A closer look at the effects of within-school ability grouping in
secondary schools: how are different performers affected?1
An investigation in four OECD countries
Miroslav Beblavy, Anna-Elisabeth Thum, Galina Potjagailo, Marten von Werder
Abstract
The relationship between ability grouping and equality of educational opportunities has been
subject of much debate. We examine this relation for differently performing students in 4 OECD
countries. Our paper shows that, depending on the student’s relative location on the PISA
performance distribution, within-school ability can increase or decrease educational equality to a
different extent. Quantile regression results show that in Belgium low performers are affected
most by inequality effects, whereas in Austria high performers are most affected. In Finland we
do not find a significant relationship. In Hungary, we find that high performers are most affected
by equality effects.
Keywords: ability grouping, quantile regression, PISA scores
JEL classification: I24, J24, Z18
1
The working paper version was funded by the European Commission within the Seventh Framework Programme
under the project NEUJOBS (www.neujobs.eu). The funding source played no further role than provision of funding
and peer review.
1
1) Introduction
Reducing or postponing ability grouping has been widely recognized as a policy measure that
contributes to increasing the equality of educational opportunities (Causa and Chapuis, 2009;
OECD, 2004, 2007, 2010; Schuetz, Ursprung and Woesmann, 2005; Hanushek and Woessmann,
2005; Woessmann, 2009). However, while effects of between-school ability grouping are widely
studied, the effects of within-school ability grouping on equality of educational opportunities are
less frequently researched in the economic literature as scholars focus more strongly on acrossschool ability grouping, also termed tracking.
In this paper we examine how within-school ability grouping can be associated with
equality of educational opportunity measured by the socio-economic gradient2 in four OECD
countries. Using quantile regressions we take a closer look at this effect by disaggregating it
across the performance distribution in the OECD’ Programme for International Student
Assessment (PISA). Distributional effects can be of interest as they are not necessarily equal:
students performing less well might benefit more than higher performers from non-grouped
lessons due to peer effects, whereas higher performers might not gain as much in terms of
educational performance3. To our knowledge this is the first paper that examines the relation
between education policies and equality of educational opportunities across the ability
distribution in an international comparison.
Quantile regression results show that the four countries differ in terms of how withinschool ability grouping is associated with equality of educational opportunities across the
performance distribution. In Belgium low performers are more strongly concerned by inequality
effects of within-school ability grouping, whereas in Austria high performers are more
concerned. In Hungary, we find that within-school ability grouping is positively correlated with
equality and more so for the high performers. In Finland within-school ability grouping does not
show to have a significant relation with equality. Our paper shows that, depending on the
student’s relative location on the PISA performance distribution, within-school ability can
2
We use ‘socio-economic gradient’ interchangeably with ‘equality of educational opportunity’, ‘educational
equality’ and ‘equality in education’. The socio-economic gradient is usually understood as the dependence of
educational outcomes on family background (Causa and Chapuis 2009). Section 3 discusses this concept in more
detail.
3
Hanushek and Woessmann (2006) speak of non-linear peer effects.
2
increase or decrease educational equality to a different extent. It further shows that a regression
at the average of the PISA distribution may not show the entire picture.
The remainder of the paper is organised as follows: Section 2 analyses the previous literature;
Section 3 describes our dataset; Section 4 presents descriptive statistics and Section 5 explains
the econometric model. In Section 6 we present the results and Section 7 concludes.
2) Previous literature
In the economic literature equality of educational opportunities is usually based on Roemer
(1998, 2008), who defines educational equality as the independence of educational outcome from
social origins. This understanding is strongly linked to the notion of the socio-economic gradient,
which is defined as the correlation between educational outcomes and family background.
Studies examining Roemer’s concept of equality of educational opportunities and their
determinants can be grouped into those focussing on (1) the effect of family background on test
scores; (2) the effect of policies on the socio-economic gradient; (3) the effect of school
characteristics on the socio-economic gradient or on test scores and (4) peer effects and equality.
Woessmann (2004) examines the effect of family-background on student achievement in 18
countries using quantile regression through the impact of socio-economic variables on Third
International Mathematics and Science Study (TIMSS) test results across the TIMSS
performance distribution. He finds that the impact of family background on scores is increasing
along quantiles for some countries and decreasing for a few others. Fertig (2003) uses quantile
regressions to study the effect of family background on PISA test scores in Germany. Also using
a quantile regression approach Hindricks, Verschelde, Rayp, Schoors (2010) find that the school
environment has a more heterogeneous effect for pupils of lower performance in Belgium.
Hanushek and Luque (2004) find that in the most OECD countries the effect of family
background on test scores diminishes across students’ age.
