Wearing Glasses during Classes Evidence from a Regression Discontinuity Design
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Wearing Glasses during Classes
Evidence from a Regression Discontinuity Design
Draft of October 2011
By CHRISTOFFER SONNE-SCHMIDT*
This paper evaluates whether eyeglasses causes better test scores
for nearsighted youths by exploiting a natural experiment
discovered in the National Health Examination Survey of Youth
(NHES III, 1966-70). I argue, based on visual science, that
uncorrected vision is partly random. Then, I show that average test
scores form a V-shaped pattern over uncorrected vision, and that
this V shape kink at the selfsame point where the average youth
begins to wear glasses. I estimate the cumulative effect of visual
correction in middle and high school with a regression-
discontinuity approach near the knickpoint; it’s about 0.1 standard
deviations of the test-score distribution and as high as 0.3 standard
deviations for nearsighted youths. (JEL C36, I21)
Key words: Kinked-regression-discontinuity design; Instrumental-
variable regression; Eyesight; Academic skills.
* University of Copenhagen, Building 26, Øster Farimagsgade 5, DK-1353 Copenhagen K. (e-mail: christoffer.sonne-
schmidt@econ.ku.dk). I thank Thomas Barnebeck Andersen, Josh Angrist, Sascha Becker, Paul Bingley, Martin Browning,
Carl-Johan Dalgaard, Mette Ejrnæs, John N. Friedman, Meghan Jakobsen, Rafael Lalive, John Rand, Pablo Selaya, Finn
Tarp, Nina Torm, participants at the CAM December 2010 workshop, and participants at the third IWAEE for their keen
eye and helpful comments; it helped me shape and develop this paper and research topic.Quite unknown to many students, they have a disadvantage in classroom
learning. They are nearsighted, and without eyeglasses, they have to squint to read
on the blackboard. While their classmates see things clearly, they have to struggle
all day to follow instructions, so they are more likely to tire in class, miss out on
classroom learning, and fail in school. But, wearing a simple pair of glasses, they
may refocus the blackboard—and so refocus classroom learning—yet many of
them aren’t wearing glasses during classes. In the United States today, 35 percent
of all middle- and high-school youths have some level of nearsightedness, which
98 percent of them can correct—and almost all can reduce—with a pair of
glasses, yet less than 76 percent have a pair, and merely 60 percent actually wear
them.1 So school-vision programs that, say, provide new glasses to students who
have none or have insufficient correction or that encourage students who have
glasses to start wearing them, could surely reduce visual blur in the classroom.
But will such remedies also improve proficiency in reading and math?
A creditable answer to this question cannot simply compare reading and math
skills of those who wear and do not wear glasses. Youths who for one develop
nearsightedness and afterwards choose to wear glasses may be quite different
from those who don’t, and that—that difference, not eyeglasses—may explain
success or failure in school. To separate these effects, I rely on a natural
experiment discovered in the National Health Examination Survey (NHES III,
1966-70) of youths from the late 1960’s.
Briefly, this experiment stems from partial randomization of nearsightedness;
nearsightedness is random close to its cutoff, and nearsightedness enables youths
to get and wear eyeglasses. Graphical evidence suggests that the consequence of
1
I’ve estimated these percentages upon a subsample of 9.961 adolescents (12-19 years old) from five rounds of the
National Health and Nutrition Examination Surveys (NHANES 1999 to 2008). Nearsightedness is defined as having an
uncorrected visual acuity of worse than 20/30 (inability to read the 20/40 line in an eye chart) in the better-seeing eye or
usually wearing glasses for distance visual correction; this definition is practically similar to the one use here, below.
Similarly, best visual acuity defines as the ability to read the 20/40 line with an optimal correction.this experiment is a V-shaped pattern between students’ performance and the
level of nearsightedness. Exactly at the cutoff of nearsightedness—when the
average youth begins to wear glasses—average performance kink, and increase.
I estimate the effect of this experiment with the instrumental-variable (IV)
version of regression-discontinuity (RD) design. Due the V shape and the fuzzy
treatment (enabling the use of glasses), the exact identification strategy combines
the kinked RD design in David Card, David S. Lee, and Zhuan Pei (2009) with
the fuzzy RD design in Jinyong Hahn, Petra Todd, and Wilbert van der Klaauw
(2001).
Results show that use of glasses strongly affects arithmetic and reading scores.
The cumulative use of glasses among middle- and high-school youths improves
average test scores between 0.1 to 0.2 standard deviations of the test-score
distribution. Among nearsighted youths, the average gain is even higher,
improving test scores between 0.3 to 0.6 standard deviations. So school-health
programs aimed at schoolchildren’s eyesight appear to be an effective educational
tool.
The ensuing section considers the identification problem and a natural
experiment that randomly assigns eyeglasses. Section II graph the identification
strategy and illustrate the effect of wearing glasses. It is complemented in Section
III, IV, and V with a formal description of the identification strategy, a description
of eyesight and test-score data in NHES III, and results from a regression-
discontinuity design. Section V also analyses robustness of the results. The final
Section—concludes.
I. Identification
Consider one of, if not the first study to look at the relationship between
eyesight and schooling: James Ware’s (1813) study on the prevalence ofnearsightedness and use of glasses among British foot guards and undergraduates
at Oxford and Cambridge.2 Ware discovered that one in four college students
wore glasses for nearsightedness whereas among foot guards this was “almost
utterly unknown,” J. Ware (1813, p 32). By simply comparing mean college
attendance of young men who wear and do not wear glasses from his study, one
might conclude that use of glasses causes academic success. But clearly, nineteen-
century foot guards and undergraduates were different in several ways other than
just their use of glasses; for example, upbringing and family background may
explain some—if not all—of the differences in academic success, not glasses. In
other—more recent—words, their parents may be “more educational ambitious
than the general run and so are not only more aware of the importance of
correcting vision in their children but also have stronger incentives to do so,” J.
W. B. Douglas et al. (1967, page 480). To my knowledge, the first study within
economics that looks at the relationship between eyesight and schooling.
Another more subtle identification problem comes from the reverse effect of
schooling on eyesight. Spurred by Ware’s finding, visual scientists have carried
out several studied that try to establish the causes of nearsightedness. See Seang-
Mei Saw et al. (1996) and Ian Morgan and Kathryn Rose (2005) for a review of
that literature, and Elena Tarutta et al. (2011) for an outline of resent advances in
the field.3 Similar to Ware’s study, they find strong relationships between the
visual environment and development of nearsightedness. Visual stress increases
the probability of developing nearsightedness in both humans and animals,
regardless of the subjects’ age, gender, race, and nationality. In particular, they
conclude that assiduous reading increase the prevalence of nearsightedness for all
2
Though Ware’s errand was different (he tried to establish that use of glasses worsened eyesight), the example still
serves to illustrate the main identification problem.
3
Among these visual-science papers, Angle and Wissmann (1980) use the same dataset that I’m using here, the
National Health Examination Survey of Youths (NHES III, 1966-70).subgroups of humans, so youths who study hard may do well in school, but along
the way, they are also more likely to develop nearsightedness and therefore wear
glasses. A simple difference in mean achievements may therefore reflect nothing
more than a difference in study intensity.
