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The S-030 roadmap: Where’s this unit in the big picture?

Unit 1:

Introduction to

simple linear regression

Unit 2:

Correlation

and causality

Unit 3:

Inference for the

regression model

Building a

solid foundation

Unit 5:

Transformations

to achieve linearity

Unit 4:

Regression assumptions:

Evaluating their tenability

Mastering

the

subtleties

Adding additional predictors

Unit 6:

The basics of

multiple regression

Unit 7:

Statistical control in depth:

Correlation and collinearity

Generalizing to other types of predictors and effects

Unit 9:

Categorical predictors II: Polychotomies

Unit 8:

Categorical predictors I: Dichotomies

Unit 10:

Interaction and

quadratic effects

Pulling

it all

together

Unit 11:

Regression modeling

in practice

In this unit, we’re going to learn about…

- What happens if we fit a linear regression model to data that are nonlinearly related?
- Alternative statistical models that are useful for nonlinear relationships
- Logarithms—a brief refresher
- The effects of logarithmic transformation
- Other nonlinear relationships that can be modeled using logarithmic transformations
- What’s the difference between taking logarithms to base 2, 10 and e?
- Interpreting the regression of Y on log(X)
- Interpreting the regression of log(Y) on X
- Interpreting the regression of log(Y) on log(X)
- How should we select among alternative transformation options: The Rule of the Bulge

The 10th Grade Math MCAS: % Scoring in the “Advanced” range

Predictor

Outcome

The UNIVARIATE Procedure

Variable: HOME

Location Variability

Mean 426.7553 Std Deviation 128.62042

Median 384.1250 Variance 16543

Mode 355.0000 Range 633.00000

The UNIVARIATE Procedure

Variable: PCTADV

Location Variability

Mean 36.63636 Std Deviation 14.19603

Median 36.00000 Variance 201.52727

Mode 38.00000 Range 60.00000

Stem Leaf # Boxplot

8 8 1 *

8

7 5 1 0

7 14 2 0

6 558 3 0

6 0 1 |

5 699 3 |

5 244 3 |

4 5678 4 +-----+

4 00012222234 11 | + |

3 55566666777778888899 20 *-----*

3 001222333334444 15 +-----+

2 7 1 |

2 4 1 |

----+----+----+----+

Stem Leaf # Boxplot

7 1 1 |

6 59 2 |

6 12 2 |

5 5678 4 |

5 12344 5 |

4 57 2 |

4 1223344 7 +-----+

3 6678888999 10 *--+--*

3 011223334 9 | |

2 567778899 9 +-----+

2 011123344 9 |

1 56799 5 |

1 1 1 |

----+----+---

Do I know the difference between

% differences and %agepoint differences?

n = 66

ID DISTRICT HOME PCTADV L2HOME

1 AMESBURY 341.75 33 8.41680

2 ANDOVER 556.75 56 9.12089

3 ARLINGTON 476.00 44 8.89482

4 ASHLAND 390.00 38 8.60733

5 BELLINGHAM 308.00 15 8.26679

6 BELMONT 675.00 54 9.39874

7 BEVERLY 385.00 38 8.58871

8 BILLERICA 365.00 34 8.51175

9 BRAINTREE 375.00 39 8.55075

10 BROCKTON 269.90 11 8.07628

11 BURLINGTON 397.00 43 8.63300

12 CAMBRIDGE 749.00 23 9.54882

13 CANTON 472.50 41 8.88417

14 CHELMSFORD 360.45 38 8.49366

15 DANVERS 389.95 31 8.60715

16 DEDHAM 383.25 28 8.58214

17 DRACUT 296.25 27 8.21067

18 DUXBURY 537.50 55 9.07012

. . .

Wellesley

Cambridge

Newton, Lexington

Wellesley

Brockton

RQ: Is the percentage of 10th graders scoring in the advanced range on the math MCAS primarily a function of a district’s socioeconomic status?

What’s the relationship between PctAdv and Home Prices?

The REG Procedure

Dependent Variable: PCTADV

Root MSE 9.11982 R-Square 0.5936

Dependent Mean 36.63636 Adj R-Sq 0.5873

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.34532 3.91746 0.09 0.9300

HOME 1 0.08504 0.00879 9.67 <.0001

What happens if we set aside Cambridge?