A set of cross-country studies examines the effect of education policies on educational
inequality. Causa and Chapuis (2009) conduct an analysis for 28 OECD countries using PISA
2006 test scores in science. The authors find that within-school ability grouping increases
inequality across countries. Schuetz, Ursprung and Woessmann (2005) use the TIMSS database
and confirm that late tracking increases educational equality. Using a combination of the PISA
3
and PIRLS data, Hanushek and Woessmann (2004) apply a difference-in-difference
methodology to eliminate cross-country differences and find that early tracking increases
inequality in education and decreases the mean performance. Ammermueller (2005) follows
Hanushek and Woessmann’s (2004) approach and finds that tracking benefits students from a
better social background. Raitano and Vona (2011) find that the effects of different ability peer
groups vary across different tracking schemes.
Other studies analyse the effect of education policies on educational inequality on a national
level. Figlio and Page (2001) examine the effects of tracking policies in the United States using
the NELS database (National Educational Longitudinal Survey). They do not find a negative
effect for tracking policy for low ability children. Betts and Shkolnik (2000) examine the effects
of ability grouping on educational achievement in secondary education in the US and – contrary
to the majority of the relevant literature – do not find a significant effect. Muehlenweg (2007)
studies the effects of secondary school tracking in Germany on educational outcomes by
comparing the outcomes of those who were tracked after the 4th grade and those who were
tracked after the 6th grade (participants of the so-called orientation stage) in the state of Hessen.
She finds that the effects of later tracking are more positive for the low ability groups than for the
rest of the ability distribution.
A set of studies focusses particularly on within-school ability grouping and its effects. The
results, however, appear to be ambiguous: Based on a meta-analysis, Hattie (2009) does not find
significant effects of ability grouping on pupils performance while even checking for potential
effects for different ability groups in OECD countries. In contrast to that, a recent study by
Collins and Gan (2013) found significant improvement in math and reading scores after sorting
pupils according to their performance. The authors control for different selection procedures and
hereby point out that the applied measures indeed may provide different conditions of ability
grouping that could affect the result of streaming. Hallam (2003) collected data on ability
grouping practices in the UK but does not relate them to pupil’s performance. However, to the
best of our best knowledge, there is indeed a gap in the academic documentation providing a
systematic typology of different grouping practices.
A set of studies use quantile regression to assess the effect of school characteristics on
educational equality. Basset, Tam and Knight (2002) study the econometric impact of high
4
school characteristics on student performance using quantile regression in the US and find that
high school characteristics have differing effects on achievement at different points on the
achievement distribution. Eide and Showalter (1998) also use US data and find that performance
at the top of the conditional distribution of maths score changes is improved by a lengthened
school year while performance at the bottom of the distribution appears not to benefit from the
extra class time, other factors being considered equal. Collier and Millimet (2009) look at the
effects of institutional characteristics on test scores in the TIMMS data base and show that it is
beneficial to study the distributional effects of institutional arrangements on education outcomes,
since the effects of institutional arrangements on test scores vary across quantiles. RangvidSchindler (2007) studies the effects of a socio-economic mix on educational outcomes in
Denmark and finds that school composition effects vary across ability quantiles.
As mentioned above, Peer group effects can have a beneficial effect on educational
performance and are more likely to be present if tracking is postponed. A set of papers looks at
the effect of characteristics of peer groups on educational outcomes across ability quantiles.
Levin (2001) looks at the effects of class size and finds that a smaller class size emphasises peer
effects. Averett and McLennan (2004) provide a comprehensive report on research and empirical
findings of the effect of class size on education outcomes, drawing both from the educationalist
and economic literature. Leuven and Ronning (2011) examine the effects of multi-age grouping
within the same classroom and find that it has a positive effect on the performance of the whole
group to mix different grades but depends on the proportion of lower and higher grades.
Schneeweis and Winter-Ebmer (2007) look at peer effects in Austrian schools and find that
positive peer effects are the largest for those with lower ability. The effect of peer effects in
schools is further analysed in Toma and Zimmer (2000), Glewwe (1997) and Rivkin (2001), for
example.
As we can see, there is no consensus in the literature on effects of ability grouping, both
generally and with regard to groups of different abilities. It appears that different countries and
different groups at the ability scale react differently. Despite this ambiguity the OECD has been
strongly pushing for a policy change towards later tracking, especially with its legitimacy
buttressed by PISA. Therefore, analyzing the effects of ability grouping at a country level in a
rigorous comparative framework should be useful.
5
3) Data
We use the PISA (2009) database. The PISA programme tests 15-year old students in
terms of their cognitive skills in mathematics, reading and science in OECD countries. In these
fields PISA measures in particular students’ ability to apply their knowledge in different contexts
(Ammermueller, 2005). The PISA sampling design is based on a two-step procedure: first a set
of schools are selected and then students are drawn randomly from the resulting population
(Ammermueller, 2005 and PISA, 2009). The test scores are computed using a mixed coefficients
multinomial logit model by Adams, Wilson and Wang (1997) in order to produce an ‘ability
scale’ measuring cognitive skills of the students in the sample (see PISA 2009, chapter 9 for
more details). A generalised form of the Rasch model4 from item response theory is used in
psychometrics to evaluate ability.