In short, neither corrected nor uncorrected eyesight is random; the group of
youths who develop nearsightedness and afterwards choose to wear glasses might
be particularly studious. Creditable estimates of the effect of visual correction on
students’ achievements therefore require other identification strategies than
simple mean comparison.
The direction of the misestimation depends, as usual, on the relationship
between the unobserved factors that determine use of glasses and performance in
school. For example, glasses may reinforce grades of children who already do
good in school, so the mean difference is too large. Or parents may give glasses to
children who show poor achievements to compensate their grades, so the mean
difference is too small.
Paul Glewwe, Albert Park and Meng Zhao (2006) solve the identification
problem in a randomized controlled trial. Following an eye exam, they randomly
offered free glasses to primary-school students with poor vision who did not
already have glasses in two rural provinces of China. They conclude that wearing
eyeglasses for one year, on average, improves test scores between 0.1 and 0.2
standard deviation of the test score distribution.4+5
4
This source of randomization will estimates the potential effect of giving glasses to poor-vision students who
currently do not wear glasses. A well-known disadvantage, though, is that a randomized experiment may be specific to its
context and the effect of glasses for primary students in two rural provinces of China may not reflect the effect elsewhere.
Potentially more problematic, however, is that the follow-up test scores, one year later, do not include students who drop
out of school (drop out of the sample) and lack of glasses may affect this probability, too. Even small dropout rates may
have a large impact on average scores because it’s the worst performing students who tend to drop out. At least,
preliminary estimates here suggest that it is a potential source of bias.
5
In addition to Douglas et al. (1967) and Glewwe et al. (2010), mentioned here, three other studies—from the annals of
economics—consider the affect of eyesight and educational outcomes. Linda N. Edwards and Michael Grossman (1980)
use National Health Examination Survey from the early 1960’s (the version before the one used here); João B. Gomes-
Netto, Eric A. Hanushek, Raimundo H. Leite and Roberto C. Frota-Bezzera (1997) use Brazilian panel data; and The-WeiHere, I suggest an alternative source of randomization; rather than deliberate randomization, I rely on natural randomization in exact eyesight, what David S. Lee (2008) and D. S. Lee and T. Lemieux (2010) call local randomization. The result of an eye exam depends partly on observed and unobserved characteristics and choices of the examined youth and partly on random chance. Say, time spent reading and family history of nearsightedness will influence your measure of eyesight, but so will numerous random differences in test procedures, such as type of eye chart, legibility of eye-chart letters, exact testing distance to the eye chart, and light in the consulting room. The review by George Smith (2006) find that with similar test procedures 95 percent of repeated eye tests still differ about one eye-chart line above and below the patients average eyesight. Allowing test procedures to differ, as well, increase this uncertainty even further. Y F Yang and M Dcole (1996) find that one-half of all school-nurse examinations differ one eye-chart line above or below the same examination in a controlled environment, and another quarter of examinations differ even further. M. J. Hirsch (1956) and K. Zadnik et al. (1992) find even larger uncertainties. Essentially, this means that the exact outcome of an eye exam is random by one or two lines. For youths bordering the cutoff of nearsightedness, that makes a huge difference because the tag nearsightedness in general will enable them to get a pair of glasses. When a school nurse gives such a student an eye exam, he or she will randomly conclude that that the student is nearsighted and therefore randomly refer the student to an optometrist. Within a tight boundary off cutoff, the tag of nearsightedness is therefore as-good-as the purposefully randomized offer to get a pair of glasses. Hu (1977) use household data from a coal-mining county in Pennsylvania. But only Glewwe et al. (2010) estimate this relation from an explicit source of random variation.
II. Graphical Evidence
On the back wall of every school nurse’s consultation room hangs an eye chart
with which they measure the performance of students’ vision. As the familiar “big
E” chart in Figure 1, it depicts several lines of capital letters of decreasing size,
and in a routine eye exam, the nurse will first ask the student to name letters on
the smallest readable line without glasses and second do the same exercise with
glasses if the student brought any. The first measure, called uncorrected visual
acuity, UCVA, is a measure of intrinsic eyesight; the second, called presenting
visual acuity, PVA, is a measure of eyesight with whatever correction—none or
some—that the student brings along. Both measured on a 20/20 scale. I’ll follow
tradition of the visual science literature and refer to 20/20 acuity as normal vision.
Visual acuity of 20/30 then means the ability to see at 20 feet (at best) what
someone with normal-vision can see at 30 feet—two line nearsightedness.
FIGURE 1: THE SNELLEN EYE CHART
Notes: NHES III (1966-70) used a slightly modified version of this eye chart, the Sloan chart; while the Snellen chart
contains letter of differing legibility, the Sloan chart does not. Please see Section IV. Properly scaled, target letters at the
20/20 line will subtend 5 arcmin at 20 feet, or 20*tan(5 arcmin) = 0.029 feet, or 0.349 inches.Figure 2 illustrates the relationship between these two measures of eyesight for
youths in NHES III (1966-70). It plots average presenting visual acuity (PVA) for
each line of uncorrected visual acuity (UCVA), 95-percent confidence intervals,
and a kinked-linear prediction of PVA on UCVA. The kinked-linear prediction is
from an OLS regression that allows for different slopes, but not intercepts, on
6
either side of 20/20 acuity.
Presenting Visual Acuity over Uncorrected Visual Acuity
First stage
/12
20
/1 5
20
/1 7
20
PVA
/2 0
20
/2 5
20
/30
20
00 00 00 00 /70 0/50 0/40 0/30 0/25 0/20 0/17 0/15 0/12
0/4 20/4 20/2 20/1 20 2 2 2 2 2 2 2 2
lt 2
UCVA
Mean PVA 95% Confidence limit
Kinked-linear prediction
FIGURE 2: PRESENTING VISUAL ACUITY OVER UNCORRECTED VISUAL
ACUITY (FIRST STAGE).
