g

Addressing Alternative ExplanationsExplanations Multiple Regression

17871 Spring 2012

1

t t

Did Cli h G lDid Clinton hurt Gore example

Did Clinton hurt Gore in the 2000 election Treatment is not liking Bill Clinton Treatment is not liking Bill Clinton

2

Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer

Clinton thermometer

3

t t

Did Cli h G lDid Clinton hurt Gore example

What alternative explanations would you need to address

Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources

Letrsquos address one source party identification How could we do this

Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents

Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive

4

t tDDemocratiic piicture

Clinton thermometer

5

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t t

Did Cli h G lDid Clinton hurt Gore example

Did Clinton hurt Gore in the 2000 election Treatment is not liking Bill Clinton Treatment is not liking Bill Clinton

2

Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer

Clinton thermometer

3

t t

Did Cli h G lDid Clinton hurt Gore example

What alternative explanations would you need to address

Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources

Letrsquos address one source party identification How could we do this

Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents

Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive

4

t tDDemocratiic piicture

Clinton thermometer

5

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer

Clinton thermometer

3

t t

Did Cli h G lDid Clinton hurt Gore example

What alternative explanations would you need to address

Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources

Letrsquos address one source party identification How could we do this

Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents

Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive

4

t tDDemocratiic piicture

Clinton thermometer

5

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t t

Did Cli h G lDid Clinton hurt Gore example

What alternative explanations would you need to address

Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources

Letrsquos address one source party identification How could we do this

Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents

Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive

4

t tDDemocratiic piicture

Clinton thermometer

5

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t tDDemocratiic piicture

Clinton thermometer

5

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t tI dIndependdent piicture

Clinton thermometer

6

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

tR bli Republican piicture

Clinton thermometer

7

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t t C bi d d i Combined data picture

Clinton thermometer

8

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

Combined data picture withCombined data picture with regression bias

Clinton thermometer

9

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid

Clinton thermometer

10

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

Tempting yet wronggp g y normalizations

Subtract the Gore therm from the avg Gore therm avg Gore therm score

Clinton thermometer

Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score

Clinton thermometer

11

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

12

3D R l hi3D Relatitionship

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

13

3D Linear Relationship3D Linear Relationship

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

14

t3D R 3D Rellationshihip Clintonti Cli

100 50 0

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

15

t3D R l hi3D Relatitionship party

Rep Ind Dem

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

The Linear Relationship between ThreeThe Linear Relationship between Three Variables

ClintonGore

XXY

Clinton thermometer

Gore thermometer Party ID

iiii XXY 22110

16

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

The method of least squaresThe method of least squares (again)

Pick and to minimizePick and to minimize 0 1 2

n

((YY YY ))2 oror ii ii i1

n

(Y X X )2 i 0 1 i 2 2

i1

17

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t

nn

Th Sl C ffiThe Slope Coefficiients n n

(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn

(X X )2 (X X1 1i 1 1 )2

i i1 i1

n n

(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n

2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22

i1 i1 0 1 X1 2 XY 2

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

18

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

The Slope Coefficients MoreThe Slope Coefficients More Simply

ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)

ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer

19

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t

Th M fThe Matriix form

y1

y2

hellip

yyn

1 x11 x21 hellip xk1

1 x12 x22 hellip xk2

1 hellip hellip hellip hellip

11 xx1n xx2n hellip xxkn

(( X X )) 1 X yy 20

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

Multivariate slope coefficientsMultivariate slope coefficients

cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)

ˆ M cov(X1Y ) ˆ M

Are Gore and Party ID related

cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22

Clinton effect ( G ) i (on Gore) in

multivariate (M) regression

1 var( X1) var( X1)

Clinton effect (on Gore) in bivariate (B)bivariate (B) regression

Are Clinton and Party IDy related

cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )

X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

t t

-

------------------------------------------------------------------------------

------------------------------------------------------------------------------

Th OThe Output reg gore clinton party3

Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804

Model | 6292619162926191 22 314630955 Prob

Prob

gt F =

Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461

-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327

gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------

clinton | 5122875 0175952 2912 0000

4777776 5467975 party3 || 5770523 5594846 1031

0000 4673191 6867856

_cons | 286299 1025472 2792 0000 2661862 3064119

Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer

22

2 314630955 Prob Prob F 00000

tS iSeparate regressions

(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858

ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)

cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23

t

Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051

corr gore clinton party cov (obs=1745)

| gore clinton party3 -------------++---------------------------

gore | 660681 clinton | 549993 883182

3 | party3 | 13 7008 137008 16 905 16905 87358735

24

ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182

M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905

57705 883182 883182

corr gore clinton partycov (obs=1745) 06227 01105

| gore clinton party3 -------------+---------------------------

gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735

25

Another way of thinking aboutAnother way of thinking about this

R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)

as cov(goreclinton) cov(clinton party)

2 1 M ˆ M

var(clinton) var(clinton)

Total effect = Direct effect + indirect effect

The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11

26

D i ki d G k Lif E lDrinking and Greek Life Example

Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social

gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house

Therersquos something about the House environment itself

27

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

t

Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051

corr gore clinton party cov (obs=1745)

| gore clinton party3 -------------++---------------------------

gore | 660681 clinton | 549993 883182

3 | party3 | 13 7008 137008 16 905 16905 87358735

24

ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182

M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905

57705 883182 883182

corr gore clinton partycov (obs=1745) 06227 01105

| gore clinton party3 -------------+---------------------------

gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735

25

Another way of thinking aboutAnother way of thinking about this

R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)

as cov(goreclinton) cov(clinton party)

