unit 10: interaction and quadratic effects

© Judith D. Singer, Harvard Graduate School of Education Unit 10/Slide 1

Unit 10: Interaction and quadratic effects


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

Unit 2:Correlation

and causality

Unit 3:Inference for the regression model

Unit 4:Regression assumptions:

Evaluating their tenability

Unit 5:Transformations

to achieve linearity

Unit 6:The basics of

multiple regression

Unit 7:Statistical control in

depth:Correlation and

collinearity

Unit 10:Interaction and quadratic effects

Unit 8:Categorical predictors I:

Dichotomies

Unit 9:Categorical predictors II:

Polychotomies

Unit 11:Regression modeling

in practice

Unit 1:Introduction to

simple linear regression

Building a solid

foundation

Mastering the

subtleties

Adding additional predictors

Generalizing to other types of

predictors and effects

Pulling it all

together


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

• What is a statistical interaction and how is it different from the effects we’ve modeled so far?– Distinguishing between ordinal and disordinal statistical interactions

• Testing for the presence of a statistical interaction– Why does a cross-product term tell us about a statistical interaction?– Two different ways of summarizing interaction effects: which predictor

should you focus upon?– How does the interaction model compare to fitting separate models

within groups?

• Important caveats – Don’t confuse interaction with correlation – Don’t extrapolate beyond the range of the data!

• A very special interaction: A predictor can interact with itself– Quadratic regression models and their relationship to the logarithmic

models we fit earlier– The need to be especially careful about not extrapolating beyond the

range of X


Our “Preview” – Unit 6, Slide 25

= Lo GRE= Med GRE

= Hi GRE

= Small Schools= Medium Schools

= Large Schools

Right now, let’s assume that the main effects assumption is

correct

What does it mean if the lines aren’t parallel?

• This says that the effect of one predictor (say the effect of L2Doc) differs by levels of the other predictor (here, GRE)

• This is called a statistical interaction and in Unit 10 we’ll learn how to test for it and modify the model if necessary

Hmmm…the larger the school, the larger the effect

of GRE?

Hmmm…the better the student body, the larger the effect of

program size?


Examples of Statistical Interactions

Statistical interaction When the effect of one predictor differs by the level of another predictor

Brain Functionin

g

Hours after 8 pm

Adults

Teens

Interest level

Hours watching documentaries

Non-tax season

Tax season

Dilbert cartoon on Unit 10 Cover: Effect of 10% salary reduction on employee morale

differs by the economic climate.


Example: Does the effect of father presence differ by paternal behavior?

Source: Jaffee, SR, Moffitt, TE, Caspi, A, & Taylor, A (2003). Life with (or without) father: The benefits of living with two biological parents depend on the father’s antisocial behavior.

Child Development, 74(1) 109-126.

12.00-25 0 25 50 75 100 125

% of Child Life that Father is Resident

15.00

18.00

21.00

24.00

27.00

30.00

33.00

Child ASB

“Figure 1 plots the interaction and shows the simple slopes for the effect of father presence on child antisocial behavior at three values of the fathers’ antisocial behavior distribution (15th, 50th, and 85th percentiles). The figure shows that at low and median levels of fathers’ antisocial behavior, father presence was negatively associated with children’s antisocial behavior, such that the longer a father resided with his child, the less antisocial behavior the child had. However, at high levels of fathers’ antisocial behavior, father presence was positively associated with child antisocial behavior, such that the longer a father resided with his child, the more antisocial behavior he had.”

n =1,116 children, followed from birth through childhood

High Antisocial Father(85th %ile)

Mid Antisocial Father(50th %ile)

Low Antisocial Father(15th %ile)

“[The goal] was to determine whether the effects of father presence were uniform across families. Our hypothesis was that the fathers’ antisocial behavior would moderate the effect of father presence, such that when a father engaged in low levels of antisocial behavior, the less time he resided with his children the more behavior problems his children would have. In contrast, when a father engaged in high levels of antisocial behavior, the more time he resided with his children, the more behavior problems his children would have”


Interactions are prevalent in social science research

(1) Interactions form the basis for several quasi-experimental analytic methods:

• Regression discontinuity design (e.g., Gormley et al., 2005, on Universal Pre-K in Oklahoma)

• Difference-in-Differences (e.g., Dynarski, 2003, on College Financial Aid). (2) Interactions between variables are common in the research literature.

Not surprisingly, especially in education, the effect of one variable differs by levels of another

Castleman et al., 2009 (preliminary)

Relationship between ELA test scores and college enrollment is positive for traditional public school students, but there is no relationship for charter school students (interaction of test scores and charter schooling).

Potkay & Potkay, 1984

How much readers identify with cartoon characters of different genders differs by the gender of individual (interaction of gender of cartoon character and gender of reader).

Reardon et al., 2009

The effect of failing the mathematics high school exit examination in CA is bigger for students who also failed the English language arts examination (interaction of math and ELA test performance).

Johnson & Birkeland, 2009

The experiences of teachers in alternative certification programs depend on characteristics of the individual, the program, and their school (interaction of school, program, and person).


What is a statistical interaction?And how is it different from the effects that we’ve modeled so far?

Main effects modelThe fitted lines are parallel because the main effects model assumes that the

effect of each predictor is identical regardless of the values of all other

predictors in the model.

This model assumes that we can describe the effect of any predictor holding all other predictors in the model constant at any particular

value.