Notes: I’ve used a dummy-variable regression with robust standard errors to
estimate means and confidence intervals. The kinked regression allows for
different slopes on either side of 20/20: PVA = π + πUUCVA + πUZUCVA·Z + ε,
where Z=1[UCVAcorrection. On the other hand, for low values of UCVA, averages of PVA increase above the 45-degree line of no correction, so the average nearsighted youth do wear glasses, and much stronger glasses, too. Development of nearsightedness therefore seems to change youths’ incentives to wear glasses, as the kinked regression indicates. Even close to the cutoff, there is a noticeable change; averages nearest the 20/20 cutoff tend to kink, too. That is, among youth who essentially have quite similar levels of UCVA, the nearsighted youths are still more likely to have and wear glasses. The differences may result from school nurse practices. Today, as well as in the 1960s, school nurses perform vision screenings to detect and refer nearsighted students to an optometrist to get a pair of glasses. The guideline is, and was, to refer fourth graders and above if visual acuity in one eye is strictly below 20/30, or the difference between best and worse eye is more than two eye-chart lines apart, see M. E. Doster (1971) and American Optometric Association, (2000). In the appendix, I show that this monocular recommendation corresponds closely to the binocular cutoff of 20/20, in Figure 2. School nurses are therefore more likely to have referred students who are below the cutoff to an optometrist whereas students above the cutoff are less likely to have been referred. Also, as vision deteriorates, it becomes more and more obvious to teachers, parents, friends, and the youth, that there is a visual problem. He or she will have to squint in the classroom and will be the last one to recognize street signs or friends when driving in a car or walking on the street. This could also change incentives to wear glasses. In any circumstance, falling below the cutoff, changes the use of glasses and PVA, so if eyesight has an effect on performance in school, such a change should show up in test scores, too. Figure 3 shows average arithmetic (top) and reading (bottom) scores for each line of UCVA, 95-percent confidence intervals, and a kinked-linear prediction,
similar to Figure 2—each test score conventionally normalized by its standard
deviation. Arithmetic Score over Uncorrected Visual Acuity
Reduced form
4.5
Arithmetic Score/sd
4
3.5
3
Reading Score over Uncorrected Visual Acuity
00 00 00 00 /70 0/5Reduced
0 /40 /30 /25 /20 /17 /15 /12
0/4 20/4 20/2 20/1 20 2 20 20form20 20 20 20 20
lt 2 5
4. UCVA
Mean score 95% Confidence limit
Reading Score/sd
4 Kinked-linear prediction
3.5
3
00 00 00 00 /70 0/50 0/40 0/30 0/25 0/20 0/17 0/15 0/12
0/4 20/4 20/2 20/1 20 2 2 2 2 2 2 2 2
lt 2
UCVA
Mean score 95% Confidence limit
Kinked-linear prediction
FIGURE 3: ARITHMETIC (TOP) AND READING (BOTTOM) SCORE OVER
UNCORRECTED VISUAL ACUITY (REDUCED FORM).
Notes: I’ve divided the test score by its standard deviation (sd) before averaging,
and used a dummy-variable regression with robust standard errors to estimate
means and confidence intervals. The kinked regression allows for different slopes
on either side of 20/20: Score = γ + γUUCVA + γUZUCVA·Z + ε, where
Z=1[UCVAmore difficult to follow classroom instructions, but as vision crosses the 20/20
cutoff, the average youth begins to wear glasses, can again see the blackboard,
refocus classroom learning, and increase test scores. Similar to Figure 2, these
kinks seem persistent even for youths close to the cutoff; that is, a kinked-linear
regression on a tighter sample would also predict a kink at 20/20. Not least
because these averages are very precise, most confidence intervals are less than
one-third of a standard deviation wide.
III. Regression-Discontinuity Framework
While these graphs illustrate that use of glasses may cause better arithmetic and
reading performance of nearsighted students, and that this effect is sufficient large
to increase the average score, too, they only show a crude picture of what is going
on—size and statistical uncertainty of this effect is less clear. To refine it, I rely
on the regression-discontinuity (RD) framework, specifically, combining the
kinked-regression-discontinuity design in David Card, David S. Lee and Zhuan
Pei (2009) with the IV version of RD design—called fuzzy RD—in Jinyong
Hahn, Petra Todd, and Wilbert van der Klaauw (2001). In the kinked design,
treatment intensity gradually increases beyond some cutoff, and the outcome of
interest is a kinked function of the treatment intensity, but treatment is certain. In
the IV design, treatment isn’t certain; instead, it follows a two-stage model that
assigns an intention to treat beyond some cutoff, but students can afterwards
choose of refuse to take the treatment. Combining these, I estimating the
following two-stage model:
(1) Ai UUCVAi PVAi i
and
(2) PVAi UUCVAi UZUCVAi Zi i ,where student i’s achievements, Ai, is a linear function of the treatment, PVAi, and treatment assignment is a kinked-linear function of uncorrected eyesight, UCVAi, above and below the 20/20 cutoff, Zi = 1[UCVAi
PVAi into equation (2) and (3) might not estimate a causal effect of PVAi but for a
slightly different reason. While stairstepped RD prohibits unobserved jumps,
kinked RD prohibits unobserved kinks.7 If some unobserved determinant of
achievements (in εi) kink at the cutoff, then UCVAi·Zi might capture this kink, and
the estimate of ρπUC will be either too big or too small. If assiduous reading—one
of the determinants of nearsightedness and achievements—changes nonlinearly
over the cutoff, then UCVAi·Zi will capture not only its causal effect but also the
effect of studying hard—the nonlinear effect of reading. On the other hand, if the
unobserved change is approximately linearly, then UCVAi will proxy its effect
and the estimate of ρπUC and, therefore, ρ is causal.
Again, as in stairstepped RD, to make causal estimation more probable, one can
estimate the IV model on a tighter sample over the cutoff. As the sample gets
tighter, it first becomes more and more likely that the linear function of UCVAi
will sufficiently approximate any unobserved differences.8 Second, it becomes
more and more likely that the tag of nearsightedness is random because results of
eye exams are partly random as explained in Section I.
The empirical section follows an alternative strategy, as well. In addition to
estimation on tighter samples, I include many predetermined covariates in
equation (1) and (2). If the kink is truly random, then this should have no effect on
ρ. The appendix goes even further and includes both nonlinear functions of
uncorrected vision for tight samples and with covariates. This has no qualitative
effect on results relative to the results presented here, below.
7
In fact, the kinked model can also allow for unobserved jumps in ε by including Z as an additional control,--not as an
instrument—in the IV regression. I do this in the appendix to allow for unobserved differences between nearsighted and
non-nearsighted youths.
8
The strategy of estimation on ever smaller samples closer to a cutoff to retrieve the causal effect of a discontinuous
change on achievements is similar to the regression-discontinuity design in, among others, Joshua D. Angrist and Victor
Lavy (1999), Caroline M. Hoxby (2000), P. Bayer et al.(2007) , and Hanley Chiang (2009), what Angrist and Lavy call
discontinuity samples.IV. Data
The dataset that I use is the National Health Examination Survey of Youth
(NHES III, 1966-70). It is cross-section data of 7,514 U.S. youths between the age
of 12 and 17 years; the response rate is 90 percent; it was collected between 1966
and 1970.9+10 The main part is a physical and intellectual examination, focusing
on health of youths but also including demographic- and socioeconomic-
background information; in particular, for this study, it subjected youths to
arithmetic, reading and eye tests. To keep examination conditions constant across
the country, the examining team conducted all examinations in specially
constructed mobile clinics—approximately 12 examinations each day, six in the
morning and six in the afternoon. Because heavy snow could make roads
impassable to the mobile clinic, the survey avoided northern regions in the winter
and southern regions in the summer. For a through description of sampling
design, exact clinical examinations, validation studies, and the extensive data-
quality control see No. 8 (1969).