Regression line with Cambridge

The REG Procedure

Dependent Variable: PCTADV

Root MSE 7.37525 R-Square 0.7346

Dependent Mean 36.84615 Adj R-Sq 0.7304

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -4.86330 3.28861 -1.48 0.1442

HOME 1 0.09888 0.00743 13.20 <.0001

What happens when we fit a linear model to a non-linear relationship?

More positive residuals

(under-predicting)

More positive residuals

(under-predicting)

More negative residuals

(over-predicting)

More negative residuals

(over-predicting)

More negative residuals

(over-predicting)

More negative residuals

(over-predicting)

What alternative statistical model might be useful here?

What kind of population model would have given rise to these sample data?

As HOME prices get larger, PCTADV increases at a slower rate

The effect of HOME prices is larger when HOME prices are low and smaller when HOME prices are high

Logarithmic transformations in everyday life

1998

CPS

Octave

+110

(doubling)

1

M-Systems’ original 16 mb “disgo,”

considered the first USB flash drive

+220

(doubling)

2

Amplitude SW

Richter

+440

(doubling)

3

Greece, 1999

1,000,000

6.0

Japan, 1995

10,000,000

7.0

SF, 1906

100,000,000

8.0

Sumatra, 2004

1,000,000,000

9.0

Musical scales

Flash drives, then and now

Richter scale

Each new generation

doubles in storage capacity

16 32 64 128 256 512 1024 (1GB)

2G 4G 8G 16G …

Up 1 octave = doubling of CPS

Up 1 Richter = 10 fold ASW

Understanding Logarithms

1

2

4

8

16

32

64

Raw

Log2

0

1

2

3

4

5

6

These are the logarithms

For more on logarithms:

Dallal, Logarithms, part I

Each 1 unit increase in a base-2 logarithm represents a doubling of x

Each 1 unit increase in a base-10 logarithm represents a 10-fold increase in x

The power identifies the logarithmbase(x)

because raising the base to that power yields x

So…taking logs spreads out the distance between small values and compresses the distance between large values

Understanding the effects of logarithmic transformation in the MCAS data

Wellesley ($875 9.77)

One log unit

Westwood ($600 9.23)

One log unit

Brockton ($269.9 8.07)

Double of the raw:

512(2) = 1024

Double of the raw:

256(2) = 512

Gapminder

What happens if we regress PctAdv on log2(HOME)?

^

L2HOME

PCTADV

Home

256

8.0

13.48

362

8.5

30.28

+33.60

+33.60

47.08

512

9.0

724

9.5

63.88

1,024

10.0

80.68

Every doubling in home price is positively associated with a 33.6 percentage point difference in students scoring in the advanced range

+33.60

+33.60

The REG Procedure

Dependent Variable: PCTADV

Root MSE 6.77259 R-Square 0.7762

Dependent Mean 36.84615 Adj R-Sq 0.7726

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -255.31368 19.78409 -12.91 <.0001

L2HOME 1 33.59919 2.27994 14.78 <.0001

How would we summarize the results of this analysis?

Cambridge

- Some possible ways to describe the effect:
- The percentage of 10th graders scoring in the advanced range on the math MCAS is 33.70 points higher for every doubling of district median home prices
- As median home prices double, the percentage of 10th graders scoring in the advanced range on the math MCAS is 33.70 points higher

OECD’s Education at a Glance

RQ: What’s the relationship between GDP and PPE in OECD countries?

Countries with a GDP per capita around US$25,000 demonstrate a clear positive relationship between spending on education per student and GDP per capita. … There is considerable variation in spending on education per student among OECD countries with a GDP per capita greater than $25,000, where the higher GDP per capita, the greater the variation in expenditure devoted to students.” (OECD, 2005)

“The relationship between GDP per capita and expenditure per student is complex. Chart B1.6 shows the co-existence of two different relationships between two distinct groups of countries…

