who chooses, who refuses? learning more from students who decline private school vouchers

Who Chooses, Who Refuses? Learning More from Students Who Decline Private SchoolVouchersAuthor(s): Joshua CowenSource: American Journal of Education, Vol. 117, No. 1 (November 2010), pp. 1-24Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/10.1086/656344 .

Accessed: 26/05/2014 01:19

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Education.

http://www.jstor.org

This content downloaded from 194.29.185.12 on Mon, 26 May 2014 01:19:50 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ucpress

http://www.jstor.org/stable/10.1086/656344?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


NOVEMBER 2010 1

American Journal of Education 117 (November 2010)� 2010 by The University of Chicago. All rights reserved.0195-6744/2010/11701-0001$10.00

Who Chooses, Who Refuses? Learning Morefrom Students Who Decline Private SchoolVouchers

JOSHUA COWENUniversity of Kentucky

I argue that lottery-based school choice programs offer the opportunity to studya unique group of students: those who want to attend or are very interested inattending private school but simply cannot, even when given the chance. Thedifferences between these students and those who choose private school arecompelling education outcomes in their own right. To illustrate the argument,I analyze data from a small and little-known private school scholarship lotteryin Charlotte, North Carolina, that occurred prior to the 1999–2000 academicyear. I show that race, family structure, employment status, and religion signif-icantly predict the decision to refuse a voucher offer, as does student admissioninto a specific school of choice. I argue that models of voucher effects on studentachievement are interpretable only in the context of factors underlying the abilityto choose in the first place.

Introduction

Formal school choice programs are widespread in the United States. Withinthe public sector, students and families in many locales may select from aparticular set of traditional elementary, junior high, or high schools main-tained by the contiguous school district within which they reside. In manymore areas, districts maintain charter or magnet schools, and parents maysend their children to them with few limits beyond enrollment capacity. In-dependent charters may be granted in most states to enterprising parents,educators, or other interested parties to establish schools in accordance witheach state’s laws. Within the private sector, a diverse array of educationaloptions serves a market of parents and students opting out of the traditionalpublic system. This market includes families whose access to private education

Electronically published October 7, 2010



Who Chooses, Who Refuses?

2 American Journal of Education

is contingent on resources beyond their personal economic means. In recentyears, one such resource has been a set of scholarship programs—some fundedby public money, others with private endeavors—with the explicit purpose toopen the private market to individuals who would choose only among thecost-free public options in the absence of scholarships.

Whether these scholarships, often known as school vouchers, improve ac-ademic outcomes is the subject of intense and ongoing scholarly debate. Re-searchers from several disciplines, including political science, economics, so-ciology, and educational policy, have examined the results of voucher programsacross the United States. Among those programs that have included a lotterydesign—that is, the voucher is offered at random to a pool of interestedstudents—the evidence generally suggests small but positive achievement gainsfor students (Barnard et al. 2003; Howell et al. 2002; Peterson et al. 1999;Wolf et al. 2009; see Krueger and Zhu [2004] for contrary evidence). Rigorousobservational studies have generated mixed results, particularly in Milwaukeeand Cleveland, where researchers evaluated highly visible public programs(Metcalf et al. 2003; Peterson et al. 1999; Rouse 1998; Witte 2000; Witte etal. 2009).

All of these evaluations include analysis of the demographic and socioeco-nomic differences between families who choose private education, whetherthey accept the random offer of the scholarship or decide to seek and use thevoucher. However, such analysis is typically framed either as a systematicconsideration of the potential of key variables to confound estimates of theprogram effect on average levels of student achievement or as one of manyoutcomes ancillary to that central focus.

Apart from such considerations, a family’s decision to choose private school-ing in the first place as an outcome of interest in its own right is underdevelopedin the literature. As important, the implications that the factors operating onthis decision may have on average student achievement are largely consignedin the literature to a discussion of statistical control variables. They are, infact, substantive and critical components to a broader description of privateschool choice that can inform public policy makers. In strictly observationalsettings, such as those in Milwaukee and Florida, these factors may be studiedin public school students who are observed in the population of families whoare also eligible for the voucher. Differences between these students and thosewho seek and use the voucher may be documented and taken into accountin statistical models of achievement. In lottery settings, however, the moregeneral attempt to balance unobservable influences on achievement between

JOSHUA COWEN is an assistant professor in the Martin School of PublicPolicy and Administration at the University of Kentucky. His current researchfocuses on school choice policies and the evaluation of public programs.



Cowen

NOVEMBER 2010 3

the treatment (lottery winners) and control (lottery losers) groups offers anadditional analytical benefit. Because all participants in such studies attemptedto take initial advantage of the voucher opportunity by placing their namesin the lottery drawing, we may uniquely observe a group that approximatesthose students who “want to” take a voucher but simply cannot, even whengiven the chance.

In this article, I focus on such voucher decliners. To fix the argumentempirically, I analyze data from a small lottery of private school scholarshipsin Charlotte, North Carolina, that occurred just prior to the 1999–2000 ac-ademic year. I note that race, family structure, employment status, and religionare all significant predictors of the family’s decision to refuse a voucher offer.The factors contributing to students’ ability to attend a particular school arealso significant, such as admission to that school or the ability to pay tuitionbeyond the size of the voucher to their preferred school. I include thesecharacteristics in a model of voucher acceptance, where acceptance in thecontrol group (those not offered the scholarship) is treated as missing data.Then, using a maximum-likelihood approach suggested in other literatures, Iestimate the differences between all potential voucher users and nonusers inboth the treatment and control groups on a variety of attributes, includingstudent achievement outcomes. In the Charlotte data, scholarship use is relatedto slightly higher levels of achievement, especially in reading, even after con-trolling for student background characteristics. However, the relationship be-tween voucher use and achievement outcomes is considerably weaker thanthe relationship between achievement outcomes and the factors determiningvoucher use in the first place.

Private School Choice: What Does the Literature Say?