The precise wearing pattern of eyeglasses is for obvious reasons unobserved—I
do not know when, where, and how often nearsighted students wear glasses. But
survey staff would typically transport examinees to and from the examination
centre, picking up morning examinees at home and dropping them off at school
and the other way round for afternoon examinees. So students who normally wear
their eyeglasses to school are also likely to bring them at the examination centre
9
In this sample, I have changed a visual loss due to glasses to a zero gain for 76 farsighted youths; otherwise, the
sample is unchanged. Farsighted people may theoretically have worse distance vision when using their glasses (have PVA
smaller than UCVA), and in the sample they do have that. But in practice they probably will not; it is likely just an artifact
of definitions. In school, or in the classroom, when looking at distant objects like the blackboard, farsighted students could
either look over the rim of their glasses or not use them at all. So in practice, they will also have PVA larger than or equal
to UCVA. This change doesn’t affect results, see the appendix.
10
Sampling was based on the same sampling design as the previous, second cycle, of the survey, with some youths
examined in the Cycle II also examined in Cycle III. Cycle III sampled from the same 40 sampling areas as Cycle II, but
when time permitted, were previously (Cycle II) examined youths re-scheduled for examination as well. The selected
sample is, because of this non-randomness in sampling, not completely unbiased of the U.S. population in the late 1960’s.because they attended school either before or after the examination; except,
perhaps, for youth surveyed during their summer holydays.11
The arithmetic- and reading-achievements tests are subtests of the Wide Range
Achievement Test, WRAT. Psychologists, even today, use this psychometric tool
to measure basic academic skills such as reading and spelling of words and
performance of basic mathematical calculations. The complete battery of tests
lasts about 70 minutes, with the reading and arithmetic subtests always
administered as the first tests in each session. To increase accuracy of the test
score, a psychologist with at least a master’s degree and experience in
administering the WRAT test to adolescents gave the test. The reading test
consists of recognizing and naming letters and pronouncing words arranged in
order of increasing difficulty. The arithmetic test requires counting, reading
number symbols, solving oral problems, and performing written computations
usually taught in school. Initial validation studies compared the WRAT scores to
standardized measures of 7 to 12 graders school achievements in West Virginia,
Wisconsin, California, and Colorado.12 Both tests have reasonably high
correlations with these standardized tests, ranging from 0.47 to 0.84 conditional
on grade levels and geographic regions and with a slightly higher accuracy for the
arithmetic test. See D. C. Hitchcock and G. D. Pinde (1974). The raw arithmetic
score range from 0 to 56 points with mean of 23 and standard deviation of 6.9,
and the reading score range from 0 to 89 with mean and standard deviation of 48
and 14. Here, I follow common practice and divide each test score with its
standard deviations.
11
Before the examination households were given a pamphlet describing the program, among others outlining that there
would be “tests of vision and visual acuity”, No. 8 (1969, page 17); hence, some students who normally do not wear
glasses to school may have brought them particularly for the purpose of the visual examination. Since academic outcomes
of these students are unaffected by the treatment, estimated treatment effects will tend to understate the effect of wearing
glasses. If they had worn glasses as they “claim”, estimated treatment effects would have been larger.
12
These standardized tests were The Stanford Achievement Test for junior high school students (grades 7-9) and
Metropolitan Achievement Tests for senior high school students.The eye examination includes tests of eye diseases, color vision, and binocular
and monocular (one eye at a time) vision. It tested for nearsightedness,
farsightedness, and astigmatism and determined current eyeglasses prescription
by a lensometer reading. Most important for this paper, however, it includes eye-
chart tests for nearsightedness both with and without current glasses; that is, tests
of PVA and UCVA. It measured eyesight on an eye chart with four lines at or
above normal vision and eight lines below, giving thirteen possible outcomes of
the visual acuity tests, from less than 20/400 (unable to read any line) to 20/12
(you can read all lines).13 Again, to keep test procedures constant and ensure
consistency of test results, the mobile clinic had a specially designed room for the
eye test with stabile light and test distance, and throughout the survey, equipment
was periodically checked. All examiners received thorough training and practice
in vision testing techniques, and a team of ophthalmologists conducted a pilot
study prior to the survey and a further validation study midway through the
survey. See J. Roberts and D. Slaby (1973), J. Roberts and D. Slaby (1974), and J.
Roberts (1975) for more detail.
As described earlier, the eye-chart measure of visual acuity is on the 20/20
scale, and this scale has an inherent nonlinearity that may approximate
nonlinearity in the assignment variable. Technically, this scale determine the
patients viewing angle at 20 feet; that is, viewing angle standing 20 feet from the
chart—a larger viewing angle (larger letters) meaning poorer vision. To see this
use the formula for right-angled triangles: tan(patients viewing angle) = height of
letter / 20 feet. Essentially, the denominators in the 20/20 scale indicate the
viewing angle because all letters subtend a viewing angle of 5 arc minutes
(arcmin) at altering viewing distances. For example, letters at the 20/200 line
13
The actual eye chart use in NHES III is the Sloan chart, a slight modification of the eye chart in Figure 1, see L. L.
Sloan (1959). Concerned about the equal legibility of lines and letters, Sloan introduced five letters on each line, selected
from ten capital letters of equal legibility.subtend 5 arcmin at 200 feet, the smaller letters at 20/50 subtend 5 arcmin at 50
feet, and the 20/20 line subtend 5 arcmin at 20 feet, and so on, so in non-linear
estimations I can simply use the denominator instead of viewing angle. 14 On the
eye chart, each line jump is approximately associated with a 0.1 change in log of
the patients viewing angle (or equivalently log of the denominator), so it’s
approximately 26 percent more difficult to read the line below. For example,
log10(25)-log10(20) ≈ log10(1+0.26) = 0.1, N. B. Carlson and D. Kurtz (2003, page
23) and L. L. Sloan (1959). This inherited nonlinearity in measurement of
eyesight has two important implications for the empirical model. First, the
estimated treatment effect, ρ, estimates the effect on achievements of a
proportional increase in eyesight of about 26 percent. Second, I can use it to allow
for a more flexible specification of the treatment assigning variable, UCVAi, by
including both the viewing distance in feet (the denominator) and its approximate
log10 transformation (the line number on the eye chart). The appendix pursues this
strategy in detail. Results from these nonlinear regressions are not qualitatively
different from results presented here.
Finally, to minimize transportation costs when picking up or dropping off
examinees, the daily consultation would typical schedule youths that either lived
in the same neighborhood or attended the same school, or both. So a variable for
day of examination likely proxy for school and neighborhood characteristics; say,
they may share the same school nurse, teachers, health care services, and
14
The denominator always indicating the distance at which a normal-vision person can barely read the line, where
normal is defined as a viewing angle of 5 arcmin at 20 feet, a 20/20 vision person. Hermann Snellen, the farther of the eye-
chart, also recognized this normalization of his measure: “Die Bestimmung der Sehschärfe had keinen absoluten sondern
nur einen relativen Werth. [w]enn man nur als Ausgangspunkt für den Vergleich einen bestimten Sehwinkel wählt. Als
solchen haben wir einen Winkel von 5’ angenommen, weil diser durchschnittlich der kleinste Sehwinkel ist, unter welchem
normale Augen unsere Buchstaben erkennen.” Snellen 1863, page 4. Where, use of the term “normal eyes” dates back as
far as Hooke’s experiments, in 1679, on the resolving power of the eye: “tis hardly possible for any unarmed eye well to
distinguish any Angle much smaller then that of a minute: and where two objects are not farther distant then a minute
[t]hey coalesce and appear one.” Levene 1977, page 43. Note that if the letter E subtends five arcminutes then each vertical
line, in the E, is separated by one arcminute.environment.15 As a robustness check, the empirical model among others include
day-of-examination fixed effects. Again, the appendix explores this issue in
further detail.
