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Unit 4: Regression assumptions: Evaluating their tenability

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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

- Reprise of the assumptions required for least squares estimation and inference
- The four major types of model violations:
- Outliers
- Nonlinearity
- Heteroscedasticity
- Non-independence of errors

- Determining whether the regression assumptions hold—strategies and rationale
- Why residuals provide a powerful lens for evaluating regression assumptions
- Residuals as controlled observations
- Raw residuals and studentized residuals
- Residual plots: How to construct them and what to look for
- What should we do if we find an outlier or other unusual observation?

- How would we summarize our results?

Y

Y|x3

Y|x2

Y|x1

X

…

x1

x2

x3

- At each value of X, there is a distribution of Y. These distributions have a mean µY|X and a variance of σ2Y|X

- The straight line model is correct. The means of each of these distributions, the µY|X‘s, may be joined by a straight line.

- Homoscedasticity. The variances of each of these distributions, the σ2Y|X’s, are identical.

- Independence of observations.
- At each given value of X (at each xi), the values of Y (the yi’s) are independent of each other.

5.Normality. At each given value of X (at each xi), the values of Y (the yi’s) are normally distributed

Assumptions ARE NOT about the sample, X, or Y overall.

Assumptions ARE about the

behavior of Y

at each X

in the population

Data Set III

x y

10 7.46

8 6.77

13 12.74

9 7.11

11 7.81

14 8.84

6 6.08

4 5.39

12 8.15

7 6.42

5 5.73

Data Set I

x y

10 8.04

8 6.95

13 7.58

9 8.81

11 8.33

14 9.96

6 7.24

4 4.26

12 10.84

7 4.82

5 5.68

Data Set IV

x y

8 6.58

8 5.76

8 7.71

8 8.84

8 8.47

8 7.04

8 5.25

8 5.56

8 7.91

8 6.89

19 12.50

Data Set II

x y

10 9.14

8 8.14

13 8.74

9 8.77

11 9.26

14 8.10

6 6.13

4 3.10

12 9.13

7 7.26

5 4.74

Heteroscedasticity. The variance of Y varies as a function of X

Nonlinearity. There’s a relationship between Y and X, but it’s not best summarized by a straight line

Outliers. Extreme observations that don’t fit the general pattern

Non-independence of errors. Observations within the data set are clustered (or otherwise related)**

A cautionary tale….

**Remember we can’t see this one visually

Predictor

Outcome

The UNIVARIATE Procedure

Variable: COST

Location Variability

Mean 38.52632 Std Deviation 10.54574

Median 38.00000 Variance 111.21263

Mode 38.00000 Range 50.00000

The UNIVARIATE Procedure

Variable: RATING

Location Variability

Mean 20.98684 Std Deviation 2.49589

Median 21.00000 Variance 6.22945

Mode 19.66667 Range 9.33333

Stem Leaf # Boxplot

68 0 1 0

66

64 0 1 |

62 0 1 |

60 |

58 0 1 |

56 0 1 |

54 00 2 |

52 000 3 |

50 0000 4 |

48 |

46 00 2 |

44 00000 5 +-----+

42 00000 5 | |

40 000 3 | |

38 0000000000 10 *--+--*

36 0000 4 | |

34 000000 6 | |

32 0000 4 | |

30 0000000 7 +-----+

28 000000 6 |

26 00000 5 |

24 000 3 |

22 0 1 |

20 |

18 0 1 |

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

Stem Leaf # Boxplot

25 777 3 |

25 333 3 |

24 |

24 0000033 7 |

23 7777 4 |

23 03 2 |

22 777 3 +-----+

22 0000333 7 | |

21 7 1 | |

21 000003333 9 *--+--*

20 777 3 | |

20 00033 5 | |

19 77777 5 | |

19 0003333 7 +-----+

18 777 3 |

18 00333 5 |

17 777 3 |

17 00033 5 |

16 |

16 3 1 |

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

Residuals as “controlled” observations

n = 76

NAME COST RATING

Jacob Wirth 26 16.3333

Grafton St. Pub 24 17.0000

29 Newbury 38 17.0000

Vox Populi 31 17.0000

Avenue One 35 17.3333

Orleans 28 17.3333

Daedalus 26 17.6667

...