2.50

2.75

3.00

3.25

3.50

3.75

1997 1998 1999 2000 2001 2002

Vintage

Loge(Price)

Burgundy

BordeauxRhone

Languedoc

Models to date: The fitted lines have been

parallel

12.00

15.00

18.00

21.00

24.00

27.00

30.00

33.00

-25 0 25 50 75 100 125


Child ASB

High Antisocial

Father

Mid Antisocial Father

Low Antisocial Father

New model: The fitted lines are not parallel

Statistical interaction modelThe fitted lines are NOT assumed to be parallel because the interaction model

allows the effect of each predictor to differ by values of all other predictors in the

model.

This model assumes that we cannot simply describe the effect of any

predictor holding all other predictors in the model constant because the effect may differ according to those values.


Two types of statistical interactions: Ordinal and Disordinal

Ordinal interactionThe direction of each predictor’s

effect is consistent across levels of the other predictor, but its

magnitude differs

Disordinal interactionThe direction of the effect of one

predictor differs across levels of the other predictor

Lung Cancer

Asbestos Exposure

Non-smokers

ActivityLevel

Dose of Ritalin

Non ADHD

ADHD

Whether ordinal or disordinal, all interactions share a common feature

—non parallel lines—So the test that detects a statistical interaction is often called a test of

parallelism

Q: If the lines in an ordinal interaction aren’t parallel,

won’t they eventually cross, making all ordinal interactions

disordinal?Always graph your fitted

model within the range of the data.

Smokers


Introducing the case study: Sector differences in college graduation rates

April 23, 2006

Questions you might want to ask:

• Okay, there’s a difference by sector, but are all types of private colleges equally effective at graduating a larger proportion of their entering freshmen?

•From our earlier work, we might ask: Does this public-private sector differential change if we control statistically for other factors associated with graduation rates?

•E.g., if we control for the quality of the student body & amount of financial aid, does the public-private gap change?

•Today, we’ll learn how to ask: Does the public/private sector differential differ by the levels of other school characteristics?

•E.g,. are selective public schools (like UC Berkeley) similar to other public schools or are they more like private schools?

•In other words, does the “effect” of sector differ at different values of these other predictors?

•In other words, might there be a statistical interaction between sector and other predictors of college graduation rates?


Data to address these research questions

Source: Scott, M, Bailey, T, & Kienzl, G (2006). Relative success: Determinants of college graduation rates in public and private colleges in the US, Research in Higher Education,

47(1), 249-279

ID Name PctGrad Public SAT SATDiff FinAid

1 ABILENE CHRISTIAN UNIVERSITY 53.0000 0 11.00 3.20 75 2 ACADEMY OF THE NEW CHURCH 55.4425 0 11.70 3.20 62 3 ADAMS STATE COLLEGE 43.0000 1 10.10 3.10 84 4 ADELPHI UNIVERSITY 46.9892 0 10.20 2.90 78 5 ADRIAN COLLEGE 46.0000 0 9.70 1.80 85 6 AGNES SCOTT COLLEGE 61.0000 0 11.90 2.80 94 7 ALABAMA A&M UNIVERSITY 34.0000 1 8.40 2.30 60 8 ALABAMA STATE UNIVERSITY 24.1702 1 7.50 2.00 90 9 ALASKA BIBLE COLLEGE 35.5643 0 10.10 5.00 6710 ALASKA PACIFIC UNIVERSITY 37.1983 0 10.40 3.30 6911 ALBANY COLLEGE OF PHARMACY 76.0000 0 11.10 1.60 7912 ALBANY STATE COLLEGE 33.4602 1 8.00 1.50 8813 ALBERTSON COLLEGE OF IDAHO 66.4433 0 10.40 1.00 8014 ALBERTUS MAGNUS COLLEGE 44.0000 0 10.10 .60 7415 ALBION COLLEGE 68.0000 0 11.30 1.60 8016 ALBRIGHT COLLEGE 63.0000 0 12.00 2.00 6717 ALCORN STATE UNIVERSITY 34.0000 1 9.70 4.10 9218 ALDERSON BROADDUS COLLEGE 44.8082 0 11.00 3.00 9119 ALFRED UNIVERSITY 67.0000 0 12.30 2.40 90

RQ 1: Besides sector,

what else predicts college

graduation rates?

RQ 2: Do the effects of these other predictors

differ by sector?

Histogram of PctGRAD # Boxplot97.5+*** 11 | .****** 21 | .******** 30 | .********** 37 | .************** 55 | .********************* 83 | .**************************** 111 | .******************************* 121 +-----+ .************************************ 143 | |52.5+****************************************** 168 | | .***************************************** 162 *--+--* .***************************************** 161 | | .************************************** 151 +-----+ .************************************ 142 | .************************** 102 | .********************* 82 | .****** 22 | .**** 15 | 7.5+* 4 | ----+----+----+----+----+----+----+----+-- * may represent up to 4 counts

)0001.0(95.9,16.9

65.43,81.52

546,1075

ptdiff

yy

nn

publicprivate

publicprivate

n = 1621

RQ 2: Does the effect of sector differ by

these other predictors?

RQ 2: In other words, is there an interaction? Does school selectivity or financial aid have a different effect in public schools than it does in

private schools?