V. Regression Evidence
In Section II, reduced-form graphs illustrate that something causes test scores to
increase as visual acuity crosses normal vision, and the first-stage graph suggest
that it’s a change in presenting eyesight. This section refines this evidence by
estimating IV regressions in the kinked-regression-discontinuity design, first
without covariates, then with. The appendix provides a comprehensive set of
estimations while this section extracts key results from the appendix.
Panel A of Table 1 presents IV estimates of PVA on test scores, and Panel B
and C present associated reduced-form and first-stage estimates. Arithmetic
scores on the left-hand side, and reading scores on the right. For this table, I’ve
selected two tight samples: 20/30 to 20/17 and 20/50 to 20/12, and the full
sample.16 Figure 4 reproduces the estimates in Panel A, together with similar
estimates for gradually wider samples; each dot represents an estimate of ρ, in
equation (1), but on a different sample size. The second, the sixth, and the final
dot in Figure 4 correspond to estimates in Panel A.
Starting from the bottom, all first-stage estimates on UCVA·Z show significant
changes in visual correction at the cutoff of nearsightedness. That is, development
of nearsightedness is highly associated with the decision to wear glasses and the
level of correction. For tighter samples, the size of the effect is obviously smaller
than in the larger samples because the probability of wearing glasses and the level
of correction is smaller when youth only have slight nearsightedness. The F-
15
Thomas Barnebeck Andersen et al. (2010) show that UV radiation may affect development of poor vision.
16
The Appendix presents a table with estimates of all discontinuity samples.statistic of this effect is nevertheless close to 50 in the small sample and reaches
well above 2000 in the full sample.17 But then, it’s probably not too surprising
that development and the level of nearsightedness is a—if not the—key
determinant in the decision to wear glasses and the level of visual correction. That
is, the first-stage seems strong.
TABLE 1—KINKED-RD ESTIMATES WITHOUT COVARIATES
(1) (2) (3) (4) (5) (6)
20/30 to 20/50 to Full 20/30 to 20/50 to Full
20/17 20/12 sample 20/17 20/12 sample
Arithmetic score Reading score
Panel A. Instrumental-variable regressions.
PVA 0.293 0.402 0.242 0.108 0.409 0.234
(0.14) (0.04) (0.02) (0.14) (0.04) (0.02)
[2.15] [9.55] [14.56] [0.77] [9.26] [13.84]
UCVA -0.213 -0.234 -0.108 -0.101 -0.241 -0.107
(0.08) (0.03) (0.01) (0.08) (0.03) (0.01)
[-2.62] [-8.02] [-18.23] [-1.24] [-7.91] [-18.72]
Panel B. Reduced-form regressions.
UCVA 0.046 0.142 0.142 -0.006 0.142 0.135
(0.05) (0.01) (0.01) (0.05) (0.01) (0.01)
[0.95] [10.14] [10.67] [-0.11] [9.71] [9.86]
UCVA∙Z -0.164 -0.268 -0.255 -0.061 -0.272 -0.247
(0.08) (0.03) (0.02) (0.08) (0.03) (0.02)
[-2.12] [-10.25] [-14.32] [-0.75] [-10.03] [-13.65]
Panel C. First-stage regressions PVA
UCVA 0.884 0.936 1.032
(0.03) (0.01) (0.01)
[28.49] [95.25] [82.50]
UCVA∙Z -0.559 -0.666 -1.055
(0.08) (0.04) (0.02)
[-7.06] [-15.82] [-50.01]
Observations 2,055 5,636 6,757 2,055 5,636 6,757
F-test of instrument 49.91 250.4 2501
The average gain of eye-chart lines from wearing glasses, and the share who uses glasses (in parenthesis):
—in the sample 0.38 0.32 1.38
(0.23) (0.15) (0.28)
—in the sample of nearsighted youths 1.06 1.73 4.54
(0.43) (0.50) (0.73)
Notes: PVA is presenting visual acuity and UCVA is uncorrected visual acuity both measured in eye-chart
lines. Test scores are divided by their standard deviation, UCVA is centered at 20/20, and Z=1[20/20Reduced-form estimates reflect this pattern; when visual correction is small so
is the reduced-form gain in test scores, and as the first-stage effects increase they
push up reduced-form effects, too. In the tighter samples, estimates are less
precise (less significant) but as sample size increases so do the level of
precision—as expected.
The Fuzzy-RD Effect of PVA on Arithmetic Scores
Kinked-linear function
1 1
in test score
in test score
Std. gain
Std. gain
.5 .497 .454 .5
.408 .38 .402
.293 .282 .261 .242 .242
.234
0
0
The
7
Fuzzy-RD
7 5 5
Effect 2
of2PVA2 on Reading 2 2
Scores
2 /12
/ 1 / 1 / 1 / 1 / 1 / 1 / 1 /1 /1 /1
20 o 20 o 20 o 20 oKinked-linear 20 o 20 o 20function 20 o 20 o 20 o 20
to t t t t t t to t t t
/25 0/130 0/30 0/40 0/40 0/50 0/70 /100 /200 /400 /400
20 2 2 2 2 2 2 2 0 2 0 20 lt 20 1
UCVA samples .8
in test score
in test score
Std. gain
Std. gain
.5 .451 .409
.313 .322 Coef. .371 .31 .6
on PVA .277 .243 .235 .234
.108
95% confidence limit .4
0
Average gain in sample (2nd axis) .2
Av. gain for nearsighted (2nd axis) 0
-.5
/ 1 7 /17 /15 /15 /12 /12 /12 / 1 2 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
to t t t t t t t t t t
/25 0/30 0/30 0/40 0/40 0/50 0/70 /100 /200 /400 /400
20 2 2 2 2 2 2 20 20 20 lt 20
UCVA samples
Coef. on PVA
95% confidence limit
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis)
FIGURE 4. IV ESTIMATES OF PVA ON ARITHMETIC (TOP) AND READING (BOTTOM)
SCORES WITHOUT COVARIATES
Combined—dividing reduced form by first stage—IV estimates suggest that
visual correction is very important for nearsighted students’ achievements. A one-
line visual correction improves test scores between 0.1 and 0.4 standard
deviations of the test score distribution. Close to the cutoff, effects are larger—
although less precise—while in larger samples effects become smaller but
stabilize just above 0.2 standard deviations per line of correction.These estimate the average effect of correcting nearsightedness one eye-chart
line, but nearsighted youths may correct their eyesight more or less than one line,
on average. The bottom part of Table 1 lists the share of nearsighted youths who
wear glasses as well as the average correction. For example, 43 percent of
nearsighted youths in column 4 wear glasses, and the average level of correction
is slightly above one eye-chart line, 1.06 lines.18 Averaging up, use of eyeglasses
therefore improves nearsighted youths’ arithmetic scores by 0.29·1.06 = 0.31
standard deviations. The upper (orange) line in Figure 4 draws similar averages
for all samples and both test scores.
