Talkin’ Hard Luck

“I’ve been cussed and discussed,Talked to and talked about,Lied to and lied about,Held up and hung upAnd doggone nigh murdered,And the only reason I’m hangin’ aroundIs to see what in the heck is going to happen next.”

(New Lost City Ramblers, 1960)

Potential Trajectories of Developmentand Value Added

Potential Trajectories

• Relates to potential outcomes model for causal inference (Rubin, 1978)

• Each person possesses a collection of potential trajectories under alternative possible sequences of treatments

• Figure 1. Selection Model

• π0i = initial status of student i

• π1i = annual growth rate given “average” teachers

• So π0i, π1i are governed by selection, not value added.

y

Child 1

Child 2

11

12

t

01

Li Li+1 Li+2 Li+ 3

π01

π12

π11

π02

Time

Test

Score

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

Li Li+1 Li+2 Li+3

Test

Score

Time

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

ŷ0i

= π0i

π0i

Li Li+1 Li+2 Li+3

Test

Score

Time

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

π1i π0i

ŷ0i = π0i

ŷli = π0i + πli +

Li Li+1 Li+2 Li+3

Test

Score

Time

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

c1i

ŷ0i = π0i

ŷli = π0i + πli + cli

π1i π0i cti=combined effect of school and grade at time t

Li Li+1 Li+2 Li+3

Test

Score

Time

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

c2i

c1i

ŷ0i = π0i

ŷli = π0i + πli + cli

ŷ2i = π0i + 2πli + cli + c2i

π1i π0i

Li Li+1 Li+2 Li+3

Test

Score

Time

cti=combined effect of school and grade at time t

y

t

Figure 2. Value Added Model: An Example of a Fortunate Student

c3i

c2i

c1i

ŷ0i = π0i

ŷli = π0i + πli + cli

ŷ2i = π0i + 2πli + cli + c2i

ŷ3i = π0i + 3πli + cli + c2i+ c3i

Gain from year t -1 to t = πli +ctiπ1i π0i

Li Li+1 Li+2 Li+3Time

Test

Score

cti=teacher effect at time t

y

t

Figure 3. Value Added Model: An Example of an Unfortunate Student

c3i

c2i c1i

ŷ0i = π0i

ŷli= π0i + πli + cli

ŷ2i= π0i + 2πli + cli + c2i

ŷ3i= π0i + 3πli + cli + c2i+ c3i

Gain from year t -1 to t = πli + cti

π1i

π0i

Li Li+1 Li+2 Li+3Time

Test

Score

Special Crossed and Nested Structure

Child School 1 School 2 …

Teacher 1 Teacher 2 … Teacher 1 Teacher 2 …

1 X X

2 X X

.

.

.

N X X

Cross-classified Random Effects Model

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Cross-classified Random Effects Model

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Cumulative Random Effects of Teachers

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Some Key Questions

• Role of the school

• Ignorable assignment of kids to teachers?• Sequential randomization?

• Reliability

• Stability over years • teacher v classroom effect

• See Carnegie Foundation Knowledge Briefs