There is a well-developed literature on school choice determinants, includingchoice in a voucher system. Some of this scholarship is located in evaluationsof voucher effects on student achievement (e.g., Howell et al. 2002; Witte2000; Witte et al. 2009; Wolf et al. 2009), while other studies are more targetedefforts to study differences between private and public school attendees (Bettsand Fairlie 2001; Figlio and Stone 2001; Howell 2004; Long and Toma 1988).There is evidence that white students are far more likely to actually attendprivate school than are African American or Hispanic students (Betts andFairlie 2001; Long and Toma 1988), a condition that may exacerbate schoolsegregation (Lankford and Wyckoff 2001). Further, students whose parents arebetter educated or have higher incomes are more likely to choose privateschools (Betts and Fairlie 2001; Figlio and Stone 2001; Lankford and Wyckoff1992; Long and Toma 1988), particularly in areas where there are fewer





public school options or where learning conditions, such as student-teacherratios, are worse (Figlio and Stone 2001). Similar factors have been identifiedin studies of specific voucher programs (Campbell et al. 2005; Howell et al.2002; Witte 2000; Witte and Thorn 1996), with minority families less likelyto attend given the chance (Campbell et al. 2005) and those with less residentialstability less likely to remain in such programs even after attending (Howell2004). As many private schools have some religious affiliation, family religiouscharacteristics, especially church attendance and Catholicism, have also beenshown to be important determinants of private school choice (Campbell etal. 2005; Cohen-Zada and Justman 2005; Howell et al. 2002; Long and Toma1988).

These studies are important and should be considered part of the centralscholarship on the relationship between private schooling and educationaldifferences between various groups of American schoolchildren. They largelyprovide compelling evidence for the characteristics of students who use privateeducation. What most lack, however, is a focus on a subgroup of students whoare critical to the debate over whether private schooling can ultimately beoffered as a policy alternative to mediate some of those educational differences.In the next section, I introduce this subgroup and explain its importance.

The Importance of Voucher Decliners

In theory, the analytical appeal of lottery-based voucher programs is straight-forward. Such programs represent natural randomized field trials, in whichthe treatment (the offer of the voucher) is administered on a random basis toa group of applicants. The effect of other characteristics on the outcome ofinterest (typically test scores) is balanced in expectation between the treatmentand control groups, yielding an unbiased estimate of the difference in outcomesbetween the two groups. In practice, however, families and students are freeto decline the offer, as noted above, and estimates of the difference betweenstudents who use the scholarship and those to whom it was not offered arelikely biased. In much of the voucher literature, the focus on obtaining un-biased estimates of these outcome differences often obscures—or at least con-signs to the mundane category of “descriptive statistics”—a basic set of em-pirical information that can predict who is most likely to benefit from a formalschool choice program.

Major studies involving lottery-based distribution of the voucher offer (esp.Campbell et al. 2005; Howell et al. 2002; Wolf et al. 2005) have examinedin detail the differences between families who accepted the offer and thosewho turned it down. Such evidence suggests that, depending on the field sitein question, factors such as race, mother’s education, religious observance,



Cowen

NOVEMBER 2010 5

and public welfare receipt are among those related to accepting the scholarship.Several observational studies have found these characteristics among thosethat influence students to seek out nonlottery scholarships as well (e.g., Figlio2008; Witte 2000; Witte and Thorn 1996; Witte et al. 2008).

When the outcome of interest is some measure of student achievement,lottery designs have several appealing attributes with respect to the preser-vation of internal validity. When the scholarship offer is accepted in the firstplace, such studies are valuable because the key benefit, randomization, ex-tends to the offer but not to acceptance, while at the same time a group ofsubjects who initially expressed interest in the scholarship is observed. In otherdesigns, researchers attempt to control for unobserved differences influencingthe decision to simultaneously seek and use a voucher. In lottery designs, aparticularly interesting group of students, from the standpoint of educationalchoice, is uniquely visible. These are students who share with all study par-ticipants an interest in applying for the scholarship (i.e., interest in seeking avoucher) and have the same externally derived opportunity to attend privateschool as do scholarship users but may possess their own unique attributesthat influence the decision not to choose private education in a variety ofways when private education is, at least theoretically, a viable choice.

This decision to reject the voucher offer may be deliberate (i.e., certainparents simply decide not to accept) or indeliberate to the extent that certainschools may reject the application of some voucher winners for admission. Ineither case, if the relationship between voucher acceptance and student char-acteristics is at all systematic, we should be able to see them borne out asdifferences between those who accept the scholarship and those who do not.If scholarship users allow analysts to determine who chooses to leave publiceducation when given the chance, the decliners provide evidence for thosewho will, will not, or still cannot leave the public sector. More generally, it iscritical for policy makers to determine the sorts of students for whom suchan intervention is most promising. To develop this argument in the sectionsthat follow, I analyze a small set of data from a scholarship lottery in Charlottethat occurred prior to the 1999–2000 academic year.

Data: A Lottery of Private School Scholarships in Charlotte

Greene (2000, 2001) detailed the general data collection process for the Char-lotte lottery, and Cowen (2008) verified that no major systematic differencesappeared between the winning and losing groups of students, even among thesubsample who participated in the study at year’s end. The final sample,which includes posttest, demographic, and survey data, contained 347 unique





student observations across voucher-accepting, voucher-rejecting, and lottery-losing students (see the appendix for more detailed descriptive statistics).

No pretest was administered before the vouchers were distributed, andalthough the lack of a pretest does not prevent an unbiased estimation ofprogram effects in a randomized design, it does mean that the use of thepretest as a covariate to improve precision of the estimate is not possible. Onthe other hand, the sample of Charlotte data is small and took place in asingle setting. These features may limit generalizability even without the pretestcovariate, which itself may impose further limitations. These aside, it is im-portant to stress that data from the larger voucher evaluations cited here arenot currently available to most public researchers. The Charlotte data, whileof limited use as evidence for whether vouchers can “work” on a larger scaleor over time, are nevertheless valuable. The data represent a group of students,several of whom were randomly offered a school voucher that some acceptedand some rejected. Such value meets the purposes of this article, which con-cerns voucher research more than programmatic effects.

Results

In this section I discuss the characteristics of voucher users and decliners andthe implications of their differences for student outcomes. In part A, I examinethe characteristics identified in the literature discussed above as importantdeterminants of private school choice in general: race, religion, gender, familystability and income, and mother’s education and employment history. I con-clude this initial examination by estimating a model of voucher acceptanceconditional on the impact of these characteristics. In part B, I expand thefocus to consider how the relationship between student background and thedecision to accept a voucher concerns student outcomes.