224 Boston St. 31 21.3333

West Side Lounge 30 21.3333

Fava 43 21.6667

...

Harvest 46 23.6667

Square Café 39 23.6667

Troquet 54 23.6667

TW Foods 39 24.0000

flora 40 24.0000

...

Hamersley's 56 25.3333

Icarus 54 25.3333

Excelsior 65 25.6667

Federalist 69 25.6667

Meritage 63 25.6667

RQ: Do you get what you pay for?:What’s the relationship between a restaurant’s rating and its cost?

RQ: Which restaurants are good values?:Given what you’re paying (controlling for price), where do you get the best food?

Effect is strong: 61.6% of the variation in ratings is associated with cost

The REG Procedure

Dependent Variable: RATING

Analysis of Variance

Sum of Mean

Source DF Squares Square

Model 1 287.86080 287.86080

Error 74 179.34826 2.42363

Corrected Total 75 467.20906

Root MSE 1.55680 R-Square 0.6161

Dependent Mean 20.98684 Adj R-Sq 0.6109

Coeff Var 7.41798

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 13.82968 0.68057 20.32 <.0001

COST 1 0.18577 0.01705 10.90 <.0001

Estimated effect is quite precise: Narrow 95% CI (0.1523, 0.2193)

Estimated effect is large†: Each $10.00 difference in cost is positively associated with a 1.9 difference in ratings

Effect is statistically significant: Unlikely that in the population of Boston restaurants, there’s no relationship between price & ratings

†How large is large? Compare to SD of outcome (≈ 2.5)

^

-Y

Y

Y

Y|x3

Y|x2

Y|x1

X

X

…

x1

x2

x3

…

x1

x2

x3

0

What does it mean to refer to the “values of Y at each X?”

residuals

- Assumptions we can examine:
- Normality
- Homoscedasticity
- Linearity
- which all refer to the values of Y at each X

- So the residual distributions should be:
- Normal
- Homoscedastic
- Totally unrelated to X

Go to Put Points

The hope? To see random scatter in the residual plot

Dependent Predicted

Obs NAME Variable Value Residual

6 Bambara 20.6667 21.6322 -0.9655

7 Birch St. Bistro 19.3333 19.4029 -0.0695

8 Blarney Stone 18.3333 17.9167 0.4166

9 blu 23.6667 23.3041 0.3625

10 Brenden Crocker' 22.0000 20.7033 1.2967

11 Bristol 25.3333 22.1895 3.1439

12 B-Side Lounge 18.6667 18.6598 0.006881

13 Central Kitchen 20.0000 17.3594 2.6406

14 Daedalus 17.6667 18.6598 -0.9931

15 Dalia's Bistro 18.0000 19.5887 -1.5887

16 Dalya's 21.0000 21.0748 -0.0748

17 Dedo Lounge 19.6667 20.1460 -0.4793

18 Devlin's 20.3333 18.4740 1.8593

19 Excelsior 25.6667 25.9049 -0.2383

20 Fava 21.6667 21.8179 -0.1513

21 Federalist 25.6667 26.6480 -0.9814

22 flora 24.0000 21.2606 2.7394

23 Franklin Café 21.0000 19.7744 1.2256

24 Gardner Museum 19.0000 18.4740 0.5260

25 Gargoyles 22.6667 20.7033 1.9634

26 Grafton St. Pub 17.0000 18.2882 -1.2882

27 Grapevine 22.6667 20.8891 1.7776

28 Green St. Grill 19.3333 19.0313 0.3020

29 Hamersley's Bist 25.3333 24.2330 1.1003

30 Harvest 23.6667 22.3753 1.2914

. . .