In addition to test scores, the PISA database contains background information for both
students and schools. The following table gives an overview of the variables used in our analysis.
Table 1: Variables used in the empirical analysis
Variable
Description
PISA score
(mathematics, science
and reading)
ESCS score (economic,
cultural and social
score)
Ability grouping
The PISA test scores are computed as described above, scores are
distributed around an international mean of 500 and a standard
deviation of 100 test score points.
The ESCS score comprises the students’ answers on home
possessions, books in the home, parental occupation and parental
education expressed as years of schooling.
This variable is retrieved from the PISA school questionnaire. Ability
grouping is measured through the school principals’ indications on
whether ability grouping between classes exists in their school and if it
is conducted for all pupils or only some. For this study we redefine the
variable and only distinguish between schools which ability grouping
takes place and schools in which ability grouping is not conducted.
Table A1 shows the shares of grouped and non-grouped students per
country.
The mean of the ESCS scores over all students within each student’s
school, which should account for school environment effects.
 Gender
 Migration background first generation
 Migration background second generation (indicated by the
ESCS school-mean
Control variables
4
For a comprehensive overview of the Rasch model and other psychometric models, see Rabe-Hesketh and
Skondral (2004), Generalised latent variable modeling: multilevel, longitudinal and structural equation models,
Boca-Raton, FL, Chapman and Hall/CRC.
6



father only)
Foreign language spoken at home
Mother at home
Father at home
4) Descriptive statistics and country selection
In order to understand how within-school ability grouping might affect the socioeconomic gradient for different PISA score performers, we assess conditional PISA performance
distributions for 27 European OECD countries5 using the PISA maths score (see Figure A1). The
conditional distributions depict the PISA performance by socio-economic background and ability
grouping. We assign pupils to three ranked and equally sized groups according to the score on
the PISA Economic, Social and Cultural Score6. We then compare the performance distribution
of pupils belonging to the lowest and the highest ESCS score group. We then examine whether
the differences in math scores between high- and low-ranking pupils differs between the
subsamples of ability grouped pupils and non-grouped pupils. We do so by using (meanstandardized) QQ-Plots and kernel density estimates (see Figures 1-4 and A1).
Based on how the conditional PISA performance distributions react to ability grouping,
we select four countries which represent typical cases of the effect of within-school ability
grouping on the socio-economic gradient. In these countries a higher social status as measured
by the ESCS score entails ceteris paribus better results in the PISA tests but the effect of ability
grouping in fact differs considerably across countries. We select Finland, Austria, Belgium and
Hungary.
In Finland (see Figures 1 and A1), the mean standardized kernel densities reveal almost
identical shapes for the distributions of math scores across ability grouping and non-grouping
even though there are level differences in scores according to the ESCS score. We therefore
conclude that ability grouping in Finland affects pupils of high and low status in a very similar
way and also affects pupils of different ability levels almost equally. In the QQ-Plot below this
5
These countries are Austria, Belgium, Bulgaria, Czech Republic, Denmark, Estonia, Finland, Germany, Greece,
Hungary, Ireland, Italy, Latvia, Lithuania, Luxemburg, Netherlands, Norway, Poland, Portugal, Romania, Slovakia,
Slovenia, Spain, Sweden, Switzerland, the United Kingdom and the United States. We exclude France as a variable
for ability grouping is not available.
6
We restrict the observations for the following figures to that third of observations with the highest (“high status”)
and the third of observations with the lowest (“low status”) ESCS score, i.e. we omit the third of the observations
with a medium score in ESCS.
7
lack of effect is depicted by the fact that almost all dots lie on the diagonal. Some other countries
show very similar results to the Finish case: Italy, Denmark, Sweden, Latvia, Lithuania, Estonia
and Norway. It is apparent that these countries track pupils according to performance generally
late (see Table A1) so that a potentially differentiating effect of tracking cannot be compounded
by additional ability grouping.
[Figure 1 here]
The second selected country is Austria (see Figures 2 and A1), where ability grouping
takes a differentiating effect: For pupils with a high social status that did not witness ability
grouping, the performance distribution is much narrower than the distribution for grouped pupils
and no ability grouping equalises performance at least for high status pupils. The same pattern,
while being less pronounced, is visible for Germany and the Netherlands. A second observation
is raised by non-mean-standardized QQ-Plots: The difference in test results between high and
low status pupils are smaller in the non-grouping subsample. In particular high achieving low
status pupils seem to benefit from the absence of ability grouping. Correspondingly, the nonmean standardized kernel densities show that the non-grouping almost fully compensates low
status pupils for their status disadvantage: They score similarly well as high status grouped
pupils. These differences would not be identified by simple OLS regression techniques, which
justify the application of quantile regressions later in this paper. Both observations suggest that
ability grouping enlarges performance differences.