A. The Characteristics of Voucher Users and Decliners

Basic differences between scholarship users and decliners.—The 54 study participantswho won the lottery in Charlotte but declined to accept the scholarship weredifferent, in several important ways, from those who accepted the scholarship.Table 1 reports these differences, of which there are three basic characteri-zations: family background, school access, and mother’s employment.

1. Family background.—The first difference concerns family background. Stu-dents who declined the offer were considerably more likely to be AfricanAmerican (defined here as the race of the mother), despite the fact that studyparticipants in the sample were overwhelmingly African American. They were



Cowen

NOVEMBER 2010 7

TABLE 1

Differences between Lottery Winners Accepting and Declining the Scholarship

Decline Acceptp

(Difference)

Background:African American mother .93 .77 .01Male .44 .47 .70Two parents at home .15 .41 .00Parents’ marital status:

Never married .37 .24 .06Married .15 .42 .00Divorced/separated .44 .25 .01Widowed .04 .07 .32No response (married) .00 .01 .41

Baptist .52 .32 .01Family income:

!$20,000 .31 .28 .68$20,000–$40,000 .59 .61 .781$40,000 .09 .10 .85

Mother’s education:High school dropout .07 .03 .18High school graduate .31 .23 .20Some college .31 .50 .02College graduate .30 .24 .42

Mother’s employment:Full-time .80 .54 .00Part-time .04 .16 .02Looking for work .11 .08 .42Not working .00 .22 .00Other employment .06 .01 .02

School access:Admitted to school of choice .50 .87 .00Could not afford school of choice .31 .09 .00

N 54 158

NOTE.—Significance tests are difference-of-proportions tests based on approximatenormal distribution. Reported p-values are based on two-tailed tests where the nullhypothesis is that the difference between the proportion of nonusers with a givencharacteristic and users with that characteristic is zero.

also far more likely to be Baptist—50 percent of decliners identified themselveswith that denomination—than were voucher users or study participants ingeneral. In addition, the decliners’ home environment can be considered morefractured than that of scholarship users. Decliners were far more likely to comefrom homes in which only one parent lived or where parents were divorcedor never married in the first place. Correspondingly, more than 40 percentof students who accepted the voucher came from families in which parents





TABLE 2

Mother’s Employment Status and Family Structure for Scholarship Users and Decliners

Decline Acceptp

(Difference)

Full-time and two parents .11 .13 .76Full-time and one parent .69 .41 .00Part-time and two parents .00 .10 .02Part-time and one parent .04 .06 .47Not working and two parents .04 .18 .01Not working and one parent .07 .11 .41

NOTE.—Significance tests are difference-of-proportions tests based on approximatenormal distribution. Reported p-values are based on two-tailed tests where the nullhypothesis is that the difference between the proportion of nonusers with a givencharacteristic and users with that characteristic is zero.

were married and living at home.1 There do not appear to be any majordifferences associated with income, which is unsurprising given that partici-pation in the lottery in the first place was conditional on a means test (seeGreene 2001).2

2. School access.—Only half of decliners were admitted to their chosen schools,while nearly 90 percent of voucher users were accepted. Although the “glasshalf full” conclusion may be that half of the decliners were admitted (i.e.,admission itself cannot explain the declined offer for half of the students),clearly admission did play a role for some students. Moreover, while nearlya third of decliners indicated that, despite the scholarship offer, they couldnot afford tuition at their preferred school, less than 10 percent of voucherusers said the same.

3. Mother’s employment.—The third difference between those who acceptedand those who declined a voucher offer is related to the mother’s employment.Although income levels were nearly identical between both groups, the sourcesof income varied considerably. Perhaps as a direct consequence of familystructure, decliners were far more likely to come from homes in which themother was employed full-time. Voucher users were somewhat more likely tohave mothers employed part-time, and both groups had similar proportionsof mothers looking for work. More than 20 percent of mothers whose childrenaccepted the voucher did not work at all. The major source of difference inemployment is between full and no employment whatsoever.

The relationship between employment and family structure as it differs withrejecting the scholarship offer is particularly worth noting. Table 2 providesseparate cross-tabulations of employment and family structure status for schol-arship users and decliners. To fix the analysis more concretely on the interplaybetween one type of home environment and another, I collapsed the marital



Cowen

NOVEMBER 2010 9

categories into an indicator for whether the student had both parents livingat home, to reflect not only marriage but also marital stability.3

The table indicates that students who declined the voucher were dispro-portionately more likely to come from single-parent homes where the motherworked full-time; students who accepted the offer were more likely to comefrom homes with two parents where the mother either worked only part-timeor was unemployed. Taken with the general finding above that scholarshipusers were far more likely to come from married or two-parent homes, thesestatistics suggest that voucher users—despite the lack of apparent incomedifferences between the groups—may come disproportionately from familiesmore willing or able to make use of the voucher through a commitment ofother resources, such as one parent’s time.4 It also may also reflect the im-portance of family stability in promoting alternative educational choices.

Predicting scholarship use.—Table 2 provides initial indications that student andfamily characteristics may operate together to determine students’ scholarshipacceptance. To develop this pattern, I specify a simple model that conditionsvoucher acceptance on X, a vector of those previously discussed characteristicsidentified in the literature as important determinants of private school atten-dance:

1P (acceptFoffer) p . (1)

�Xb1 � e

Table 3 reports the results of estimates of this logit model, as well as oddsratios based on the estimated coefficients. Because some proportions of de-cliners were tiny or nonexistent (e.g., the “not working” category had nodecliners) in some categories, I consolidated the employment variables tocompare the two largest categories (full-time and part-time) against the ref-erence categories (not working and single-parent families). Finally, I estimatetwo models: one with an indicator variable and one without, to show whetherstudents were actually admitted. If the relationship between demographiccharacteristics and voucher acceptance changes considerably between thesetwo specifications, it is evidence that whether the student is admitted maydrive some of the relationship between other factors and acceptance. Forexample, if for whatever reason schools are more likely to admit students withtwo parents at home, a model without the admission indicator may incorrectlyattribute family stability to a direct decision to accept a voucher offer.

The admission and affordability indicators provide additional leverage fora set of variables absent from the model: namely, those directly measuring theunobserved ways in which individual private schools select their students. Forstudents who seek to use the voucher to attend a particular private school (asopposed to simply a private option), the inability to gain admission or to affordthe school with the voucher should influence the students’ decisions to accept.