Financial aid appears to

have no relationship with PctGrad (uncontrolled

at least)

Predicting college graduation rates (a first look…)

r = 0.75***

r = -0.27***

r = 0.03 (ns)

Pearson Correlation Coefficients, N = 1621 Prob > |r| under H0: Rho=0

PctGrad SAT SATDiff FinAid Public

PctGrad 1.00000 0.75150 -0.26589 0.02812 -0.24014 <.0001 <.0001 0.2578 <.0001

SAT 0.75150 1.00000 0.00533 -0.07362 -0.10078 <.0001 0.8302 0.0030 <.0001

SATDiff -0.26589 0.00533 1.00000 0.07059 -0.00402 <.0001 0.8302 0.0045 0.8715

FinAid 0.02812 -0.07362 0.07059 1.00000 -0.45010 0.2578 0.0030 0.0045 <.0001

Public -0.24014 -0.10078 -0.00402 -0.45010 1.00000 <.0001 <.0001 0.8715 <.0001

Graduation rates are

In contrast to private colleges, public colleges:

We already know how to address RQ1 about the

predictors’ effects, but how do we address RQ2 about the

statistical interaction?

• Have lower graduation rates

• Are indistinguishable in terms of student body heterogeneity

• Have somewhat less strong student bodies

• Have smaller percentages of students on financial aid

Graduation rates are

These 3 predictors are

higher in schools with

stronger student bodies

lower in schools with

more heterogeneou

s student bodies

relatively uncorrelated (although the r’s of |.07| are stat sig because of the large sample size)


How do we test for the presence of a statistical interaction?

ID Name PctGrad Public SAT PSAT

1 ABILENE CHRISTIAN UNIVERSITY 53.0000 0 11.00 0 2 ACADEMY OF THE NEW CHURCH 55.4425 0 11.70 0 3 ADAMS STATE COLLEGE 43.0000 1 10.10 10.10 4 ADELPHI UNIVERSITY 46.9892 0 10.20 0 5 ADRIAN COLLEGE 46.0000 0 9.70 0 6 AGNES SCOTT COLLEGE 61.0000 0 11.90 0 7 ALABAMA A&M UNIVERSITY 34.0000 1 8.40 8.40 8 ALABAMA STATE UNIVERSITY 24.1702 1 7.50 7.50 9 ALASKA BIBLE COLLEGE 35.5643 0 10.10 010 ALASKA PACIFIC UNIVERSITY 37.1983 0 10.40 011 ALBANY COLLEGE OF PHARMACY 76.0000 0 11.10 012 ALBANY STATE COLLEGE 33.4602 1 8.00 8.0013 ALBERTSON COLLEGE OF IDAHO 66.4433 0 10.40 014 ALBERTUS MAGNUS COLLEGE 44.0000 0 10.10 015 ALBION COLLEGE 68.0000 0 11.30 016 ALBRIGHT COLLEGE 63.0000 0 12.00 017 ALCORN STATE UNIVERSITY 34.0000 1 9.70 9.7018 ALDERSON BROADDUS COLLEGE 44.8082 0 11.00 019 ALFRED UNIVERSITY 67.0000 0 12.30 0

Step 1: Create a cross-product term, which is literally the product of the two

predictors whose interaction you want to test

(e.g, PSAT=SAT*Public)(NOTE: In your data step in SAS, just write PSAT =

SAT*public;)

schools publicin SATSAT)*1(

schools privatein 0SAT)*(0

PSAT

PSATPublicSATY 3210

Step 2: Include the cross-product in a MR model that also includes the constituent

main effects (NOTE: You should always include the main

effects)

Step 3: Test H0: cross-product = 0If the individual t-test rejects H0, the

two predictors interact; if not, there is no statistical interaction between these

two predictors

There’s an easy, three-step process!

PSATPublicSATY 3210

NOTE on terminology: Main Effects Cross-Product Term


Remembering dichotomous predictors: Main effects model

PublicSATY 210ˆˆˆˆ

)0(ˆˆˆˆ210 SATY

SATY 10ˆˆˆ

schools) Private (i.e., 0PublicWhen

Schools) Public (i.e., 1PublicWhen

SATY 120ˆ)ˆˆ(ˆ

)1(ˆˆˆˆ210 SATY

intercept in the differenceˆ2

Public

Private

Our simple, main effects model:


Why does a cross-product term tell us about a statistical interaction?

PSATPublicSATY 3210ˆˆˆˆˆ

SATPublicPublicSATY *ˆˆˆˆˆ3210

SATSATY *)0(ˆ)0(ˆˆˆˆ3210

SATY 10ˆˆˆ

schools) Private (i.e., 0PublicWhen

schools) Public (i.e., 1PublicWhen

)ˆˆ(ˆ20 Y SAT)ˆˆ( 31

SATSATY *)1(ˆ)1(ˆˆˆˆ3210

intercept indiff 2̂ slope indiff 3̂

Public

Private

Note the inclusion of main effects

data) the of range the outside way is (which

0 SAT whenintercept in diff 2̂ schoolsprivate vs. public in

SAT)of effect the in (diff slopein diff3̂

The parameter estimate for the interaction term

tells us how much steeper the slope on

SAT is for public schools than for private schools

1

Our interaction model with cross-product term


Is there a statistically significant interaction between SECTOR and SAT?

