadratic is most flexible because A quadratic is most flexible because

it uses two parameters to fit the it uses two parameters to fit the relationship between to fit the relationship between to fit the relationship between relationship between yy and and xx..

Some further analysis is in Chapter 6 Some further analysis is in Chapter 6 where tests for nonlinearity are where tests for nonlinearity are discussed.discussed.


5.2.5 Fitting Curved Trends5.2.5 Fitting Curved Trends

If the data is collected over time, we If the data is collected over time, we may want to consider variations on may want to consider variations on the linear trend model of Chapter 3.the linear trend model of Chapter 3.

Another is the S-Curve trend:Another is the S-Curve trend:

tt etty 2210:trendQuadratic

tt et

y1

exp 10


S Curve ModelS Curve Model

Many products have a demand curve Many products have a demand curve like this.like this.

1.1. Initial demand increases slowlyInitial demand increases slowly

2.2. As product matures, demand picks up As product matures, demand picks up and steadily grows.and steadily grows.

3.3. At some saturation point demand At some saturation point demand levels off.levels off.


Exponential Growth ModelExponential Growth Model

Another alternative is an exponential Another alternative is an exponential trend:trend:

This can be fit by least squares if you This can be fit by least squares if you model ln(model ln(yy).).

tt ety 10exp

Documents

Fitting Curves to Data 1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 5: Fitting Curves to Data Terry Dielman Applied Regression