TA

BL

E3

Pre

dict

ing

Vou

cher

Use

amon

gL

otte

ryW

inne

rs

WIT

HA

DM

ISS

ION

IND

ICA

TO

RV

AR

IAB

LE

WIT

HO

UT

AD

MIS

SIO

NIN

DIC

AT

OR

VA

RIA

BL

E

VA

RIA

BL

EE

stim

ate

(SE

)O

dds

Rat

ioE

stim

ate

(SE

)O

dds

Rat

io

Bac

kgro

und:

Afr

ican

Am

eric

anm

othe

r�

1.06

(.66)

�.3

5�

1.14

(.70)

�.3

2M

ale

.12

(.39)

1.12

�.0

3(.4

1).9

7T

wo

pare

nts

atho

me

1.41

(.56)

**4.

081.

43(.5

8)**

4.19

Bap

tist

�.8

4(.3

8)*

.43

�.9

2(.4

0)*

.40

Fam

ilyin

com

e:!

$20,

000

.50

(.49)

1.64

.47

(.51)

1.59

1$4

0,00

0�

.59

(.69)

.56

�.7

3(.7

2).4

8M

othe

r’s

educ

atio

n:H

igh

scho

oldr

opou

t�

1.33

(.89)

a.2

6�

1.67

(.91)

�.1

9So

me

colle

ge1.

04(.4

9)*

2.82

1.13

(.52)

*3.

08C

olle

gegr

adua

te.2

1(.5

5)1.

23.0

6(.5

7)1.

07M

othe

r’s

empl

oym

ent:

Full-

time

�.6

9(.5

1).5

0�

.53

(.53)

.59

Part

-tim

e.9

0(.9

0)2.

44.7

8(.9

2)2.

18Sc

hool

acce

ss:

Adm

itted

tosc

hool

ofch

oice

1.89

(.54)

**6.

62C

ould

not

affo

rdsc

hool

ofch

oice

�2.

21(.5

3)**

.11

�.9

4(.6

6).3

9N

212

212

Chi

-squ

are

58.4

8**

71.4

7**

Tot

al%

corr

ectly

clas

sified

80.6

682

.55

Scho

lars

hip

use

93.6

794

.30

Scho

lars

hip

refu

sal

42.5

948

.15

NO

TE

.—E

stim

ates

are

logi

tco

effic

ient

spr

edic

ting

clas

sm

embe

rshi

p.a p

!.1

5.�

p!

.10.

*p

!.0

5.**

p!

.01.



Cowen

NOVEMBER 2010 11

The inability to measure school-specific factors that determine admission is alimitation in most voucher research but is mitigated by the fact that, as notedearlier, research on the determinants of school choice demonstrates systematicrelationships between student characteristics and private school enrollment.In the model estimated in table 3, student characteristics include race, gender,religion, family structure (whether two parents live at home), income, andmother’s education and employment status. Although we cannot measureschool selection criteria (such as a student’s application packet or an interviewresult), it seems unlikely that these criteria are entirely unrelated to factorsaccounted for by these student-level (and, critically, specific combinations ofthese student-level) characteristics. More important, the school selection cri-teria are implicitly captured by the admission and affordability indicators thataccount for any remaining particular school characteristics that may relateschool access to the decision to accept the voucher offer.

That many of the estimates in table 3 are statistically insignificant at con-ventional levels is likely an indication of collinearity between several of theexplanatory variables, which underscores the difficulty of separating each re-lationship with the decision to choose from the other. This caveat notwith-standing, the results in table 3 indicate that students with two parents at homewere more likely to accept the scholarship and that African American studentswere less likely, although the latter result is only significant at p ! .10. Comparedwith students whose mother’s highest level of education was a high schooldiploma, students whose mothers were dropouts were less likely to accept thevoucher. Students whose mothers had some college were considerably morelikely. Baptist students were far less likely to accept the offer than were non-Baptists, suggesting a strong religious element to choice within the sample.These results are nearly identical between the specifications with and withoutthe admission variable. The exception, at least as far as statistical significanceis concerned, is the indicator for inability to afford the school (even after thevoucher). Unsurprisingly, in both models, parents who say they are unable toafford tuition at their preferred school are far more likely to reject the voucheroffer than not to reject it. In the model with the admission indicator, the no-afford indicator has a two-sided p-value of .16. This is evidence that theadmission and affordability indicators may capture some similar dynamic.5

Taken together, the results in tables 1–3 provide compelling evidence thateven among families who may be classified as “at risk,” such as those partic-ipating in the Charlotte study (urban, largely minority, high rates of singleparenthood, low income), there are distinctive groups who may be far morelikely to accept so dramatic an educational intervention as a relocation betweenschool sectors. This despite the fact that the distinctive groups in this caseshared at least some basic interest in switching schools, insofar as interest canbe measured by participation in the scholarship lottery and subsequent study.





Perhaps most important, these results provide some evidence that even aftercontrolling for school factors such as selectivity and cost, student backgroundcharacteristics still predict voucher acceptance. This evidence suggests thatthese characteristics may yet pose barriers to school choice beyond the simplequestion of school access.

B. Scholarship Rejection and Student Outcomes

As discussed above, the overall focus of most of the leading voucher researchconcerns the identification of voucher effects on average levels of studentachievement. The small sample of study participants in Charlotte has beenanalyzed independently in two studies (Cowen 2008; Greene 2000), whichgenerally found that scholarship users had slightly higher math and readingtest scores at the end of the academic year, although the more recent of thetwo studies cautioned that the data themselves were “hardly ideal” as moregeneralizable evidence of net gains for voucher students (Cowen 2008, 313).Both of these studies and studies concerning other lottery-based analyses ofvoucher effects included intention-to-treat (ITT) and instrumental variables(IV) analyses as separate attempts to account for nonrandom selection withinthe group of lottery winners. The IV approach in particular makes use of thelottery as an appealing instrument that is related to voucher acceptance be-cause students cannot accept an unoffered scholarship; but because it is ran-dom, it is unrelated to the outcome. As such, the IV approach can provideunbiased estimates of the effect of vouchers on students who use them, evenin the presence of selection bias caused by just the sort of refusal to acceptthe scholarship offer described above.