Let’s examine the OECD data for ourselves…

Predictor

Outcome

The UNIVARIATE Procedure

Variable: GDP

Location Variability

Mean 24.26592 Std Deviation 7.48505

Median 27.08150 Variance 56.02600

Mode . Range 26.9060

The UNIVARIATE Procedure

Variable: PPE

Location Variability

Mean 10.65453 Std Deviation 4.67585

Median 10.40407 Variance 21.86353

Mode . Range 18.98350

Stem Leaf # Boxplot

22 7 1 0

20 5 1 |

18 |

16 |

14 27 2 |

12 04417 5 +-----+

10 0788 4 *--+--*

8 023638 6 | |

6 0120 4 +-----+

4 788 3 |

----+----

Stem Leaf # Boxplot

3 56 2 |

3 03 2 |

2 6667778888999 13 +-----+

2 2 1 | + |

1 5788 4 +-----+

1 124 3 |

0 9 1 |

----+----+----

n = 26

country GDP PPE L2PPE

Mexico 9.215 6.0737 2.60258

Poland 10.846 4.8342 2.27327

Slovak 12.255 4.7556 2.24964

Hungary 13.894 8.2048 3.03647

Czech Re 15.102 6.2355 2.64051

Korea 17.016 6.0467 2.59614

Portugal 18.434 6.9602 2.79913

Greece 18.439 4.7306 2.24204

Spain 22.406 8.0205 3.00369

Italy 25.568 8.6357 3.11032

Germany 25.917 10.9990 3.45930

Finland 26.495 11.7676 3.55675

Japan 26.954 11.7158 3.55039

Sweden 27.209 15.7151 3.97408

France 27.217 9.2764 3.21356

Belgium 27.716 12.0187 3.58720

UnK 27.948 11.8222 3.56342

Australia 28.068 12.4160 3.63412

Iceland 28.399 8.2505 3.04449

Austria 28.872 12.4475 3.63778

Nether 29.009 13.1011 3.71162

Denmark 29.231 15.1830 3.92438

Switzerl 30.455 23.7141 4.56768

Ireland 32.646 9.8091 3.29413

Norway 35.482 13.7387 3.78018

US 36.121 20.5454 4.36074

Switzerland

US

What’s the relationship between PPE and GDP?

More positive residuals

(under-predicting)

More positive residuals

(under-predicting)

More negative residuals

(over-predicting)

The REG Procedure

Dependent Variable: PPE

Root MSE 3.09519 R-Square 0.5793

Dependent Mean 10.65453 Adj R-Sq 0.5618

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 -0.88349 2.09666 -0.42 0.6772

GDP 1 0.47548 0.08270 5.75 <.0001

What alternative statistical model might be useful here?

What kind of population model would have given rise to these sample data?

The effect of GDP is relative to its magnitude: its effect is larger when GDP is larger and

smaller when GDP is smaller

How do we fit and interpret the exponential growth model?

2 key properties of logs

1. log(xy)=log(x)+log(y)

2. log(xp)=p*log(x)

So just regress log2(Y) on X and substitute the estimated slope into the equation for the percentage growth rate to obtain the estimated percentage growth rate per unit change in X

What happens if we regress log2(PPE) on GDP?

+.0699

+.0699

+.0699

^

^

GDP

L2PPE

PPE

2.2874

4881.75

10

2.3573

5124.11

11

2.9864

7924.94

20

21

3.0563

8318.37

3.6854

12865.18

30

3.7553

13503.86

31

The REG Procedure

Dependent Variable: L2PPE

Root MSE 0.34547 R-Square 0.7051

Dependent Mean 3.28514 Adj R-Sq 0.6928

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 1.58837 0.23402 6.79 <.0001

GDP 1 0.06992 0.00923 7.58 <.0001

+ $244 ≈ 5%

+ $393 ≈ 5%

+ $638 ≈ 5%

For each $1,000 of GDP, PPE is 5% higher

OECD text

What’s a natural logarithm (and why would we ever use it)?

In its IPO, Google announced its intention to raise $2,718,281,828 (e billion dollars)

For more about e and natural logs

The REG Procedure

Dependent Variable: LnPPE

Root MSE 0.23946 R-Square 0.7051

Dependent Mean 2.27708 Adj R-Sq 0.6928

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 1.10098 0.16221 6.79 <.0001

GDP 1 0.04847 0.00640 7.58 <.0001

Go back to regression of L2PPE on GDP

Go to understanding the relationship between natural and base 2 logarithms using the OECD data

How would we summarize the results of this analysis?