57 Square Café 23.6667 21.0748 2.5918

58 Stanhope Grille 18.6667 21.8179 -3.1513

59 Stephanie's 18.6667 19.9602 -1.2935

60 Temple Bar 17.6667 18.8456 -1.1789

61 Ten Tables 23.0000 20.8891 2.1109

62 33 Restaurant 19.3333 21.6322 -2.2988

63 Top of the Hub 22.0000 23.1183 -1.1183

64 Tremont 647 19.6667 20.3317 -0.6651

65 Troquet 23.6667 23.8614 -0.1948

66 Tryst 20.6667 21.2606 -0.5939

67 29 Newbury 17.0000 20.8891 -3.8891

The UNIVARIATE Procedure

Variable: rawres1 (Residual)

Basic Statistical Measures

Location Variability

Mean 0.000000 Std Deviation 1.54639

Median 0.001587 Variance 2.39131

Mode . Range 7.03292

Interquartile Range 2.14756

Stem Leaf # Boxplot

3 1 1 |

2 6679 4 |

2 0013 4 |

1 57899 5 |

1 00112334 8 +-----+

0 5566778 7 | |

0 011133344 9 *--+--*

-0 443222110 9 | |

-0 977665 6 | |

-1 33322110000 11 +-----+

-1 765 3 |

-2 3311 4 |

-2 65 2 |

-3 20 2 |

-3 9 1 |

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

positive

negative

67%

1

3 stdres > 2 is well within our expectations when n= 76

95% (1.96)

-4 -3 -2 -1 0 +1 +2 +3 +4

99% ( 2.58)

The UNIVARIATE Procedure

Variable: stdres (Studentized Residual)

Basic Statistical Measures

Location Variability

Mean -0.00016 Std Deviation 1.00425

Median 0.00105 Variance 1.00852

Mode . Range 4.55280

Interquartile Range 1.40607

Stem Leaf # Boxplot

20 4 1 |

18 9 1 |

16 857 3 |

14 6 1 |

12 25786 5 |

10 35 2 |

8 4437 4 |

6 67239 5 +-----+

4 1353 4 | |

2 0147346 7 | |

0 08997 5 *-----*

-0 630550 6 | + |

-2 81961 5 | |

-4 7530 4 | |

-6 97307652 8 +-----+

-8 9644 4 |

-10 13 2 |

-12 97 2 |

-14 29 2 |

-16 83 2 |

-18 4 1 |

-20 4 1 |

-22 |

-24 1 1 |

- Rule of thumb: If studentized residuals are normally distributed, we expect:
- 5% > 2
- 1% > 2.5

- Flag & examine stdres > 2

Bristol

TWFoods

flora

Bristol

Central Kitchen

TWFoods

Central Kitchen

Square Cafe

flora

Square Cafe

Jer ne

Vox Populi

Jer ne

Vox Populi

Avenue One

Stanhope Grille

Avenue One

Stanhope Grille

29 Newbury

29 Newbury

Raw residuals

Studentized residuals

5 best places to eat (controlling for cost)

NAME COST RATING RESIDUAL

Square Café 39 23.6667 2.59183

Central Kitchen 19 20.0000 2.64063

flora 40 24.0000 2.73939

TWFoods 39 24.0000 2.92516

Bristol 45 25.3333 3.14385

5 most over-rated (controlling for cost)

NAME COST RATING RESIDUAL

Jer-Ne 52 21.0000 -2.48989

Vox Populi 31 17.0000 -2.58865

Avenue One 35 17.3333 -2.99841

Stanhope Grille 43 18.6667 -3.15127

29 Newbury 38 17.0000 -3.88907

Bristol

•

TWFoods

flora

CentralKitchen

Bristol

Square Cafe

•

•

•

TWFoods

flora

SquareCafe

•

Jer ne

•

CentralKitchen

•

Stanhope Grille

•

•

Avenue One

•

29 Newbury

Vox Populi

Raw residuals

Where were these on the original scatterplot?