The focus in this article is not on programmatic scholarship effects onstudent achievement but on the more general characteristics of the nonusersthemselves. However, achievement itself is among those characteristics, andpotential achievement differences are emphasized in the broader voucherliterature. As such, achievement is worth attention here, particularly in thecontext of the relationship between the same qualities that determine voucheracceptance and student outcomes. I stipulate that the limited evidence fromthe Charlotte program indicates a small, positive effect for voucher users (seethe appendix), although standard errors of the estimates of this effect aresensitive to several different model specifications, including the use of covar-iates in outcome models (Cowen 2008). In the context of scholarship decliners,the more interesting question concerns not overall average differences inachievement but rather differences in the distribution of achievement betweenthat group and those who accept the offer.

Some scholars consider school vouchers to be among the potential remedies



Cowen

NOVEMBER 2010 13

for persistent achievement “gaps” between higher-income (often white) stu-dents and lower-income (typically minority) students. If choice is among thesolutions to this problem, however, it is reasonable to examine whether theentire distribution of poor-performing students shifts forward along with anaverage change or whether the average change is caused simply by a dispro-portionate number of high or low scores among either the choice or publicschool students. In the latter case, if the differences are at all related to someof the same characteristics determining who chooses the private option whengiven the chance—the characteristics examined above, for example—re-searchers may gain some analytical leverage as they speculate on the potentialfor widespread choice as a policy option.

Such an exercise necessitates a counterfactual scenario. As I have shown,among students who won the voucher lottery, those who had certain char-acteristics (e.g., had two parents at home, had mothers who were not workingfull-time, were not African American) were considerably more likely to acceptthe voucher offer. How might differences in relevant student outcomes appearfor all students of similar characteristics, including those who lost the lotterybut would have accepted (or refused) the offer given the chance? To addressthis question, I employ a latent class approach, in which student membershipin one of two classes is observed among lottery winners but is unobserved—latent—among lottery losers.

In other studies, including at least three dealing with education interventions(Barnard et al. 2002; Borman and Dowling 2006; Cowen 2008)—one of whichconcerned the Charlotte lottery—the object was to identify and estimate aparticular evaluation parameter, the “complier average causal effect” (CACE),which is the average difference in outcomes between those who would acceptthe randomized treatment (compliers, or in the present context, voucher users)when offered and those who would not (noncompliers, or in the presentcontext, nonusers).6 This approach considers the potential outcomes of studyparticipants across all treatment conditions and rests on a series of theoreticalassumptions. Random assignment is critical among these assumptions. See theappendix for a brief discussion of the other underlying assumptions. AlthoughI employ precisely the same estimation strategy used in such CACE modelingin this article, the focus is less on the estimated value of the CACE parameter(i.e., the estimated voucher effect referenced below) and more on the differ-ences in the variation of the outcome between each latent class.

The basic approach, following Borman and Dowling (2006), Jo (2002), andJo et al. (2008), begins with a latent variable mixture model:

Y p a d � a c � g c Z � l d X � l c X � e � e , (2)i d i c i c i i d i i c i i di ci

where Yi is the outcome of interest (here, student achievement); is the averagegc

effect of the voucher for all compliers (those who would accept); and ld andlc are the different effects of covariates for decliners and compliers, which I





constrained in this model to have similar effects on the outcome. The prob-ability of compliance—using the scholarship when offered—is predicted vialogistic regression:

P (c p 1FX ) p pi i ci

P (c p 0FX ) p 1 � p p p (3)i i ci di

logit(p ) p b p b X ,ci 0 1 i

where b0 is the logit intercept and b1 represents the effect of each covariateon compliance (accepting the scholarship). The model is estimated with max-imum likelihood via the expectation-maximation (EM) algorithm, where inthe E step the probability of compliance conditional on the X covariates iscomputed, given the observed data and parameters, which correspond to thosein the same general model whose estimates are reported in table 3 for lotterywinners only. In the M step, new parameters relating acceptance of the schol-arship and the other covariates to the outcome are generated. The softwareMplus 5.0 (Muthen and Muthen 2007) is used for these procedures to analyzethe data.

Table 4 provides basic model estimates for math and reading and for thelatent class predictions.7 Estimates of the achievement model indicate thatpositive user and nonuser distinctions still remain, although neither are sig-nificant at traditional levels. Covariate specifications follow Greene (2000,2001) and Cowen (2008). It is also important to note that estimates of thelatent class model are very similar to those reported in table 3, where thesample was limited to lottery winners. This is as it should be because, asanticipated, it is precisely the relationship between these variables and thedecision to accept the voucher in the treatment group that is used to generateprobabilities for those in the control group. Mother’s employment, race, thepresence of two parents at home, and mother’s education levels are all pre-dictors of user class status. The table also indicates average latent class prob-abilities, which are greater than 0.90 for both classes. These indicate that themodels provide far better predictions of class membership than, for example,randomly assigning control group members to the scholarship user class basedon the same proportion observed to be users in the treatment group (0.75).

As part of this estimation, observations are assigned to their “most likely”class. For lottery winners, this is known with a probability of 1.0 because theircompliance class is observed. All descriptive statistics are weighted by theprobability of class membership, so it is possible to analyze the groups notonly on the basis of an overall difference in potential outcomes (i.e., CACE)for those who would and would not accept the scholarship but also withattention to other features such as the range and variance of the outcome.Table 5 reports these weighted statistics for both potential users and nonusers



TABLE 4

CACE and Latent Class Model Results

Variable Math Estimate (SE) Reading Estimate (SE)

Outcome model:Choice 4.58 (3.49) 5.86 (3.72)a

African American mother �12.74 (4.03)** �10.55 (3.93)**Male �.13 (2.66) �6.35 (2.71)*Mother works full-time .93 (3.96) �5.85 (3.65)a

Mother works part-time �2.72 (4.46) �6.52 (4.48)Two parents at home 4.72 (3.28)a 11.44 (3.65)**Baptist .64 (2.78) �.21 (2.78)Income !$20,000 �6.14 (3.34)� �2.19 (3.58)Income 1$40,000 1.04 (4.72) .49 (4.29)Mother is high school dropout �12.71 (4.96)** �21.68 (5.06)**Mother has some college education 2.81 (3.04) .19 (3.21)Mother is college graduate 10.01 (3.92)** 14.20 (3.6)**Admitted to school of choice 1.82 (3.93) �2.20 (4.77)Could not afford school of choice 5.19 (4.07) �3.34 (4.95)Intercept 36.02 (6.27)** 47.64 (7.01)**