[Figure 2 here]
In Belgium (see Figures 3 and A1) the kernel densities reveal that generally the standard
deviation in the performance of low status pupils is higher than for high status pupils. But it is
also visible, that for these low status pupils ability grouping seems to even enlarge the dispersion
of the performance record. Ability grouping therefore differentiates the performance of low
status pupils. The mean standardized QQ-Plots confirm this finding of slightly more pronounced
tails in the ability grouped subsamples. The non-standardized QQ Plots then show that for low
and medium achieving low status pupils the achievement deficit compared to high status pupils
is much smaller in the non-grouped subsample. Hence, the Belgian case depicts somehow the
pendant to the Austrian one: In Austria high status pupils witness a strong differentiation through
8
ability grouping, in Belgium this effect can be observed for low status pupils. The Polish case
has some similarities with the Belgian.
[Figure 3 here]
The fourth selected country is Hungary (see Figures 4 and A1), where ability grouping
show to equalise achievements in test scores for high status pupils. This is a unique finding and
contradicts the expected effect of ability grouping. The mean standardized kernel density shows
a much narrower form for high status pupils in ability grouping. The mean standardized QQ
Plots correspondingly show fatter tails for low status pupils in ability grouping. The nonstandardized QQ Plots in turn reveal that particularly high achieving low status pupils seem to
fall behind in the non-grouped subsample.
[Figure 4 here]
These four countries depict the main effects ability grouping has taken in the 27 OECD
countries. In the following section we will shed further light on these cases by estimating
quantile regressions.
5) Econometric Methodology
We base our econometric analysis on Roemer’s definition of educational equality (see
Section 2). The educational achievement of a child, in Roemer’s view, should be independent
from social background. We set up an econometric model in which we test whether ability
grouping reduces dependence between educational achievement and parental background. The
model contains an interaction term, which measures the effect of ability grouping on the socioeconomic gradient.
5.1 Econometric model
Our econometric analysis builds on Woessmann (2004), who applies a quantile
regression approach to individual-level data in a set of countries to analyse the effect of social
background on achievement across the PISA performance distribution. The paper is also related
to work by Causa and Chapuis (2009) who conduct a cross-country panel data analysis of the
role of education policies – including ability grouping - in equality of educational opportunities.
In this paper, for each country the following econometric model is tested:
9
∑
where
is the individual PISA maths score;
is the individual family background score;
indicates whether within-school ability grouping was present at the school the individual
attended and
represents the set of control variables (immigration background, gender,
language spoken at home and an indicator of whether the individual’s mother and father live at
home). ∑
represents the mean ESCS score across individuals in school s of individual i.
We include this term in order to account for school environment effects7.
denotes an error
term.
We are principally interested in the estimation of
, which represents the interaction
effect between the ability grouping indicator and the family background. It is a measure of the
effect of within-school ability grouping on the socio-economic gradient, which measures the
effect of the family background on the PISA score.
We are interested in how the effect of within-school ability grouping on the socioeconomic gradient varies over the conditional PISA performance distribution. We estimate the
equation above for each country by quantile regression, which allows assessing how independent
variables affect the shape of the conditional response distribution (Koenker, 2005). The
distributional effects can provide evidence of whether within-school ability grouping has a
heterogeneous effect across the conditional performance distribution. To compute the quantile
regression coefficients the absolute sum of errors is minimized. To produce estimates at different
ends of the error distribution the absolute sum is weighted: to compute coefficients for the lower
quantiles, the lower end of the empirical conditional distribution receives more weight and for
the higher quantiles the higher end of the conditional distribution receives more weight.
7
This measure is created by averaging the ESCS across individuals in the respective school of the respective
individual.
10
5.2 Endogeneity issues
As mentioned in Section 3, as indicator of within-school ability grouping we use a
variable measuring the presence of ability grouping at the school level reported by the principal.
The decision to perform within-school ability grouping is not prescribed by a nationwide policy
in the four countries we examine (see Section 2) and the decision of a principal to conduct
within-school ability grouping could therefore be correlated with school characteristics: an
unfavourable school environment could lead to the introduction of within-school ability grouping
and simultaneously affect PISA performance. We account for this possibility by including the
average level of the social background variable (ESCS) over all students from their respective
school. The school average of the ESCS variable should capture school environment effects by
accounting for the socio-economic composition of the students’ school and for peer effects.
Nevertheless, we cannot exclude the possibility that there are some school characteristics, other
than the school environment effect stemming from socio-economic background, that drive both
the presence of within-school ability grouping and the achievement of students.
A further source of endogeneity might arise on the individual level: the probability of visiting
a school with or without within-school ability grouping could be correlated with innate ability,
which also affects PISA performance. The question therefore arises whether the principal’s
decision to practice within-school ability grouping is related to the level innate ability. We argue
firstly, that there are other and stronger factors at play in this decision such as socio-cultural
factors of the community, elitism, pressure from different groups and preferences. Secondly,
when believing in ability level as a determinant in the principal’s decision of whether or not to
conduct within-school ability grouping, one can argue both ways: one could either argue that
schools with mainly students of low levels of innate ability may decide to conduct ability
grouping in order to help students in their specific needs or vice versa, that within-school ability
grouping would be conducted in institutions with students of high innate ability to select the best
and enhance their competences to a maximum.