Comparing main effects and interaction models

Interaction effect model Sum of MeanSource DF Squares Square F Value Pr > F

Model 3 312657 104219 787.89 <.0001Error 1617 213889 132.27538Corrected Total 1620 526546

Root MSE 11.50110 R-Square 0.5938Dependent Mean 49.72992 Adj R-Sq 0.5930Coeff Var 23.12713 Parameter StandardVariable DF Estimate Error t Value Pr > |t|

Intercept 1 -41.32250 2.65428 -15.57 <.0001SAT 1 8.85605 0.24751 35.78 <.0001Public 1 -17.87644 4.43583 -4.03 <.0001PSAT 1 1.10674 0.42131 2.63 0.0087

Main effects model Sum of MeanSource DF Squares Square F Value Pr > F

Model 2 311744 155872 1174.11 <.0001Error 1618 214802 132.75777Corrected Total 1620 526546


Intercept 1 -45.38289 2.16172 -20.99 <.0001SAT 1 9.23804 0.20066 46.04 <.0001Public 1 -6.33366 0.60861 -10.41 <.0001

We can reject the null hypothesis that all predictors in each model have no

effect

Reasonably high R2 in both models (although the interaction model seems

only trivially better than the main effects model)

Both SAT and Public each have statistically significant main effects

There is a statistically significant interaction between SAT and

Public (in this simple uncontrolled model)

The main effects assumption doesn’t hold:

The effect of SAT scores differs by sector

Built in assumptions: The effects of SAT are identical in private and public colleges and the private-public differential is identical

across the full range of SAT scores

The public/private differential differs by selectivity of school.

Until we “do the math” we don’t know how it differs, we just know that it does.

SATPublicPublicSATY *11.188.1786.832.41ˆ

CAUTION: Do NOT try to interpret

the main effects by

themselves


Graphing and interpreting a model with a statistical interaction

SATPublicPublicSATY *11.188.1786.832.41ˆ

Public

Private

SATY

SATSATY

86.832.41ˆ

*)0(11.1)0(88.1786.832.41ˆ

0Public when

SATY

SATY

SATSATY

97.920.59ˆ

)11.186.8()88.1732.41(ˆ

*)1(11.1)1(88.1786.832.41ˆ

1Public when

-41.32, the y-intercept, is the

predicted grad rate in private schools

with SAT=0(rarely interpreted)

8.86, the coefficient for SAT, is the slope (“effect” of SAT)

when Public=0 (in private schools)

-17.88, the coefficient for Public,

assesses the public/private

difference when SAT=0 (not

interesting here)

1.11, the coefficient for the cross-product, is the increment to the slope (“effect” of

SAT) in public schools


All interaction effects can be summarized in (at least) two different ways

depending on which predictor you choose to focus upon

Interpretation #1The effect of SAT scores

differs by sector The effect is larger (the slope is steeper)

among public colleges than among private colleges

Interpretation #2The effect of sector (the

public/private differential) differs by SAT scores

The weaker the student body, the larger the differential; when the student body of a

school is very strong, there is no differential

12.00

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27.00

30.00

33.00

-25 0 25 50 75 100 125


Child ASB

High Antisocial Father



Interpretation #1The effect of father presence differs by levels of paternal

antisocial behaviorThe effect is positive for low antisocial fathers and negative for high antisocial

fathersInterpretation #2

The effect of paternal antisocial behavior differs by the %age of a

child’s life that the father is resident

The more time the child lives with the father, the larger the effect

0.00

25.00

50.00

75.00

100.00

6 8 10 12 14 16

75th %ile of SAT (in 100s)

PctGrad

SATY 97.920.59ˆ

Public

SATY 86.832.41ˆ

Private


Comparing the interaction model to separate models fit within SECTOR

SATY 8561.83225.41ˆ

0)(Public Private

Interaction effect modelNumber of Observations Used 1621


Intercept 1 -41.32250 2.65428 -15.57 <.0001SAT 1 8.85605 0.24751 35.78 <.0001Public 1 -17.87644 4.43583 -4.03 <.0001PSAT 1 1.10674 0.42131 2.63 0.0087

Effect of SAT in public colleges Public=1

Number of Observations Used 546

Root MSE 10.83712 R-Square 0.6387Dependent Mean 43.65689 Adj R-Sq 0.6381Coeff Var 24.82338

Parameter Estimates

Parameter StandardVariable DF Estimate Error t Value Pr > |t|

Intercept 1 -59.19894 3.34888 -17.68 <.0001SAT 1 9.96280 0.32125 31.01 <.0001

Effect of SAT in private colleges Public=0

Number of Observations Used 1075

Root MSE 11.82350 R-Square 0.5303Dependent Mean 52.81446 Adj R-Sq 0.5298Coeff Var 22.38687

Parameter Estimates


Intercept 1 -41.32250 2.72868 -15.14 <.0001SAT 1 8.85606 0.25445 34.80 <.0001

SATY 9628.91989.59ˆ

1) (Public Public

Advantages of interaction models1. Interaction models can be fit whether the

predictors are dichotomous or continuous

2. Interaction models provide an easy statistical test of whether the slopes differ across groups (or across levels of a continuous predictor).

3. Interaction models keep the sample intact (you don’t need to break it down into many different groups).

Why can’t we just fit separate models for public and private schools?


Are there interactions between the other predictors and SECTOR?