Latent class model:African American mother �1.10 (.60)� �1.05 (.59)�

Male �.12 (.43) �.12 (.42)Mother works full-time �.52 (.51) �.51 (.52)Mother works part-time .73 (.96) .71 (.96)Two parents at home 1.54 (.68)* 1.52 (.67)*Baptist �.93 (.40)* �.95 (.40)*Income !$20,000 .50 (.56) .50 (.55)Income 1$40,000 �.65 (.79) �.80 (.76)Mother is high school dropout �1.78 (.89)* �1.54 (.92)�

Mother has some college education 1.14 (.58)* 1.18 (.55)*Mother is college graduate .07 (.57) .13 (.55)Admitted to school of choice 1.94 (.53)** 1.83 (.55)**Could not afford school of choice �.92 (.68) �1.10 (.67)a

N 347 347Average class probability (user class) .94 .94Average class probability (nonclass) .92 .91

NOTE.—CACE p complier average causal effect. Estimates for latent class modelare logit coefficients predicting class membership.

ap ! .15.�p ! .10.* p ! .05.** p ! .01.





TABLE 5

Estimated Sample Statistics by Scholarship User/Nonuser Latent Class

Nonuser User

Math (SD) 25.51 (21.67) 35.20 (27.61)Reading (SD) 30.87 (23.83) 40.62 (29.31)African American mother .91 .77Male .49 .43Mother works full-time .80 .57Mother works part-time .04 .18Two parents at home .15 .42Baptist .56 .33Income !$20,000 .33 .26Income 1$40,000 .07 .13Mother is high school dropout .13 .03Mother has some college education .28 .50Mother is college graduate .25 .22Admitted to school of choice .43 .86Could not afford school of choice .39 .08

NOTE.—Statistics are weighted by probability of class membership. All figuresare weighted proportions except math and reading, where weighted means arereported.

for both reading and math. These results have similar substantive implicationsto their raw measures in comparisons between the observed users and nonusersin the treatment group. The standard deviations for both math and readingoutcomes for the user (complier) class are higher, and—as the kernel densityestimates in figures 1 and 2 indicate—there are comparably fewer noncom-plying students at the high end of the achievement distribution. This suggeststhat overall differences in mean achievement outcomes may tell only part ofthe story.

Discussion

The focus of this article has examined the differences between students whoaccepted a private school scholarship when offered it and those who refusedthe opportunity. I have argued that lottery-based research designs that ran-domly assign students to receive or not receive an offer are particularly usefulin applying such a focus because they are the only contexts under whichanalysts can guarantee that a scholarship has actually been refused. In a strictlyobservational setting, all nonusers may in some sense have “declined” theopportunity, notably in settings where publicly distributed private school op-tions are widely available, such as in Milwaukee (Witte et al. 2008) and Florida



Cowen

NOVEMBER 2010 17

FIG. 1.—Kernel density estimates for math

(Figlio 2008). However, it is difficult to determine the degree to which infor-mation about these programs exists in such settings. Some research indicatesthat many public school parents have simply not heard of the private schooloptions for which their children may qualify (Witte et al. 2008). In the lotterydesigns, the differences between those who subsequently decline the offer maybe attributed to characteristics not confounded by their relationship with otherpolicy-relevant factors because parents have volunteered to participate andtry their chances for a scholarship.

In the scholarship program under study here, students in Charlotte whoentered the lottery and won were less likely to use their scholarship if theyexhibited one or a combination of personal and family attributes. Althoughthe Charlotte data are not ideal, for several reasons mentioned earlier, theevidence drawn from their examination may nevertheless suggest more gen-eralized relationships that have been and should continue to be a focal pointof ongoing research and debate. Unfortunately, data from full-scale voucherfield trials are not widely available to most researchers, so evidence such asthat from Charlotte can only add to these efforts in a limited way.

In this case, a student whose parents did not live together, whose motherwas fully employed or had lower levels of education, who was African Amer-ican, who held a specific religious affiliation, who was not accepted at thedesired school, or who could not afford tuition there (net of the scholarship)was less likely to make the private school choice. These characteristics are





FIG. 2.—Kernel density estimates for reading

similar to those identified in previous scholarship, but I argue that they areoften considered in this literature to be preliminary components of the analysisof voucher effects on the outcome, rather than an outcome of interest in theirown right.

When I employed these characteristics in a statistical model predicting theprobability of scholarship acceptance among lottery losers to estimate potentialoutcomes conditional on user/nonuser classifications, I predicted average dif-ferences in student achievement to be higher for users than for nonusers.However, the difference attributable to the scholarship alone was smaller thanthose generated by more traditional estimates of the treatment effect. More-over, the average differences alone do not tell the complete story. In the caseof the Charlotte scholarships, the distribution of outcomes was wider for usersthan for nonusers, with perhaps a slightly larger grouping of test scores at thehigh end of the distribution. The potential for such an effect on the variationin outcomes is understudied in the voucher-evaluation literature.

Researchers need not be familiar with the entire body of scholarship oneducational outcomes to understand intuitively that students performingpoorly, irrespective of school sector, may exhibit several of the characteristicsidentified above as determinants of scholarship acceptance in the first place.If such characteristics condition, or even partly condition, acceptance of anintervention like the movement to a private school, this would suggest thatvouchers might be targeted beyond a simple test of economic means if they



Cowen

NOVEMBER 2010 19

are to reach a population of students who are most at risk. In such a framework,the size of the voucher may be adjusted according to not only tuition atpreferred schools to which students gain admission but also costs associatedwith transportation and other barriers to accepting the scholarship.

Considerable and commendable research has stressed average differencesin student achievement as the key outcome of interest in the majority of schoolvoucher studies and, indeed, in the majority of school choice studies as awhole. Most of this work considers, at least in an ancillary fashion, several ofthe questions I have raised here. But determining who will accept the op-portunity to change educational environments if given the chance is not simplyone step among many in estimating the valid effects of these programs. Thedistinction is critical: those families who will not or cannot choose an alter-native educational sector over public schooling are by definition those forwhom choice programs are not a realistic solution to persistent educationalinequality.