6) Results
The results are displayed in Table A2 and summarized in Table 2.
11
Table 2: Estimates of Interaction term from quantile Regressions, per country
Estimates for Interaction term
q20
q50
q80
1.567
11.96*** 15.26***
Austria
(3.293)
(2.783)
(1.840)
Belgium 9.333*** 6.167*** 5.215***
(2.218)
(2.077)
(1.938)
-1.110
1.442
-2.977
Finland
(4.284)
(3.009)
(2.754)
-4.919** 7.028***
Hungary -2.198
(2.458)
(2.197)
(2.678)
In Austria within-school ability grouping is positively correlated with the effect of the
socio-economic background on the PISA score, which means that within-school ability grouping
will reinforce the effect of the socio-economic background and decrease educational equality.
This social stratification is reinforced by ability grouping most strongly for the better performing
students and socio-economic background has a stronger effect for the more able than for the less
able.
In Belgium, on the other hand, ability grouping reinforces stratification more for the less
able. Belgium can be seen as a ‘polarising’ country: ability grouping stratifies and does so mostly
for those at the lower end of the performance distribution.
In Hungary, ability grouping is associated with higher equality. The negative estimates
indicate that the total effect of the social background on PISA performance is smaller for pupils
at schools with ability grouping. While being insignificant for the less able, the equalising effect
of ability grouping in Hungary exists for medium performing pupils and is particularly
pronounced for high achieving pupils. However, the direction of the effect is most uncommon. In
our sample of 27 countries, negative estimates for the interaction effect are generally rare and
only Slovakia and Hungary show negative estimates for the interaction term over the whole
performance distribution.
The estimates for Finland do not indicate a certain effect of ability grouping. First, the
estimates are insignificantly different from zero. Second, the changing signs don’t even allow
recognizing a tendency for the entire performance distribution. Ability grouping does not seem to
12
matter for the performance in the PISA test and has no impact on the educational equality in
Finland.
It would be useful to link these empirical findings to a documentation of within-school
ability grouping measures (as in e.g. Hallam et al., 2003 for the UK) in these four countries.
However, very few sources examine the practice of within-school ability grouping in a
systematic way. Only for Finland we find suitable sources: Kupiainen et al. (2012) and OECD
(2012) document that within-school ability grouping as a mean to meet the needs of pupils with
rather different performance records is most uncommon. Ability grouping only occurs during the
first 6 grades by selective classes, i.e. in music or if certain languages are chosen. These findings
in fact match our descriptive statistics8 and regression results which show that there is virtually
no difference in math scores across schools with or without ability grouping. Even in the
simplest specification, the estimates for ability grouping and the interaction term remain
insignificant.
6) Conclusion
In this paper we set out to investigate the relationship of within-school ability grouping
with equality of educational opportunities across the PISA performance distribution. We
examined the distributional effect of an interaction term between a family background indicator
and an indicator for within-school ability grouping. The paper reveals that the effects of ability
grouping remain highly controversial. This holds for the general effect of ability grouping just as
for the effect over different ability groups. We observe that within-school ability grouping
evokes different reactions across countries and different reactions across the ability scale. The
OECD nevertheless has been a strong campaigner for a policy change towards less abilitygrouping and particularly invoked PISA results. Therefore, a more systematic and comparative
analysis of the effects of ability grouping at the country level should be implemented.
In particular, we find that the effect of within-school ability grouping on equality of
educational opportunity is heterogeneous across the four countries we study and that the effect of
within-school ability grouping on equality of educational opportunities is not the same across the
ability distribution. Quantile regression results indicate that in Austria within-school ability
8
A Kolmogorov-Smirnov equlity of distribution test confirms the finding and attributes the slight differences to
chance.
13
grouping is associated with stronger inequality, especially for more able children: talent might be
wasted if the socio-economic background is low. In Belgium within-school ability grouping is
associated with higher educational inequality, especially for less able children: stratification is
reinforced for the less able. In Hungary within-school ability grouping is correlated with a higher
equality especially for the more able: the policy has a beneficial effect especially for the more
able. In Finland, we do not find within-school ability grouping to have a significant effect on
educational equality.
Policy-makers should take into account that an educational policy might have a different
effect for different children. Our paper further shows that a regression at the average of the PISA
distribution may not show the entire picture. Our findings imply that within-school ability
grouping might have a different effect for different children and the effect is further mediated by
institutional effects of a country educational system. Therefore, policy-makers should be careful
in application of international one-size-fits-all recommendations. Understanding one’s own
education system and how it interacts with earlier or later tracking is essential for policy-making,
regardless of political objectives.