Alternatively, to compare this effect to other school interventions—say, a class
size reduction, which potentially affects every student—requires averaging over
everybody, both nearsighted and non-nearsighted youths, alike. The bottom part
of Table 1 also lists the share of all youths (in each sub-sample) who wear glasses
as well as the average correction. For example, 23 percent of youths in column 4
wear glasses, and the average level of correction is 0.38 lines. Averaging down,
use of eyeglasses therefore improves average arithmetic scores by 0.29·0.38 =
0.11 standard deviations. The lower (green) line in Figure 4 draws similarly
averaged effects for all samples and both test scores.
Overall, these effects are large, and for the full sample, maybe too large. As
explained above, if reduced-form estimates capture the effect of, say, being
studious, that might lead to overestimation. For example, if assiduous reading is
not approximately linear in equation (3) then UCVAi will not sufficient proxy
study intensity, and UCVAi·Zi might capture the non-linearity so the estimate of
ρπUZ is too large. In particular, the wider samples might overestimate the effect
while the tighter samples should more accurately estimate ρπUZ and therefore also
ρ.
18
Nearsightedness is defined as UCVAA. Regression with covariates Following, I explore validity of the identification strategy as sample size increases over the cutoff. I present three sets of figures with covariate-adjusted estimates and compare these to the non-adjusted estimates in Figure 4. Conditioning on predetermined characteristics should not affect estimates of ρ in samples where the kink (the instrument) captures random variation. On the other hand, if the kink is related to observed determinants of student performance, then conditioning on them will affect the estimate of ρ—and if observed characteristics can affect the estimate, then maybe so can unobserved characteristics. Covariate- adjusted estimates can therefore illustrate how far from the cutoff that the estimate of ρ is causal. The appendix presents several other robustness analyses, among others, regressing characteristics on the kink; these reach the same conclusion.
The Fuzzy-RD Effect of PVA on Arithmetic Scores
Kinked-linear function
.8
1.5
.6
in test score
in test score
Std. gain
Std. gain
1
.4
.606
.5 .2
.347 .306 .324 .35 .3
.18 .175 .157 .167 .167
0 The Fuzzy-RD Effect of PVA on Reading Scores 0
/17 0/17 0/15 0/15Kinked-linear /12 0/12 0/12 0/12 0/12
/12 0/12 0function
20 2 2 2 20 2 2 2 2 2 2
5 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to .8
20
/2
20
1/3
20
/3 0/4
2 20
/4 0/5 0/7
2 2 /10 0/20 0/40 0/40
20 2 2 lt 2
.6
in test score
in test score
UCVA samples
Std. gain
Std. gain
.5
.346 .337 .328
.191
.279
.243 Coef. on PVA .228 .209 .4 .182 .173 .172
0 95% confidence limit
.2
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis) 0
-.5
/17 0/17 0/15 0/15 0/12 0/12 0/12 0/12 0/12 0/12 0/12
20 2 2 2 2 2 2 2 2 2 2
5 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to
20
/2
20
/3
20
/3 0/4 0/4 0/5 0/7
2 2 2 2 0 /10 0/20 0/40 0/40
2 2 2 lt 2
UCVA samples
Coef. on PVA
95% confidence limit
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis)
FIGURE 5. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES
The Fuzzy-RD
WITH YEffect
OUTHS of PVA on .Arithmetic Scores
COVARIATES
Kinked-linear function
1.5
in test score
1
Std. gain
.606
.5
.347 .306 .324 .35 .3
.18 .175 .157 .167 .167
0 The Fuzzy-RD Effect of PVA on Reading Scores
7 7 5 5 2 2 2 2 2 2 2
0/1 20/1 20/1 20/1 Kinked-linear
0/1 20/1 2function
0/1 20/1 20/1 20/1 20/1
o2 o o o o2 o o o o o o
/2 5 t 1 /30 t /30 t /40 t /40 t /50 t /70 t 00t 00t 00t 0 0t
20 20 20 20 20 20 20 /1 /2 /4 /4
20 20 20 lt 20
UCVA samples
in test score
.5
Std. gain
.346 .337
.243 .279 Coef. on.328
PVA .228 .209
.191 .182 .173 .172
95% confidence limit
0
Unconditional estimate
-.5
/17 /17 /15 /15 /12 /12 /12 /12 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
5 to 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t
20
/2
20
/3
20
/3
20
/4
20
/4
20
/5
20
/7 /10 0/20 0/40 0/40
20 2 2 lt 2
UCVA samples
Coef. on PVA
95% confidence limit
Unconditional estimate
FIGURE 6. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES
WITH YOUTHS COVARIATES.The Fuzzy-RD Effect of PVA on Arithmetic Scores
Kinked-linear function
.6
1.5
in test score
in test score
.4
Std. gain
Std. gain
1
.611
.5 .2
.361 .314 .315 .304 .26
.149 .14 .126 .136 .134
0 The Fuzzy-RD Effect of PVA on Reading Scores 0
/17 0/17 0/15 0/15Kinked-linear /12 0/12 0/12 0/12 0/12
/12 0/12 0function
20 2 2 2 20 2 2 2 2 2 2
5 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to .6
/2 1/3 /3 0/4 /4 0/5 0/7 /10 0/20 0/40 0/40
20 20 20 2 20 2 2 20 2 2 2
lt
in test score
in test score
UCVA samples .4
Std. gain
Std. gain
.5
.217 .156
.246 on
Coef..205 PVA.269 .188 .164 .142 .131 .128
.266
0 95% confidence limit .2
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis) 0
-.5
/ 17 /17 /15 /15 /12 /12 /12 /1 2 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
to t t t t t t t t t t
/25 0/30 0/30 0/40 0/40 0/50 0/70 /100 /200 /400 /400
20 2 2 2 2 2 2 20 20 20 lt 20
UCVA samples
Coef. on PVA
95% confidence limit
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis)
FIGURE 7. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES
TheWITH
Fuzzy-RD
YOUTHS Effect of PVACon
AND PARENTAL Arithmetic
OVARIATES. Scores
Kinked-linear function
1.5
in test score
1
Std. gain
.611
.5
.361 .314 .315 .304 .26
.149 .14 .126 .136 .134
0 The Fuzzy-RD Effect of PVA on Reading Scores
7 7 5 5 2 2 2 2 2 2 2
0/1 20/1 20/1 20/1 Kinked-linear
0/1 20/1 2function
0/1 20/1 20/1 20/1 20/1
o2 o o o o2 o o o o o o
/2 5 t 1 /30 t /30 t /40 t /40 t /50 t /70 t 00t 00t 00t 0 0t
20 20 20 20 20 20 20 /1 /2 /4 / 4
20 20 20 lt 20
UCVA samples
in test score
.5
Std. gain
.217 .205 .246 Coef.