Appendix

Identifying Assumptions for CACE Estimation

The complier average causal effect (CACE) parameter, known as the localaverage treatment effect (LATE) in the econometric field, is identified undera general set of assumptions. See Angrist et al. (1996), Imbens and Angrist(1994), Imbens and Rubin (1997), Jo (2002), Jo and Muthen (2001), Little andYau (1998), and Yau and Little (2001) for an extended discussion. In brief,the assumptions are as follows:

1. Randomization: observations are assigned a treatment condition atrandom.

2. Stable unit treatment value assumption: the potential outcome for anyobservation under study is unrelated to the treatment condition of theother observations.

3. Exclusion restriction: there is no effect of the treatment assignment onnoncompliers. In the case of the scholarship offer, this restricts thedefinition of the treatment to be the use of the scholarship to attendprivate school.

4. Monotonicity: there are no treatment “defiers” under study. In this case,there are no observations of who would attend private school only afterlosing the random lottery but would attend public school only afterwinning the lottery.



TABLE A1

Intention-to-Treat and Instrumental Variables Estimates of Average Scholarship Effects

ITT IV

Math (SE) Reading (SE) Math (SE) Reading (SE)

Scholarship 4.71 (2.76)� 5.66 (2.84)* 6.45 (3.72)� 7.75 (3.83)*Intercept 36.45 (4.76)** 39.32 (4.90)** 36.71 (4.65)** 39.64 (4.79)**R2 .15 .21 .15 .21N 347 347 347 347

NOTE.—Models include the same covariates as specified in table 4. Estimates of bfor these covariates as well as standard errors are reported for comparison to table 4and Greene (2000, 2001) and Cowen (2008). Intention-to-treat (ITT) estimates are theaverage differences between those assigned the scholarship offer and those in the controlgroup. Instrumental variables (IV) estimates use this lottery assignment as the instrumentfor choosing to accept the offer.

�p ! .10.* p ! .05.** p ! .01.

TABLE A2

Differences between Lottery Winners and Losers

Losers Winnersp

(Difference)

African American mother .80 .81 .79Male .41 .47 .34Two parents at home .33 .34 .79Parents’ marital status:

Never married .30 .27 .65Married .35 .35 .90Divorced/separated .26 .30 .39Widowed .06 .06 .80No response (married) .00 .01 .26

Baptist .43 .37 .29Family income:

!$20,000 .26 .29 .50$20,000–$40,000 .61 .61 .911$40,000 .13 .10 .43

Mother’s education:High school dropout .08 .04 .13High school graduate .32 .25 .16Some college .41 .45 .49College graduate .19 .25 .13



Cowen

NOVEMBER 2010 21

TABLE A2 (Continued)

Losers Winnersp

(Difference)

Mother’s employment:Full-time .69 .60 .11Part-time .14 .13 .82Looking for work .09 .08 .90Not working .08 .16 .03Other employment .00 .02 .11On public assistance .26 .31 .30

N 135 212

NOTE.—Significance tests are difference-of-proportions tests based on approximatenormal distribution. Reported p-values are based on two-tailed tests where the nullhypothesis is that the difference between the proportion of losers with a given char-acteristic and winners with that characteristic is zero.

Notes

1. The surveys do not indicate whether both parents living at home were biological.2. Although mother’s education, race, and family stability have been shown in the

literature described above to be important predictors of school choice, one importantlimitation of the Charlotte data is that an additional determinant—whether a studenthas a sibling in a private school—is unobserved here. Part of the difficulty is that thevoucher was distributed at random (i.e., not conditional on sibling status), and the dataare linked to students, not families. Therefore, sibling lottery outcomes are not acomponent of family background and are unavailable in the data.

3. Seven of the 122 students with parents reported as married are classified as havingonly one parent in the home. All of the students reporting two parents at home haveparents reported as married. Thus the “two parents” indicator refines the marriagemeasure to indicate a traditional two-parent home. I thank one of the anonymousreviewers for this suggestion.

4. Although the data do not include information on transportation costs and vol-unteer activities, previous research on public/private differences has shown that privateschool parents are largely left to themselves to transport their child to and from schooland are more likely to be asked or required to participate in school activities. If suchwas the case for private schools in the Charlotte lottery, it may explain why acceptingstudents were more likely to have both parents at home and have mothers who hadto work less (and thus more time to transport and volunteer).

5. Unreported interaction models indicate no differential effect of admission oraffordability by the significant demographic characteristics reported in table 3.

6. For extended development of the principles of identification and estimation ofthis parameter, see Angrist et al. (1996), Imbens and Angrist (1994), Imbens and Rubin(1997), Jo (2002), Jo and Muthen (2001), Little and Yau (1998), and Yau and Little(2001). In the econometric literature, CACE is known as the “local average treatment





effect” and, given the assumptions discussed in the literature cited here, is the parameterestimated in the IV approach noted in the text.

7. Outcomes are the results of the Iowa Test of Basic Skills examinations admin-istered to students in the spring of their scholarship year (1999–2000). See Greene(2000, 2001) for details of the testing.

References

Angrist, Joshua D., Guido W. Imbens, and Donald B. Rubin. 1996. “Identification ofCausal Effects Using Instrumental Variables.” Journal of the American Statistical Asso-ciation 91 (434): 444–45.

Barnard, John, Constantine E. Frangakis, Jennifer L. Hill, and Donald B. Rubin. 2003.“Principal Stratification Approach to Broken Randomized Experiments: A CaseStudy of School Choice Vouchers in New York City.” Journal of the American StatisticalAssociation 98 (462): 299–311.

Betts, Julian, and Robert W. Fairlie. 2001. “Explaining Ethnic, Racial and ImmigrantDifferences in Private School Attendance.” Journal of Urban Economics 50 (1): 26–51.

Borman, Geoffrey D., and N. Maritza Dowling. 2006. “Longitudinal AchievementEffects of Multiyear Summer School: Evidence from the Teach Baltimore Random-ized Field Trial.” Educational Evaluation and Policy Analysis 28 (1): 25–48.

Campbell, David E., Martin R. West, and Paul E. Peterson. 2005. “Participation ina National, Means-Tested Voucher Program.” Journal of Policy Analysis and Management24 (3): 523–41.

Cohen-Zada, Danny, and Moshe Justman. 2005. “The Religious Factor in PrivateEducation.” Journal of Urban Economics 57:391–418.

Cowen, Joshua M. 2008. “School Choice as a Latent Variable: Estimating the ‘Com-plier Average Causal Effect’ of Vouchers in Charlotte.” Policy Studies Journal 36 (2):301–15.