14
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18
Appendix
Table A1: Tracking age and share of grouped students by country
Tracking age
Share of non-gouped pupils
1 Austria
10
58.06
2 Belgium
12
53.2
3 Bulgaria
45.81
4 Czech Republic
11
28.24
5 Denmark
16
53.08
6 Estonia
15
41.27
7 Finland
16
39.78
8 Germany
10
49.12
9 Greece
15
86.08
10 Hungary
11
32.05
11 Ireland
15
3.27
12 Italy
14
43.71
13 Latvia
52.99
14 Lithuania
23.1
15 Luxembourg
13
33.13
16 Netherlands
12
22.04
17 Poland
16
52.1
18 Portugal
15
66.4
19 Romania
27.05
20 Slovakia
11
25.19
21 Slovenia
14
47.09
22 Spain
16
45.76
23 Sweden
16
25.9
24 UK
16
1.31
25 Norway
16
26.54
26 Switzerland
12
16.72
27 United States
16
10.19
Note: The dataset does not provide data for ability grouping in France; the tracking information are taken from:
OECD (2012)
19
Table A2: Regression Estimates for Austria, Belgium, Finland and Hungary
dependent:
Math Score
ESCS
Quantile Regression I
q20
q50
q80
30.25*** 33.15*** 33.50***
(1.490)
(1.597)
(1.383)
m_ESCS
ESCS*
ABGROUP
ABGROUP -54.28***
(2.535)
Female
-55.12***
(2.414)
-43.81***
(3.380)
449.3*** 520.7***
(1.874)
(1.499)
Observations
6174
6174
R-squared
Standard errors in parentheses
585.4***
(1.922)
6174
Mother at
home
Father at
home
Foreign
born
Foreign
born father
language at
home
Constant
dependent:
Math Score
ESCS
Quantile Regression I
q20
q50
q80
55.83*** 49.22*** 40.50***
(2.065)
(1.409)
(1.954)
2.347
(3.383)
9.146***
(2.351)
15.17***
(2.600)
Female
Mother at
home
Father at
home
Foreign
born
Foreign
born father
Language at
home
Constant
432.2*** 513.0***
(1.803)
(1.985)
Observations
8258
8258
R-squared
Standard errors in parentheses
-37.92***
(4.438)
-28.09***
(2.479)
-32.03***
(8.099)
-1.631
(3.441)
-0.755
(5.840)
-22.05***
(6.979)
-15.51**
(6.699)
503.3***
(10.65)
5460
-32.61***
(2.762)
-31.79***
(2.189)
-17.91*
(9.915)
-5.592
(3.604)
1.311
(5.745)
-28.30***
(5.599)
-10.29*
(6.023)
556.0***
(9.534)
5460
-27.66***
(3.809)
-31.25***
(3.111)
-24.79***
(9.362)
-0.447
(5.040)
4.625
(5.722)
-22.26***
(6.249)
-8.283
(5.797)
614.1***
(12.17)
5460
Quantile Regression III
OLS
q20
q50
q80
PV1MATH
7.630***
1.067
-0.932
2.407
(2.072)
(2.078)
(2.124)
(1.768)
79.27*** 96.28*** 94.19*** 89.80***
(3.823)
(3.693)
(3.922)
(3.025)
1.567
11.96*** 15.26*** 8.913***
(3.293)
(2.783)
(1.840)
(2.173)
-38.57*** -33.96*** -30.98*** -30.99***
(2.966)
(2.456)
(2.419)
(2.253)
-28.24*** -32.88*** -31.96*** -32.10***
(2.277)
(2.296)
(2.908)
(2.049)
-31.67***
-20.45*
-25.61*** -23.91***
(7.539)
(12.13)
(8.152)
(7.787)
-1.185
-6.901**
-1.292
-3.431
(4.311)
(3.359)
(4.162)
(3.005)
-0.967
0.0767
4.201
1.356
(5.258)
(5.501)
(8.179)
(5.232)
-21.32*** -26.96*** -19.27*** -22.43***
(5.136)
(4.700)
(5.733)
(4.037)
-15.37**
-9.902**
-11.77** -13.26***
(6.107)
(4.995)
(5.659)
(4.762)
502.4*** 561.8*** 618.6*** 557.6***
(8.579)
(12.54)
(10.94)
(8.637)
5460
5460
5460
5460
0.321
*** p<0.01, ** p<0.05, * p<0.1
m_ESCS
ESCS*
ABGROUP
ABGROUP
Regression Estimates Austria
Quantile Regression II
q20
q50
q80
8.005***
5.527**
6.722***
(2.114)
(2.733)
(2.588)
79.40*** 96.67*** 97.89***
(4.844)
(4.552)
(5.669)
587.3***
(1.754)
8258
Regression Estimates Belgium
Quantile Regression II
q20
q50
q80
15.10*** 13.32*** 12.16***
(1.966)
(1.324)
(1.612)
105.9*** 94.13*** 81.12***
(3.499)
(3.468)
(3.905)
22.54***
(2.995)