.266 on PVA
.269
.156 .188 .164 .142
95% confidence limit .131 .128
0
Unconditional estimate
-. 5
/17 /17 /15 /15 /12 /12 /12 /12 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
5 to 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t
/2 /3 /3 /4 /4 /5 /7 /1 0 /2 0 /4 0 /40
20 20 20 20 20 20 20 20 20 20 lt 20
UCVA samples
Coef. on PVA
95% confidence limit
Unconditional estimate
FIGURE 8. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES
WITH YOUTH AND PARENTAL COVARIATES.The Fuzzy-RD Effect of PVA on Arithmetic Scores
Kinked-linear function
.6
1
in test score
in test score
.4
Std. gain
Std. gain
.5 .364 .381
.327 .317 .252 .202 .139 .126 .114 .127 .129 2
.
0
-.5 The Fuzzy-RD Effect of PVA on Reading Scores 0
0/17 0/17 0/15 0/15 Kinked-linear /12 0/12 0/12 0/12 0/12
/12 0/12 0function
2 2 2 2 20 2 2 2 2 2 2
5 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to 0 to .6
/2 /3 /3 /4 /4 /5 /7 0 0 0 0
20 201 20 20 20 20 20 20/1 20/2 20/4 20/4
lt
in test score
in test score
UCVA samples .4
Std. gain
Std. gain
.5
.323
.186 .238 Coef.
.227 on PVA
.206 .215 .178 .154 .135 .127 .126 2
0 95% confidence limit .
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis) 0
-.5
/ 17 /17 /15 /15 /12 /12 /12 /1 2 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
to t t t t t t t t t t
/25 0/30 0/30 0/40 0/40 0/50 0/70 /100 /200 /400 /400
20 2 2 2 2 2 2 20 20 20 lt 20
UCVA samples
Coef. on PVA
95% confidence limit
Average gain in sample (2nd axis)
Av. gain for nearsighted (2nd axis)
FIGURE 9. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM) SCORES
The Fuzzy-RD Effect of PVA on Arithmetic Scores
WITH YOUTH, PARENTAL, AND NEIGHBORHOOD COVARIATES.
Kinked-linear function
1
in test score
Std. gain
.5 .381
.364 .327 .317
.252 .202
.139 .126 .114 .127 .129
0
-.5 The Fuzzy-RD Effect of PVA on Reading Scores
/17 /17 /15 /15 0/12 0/12 function /12 /12 /12 /12 /12
20 o 20 o 20 o 20 o Kinked-linear 2 2 20 o 20 o 20 o 20 o 20
5 to 0 t 0 t 0 t 0 t 0 to 0 to 0 t 0 t 0 t 0 t
20
/2
210
/3
20
/3
20
/4
20
/4
20
/5
20
/7 /10 0/20 0/40 0/40
20 2 2 lt 2
UCVA samples
in test score
Std. gain
.5
.323 Coef. on PVA
.186 .238 .227 .206 .215 .178
95% confidence limit .154 .135 .127 .126
0 Unconditional estimate
-. 5
/17 /17 /15 /15 /12 /12 /12 /12 /12 /12 /12
20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20 o 20
5 to 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t 0 t
/2 /3 /3 /4 /4 /5 /7 /1 0 /2 0 /4 0 /40
20 20 20 20 20 20 20 20 20 20 lt 20
UCVA samples
Coef. on PVA
95% confidence limit
Unconditional estimate
FIGURE 10. IV ESTIMATES FOR ARITHMETIC (TOP) AND READING (BOTTOM)
SCORES WITH YOUTH, PARENTAL, AND NEIGHBORHOOD COVARIATES.Figure 5, Figure 7, and Figure 9 plot point estimates of PVA on test scores similar to Figure 4, except; these figures condition on covariates. And Figure 6, Figure 8, and Figure 10 compare these adjusted estimates to the non-adjusted estimates in Figure 4. Estimates in Figure 5 and Figure 6 condition on youths’ characteristics: the age measured in months, gender, race measured as white or non-white, birth weight measured in grams, gestation period measured in weeks, and birth order of youth. In addition to these covariates, Figure 7 and Figure 8 also include household and parental characteristics given by the number of children in the household, single parent home, household income, years of parental education, and father and mothers age at birth. And finally, Figure 9 and Figure 10 add a set of indicator variables for the day of examination. As explained in Section IV, youths examined on the same day come from the same neighborhood and attend the same school, so day-of-examination fixed effects will proxy for differences in school and neighborhood characteristics. For example, differences in school-nurse praxises. The appendix gives a more detailed description of all these covariates as well as the complete set of estimation tables. Overall, covariate adjustments do not change the pattern in Figure 4: larger effects initially that tend to reduce in wider samples. Close to the cutoff, the kink—the natural experiment—is as good as a controlled experiment while further from the cutoff unadjusted estimates may overestimate the causal effect of wearing glasses. Point estimates from the tighter samples, the first two to three estimates, still fluctuate about an effect size of a quarter of a standard deviation (and are not statistically different from unadjusted estimates). The next three to four estimates become smaller than their unadjusted kind but stabilize at the level of the tighter samples, reducing some of the hump shape in Figure 4. The final estimates are still very precise but smaller by about 0.1 standard deviations, and stabilize at an effect of just above 0.1 standard deviations of the test score distributions. That is, close to the cutoff, the kink is unrelated to covariates, so
estimation of equation (1) and (2), based on a tight sample, will retrieve and
estimate ρ. As the sample size increase, however, and include youths who are
more nearsighted, the unadjusted kinked-linear model begins to capture other
determinants of test scores. The figures suggest, however, that covariate-adjusted
estimates based on intermediate samples can also estimate ρ because these
estimates tend to stabilize at the same level as the tighter samples. Further from
the cutoff, covariate adjustments make a difference, so it is less reliable to draw
causal conclusions from these estimates.
Averaged effects are also more stabile in regressions with covariates. The
sample average is about 0.1 standard deviations, perhaps slightly lower for
reading scores, although not statistically. In the larger samples that include the
effect of glasses for very nearsighted youth, the effect is higher. Averaged over
nearsighted youth—the upper (orange) lines—the effect of wearing glasses seem
to stabilize around 0.3 close to the cutoff. That is, nearsighted youth gain about
0.3 standard deviations on their test scores if they are able to see clearly. Again,
this effect is slightly higher for arithmetic scores and slightly lower for reading
scores, but again not statistically different. Including youths with severe
nearsightedness, the effect of wearing glasses approaches 0.6 standard deviations
of the test score distribution for both reading and arithmetic scores, but still these
effects are less reliable.