Jerne

Vox Populi

Avenue One

Stanhope Grille

29 Newbury

5 best places to eat (controlling for cost)

NAME COST RATING RESIDUAL

Square Café 39 23.6667 2.59183

Central Kitchen 19 20.0000 2.64063

flora 40 24.0000 2.73939

TWFoods 39 24.0000 2.92516

Bristol 45 25.3333 3.14385

5 most over-rated (controlling for cost)

NAME COST RATING RESIDUAL

Jer-Ne 52 21.0000 -2.48989

Vox Populi 31 17.0000 -2.58865

Avenue One 35 17.3333 -2.99841

Stanhope Grille 43 18.6667 -3.15127

29 Newbury 38 17.0000 -3.88907

Boston Restaurant Weeks, March 9th – 14th & 16th to 21st

Three-course Prix-fixe Lunch Menu: $20.08Three-course Prix-fixe Dinner Menu: $33.08

If you punch the second hole, you are voting for the Reform party (ie, Pat Buchanan)

Although Democrats are listed second in the left hand column, you vote Democratic by punching the third hole

Poly-CY, Internet Resources for political science has much more information

on the statistical analysis of the 2000 Presidential election results

Of the nearly 6 million votes cast in Florida, the official tally has Bush beating Gore by 537 votes

RQ: In the 2000 Presidential election, did Buchanan get more votes than we “would have expected?”

Pinellas

Hillsborough

Broward, Duvall

ID COUNTY BUCH REGREF

1 ALACHUA 262 91

2 BAKER 73 4

3 BAY 248 55

4 BRADFORD 65 3

5 BREVARD 570 148

6 BROWARD 789 332

. . .

48 ORANGE 446 199

49 OSCEOLA 145 62

50 PALM BEACH 3407 337

51 PASCO 570 167

52 PINELLAS 1010 425

. . .

64 VOLUSIA 396 176

65 WAKULLA 46 7

66 WALTON 120 22

67 WASHINGTON 88 9

The UNIVARIATE Procedure

Variable: BUCH

Location Variability

Mean 258.6119 Std Deviation 449.48775

Median 114.0000 Variance 202039

Mode 29.0000 Range 3398

n = 67

Stem Leaf # Boxplot

34 1 1 *

32

30

28

26

24

22

20

18

16

14

12

10 1 1 0

8 4 1 0

6 59 2 0

4 05046677 8 |

2 345677789011 12 +--+--+

0 112233333333444455677788999900011122245899 42 *-----*

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

Multiply Stem.Leaf by 10**+2

10 Nov 2000: “The Bush campaign claims that the number of votes for Buchanan in Palm Beach County is perfectly accurate. ‘New information has come to our attention that puts in perspective the results of the vote in Palm Beach County,’ Bush spokesman Ari Fleischer said on Thursday. ‘Palm Beach County is a Pat Buchanan stronghold and that's why Pat Buchanan received 3,407 votes there.’” (Salon.com) View Article

Effect is strong: 55.6% of the variation in Buchanan votes is associated with Reform party registration

Palm Beach

Estimated effect is somewhat precise: 95% CI (2.87, 4.50)

Estimated effect is large: Each registered Reform party member is associated with 3.69 votes for Buchanan

Effect is statistically significant: Unlikely that in the population of Florida counties, there’s no relationship between reform party registration and Buchanan votes

The REG Procedure

Dependent Variable: BUCH

Analysis of Variance

Sum of Mean

Source DF Squares Square

Model 1 7412114 7412114

Error 65 5922476 91115

Corrected Total 66 467.20906

Root MSE 301.85265 R-Square 0.5559

Dependent Mean 258.61194 Adj R-Sq 0.5490

Coeff Var 116.72031

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 1.53252 46.60847 0.03 0.9739

REGREF 1 3.68671 0.40876 9.02 <.0001

10 Nov 2000: When asked about the Bush campaign's statement, Buchanan's Florida coordinator, Jim McConnell, responded: "That's nonsense.“ He estimate[s] the number of Buchanan activists in the county to be between 300 and 500 -- nowhere near the 3,407 who voted for him. (Salon.com) View Article