-27.91***
(2.669)
-34.16***
(9.043)
-2.908
(5.091)
-15.05**
(6.321)
-24.81***
(3.745)
2.210
(3.204)
486.7***
(13.00)
7102
20.68***
(2.299)
-25.01***
(2.374)
-28.59***
(6.738)
-2.460
(3.378)
-11.13**
(4.368)
-27.22***
(2.968)
-1.014
(2.736)
549.7***
(9.651)
7102
*** p<0.01, ** p<0.05, * p<0.1
20
15.49***
(2.585)
-28.89***
(2.544)
-20.70*
(11.93)
-0.266
(3.802)
-8.537*
(4.484)
-29.50***
(3.114)
0.898
(2.877)
609.8***
(14.97)
7102
Quantile Regression III
OLS
q20
q50
q80
PV1MATH
9.863*** 9.828*** 9.333*** 9.873***
(1.838)
(2.109)
(2.001)
(1.475)
107.0*** 95.16*** 81.50*** 93.97***
(3.646)
(3.047)
(3.460)
(2.229)
9.333*** 6.167*** 5.215*** 6.148***
(2.218)
(2.077)
(1.938)
(1.348)
19.53*** 18.08*** 12.71*** 15.41***
(2.529)
(2.628)
(2.180)
(1.897)
-27.22*** -25.74*** -29.02*** -27.20***
(2.791)
(1.976)
(3.280)
(1.808)
-36.78*** -28.39*** -24.04*** -27.69***
(5.062)
(5.782)
(7.915)
(5.889)
-0.986
-3.642
0.923
-1.810
(4.091)
(3.145)
(3.761)
(2.826)
-15.33*** -13.05*** -11.84** -10.52***
(4.584)
(3.109)
(4.731)
(3.502)
-25.94*** -25.66*** -27.84*** -28.12***
(2.692)
(2.224)
(3.315)
(2.645)
2.617
0.935
2.792
1.449
(2.301)
(2.161)
(2.904)
(2.248)
487.9*** 551.0*** 612.6*** 548.9***
(8.617)
(7.577)
(10.56)
(6.820)
7102
7102
7102
7102
0.372
dependent:
Math Score
ESCS
Quantile Regression I
q20
q50
q80
26.92*** 27.48*** 27.73***
(2.439)
(2.244)
(2.087)
m_ESCS
ESCS*
ABGROUP
ABGROUP
0.535
(2.132)
-1.590
(1.972)
-4.420*
(2.517)
Female
Mother at
home
Father at
home
Foreign born
Foreign born
father
Language at
home
Constant
460.4*** 530.5***
(2.177)
(2.417)
Observations
5774
5774
R-squared
Standard errors in parentheses
dependent:
Math Score
ESCS
12.35***
(2.036)
11.23***
(2.889)
Female
student
Mother at
home
Father at
home
Foreign born
Foreign born
father
Language at
home
Constant
431.7*** 497.3***
(1.802)
(2.162)
Observations
4582
4582
R-squared
Standard errors in parentheses
-1.321
(2.898)
5.036
(3.496)
-10.13
(12.08)
-13.42***
(4.506)
-16.97
(16.42)
-24.01***
(8.908)
-18.92***
(7.055)
488.1***
(13.82)
5452
-0.504
(2.193)
-4.201**
(2.089)
-11.29**
(5.515)
-14.81***
(4.963)
-12.55
(9.467)
-22.09***
(7.327)
-16.89***
(5.477)
563.9***
(7.506)
5452
-2.229
(2.311)
-9.891***
(2.924)
-13.10**
(6.578)
-9.095***
(3.062)
2.132
(13.90)
-13.48
(8.214)
-18.81**
(7.691)
623.4***
(8.913)
5452
Quantile Regression III
OLS
q20
q50
q80
PV1MATH
23.15*** 23.99*** 27.04*** 23.14***
(3.765)
(2.747)
(2.717)
(2.220)
11.00
4.929**
10.78**
8.751**
(7.104)
(1.964)
(5.344)
(3.810)
-1.110
1.442
-2.977
0.872
(4.284)
(3.009)
(2.754)
(2.569)
-0.880
-1.339
-1.487
-1.891
(4.347)
(4.254)
(3.280)
(2.445)
5.540*
-4.145
-9.211*** -4.645**
(3.005)
(2.750)
(2.611)
(2.109)
-10.04
-10.12
-13.62*
-10.44*
(7.692)
(6.791)
(7.037)
(5.489)
-13.02*** -14.23*** -7.949*** -11.79***
(3.829)
(3.532)
(2.732)
(3.062)
-16.47
-11.08
-1.708
-13.28*
(12.57)
(13.44)
(15.35)
(7.360)
-25.15*** -22.07***
-12.66
-15.55***
(7.866)
(6.591)
(8.561)
(5.450)
-19.31*** -16.69*** -18.32*** -18.84***
(5.708)
(4.103)
(5.290)
(4.454)
486.8*** 562.4*** 622.1*** 556.7***
(10.75)
(8.345)
(9.341)
(7.284)
5452
5452
5452
5452
0.076
*** p<0.01, ** p<0.05, * p<0.1
Quantile Regression I
q20
q50
q80
46.16*** 48.32*** 45.52***
(1.314)
(1.545)
(2.252)
m_ESCS
ESCS*
ABGROUP
ABGROUP
596.4***
(2.579)
5774
Regression Estimates Finland
Quantile Regression II
q20
q50
q80