VI. Conclusion
Nearsighted students who do not wear glasses will fall behind in school. The
evidence presented here suggests that nearsighted youths who wear glasses do
much better in school than nearsighted youths who do not wear glasses. For all
middle- and high-school aged youths, the cumulative effect of eyeglasses is about0.1 standard deviation of the test score distribution, and for nearsighted youths
alone it is 0.3 standard deviations.
Essentially, the identification strategy goes as follows: nearsightedness is partly
random near its cutoff, but youths who develop nearsightedness are more likely to
wear glasses because school nurses refer them to an optometrist, and nearsighted
youths who wear glasses can follow classroom instructions and improve their
grades. So comparing performance of nearsighted youths to non-nearsighted
youths will estimate the causal effect of glasses.
Interpretation of these results requires some caution because I rely on
instrumental-variable estimation near the cutoff. First of all, I estimate the effect
of eyeglasses for youths who decide to wear glasses as they become nearsighted.
That is, the effect for current users of glasses. A vision-health program that
increases the use of glasses among current non-users may have a different effect.
Second, the estimate is only causal close to the cutoff of nearsightedness. That is,
the effect is not for youths who have severe levels of nearsightedness. On the
other hand, most people who develop severe nearsightedness will have crossed
the cutoff at some point (few children below the age of twelve are severely
nearsighted). Probably, severely nearsighted youths will have even larger benefits
of wearing glasses.
Overall, these findings suggest that use of glasses is an efficient way of
improving reading and math performance in school. Compared to the Tennessee
STAR experiment (Alan B. Krueger, 1999) or the class size reduction from
Maimonides’ rule in Israeli school (Joshua D. Angrist and Victor Lavy, 1999),
this effect of visual correction is about half the size of a seven student reduction.
But the average cost of the STAR experiment was about $1035 per student in the
1980’s (Alan B. Krueger, 1999)—clearly more than double the price of visual
correction even today.
***“Quite unknown to myself, I was, while a boy, under a hopeless disadvantage in
studying nature. I was very nearsighted, so that the only things I could study were
those I ran against or stumbled over. It was that summer, [when I was about
thirteen,] that I got my first gun, and it puzzled me to find that my companions
seemed to see things to shoot at which I could not see at all. One day they read
aloud an advertisement in huge letters on a distant billboard, and I then realized
that something was the matter, for not only was I unable to read the sign but I
could not even see the letters. I spoke of this to my father, and soon afterwards got
my first pair of spectacles, which literally opened an entire new world to me. I
had no idea how beautiful the world was until I got those spectacles. I had been a
clumsy and awkward little boy, [a good deal,] due to the fact I could not see and
yet was wholly ignorant that I was not seeing.” Narrative by 26th U.S. President,
Theodore Roosevelt (2010, page 16), from ca. 1871.
REFERENCES
8, No. 1969. "Plan and Operation of a Health Examination Survey of U.S. Youths
12-17 Years of Age." Vital Health Stat 1, (8), pp. 1-80.
Andersen, Thomas Barnebeck; Carl-Johan Dalgaard and Pablo Selaya. 2010.
"Eye Disease and Development," In Unpublished paper. University of
Copenhagen.
Angrist, Joshua D. and Victor Lavy. 1999. "Using Maimonides' Rule to
Estimate the Effect of Class Size on Scholastic Achievement*." Quarterly
Journal of Economics, 114(2), pp. 533-75.
Association, American Optometric. 2000. "A School Nurse's Guide to Vision
Screening and Ocular Emergencies—Part 3." School Nurse News, 17(3), pp. 14-9.
Bayer, P.; F. Ferreira and R. McMillan. 2007. "A Unified Framework for
Measuring Preferences for Schools and Neighborhoods." Journal of Political
Economy, 115(4), pp. 588-638.
Card, David; David S. Lee and Zhuan Pei. 2009. "Quasi-Experimental
Identification and Estimation in the Regression Kink Design," In Working paper.
Princeton University.
Carlson, N.B. and D. Kurtz. 2003. Clinical Procedures for Ocular Examination.
McGraw-Hill.
Chiang, Hanley. 2009. "How Accountability Pressure on Failing Schools AffectsStudent Achievement." Journal of Public Economics, 93(9-10), pp. 1045-57. Doster, M. E. 1971. "Vision Screening in Schools--Why, What, How and When?" Clin Pediatr (Phila), 10(11), pp. 662-5. Douglas, J. W. B.; J. M. Ross and H. R. Simpson. 1967. "The Ability and Attainment of Short-Sighted Pupils." Journal of the Royal Statistical Society. Series A (General), 130(4), pp. 479-504. Glewwe, Paul.; Albert. Park and Meng. Zhao. 2006. "The Impact of Eyeglasses on the Academic Performance of Primary School Students: Evidence from a Randomized Trial in Rural China," In. University of Minnesota and University of Michigan. Hahn, Jinyong; Petra Todd and Wilbert Van der Klaauw. 2001. "Identification and Estimation of Treatment Effects with a Regression- Discontinuity Design." Econometrica, 69(1), pp. 201-09. Hirsch, M. J. 1956. "The Variability of Retinoscopic Measurements When Applied to Large Groups of Children under Visual Screening Conditions." Am J Optom Arch Am Acad Optom, 33(8), pp. 410-6. Hitchcock, D. C. and G. D. Pinder. 1974. "Reading and Arithmetic Achievement among Youths 12-17 Years as Measured by the Wide Range Achievement Test: United States." Vital Health Stat 11, (136), pp. 1-32. Hoxby, Caroline M. 2000. "The Effects of Class Size on Student Achievement: New Evidence from Population Variation." The Quarterly Journal of Economics, 115(4), pp. 1239-85. Krueger, Alan B. 1999. "Experimental Estimates of Education Production Functions*." Quarterly Journal of Economics, 114(2), pp. 497-532. Lee, D. S. and T. Lemieux. 2010. "Regression Discontinuity Designs in Economics." Journal of Economic Literature, 48(2), pp. 281-355. Lee, David S. 2008. "Randomized Experiments from Non-Random Selection in U.S. House Elections." Journal of Econometrics, 142(2), pp. 675-97. Linda N., Edwards and Michael Grossman. 1980. "The Relationship between Children's Health and Intellectual Development." NBER Working paper, (w0213). Morgan, Ian and Kathryn Rose. 2005. "How Genetic Is School Myopia?" Progress in Retinal and Eye Research, 24(1), pp. 1-38. Roberts, J. 1975. "Eye Examination Findings among Youths Aged 12-17 Years. United States." Vital Health Stat 11, (155), pp. 1-75. Roberts, J. and D. Slaby. 1974. "Refraction Status of Youths 12-17 Years, United States." Vital Health Stat 11, (148), pp. 1-55. ____. 1973. "Visual Acuity of Youths 12-17 Years. United States." Vital Health Stat 11, (127), pp. 1-44. Saw, Seang-Mei; Joanne Katz; Oliver D. Schein; Sek-Jin Chew and Tat- Keong Chan. 1996. "Epidemiology of Myopia." Epidemiol Rev, 18(2), pp. 175- 87. Sloan, L. L. 1959. "New Test Charts for the Measurement of Visual Acuity at Far
You can also read