Model A Model B Model C

Intercept-41.32***

(2.65)67.20***

(1.82)70.78***

(2.46)

Public-17.88***

(4.44)-2.51(3.20)

-33.33***(3.34)

SAT8.86***(0.25)

Public*SAT

1.11**(0.42)

SATDiff-5.81***(0.71)

Public* SATDiff

-2.71*(1.24)

FinAid-0.24***(0.03)

Public*FinAid

0.35***(0.05)

R2 59.4% 13.1% 9.4%

If we test each interaction on its own, not controlling for any other

predictors (or interactions), all three tests reject the null

hypothesis (!) % students on financial aid

0.00

25.00

50.00

75.00

100.00

0 25 50 75 100

PctGrad

FinAidY 11.045.37ˆ

Public

FinAidY 24.078.70ˆ

Private

0.00

25.00

50.00

75.00

100.00

0 1 2 3 4 5

IQR of SAT (in 100s)

PctGrad

SATDiffY 52.869.64ˆ

Public

SATDiffY 81.520.67ˆ

Private

What’s the effect of heterogeneity (SATDiff)

in private schools?

How does the effect of heterogeneity differ by sector?

What’s the effect of heterogeneity (SATDiff)

in public schools?

How does the effect of sector differ by heterogeneity?

What’s the effect of financial aid in private schools?

How does the effect of financial aid differ by sector?

What’s the effect of financial aidin public schools?

How does the effect of sector differ by financial aid?

81.5ˆ: SATDiffPRIVATE

52.8)71.281.5(ˆ: SATDiffPUBLIC

Heterogeneity has a larger effect in public than it does in private schools

The more homogeneous the student body, the smaller the sector

differential

24.0ˆ: FinAidPRIVATE

11.0)35.024.0(ˆ: FinAidPUBLIC

The effect of FinAid is positive in public schools and negative in private

ones

The more financial aid offered, the smaller the sector differential


Do the interactions remain when we control for all main effects?

Model D Model E Model F Model G

Intercept-31.5***(2.52)

-29.52***(2.92)

-31.36***(2.54)

-27.60***(2.74)

Public-5.78***(0.62)

-11.20**(4.09)

-6.68***(1.98)

-12.80***(2.06)

SAT9.31***(0.18)

9.12***(0.23)

9.31***(0.18)

9.23***(0.18)

Public*SAT

0.52(ns)(0.38)

SATDiff-6.84***(0.36)

-6.80***(0.36)

-6.96***(0.44)

-6.67***(0.36)

Public* SATDiff

0.37(ns)(0.77)

FinAid0.03*(0.01)

0.03~(0.02)

0.03*(0.01)

-0.02(ns)(0.02)

Public*FinAid

0.11***(0.03)

R2 66.6% 66.7% 66.6% 66.9%

[Public]

[Private]

FinAidSATdiffSATY

FinAidSATdiffSATY

FinAidPublicPublicFinAidSATdiffSATY

09.067.623.940.40ˆ

02.067.623.960.27ˆ

*11.080.1202.067.623.960.27ˆ

From Model D (the main effects model) we conclude that

These equations result from substituting in the

values for PUBLIC (0=private, 1=public)

From Model E we conclude that

From Model F we conclude that

From Model G we conclude that

all three main effects—SAT, SATDiff and FinAid—are statistically significant and should be included in our model; controlling for these three, there’s a main effect of SECTOR

when we control statistically for the main effects of SATDiff and FinAid, there is no interaction between SAT and SECTOR. Thus, we do not need to keep the cross-product term (PUBLIC*SAT) in our model.

when we control statistically for the main effects of SAT and FinAid, there is no interaction between SATdiff and SECTOR. Thus, we do not need to keep the cross-product term (PUBLIC*SATDiff) in our model.

when we control statistically for the main effects of SAT and SATDiff, the interaction between FinAid and SECTOR persists

What about the n.s. main effect of FinAid in Model G?The effect of FinAid is non-significant in Private schools. But…..Never delete main effects that are components of a statistically significant interaction EVEN IF THEY ARE NON-SIGNFICANT! NEVER!*

*or at least not yet


How might we summarize our findings, including the interaction?

[Public]

[Private]

FinAidSATdiffSATY

FinAidSATdiffSATY

09.067.623.940.40ˆ

02.067.623.960.27ˆ

SAT 10th & 90th %iles = 8.8 and 12.3 SATdiff 10th & 90th %iles = 1.7 and 3.4

PctGrad

10.00

30.00

50.00

70.00

90.00

% students on financial aid

0 25 50 75 100

Very strong SAT scores

Very weak SAT scores

Private

Private

Private

Private

Public

Public

Public

Public

Homogeneous scores

Heterogeneous scores

Homogeneous scores

Heterogeneous scores

What’s the effect of SAT scores?

What’s the effect of SAT homogeneity?

What’s the effect of Financial Aid?

Controlling for SAT homogeneity, financial aid and sector, the higher the 75th %ile of SAT scores, the higher the graduation rate

The more homogeneous the SAT scores, the higher the graduation rate (holding constant the 75th%ile of SAT scores, financial aid and sector)

Controlling for SAT scores (both 75th %ile and IQR) the relationship between financial aid and graduation rates differs by SECTOR. In private schools, AID has almost no effect; in public schools, the larger the %age of students receiving aid, the higher the graduation rate (seen in slope of lines)

What’s the effect of SECTOR?Controlling for SAT scores (both 75th %ile and IQR) the effect of SECTOR differs by financial aid. In schools that provide most students with aid, there is no sector differential; the lower the %age of students receiving aid, the larger the public/private differential

Note: Remember what our variables mean substantively


Now that we know about interactions, what about the Ed School data? Were those lines really parallel?