22.39*** 24.43*** 24.34***
(2.098)
(2.055)
(1.650)
10.57
5.168
11.32***
(6.535)
(3.385)
(3.702)
Regression Estimates Hungary
Quantile Regression II
q20
q50
q80
5.015**
9.024*** 11.46***
(2.204)
(1.083)
(1.672)
90.89*** 88.43*** 88.11***
(3.767)
(1.832)
(2.732)
Quantile Regression III
OLS
q20
q50
q80
PV1MATH
7.087*
12.43*** 15.95*** 12.16***
(3.907)
(2.626)
(2.768)
(1.987)
89.50*** 88.15*** 87.42*** 87.98***
(2.901)
(3.068)
(3.566)
(2.121)
-2.198
-4.919** -7.028*** -4.400**
(2.458)
(2.197)
(2.678)
(2.001)
3.658
-3.148
-2.184
-1.452
-4.615*
-3.949
-2.365
-4.111*
(3.895)
(3.267)
(2.003)
(2.310)
(2.649)
(3.291)
(2.404)
(2.148)
-18.70*** -21.34*** -24.97*** -18.42*** -21.90*** -25.98*** -22.07***
(3.587)
(2.945)
(2.336)
(3.375)
(2.222)
(2.940)
(1.944)
-24.03*** -17.07*** -18.04** -24.10*** -17.30*** -14.96** -16.23***
(7.187)
(4.666)
(9.133)
(7.038)
(5.742)
(7.343)
(5.050)
-3.492
-4.093**
0.903
-4.169
-3.577
1.840
-1.661
(3.942)
(2.074)
(3.223)
(3.901)
(2.959)
(3.322)
(2.523)
5.023
-6.454
0.196
5.774
-6.440
2.107
-1.578
(15.18)
(7.334)
(14.24)
(6.930)
(6.755)
(12.04)
(8.018)
-4.903
-4.404
-10.50
-4.903
-4.012
-8.880
-2.460
(10.35)
(4.219)
(10.82)
(7.687)
(4.716)
(7.229)
(6.006)
-22.44**
-52.76**
-33.12**
-22.23
-51.74**
-27.38
-25.98**
(10.99)
(22.36)
(15.77)
(13.51)
(23.43)
(23.46)
(11.51)
564.5*** 500.1*** 548.9*** 598.3*** 501.6*** 549.8*** 595.5*** 545.9***
(3.023)
(8.564)
(6.090)
(10.92)
(8.660)
(5.310)
(6.105)
(6.053)
4582
4341
4341
4341
4341
4341
4341
4341
0.476
*** p<0.01, ** p<0.05, * p<0.1
Note: Quantile Regression I ist the simplest specification, II adds the controls, III the interaction term.
21
[Figure A1 here]
[Figure A2 here]
[Figure A3 here]
[Figure A4 here]
22
Apppendix
Figure 1: Mean standardized QQ-plots of PISA maths scores by social background and ability
grouping in Finland
Figure 2: Mean standardized QQ-plots of PISA maths scores by social background and ability
grouping in Austria
23
Figure 3: Mean standardized QQ-plots of PISA maths scores by social background and ability
grouping in Belgium
Figure 4: Mean standardized QQ-plots of PISA maths scores by social background and ability
grouping in Hungary
24
Figure A1: Mean standardized Kernel density estimates for PISA maths scores by socioeconomic status and ability grouping in Finland
Figure A2: Mean standardized Kernel density estimates for PISA maths scores by socioeconomic status and ability grouping in Austria
25
Figure A3: Mean standardized Kernel density estimates for PISA maths scores by socioeconomic status and ability grouping in Belgium
Figure A4: Mean standardized Kernel density estimates for PISA maths scores by socioeconomic status and ability grouping in Hungary
26
Figure A5: QQ-plots of PISA maths scores by social background and ability grouping in Finland
Figure A6: QQ-plots of PISA maths scores by social background and ability grouping in Austria
27
Figure A7: QQ-plots of PISA maths scores by social background and ability grouping in
Belgium
Figure A8: QQ-plots of PISA maths scores by social background and ability grouping in
Hungary
28
Figure A9: Plotted estimates from Quantile and OLS Regressions for Austria, Belgium, Finland
and Hungary
29
Figure A10: Mean standardized Kernel density estimates for PISA maths scores by socioeconomic status and ability grouping in 27 OECD countries.
30
31