2 1 M ˆ M

var(clinton) var(clinton)

Total effect = Direct effect + indirect effect

The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11

26

D i ki d G k Lif E lDrinking and Greek Life Example

Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social

gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house

Therersquos something about the House environment itself

27

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182

M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905

57705 883182 883182

corr gore clinton partycov (obs=1745) 06227 01105

| gore clinton party3 -------------+---------------------------

gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735

25

Another way of thinking aboutAnother way of thinking about this

R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)

as cov(goreclinton) cov(clinton party)

2 1 M ˆ M

var(clinton) var(clinton)

Total effect = Direct effect + indirect effect

The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11

26

D i ki d G k Lif E lDrinking and Greek Life Example

Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social

gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house

Therersquos something about the House environment itself

27

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

Another way of thinking aboutAnother way of thinking about this

R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)

as cov(goreclinton) cov(clinton party)

2 1 M ˆ M

var(clinton) var(clinton)

Total effect = Direct effect + indirect effect

The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11

26

D i ki d G k Lif E lDrinking and Greek Life Example

Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social

gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house

Therersquos something about the House environment itself

27

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

D i ki d G k Lif E lDrinking and Greek Life Example

Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social

gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house

Therersquos something about the House environment itself

27

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

Dependent variable TimesDependent variable Times Drinking in Past 30 Days

HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH

28

infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)

recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)

replace timespast30=0 if screenlt=3 (4631 real changes made)

29

tab timespast30

timespast30 | Freq Percent Cum ------------++-----------------------------------

0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204

75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000

------------+-----------------------------------Total | 13939 10000

30

t

K l bl Key explanatory variiables

Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise

Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise

31

t

Th RThree Regressiions Dependent variable number of times drinking in past 30 days

Live in fratsor house (indicator variable)

M b f f Member of fratsor (indicator variable)

Intercept

SER

R2R2

N

444 (035)

---

454 (056)

649

011 011

13876

---

2 88 288 (016) 427

(0059)

644

023023

13876

226 (038)

2 44 244 (018) 427

(0059)

644

025025

13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42

32

t

Th PiThe Picture

i k Living in frat house

=226

X

M 2

Drinks per 30 days

house

=01921 Y

X2

Member of fraternity =244

X1

M 1 y 1

33

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

tab timespast30

timespast30 | Freq Percent Cum ------------++-----------------------------------

0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204

75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000

------------+-----------------------------------Total | 13939 10000

30

t

K l bl Key explanatory variiables

Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise

Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise

31

t

Th RThree Regressiions Dependent variable number of times drinking in past 30 days

Live in fratsor house (indicator variable)

M b f f Member of fratsor (indicator variable)

Intercept

SER

R2R2

N

444 (035)

---

454 (056)

649

011 011

13876

---

2 88 288 (016) 427

(0059)

644

023023

13876

226 (038)

2 44 244 (018) 427

(0059)

644

025025

13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42

32

t

Th PiThe Picture

i k Living in frat house

=226

X

M 2

Drinks per 30 days

house

=01921 Y

X2

Member of fraternity =244

X1

M 1 y 1

33

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

t

K l bl Key explanatory variiables

Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise

Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise

31

t

Th RThree Regressiions Dependent variable number of times drinking in past 30 days

Live in fratsor house (indicator variable)

M b f f Member of fratsor (indicator variable)

Intercept

SER

R2R2

N

444 (035)

---

454 (056)

649

011 011

13876

---

2 88 288 (016) 427

(0059)

644

023023

13876

226 (038)

2 44 244 (018) 427

(0059)

644

025025

13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42

32

t

Th PiThe Picture

i k Living in frat house

=226

X

M 2

Drinks per 30 days

house

=01921 Y

X2

Member of fraternity =244

X1

M 1 y 1

33

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

t

Th RThree Regressiions Dependent variable number of times drinking in past 30 days

Live in fratsor house (indicator variable)

M b f f Member of fratsor (indicator variable)

Intercept

SER

R2R2

N

444 (035)

---

454 (056)

649

011 011

13876

---

2 88 288 (016) 427

(0059)

644

023023

13876

226 (038)

2 44 244 (018) 427

(0059)

644

025025

13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42

32

t

Th PiThe Picture

i k Living in frat house

=226

X

M 2

Drinks per 30 days

house

=01921 Y

X2

Member of fraternity =244

X1

M 1 y 1

33

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

t

Th PiThe Picture

i k Living in frat house

=226

X

M 2

Drinks per 30 days

house

=01921 Y

X2

Member of fraternity =244

X1

M 1 y 1

33

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

t t t

A ti f th l ffAccounting for the total effect

ˆ B ˆ M ˆ M 1 1 2 21

Total effect = Direct effect + indirect effect

M =226

Drinks per Living in frat house X2

2

Drinks per 30 days

=01921 Y

Member of fraternity =244

X1

M 1

34

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

t

Live in frat

Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From

accounting identity T=D+I

Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)

444 226 218 sor house (51) (49)

From bi i

From multiple regressions

bivariate regressions

35

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms

MIT OpenCourseWarehttpocwmitedu

17871 Political Science LaboratorySpring 2012

For information about citing these materials or our Terms of Use visit httpocwmiteduterms