School L2Doc GRE L2DocGRE

Harvard 5.90689 6.625 39.1332 UCLA 5.72792 5.780 33.1074 Stanford 5.24793 6.775 35.5547 TC 7.59246 6.045 45.8964 Vanderbilt 4.45943 6.605 29.4545 Northwestern 3.32193 6.770 22.4895 Berkeley 5.42626 6.050 32.8289 Penn 5.93074 6.040 35.8217 Michigan 5.24793 6.090 31.9599 Madison 6.72792 5.800 39.0219 NYU 6.80735 5.960 40.5718

Interaction effect model R2=66.8% Parameter StandardVariable DF Estimate Error t Value Pr > |t|

Intercept 1 441.63946 198.90350 2.22 0.0292PctDoc 1 0.74971 0.18698 4.01 0.0001L2Doc 1 -94.53164 40.04790 -2.36 0.0206GRE 1 -32.93383 35.38194 -0.93 0.3547L2DocGRE 1 18.93133 7.10577 2.66 0.0093

× =

= Small Schools= Medium Schools

= Large Schools

Large

Small

Medium

GREDocLGREDocLPctDocY *293.1893.32253.9475.064.441ˆ

Should we do anything about the main effect

of GRE?

NO!


Interpreting and displaying an interaction between continuous predictors

Mean GRE Scores = on the x-axis

PctDoc = 38.20 (sample mean)

Small: L2Doc = 4 (NDoc=16)Medium: L2Doc = 5 (NDoc=32)Large: L2Doc = 6 (NDoc=64)

Small

Medium

Large


GRE

GRE

GREGREY

79.4217.92

)72.7593.32()12.37865.2864.441(

)4(93.1893.32)4(53.94)20.38(75.064.441ˆ

GRE

GRE

GREGREY

72.6136.2

)65.9493.32()65.47265.2864.441(

)5(93.1893.32)5(53.94)20.38(75.064.441ˆ

GRE

GRE

GREGREY

65.8089.96

)58.11393.32()18.56765.2864.441(

)6(93.1893.32)6(53.94)20.38(75.064.441ˆ

Peer Rating

250

300

350

400

450

500

4.5Mean GRE

5.0 5.5 6.0 6.5 7.0

In our main effects model, the lines:

• Have different intercepts, but were a constant distance apart (distance = controlled effect of L2DOC)

• Have the same slope (slope = controlled effect of GRE)

In our interaction effects model, the lines:

• Have different intercepts and are not a constant distance apart (controlled effect of L2DOC still seen in distance between lines, but it varies at different levels of GRE)

• Have different slopes (controlled effect of GRE still seen in the slope, but it varies at different levels of L2DOC)


Interpreting and displaying an interaction between continuous predictors

Holding PctDoc constant, the effect of program size differs by mean GRE score (distance between lines).

• In schools with low mean GREs, there’s virtually no effect of program size; the higher a school’s mean GRE scores, the larger the effect of program size.

• For example, when mean GRE = 500, a doubling of size is associated with a 0.12 difference in peer ratings, whereas when mean GRE = 600, a doubling of size is associated with a 19.05 difference in ratings.

Holding PctDoc constant, the effect of mean GRE score differs by department size (slope of the lines).

• The larger the school, the larger the effect of GRE scores.

• For example, among small schools (L2Doc = 4) a difference of 100 points in mean GRE is associated with a 42.8 point in peer ratings, whereas among large schools (L2Doc = 6), the same 100-point difference in mean GRE scores is associated with a 80.65 difference in peer rating.

Small

Medium

Large


GRE

GRE

GREGREY

79.4217.92

)72.7593.32()12.37865.2864.441(

)4(93.1893.32)4(53.94)20.38(75.064.441ˆ

GRE

GRE

GREGREY

72.6136.2

)65.9493.32()65.47265.2864.441(

)5(93.1893.32)5(53.94)20.38(75.064.441ˆ

GRE

GRE

GREGREY

65.8089.96

)58.11393.32()18.56765.2864.441(

)6(93.1893.32)6(53.94)20.38(75.064.441ˆ

Peer Rating

250

300

350

400

450

500

4.5Mean GRE

5.0 5.5 6.0 6.5 7.0

What’s the effect of mean GRE scores?What’s the effect of program size?


Caveats: (1) Don’t confuse interaction with correlation

CorrelationX1 and X2 are associated

InteractionThe effect of X1 differs by levels

of X2

Financial aid and Public are correlated and they do interact

in predicting graduation rates

SATDiff and Public are not correlated, but they do

interact in predicting grad rates (not controlling for

other predictors)

SAT verbal and math are

correlated but they do not interact

when predicting college

achievement

SAT and SATDiff are not correlated, and

although not documented here, they

do not interact in predicting graduation

rates

Predictors do interact

Predictors do not

interact

Predictors are correlated

Predictors are not correlated

Correlation does not imply interaction

•Just because two predictors are correlated doesn’t suggest anything about whether they might interact.

•The question to ask is whether there’s reason to hypothesize that the effect of one predictor might differ according to levels of another predictor.