Palm Beach

Marion

Duval

Polk

Collier

1106

646

R2=86.4%

Palm Beach

All Florida counties

Without Palm Beach

It’s déjà vu all over again…

The results of the 2006 congressional elections in Florida

The New York Times, 24 February 2007

- Regression models invoke a series of important assumptions
- Before accepting a set of regression results, you should examine the assumptions to make sure they’re tenable
- The assumptions may well be reasonable but you can’t be sure your conclusions are correct unless you have evaluated their tenability
- Residuals are the key to evaluating regression assumptions

- 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”

- Outliers can distort regression results or be interesting on their own
- Always inspect scatterplots and residual plots to determine whether there are any unusual values that might unduly influence the fitted regression line
- If you find outliers, re-fit the regression model without those observations and compare the results
- Regardless of how you decide to handle the presence of outliers, always tell your audience about their existence and what you did about them

proc sortsorts the newly created SAS data set (named “one”). The by statement identifies the variable according to which to data is sorted

proc regallows you to add a plot statementthat produces a bivariate scatterplot of the raw and studentized residuals by the predictor (“residual” stands for raw residuals, and “student” for studentized). The syntax is identical to that of proc gplot (ie, plot y*x), (note the “.” after naming the residual, as part of the PLOT statement). The output statement creates a new (temporary) dataset, called RESDAT, that contains all the data in ONE as well as the raw and standardized residuals.

proc univariatecan be used tp analyze the new dataset RESDAT andpresents summary statistics of the residuals (e.g., means, sd’s, stem-and-leaf displays). As in any proc univariate, the var statement specifies the residuals you want analyzed; the id statementprovides identifiers for extreme values

Note that the handouts include only annotations for the new additional SAS code. For the complete program, check program “Unit 4—ZAGAT analysis” on the website.

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

Sorting observations in sample from lowest to highest

RATING

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

procsort data=one;

by rating;

procreg data=one;

title2 "Examining residuals from the regression of Rating on Cost";

model rating=cost/p;

plot residual.*cost;

plot student.*cost;

output out=resdat r=rawres student=stdres;

id name;

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

Univariate summary information on raw and studentized residuals

from OLS regression model RATING on COST

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

procunivariate data=resdat plot;

title2 "Distribution of residuals from the regr of Rating on Cost";

var rawres stdres;

id name;

The datastep here creates a new dataset called “two.”

The if-then-delete statement specifies which data points to delete.

proc reghere reads the data from the new dataset “two”. An option has been added to the model statement to produce 95% prediction intervals for individuals levels of Y at each value of X (/conf95), superimposed on the bivariate scatterplot. The other plot statement produces a bivariate scatterplot of studentized residuals by predictor. Finally, the output statement creates a new (temporary) dataset, called RESDAT2, that includes all the data in ONE and the raw and standardized residuals.

proc univariateanalyzes the new dataset RESDAT2 andpresents summary statistics of the residuals

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

Creating FLVote data subset that excludes Palm Beach County (id=50)

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

data two;

set one;

if id=50 then delete;

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

Fitting OLS regression model BUCH on REGREF, excluding Palm Beach County

Plotting BUCH vs REGREF with 95% prediction interval bands

Plotting studentized residuals on REGREF

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

procreg data=two;

title2 "Regression results and residual analysis";

model buch=regref/p;

plot buch*regref/pred95;

plot student.*regref;

output out=resdat2 r=residual student=student;

id county;

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

Univariate summary information of studentized residuals from

OLS regression model BUCH on REGREF, excluding Palm Beach County

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

procunivariate data = resdat2 plot;

title2 "Studentized Residuals";

var student;

id county;

- Assumptions
- Homoscedasticity
- Independence of observations
- Linearity
- Normality
- Outlier
- Studentized residuals