Figlio, David N. 2008. “Evaluation of Florida’s Corporate Tax Credit ScholarshipProgram: Baseline Report, Compliance and Test Scores in 2006–07.” Departmentof Economics, University of Florida, and National Bureau of Economic Research,Cambridge, MA, http://www.floridaschoolchoice.org/information/ctc/files/CTC_Baseline_Report.pdf.

Figlio, David N., and Joe A. Stone. 2001. “Can Public Policy Affect Private SchoolCream-Skimming?” Journal of Urban Economics 49:240–66.

Greene, Jay P. 2000. “The Effect of School Choice: An Evaluation of the CharlotteChildren’s Scholarship Fund.” Civic Report 12, Manhattan Institute, New York.

Greene, Jay P. 2001. “The Effect of School Choice: An Evaluation of the CharlotteChildren’s Scholarship Program.” Education Next 1 (2): 55–60.

Howell, William G. 2004. “Dynamic Selection Effects in Means-Tested, Urban SchoolVoucher Programs.” Journal of Policy Analysis and Management 23 (2): 225–50.

Howell, William G., Paul E. Peterson, Patrick J. Wolf, and David E. Campbell. 2002.The Education Gap: Vouchers and Urban Schools. Washington, DC: Brookings InstitutionPress.

Imbens, Guido, and Joshua D. Angrist. 1994. “Identification and Estimation of LocalAverage Treatment Effects.” Econometrica 62 (2): 467–75.

Imbens, Guido W., and Donald B. Rubin. 1997. “Bayesian Inference for Causal Effectsin Randomized Experiments with Noncompliance.” Annals of Statistics 25 (1): 305–27.



Cowen

NOVEMBER 2010 23

Jo, Booil. 2002. “Model Misspecification Sensitivity Analysis in Estimating CausalEffects of Interventions with Non-compliance.” Statistics in Medicine 21:3161–81.

Jo, Booil, Tihomir Asparouhov, Bengt O. Muthen, Nicholas S. Ialongo, and C. Hen-dricks Brown. 2008. “Cluster Randomized Trials with Treatment Noncompliance.”Psychological Methods 13 (1): 1–18.

Jo, Booil, and Bengt O. Muthen. 2001. “Modeling of Intervention Effects with Non-compliance: A Latent Variable Approach for Randomized Trials.” In New Developmentsand Techniques in Structural Equation Modeling, ed. George A. Marcoulides and RandallE. Schumacker. Mahwah, NJ: Erlbaum.

Krueger, Alan B., and Pei Zhu. 2004. “Another Look at the New York City VoucherExperiment.” American Behavioral Scientist 47 (5): 658–98.

Lankford, Hamilton, and James Wyckoff. 1992. “Primary and Secondary SchoolChoice among Public and Religious Alternatives.” Economics of Education Review 11:317–37.

Lankford, Hamilton, and James Wyckoff. 2001. “Who Would Be Left Behind byEnhanced Private School Choice?” Journal of Urban Economics 50:288–312.

Little, Roderick J., and Linda H. Y. Yau. 1998. “Statistical Techniques for AnalyzingData from Prevention Trials: Treatment of No-Shows Using Rubin’s Causal Model.”Psychological Methods 3 (2): 147–59.

Long, James E., and Eugenia F. Toma. 1988. “The Determinants of Private SchoolAttendance, 1970–1980.” Review of Economics and Statistics 70 (2): 351–57.

Metcalf, Kim K., Stephen D. West, Natalie A. Legan, Kelli M. Paul, and William J.Boone. 2003. “Evaluation of the Cleveland Scholarship and Tutoring Program:Summary Report, 1998–2002.” Indiana University, Bloomington.

Muthen, Linda K., and Bengt O. Muthen. 2007. Mplus User’s Guide. 5th ed. Los Angeles:Muthen & Muthen.

Peterson, Paul E., William G. Howell, and Jay P. Greene. 1999. “An Evaluation ofthe Cleveland Voucher Program after Two Years.” Program on Education Policyand Governance, Harvard University.

Rouse, Cecilia E. 1998. “Private School Vouchers and Student Achievement: AnEvaluation of the Milwaukee Parental Choice Program.” Quarterly Journal of Economics113 (2): 553–602.

Witte, John F. 2000. The Market Approach to Education: An Analysis of America’s First VoucherProgram. Princeton, NJ: Princeton University Press.

Witte, John F., and Christopher A. Thorn. 1996. “Who Chooses? Voucher and In-terdistrict Choice Programs in Milwaukee.” American Journal of Education 104 (3): 186–217.

Witte, John F., Patrick J. Wolf, Joshua M. Cowen, David J. Fleming, and JuanitiaLucas-McLean. 2008. “MPCP Longitudinal Educational Growth Study: BaselineReport.” SCDP Milwaukee Evaluation Report 5, Department of Education Reform,University of Arkansas, http://www.uark.edu/ua/der/SCDP/Milwaukee_Eval/Report_5.pdf.

Witte, John F., Patrick J. Wolf, Joshua M. Cowen, David J. Fleming, and Juanita Lucas-McLean. 2009. “MPCP Longitudinal Educational Growth Study: Second Year Re-port.” SCDP Milwaukee Evaluation Report 10, Department of Education Reform,University of Arkansas, http://www.uark.edu/ua/der/SCDP/Milwaukee_Eval/Report_10.pdf.

Wolf, Patrick, Babette Gutmann, Nada Eissa, and Michael Puma. 2005. “Evaluationof the DC Opportunity Scholarship Program: First Year Report on Participation.”U.S. Department of Education, Institute of Education Sciences, Washington, DC.

Wolf, Patrick J., Babette Gutmann, Michael Puma, Brian Kisida, Lou Rizzo, and Nada





Eissa. 2009. “Evaluation of the DC Opportunity Scholarship Program: Impacts afterThree Years.” U.S. Department of Education, Institute of Education Sciences, Wash-ington, DC.

Yau, Linda H. Y., and Roderick J. Little. 2001. “Inference for the Complier AverageCausal Effect from Longitudinal Data Subject to Noncompliance and Missing Data,with Application to a Job Training Assessment for the Unemployed.” Journal of theAmerican Statistical Association 96 (456): 1232–44.



who chooses, who refuses? learning more from students who decline private school vouchers

Documents