12.00

15.00

18.00

21.00

24.00

27.00

30.00

33.00

-25 0 25 50 75 100 125


Child ASB

High Antisocial Father



Of course this reversal makes no

sense:it’s outside the

range of our data and we shouldn’t be

predicting there!

and (2) Don’t extrapolate beyond the range of the data

Caveat 1

Caveat 2


A very special interaction: A predictor can interact with itself!

112110 * XXXY

0.00

1.00

2.00

3.00

4.00

5.00

0 2 4 6 8 10 12

X

Y211 10.020.150.0ˆ XXY

Q: What if the effect of a given predictor differed by levels of that very predictor—the “effect of X”

differed by levels of X?

212110 XXY

Quadratic modelWe allow a predictor’s effect to differaccording to levels of that predictor.

The test on 2 provides a test of whether the quadratic term (model)

is necessary

All quadratics are non-monotonic—they both rise and fall (or fall and rise)

Satisfaction

Number of Jellybeans Eaten

At low levels of X, the relationship between Y and X is relatively steep

and positive

At moderate levels of X, the

relationship between Y and X

is quite flat (almost no

relationship)

At high levels of X, the

relationship between Y and X is relatively steep

and negative


0.00

1.00

2.00

3.00

4.00

5.00

0 2 4 6 8 10 12

X

Y 211 02.004.000.1ˆ XXY

0.00

1.00

2.00

3.00

4.00

5.00

0 2 4 6 8 10 12

X

Y 211 03.003.000.4ˆ XXY

“Research has argued that the shape of the relationship between SES and health is actually curvilinear such that there are decreasing returns to health as SES increases.

Source: Finch, BK (2003). Socioeconomic gradients and low birth-weight: Empirical and policy considerations.

Health Services Research, 38(6), 1819-1841.

You’re only interested in the shape of a quadratic within the range of X

When you’re poor, $1000 has a bigger effect than when you’re rich

Thus, quadratics are another tool we can use to transform our data

and fit non-linear relationships


Residuals from linear regression on Rating

Wine ratings redux: Does quality (or at least ratings) matter?

ID Lprice Rating Rating2

1 2.54721 7 49 3 2.58238 6 36 31 3.29584 8 64 33 3.33831 8 64 35 3.39881 7 49 50 3.70658 10 100 57 3.85167 10 100 58 3.91488 9 81 60 4.19570 11 121 61 2.64820 1 1 66 3.00285 5 25 67 3.00498 4 16

Rating2 = Rating*Rating

The REG ProcedureDependent Variable: Lprice

Root MSE 0.30576 R-Square 0.5276Dependent Mean 3.06102 Adj R-Sq 0.5190Coeff Var 9.98886 Parameter Estimates


Intercept 1 2.50664 0.26885 9.32 <.0001Rating 1 -0.03372 0.07640 -0.44 0.6598Rating2 1 0.01461 0.00537 2.72 0.0075

20146.00337.05066.2ˆ RatingRatingY

2.00

3.00

4.00

5.00

0 2 4 6 8 10 12

Rating

Loge(price)

The effect of rating on price is small when ratings are low and high when ratings are

high

The t-statistic for Rating2 indicates that the linear term is not sufficient. We need

to include Rating2 in our model.


Does the effect of Rating remain quadratic after controlling for Region and Vintage?

The REG ProcedureModel: MODEL1Dependent Variable: Lprice Root MSE 0.21313 R-Square 0.7788Dependent Mean 3.06102 Adj R-Sq 0.7663Coeff Var 6.96281

Parameter Estimates


Intercept 1 2.45448 0.20852 11.77 <.0001Burgundy 1 0.72380 0.08279 8.74 <.0001Bordeaux 1 0.22813 0.05701 4.00 0.0001Rhone 1 0.17490 0.06183 2.83 0.0056Year 1 -0.09673 0.02136 -4.53 <.0001Rating 1 0.00288 0.05518 0.05 0.9584Rating2 1 0.01077 0.00380 2.83 0.0055

Set YEAR at its mean = 2.04 2.00

3.00

4.00

5.00

0 2 4 6 8 10 12Rating

Loge(price)

Burgundy

BordeauxRhone

Languedoc

RhoneBordeauxBurgundy

RatingRatingY

17490.022813.072380.0

01077.000288.025812.2ˆ 2

$7.39

$20.09

$54.60

$148.41

3.76787

3.53882

..

2.71667

2.61686

..

2.44667

Rhone

3.8211

3.59205

..

2.7699

2.67009

..

2.4999

Bordeaux

4.31677

4.08772

..

3.26557

3.16576

..

2.99557

Burgundy

3.59297

3.36392

..

2.54177

2.44196

..

2.27177

Languedoc

Predicted Log Price

11

10

..

5

4

..

1

Rating

+0.10

+0.23

+0.10

+0.23

Actual price

The effect of 1-unit difference in rating is smaller when ratings are lower…

… and bigger when ratings are higher.

After controlling for linear vintage and quadratic ratings, which regions

are significantly different?