Comparison of regression models predicting Per Pupil Expenditures in OECD countries (n=26) (OECD, 2005)

Model A: PPE

Model B: ln(PPE)

Predictor

Intercept

-0.883

(2.097)

-0.42

1.101***

(0.162)

6.79

0.048***

(0.006)

7.58

Per Capita GDP

0.475***

(0.083)

5.75

R2

57.9%

70.5%

Cell entries are estimated regression coefficients, (standard errors) and t-statistics.

*** p<.001

- Another possible way to describe the effect:
- Per capita gross domestic product (GDP) is a strong predictor of per pupil expenditures. If we compare two countries whose GDPs differ by $1,000, we predict that the richer country will have a per pupil expenditure that is 5% higher.

- When comparing models, remember that:
- You’re trying to evaluate whether the model’s assumptions are tenable
- R2 is NOT a measure of whether assumptions are tenable
- R2 statistics do not tell us which model is “better” (both in general and especially if you’ve transformed Y)

OECD text

Who’s got the biggest brain?

Source: Allison, T. & Cicchetti, D. V. (1976). Sleep in mammals: Ecological and Constitutional Correlates. Science, 194, 732-734 View Article

ID SPECIES BRAIN BODY lnBRAIN lnBODY

1 Lessershort-tailedshrew 0.14 0.01 -1.96611 -5.29832

2 Littlebrownbat 0.25 0.01 -1.38629 -4.60517

3 Bigbrownbat 0.30 0.02 -1.20397 -3.77226

4 Mouse 0.40 0.02 -0.91629 -3.77226

5 Muskshrew 0.33 0.05 -1.10866 -3.03655

6 Starnosedmole 1.00 0.06 0.00000 -2.81341

7 Easter.mericanmole 1.20 0.08 0.18232 -2.59027

8 Groundsquirrel 4.00 0.10 1.38629 -2.29263

9 Treeshrew 2.50 0.10 0.91629 -2.26336

10 Goldenhamster 1.00 0.12 0.00000 -2.12026

11 Molerat 3.00 0.12 1.09861 -2.10373

12 Galago 5.00 0.20 1.60944 -1.60944

13 Rat 1.90 0.28 0.64185 -1.27297

14 Chinchilla 6.40 0.43 1.85630 -0.85567

15 Owlmonkey 15.50 0.48 2.74084 -0.73397

. . .

47 Chimpanzee 440.00 52.16 6.08677 3.95432

48 Sheep 175.00 55.50 5.16479 4.01638

49 Giantarmadillo 81.00 60.00 4.39445 4.09434

50 Man 1320.00 62.00 7.18539 4.12713

51 Grayseal 325.00 85.00 5.78383 4.44265

52 Jaguar 157.00 100.00 5.05625 4.60517

53 Braziliantapir 169.00 160.00 5.12990 5.07517

54 Donkey 419.00 187.10 6.03787 5.23164

55 Pig 180.00 192.00 5.19296 5.25750

56 Gorilla 406.00 207.00 6.00635 5.33272

57 Okapi 490.00 250.00 6.19441 5.52146

58 Cow 423.00 465.00 6.04737 6.14204

59 Horse 655.00 521.00 6.48464 6.25575

60 Giraffe 680.00 529.00 6.52209 6.27099

61 Asianelephant 4603.00 2547.00 8.43446 7.84267

62 Africanelephant 5712.00 6654.00 8.65032 8.80297

n = 62

RQ: What’s the relationship between brain weight and body weight?

Distribution of BRAIN and BODY

Outcome

Predictor

The UNIVARIATE Procedure

Variable: BRAIN

Location Variability

Mean 283.1342 Std Deviation 930.27894

Median 17.2500 Variance 865419

Mode 1.0000 Range 5712

The UNIVARIATE Procedure

Variable: BODY

Location Variability

Mean 198.7900 Std Deviation 899.15801

Median 3.3425 Variance 808485

Mode 0.0230 Range 6654

Histogram # Boxplot

5750+* 1 *

.

.* 1 *

.

.

.

.

.

.

.* 1 *

.* 2 0

250+***************************** 57 +--0--+

----+----+----+----+----+----

* may represent up to 2 counts

Histogram # Boxplot

6750+* 1 *

.