Bonferroni multiple comparisons of REGION means (controlling for RATING, RATING2 and VINTAGE)

The GLM ProcedureLeast Squares MeansAdjustment for Multiple Comparisons: Bonferroni

Lprice LSMEANRegion LSMEAN Number

1 3.58861155 12 3.09294691 23 3.03971063 34 2.86481291 4

Least Squares Means for Effect Region t for H0: LSMean(i)=LSMean(j) / Pr > |t|

Dependent Variable: Lprice

i/j 1 2 3 4

1 5.93396 6.207648 8.742802 <.0001 <.0001 <.0001

2 -5.93396 0.942968 4.001843 <.0001 1.0000 0.0007

3 -6.20765 -0.94297 2.828796 <.0001 1.0000 0.0335

4 -8.7428 -4.00184 -2.8288 <.0001 0.0007 0.0335

The Languedoc remains significantly less expensive than all other regions, after controlling for linear vintage and the quadratic effect of rating

BurgundyBordeaux

RhoneLanguedoc

Bordeaux and the Rhone are indistinguishable after controlling for linear vintage and the quadratic effect of rating

Burgundy is significantly more expensive than all other regions after controlling for linear vintage and the quadratic effect of rating

Burgundy Bordeaux Rhone Languedoc

Burgundy

Bordeaux

Rhone

Languedoc


Hypothesizing the existence of quadratic effects: A recent example


What’s the big takeaway from this unit?

• Statistical interactions are an important type of effect– An interaction tells us that the effect of one predictor varies by levels of

another– Sometimes the magnitude of an effect will vary; other times the direction of

an effect will vary– The standard regression model, which initially assumes that there are no

interactions, can be easily modified to accommodate their presence– Many substantive theories suggest that effects will be interactive

• You test for a statistical interaction by adding a cross-product term

– The cross-product is literally the product of the two constituent variables– If it is significant in a model that also includes the constituent main effects,

you know that the two predictors interact. And never remove the main effects– Graph out the fitted model to ensure correct interpretation.

• Predictors can interact with themselves!– Quadratic models provide a flexible strategy for fitting nonlinear models,

especially those that can’t be linearized by taking logarithms– Substantive theories often suggest that a predictor’s effect may be quadratic– You test for the presence of a quadratic effect and include it in a regression

model using the same strategy used to include interaction effects


*-------------------------------------------------------------------*Input Gradrate data and name variables in datasetCreate interaction terms PSAT, PSATDIFF, & PFINAID *------------------------------------------------------------------*; data one; infile "m:\datasets\gradrate.txt"; input ID 1-4 Name $ 7-47 PctGrad 50-61 Public 64 SAT 67-70 SATdiff 73-76 FinAid 78-80; PSAT=public*sat; PSATDiff=public*satdiff; PFinAid=Public*FinAid;

Appendix: Annotated PC-SAS Code for Interactions and Quadratic Effects

Gradrate AnalysisGradrate Analysis

The data step here is used to create cross-product terms, literally the product of the two variables whose interaction you want to test.

The data step here is used to create cross-product terms, literally the product of the two variables whose interaction you want to test.

*------------------------------------------------------------------*Listing data on observations 1-19 for inspection *------------------------------------------------------------------*; proc print data=one; where 1 <= id <= 19; var id name pctgrad public sat satdiff finaid; run;

It is often helpful to add a where statement to a proc print, because it tells SAS to print out only a subset of the data (here, IDs 1 – 19). This can save reams of paper!

It is often helpful to add a where statement to a proc print, because it tells SAS to print out only a subset of the data (here, IDs 1 – 19). This can save reams of paper!

Wine AnalysisWine Analysis

data one; infile "m:\datasets\wine.txt"; input ID 1-3 Price 5-16 Region 19 Area $ 21-31 Year 34 Vintage $ 38-44 Rating 48-51; Rating2 = rating**2;

The data step is also the place to create quadratic terms. You can either multiply a variable by itself (i.e., rating2=rating*rating) or more easily just raise the variable to the second power (as done in this code).

The data step is also the place to create quadratic terms. You can either multiply a variable by itself (i.e., rating2=rating*rating) or more easily just raise the variable to the second power (as done in this code).


Glossary terms included in Unit 10

• Interactions (ordinal and disordinal)• Quadratics


Unit 6, Slide 17: Understanding the fitted MR model algebraically & graphically

DocLGRERatrPee 234.1532.6329.87ˆ

Controlled effect of GRE can be seen in the

common slope (63.32) of these lines

Controlled effect of L2Doc can be seen in the common distance

(15.34) between these lines

GRErRatePe

GRErRatePe

32.6393.25ˆ

)4(34.1532.6329.87-ˆ

4L2Doc

GRErRatePe

GRErRatePe

32.6359.10ˆ

)5(34.1532.6329.87-ˆ

5L2Doc

GRErRatePe

GRErRatePe

32.6327.41ˆ

)3(34.1532.6329.87-ˆ

3L2Doc

GRErRatePe

GRErRatePe

32.6375.4ˆ

)6(34.1532.6329.87-ˆ

6L2Doc

AlgebraicallyPlug in different values of

L2Doc

-25.93-(-

41.27)15.34

-10.59-(-

25.93)15.34

4.75-(-

10.59)15.34

GraphicallyReturn to the plot from

before

34567


Appendix: Visualizing a statistical interaction

L2DOC = 3

L2DOC = 7

PEER

L2DOC = 3

L2DOC = 7

PEER

The twist in the “plane” allows the relationship (slope) between PEER and

GRE to vary at different values of L2DOC. We can see

this twist by looking at the fitted regression

“plane” in our 3D view from two different

angles.

unit 10: interaction and quadratic effects

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