.

.

.

.

.

.

.* 1 *

.

.

.

.* 2 *

250+***************************** 58 +--0--+

----+----+----+----+----+----

* may represent up to 2 counts

Plots of BRAIN vs. BODY on several scales

African El

Asian El

African El

Asian El

African El

Asian El

Go to data slide

What’s the relationship between LnBRAIN and LnBODY?

^

BODY

BRAIN

2.72

17.81

7.39

37.71

54.60

169.02

403.43

757.48

2980.96

3394.80

22026.47

7186.79

+.75

+.75

+1

+1

^

The REG Procedure

Dependent Variable: LnBRAIN

Root MSE 0.69429 R-Square 0.9208

Dependent Mean 3.14010 Adj R-Sq 0.9195

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 2.13479 0.09604 22.23 <.0001

LnBODY 1 0.75169 0.02846 26.41 <.0001

LnBODY

LnBRAIN

2.88

1

3.63

2

+.75

+1

5.13

4

6

6.63

8.13

8

But how do we interpret the estimated regression coefficient, 0.75?

8.88

9

+.75

+1

The proportional growth model: The regression of log(Y) on log(X)

2 key properties of logs

1. log(xy)=log(x)+log(y)

2. log(xp)=p*log(x)

Economists call 1 anelasticity

So just regress loge(Y) on loge(X) and the estimated slope provides the estimated percentage change in Y per 1% change in X

- A 1% difference in bodyweight is positively associated with a 0.75% difference in brain weight
- For every 1% difference in bodyweight, animal brains differ by ¾ of a percent

Sidebar: Why couldn’t we just take the ratio of BrainWt to BodyWt?

Ground squirrel

Owl monkey

Man

Mouse

Baboon

African Elephant

Obs SPECIES BRAIN BODY RATIO

1 Africanelephant 5712.00 6654.00 0.8584

2 Cow 423.00 465.00 0.9097

3 Pig 180.00 192.00 0.9375

4 Braziliantapir 169.00 160.00 1.0563

5 Wateropossum 3.90 3.50 1.1143

6 Horse 655.00 521.00 1.2572

7 Giraffe 680.00 529.00 1.2854

8 Giantarmadillo 81.00 60.00 1.3500

9 Jaguar 157.00 100.00 1.5700

10 Kangaroo 56.00 35.00 1.6000

11 Asianelephant 4603.00 2547.00 1.8072

12 Okapi 490.00 250.00 1.9600

13 Gorilla 406.00 207.00 1.9614

14 Donkey 419.00 187.10 2.2394

15 Tenrec 2.60 0.90 2.8889

. . .

47 Vervet 58.00 4.19 13.8425

48 Chinchilla 6.40 0.43 15.0588

49 Easter.mericanmole 1.20 0.08 16.0000

50 Rockhyrax(Heterob) 12.30 0.75 16.4000

51 Starnosedmole 1.00 0.06 16.6667

52 Baboon 179.50 10.55 17.0142

53 Mouse 0.40 0.02 17.3913

54 Man 1320.00 62.00 21.2903

55 Treeshrew 2.50 0.10 24.0385

56 Molerat 3.00 0.12 24.5902

57 Littlebrownbat 0.25 0.01 25.0000

58 Galago 5.00 0.20 25.0000

59 Rhesusmonkey 179.00 6.80 26.3235

60 Lessershort-tailedshrew 0.14 0.01 28.0000

61 Owlmonkey 15.50 0.48 32.2917

62 Groundsquirrel 4.00 0.10 39.6040

Review: How to fit and interpret models using log-transformed variables

Learning Curve

Exponential Growth Model

Proportional Growth Model

Every 1% difference in X

is associated with a

difference in Y

Every doubling of X

(100% difference)

is associated with a

difference in Y

Every 1 unit difference in

X is associated with a

% difference

in Y (often interpreted as

a %age growth rate)

Helpful mnemonic device: If you’ve logarithmically transformed a variable, you’ll be modifying the interpretation of an effect by expressing differences for that variable in percentage, not unit, terms

Another helpful mnemonic:Mosteller and Tukey’s “Rule of the Bulge”

John

Tukey

Fred Mosteller

Bulge

Bulge

Bulge

Bulge

Broadly speaking, there are four general shapes that a monotonic nonlinear relationship might take:

Up in Y (e.g., Y2)

We’ll learn about this shape in Unit 10

MCAS/Brain

Up in X

(e.g., X2)

Down in X

(e.g., log(X))

- Two more important ideas about transformation:
- It’s usually “low cost” to transform X, potentially “higher cost” to transform Y
- If the range of a variable is very large, taking logarithms often helps

OECD

Down in Y (e.g., log(Y))

If you think of this display as representing plots of Y vs. X, identify the curve that most closely matches your data (and theory, hopefully) and you can linearize the relationship by choosing transformations of X, Y or both that go in the “direction of the bulge”

What’s the big takeaway from this unit?

- Check your assumptions
- Regression is a very powerful statistical technique, but its built on a set of assumptions
- Before accepting a set of regression results, you should examine the assumptions to make sure they’re tenable
- A high R2 or small p-value cannot tell you whether your assumptions hold
- Plot your data and plot your residuals
- Many relationships are nonlinear
- We often begin by assuming linearity, but we often find that the underlying relationship is nonlinear
- Transformation makes it easy to fit nonlinear models using linear regression techniques
- Models expressed using transformed variables can be easily interpreted
- Regression as statistical control
- We often want to do more than just summarize the relationship between variables
- Regression provides a straightforward strategy that allows us to statistically control for the effects of a predictor and see what’s “left over”
- Residuals can be easily interpreted as “controlled observations”

Appendix: Annotated PC-SAS Code for transforming variables

The data step can include additional statements to create new variables by transforming variables already included in the data set.

To add log base 2 transformations of variables in the sample, use the following syntax:

Newvar = log2(oldvar);

Different transformation can be used, including natural logs (log (var)), squared and cubic versions (var**2 or var**3, inverses (-1/var), and roots (var**.5).

The data step can also be repeated in the middle of the program to add additional new variables to the original data set. Note that you can keep the data set’s original name by using the same name in both the set and data statements.

Note that the handouts include only annotations for the new additional SAS code. For the complete program, check program “Unit 5—MCAS analysis” on the website.

data one;

infile \'m:\datasets\MCAS.txt\';

input ID 1-2 District $ 4-22 Home 24-29 PctAdv 33-34;

L2Home=log2(home);

“Unit 5—OECD analysis”

*-------------------------------------------------------------*

Fitting OLS regression model L2PPE on GDP

Plotting studentized residuals on GDP

*-------------------------------------------------------------*;

procreg data=one;

model L2PPE=GDP;

plot student.*GDP;

output out=resdat2 r=residual student=student;

id country;

procunivariate data = resdat2 plot;

var student;

id country;

*-----------------------------------------------------------*

Create new natural log transformation of outcome PPE: Ln(PPE)

*-----------------------------------------------------------*;

data one;

set one;

LnPPE = Log(PPE);

*-------------------------------------------------------------*

Understanding the effects of transformation in the OECD data

Remember to go back or the presentation will end!

Ln(PPE) = 0.6935*log2(PPE)

rLnPPE, L2PPE = 1.00

Appendix: Why you shouldn’t rely solely on R2 statistics to select models

A good model is a model in which your assumptions appear tenable

ID X log2(X) Y

1 45 5.4919 110.000

2 56 5.8074 55.574

3 96 6.5850 59.762

4 136 7.0875 65.318

5 176 7.4594 76.433

6 216 7.7549 90.033

7 256 8.0000 41.970

8 296 8.2095 101.890

9 336 8.3923 98.228

10 376 8.5546 89.939

11 416 8.7004 50.914

12 456 8.8329 99.551

13 496 8.9542 73.437

14 536 9.0661 133.139

15 576 9.1699 92.485

16 616 9.2668 116.767

17 656 9.3576 108.697

18 696 9.4429 102.030

19 1500 10.5507 80.000

20 1800 10.8138 100.000

21 2000 10.9658 80.000

22 3000 11.5507 100.000

23 4000 11.9658 130.000

24 6000 12.5507 120.000

25 8000 12.9658 140.000

Regression of Y on X

Regression of Y on log2(X)

Glossary terms included in Unit 5

- Logarithms
- Rule of the bulge
- Transformation

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