Discussion Papers In Economics
And Business

The effect of locally hired teachers on school outcomes (the Dose
response function estimation evidence from Kenya)

Ayako Wakano

Discussion Paper 16-15

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe effect of locally hired teachers on school outcomes (the Dose
response function estimation evidence from Kenya)

Ayako Wakano

Discussion Paper 16-15

May 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe effect of locally hired teachers on school outcomes (the Dose response function estimation
evidence from Kenya)
Ayako Wakano†

Abstract
Do locally hired teachers benefit pupils’ school achievements more than governmental employedteachers? In Republic of Kenya (below referred as Kenya), there are two types of teachers in public
primary schools. One is those employed by the government and the other is those hired by the local
school community, named “PTA teacher”. Though locally hired teachers are in general less qualified interms of educational background and paid substantially lower than that of governmentally employedteachers, past randomized experiment results show that the marginal product in terms of test score ispositive and significant when pupils are taught by PTA teachers (Duflo et al. 2012, and Bold et al.,2013).

By using a nationally representative rich data set, with the Generalized propensity score matching
method, the present study examines the effect of PTA teacher ratio (ratio of PTA teachers out of total
number of teachers) on education outcome. The question of this study is “if PTA teachers havesuperior performance, proved by the Randomized Controlled trial in Kenya, should higher PTA
teacher ratio in one school bring better educational outputs?”. With the nationally representative
dataset containing rich educational school inputs as well as individual pupils’ background andhousehold information, this paper estimates the dose response function of school average outcomes.
Provided that government teachers’ allocation and school selection by the parents can be an
endogenous to pupils’ school outcomes, this paper utilizes the generalized propensity score method by
Hirano and Imbens (2004) which enable us to estimate the function of the continuous treatment effect,
PTA teacher ratio. The result consistently shows that the PTA teacher ratio affects school outcomes
nonlinearly.

JEL code: C30, C32
Keywords: Continuous treatment, Generalized propensity score, Program evaluation†Corresponding author: Graduate School of Economics, Osaka University,
Address: 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043, JAPAN,
E-mail address: qgp802wa@student.econ.osaka-u.ac.jp

1．Introduction
One of the significant improvements in education sector for the past two decades is drastic increase inenrollment of primary school pupils in developing countries. According to the UNESCO Institute of
Statistics (UIS) reports, the out of school children in the world has greatly reduced from 100 million to
58 million as of 2012 since 2000 (UIS, 2014)1. At the same time, facing the sudden increase inenrollment of primary school pupils, supply side was not kept up with the pace, especially supply ofteachers. After the Free Primary Education policy, how to supply school teachers has become one of
the substantial topics in education sector. Within the limited budget, various developing countries wereto introduce the system of “contract teachers”. For example, in 11 west African countries, on average,
nearly 50% of primary school teachers were contract teachers (Bourdon et al., 2007). In Kenya,
though there is no official figure for the number of PTA teachers, it is on average 20% of teachers inschools employed by the PTA (Alice Muthoni, 2010, Uwezo Kenya, 2010)2.

The characteristics of contract teachers are with shorter term contract and in general with lower
qualifications than that of governmentally hired teachers (Duthilleul 2005). For example, half of the
PTA teachers in Kenya have a Primary 1 (P1) certificate as their highest qualifications, whereas 20%
had no formal qualifications. On the other hand, 79% of the governmentally employed teachers had P1
qualifications (Alice Muthoni, 2010). In terms of the cost, the PTA teachers are paid substantiallylower, which is one forth of that of governmentally employed teachers (Duflo et al., 2012, Alice
Muthoni, 2010).

Substantial reduction was found in south and west part of Asia from 2000 to 2012 and thehalf of the entire out of children as of 2012 is concentrated in sub-Sahara Africa. A significant
decline in the number of out of school children is mainly due to the “Free Primary Education”
policy introduction (UIS, 2014).
http://www.uis.unesco.org/FactSheets/Documents/fs-28-out-of-school-children-en.pdf
2 This figure varies slightly with sources. One report approximates 17.5% of teachers are paid by
the PTA (Alice Muthoni, 2010) and the other repot says on average, 1 out of 5 teachers are employedby the PTA (Uwezo Kenya, 2010).
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While it is obvious that hiring PTA teachers is less costly, past experiments show that performance is
not lower than the government employed teachers. The performance of these contract teachers is
empirically studied, based both on the randomized experiment and by observational data (Atherton
and Kingdon 2015, Bold et al., 2013, Muralidharan and Sundaraman 2013, Duflo et al., 2012,
Bourdon et al., 2010, Duthilleul 2005, and Vegas and De Laat 2003). Experimental evidence infers
that PTA teachers cause better educational outcomes measured by test score in Kenya (Duflo etal.,2012) . In India, marginal effect of randomly allocating local teacher raises test score withstatistical significance. As the experimental evidence in Kenya and India, it shows that PTA teacher is
as good as or better than the government employed teachers. Then, if PTA teachers are superior inteaching, should all teachers be replaced by PTA teachers?. The more PTA teachers in one school
would bring better educational outcomes on average?”. The cost of hiring PTA teachers is substantiallylower than government teachers, the question is important to be answered. Provided that PTA teachers’
allocation within a school could be an endogenous to pupils’ school outcomes, this paper is toinvestigate the aggregated effect of PTA teachers, that is, the ratio of PTA teachers (the number of PTA
teachers out of total teachers in one school) on pupils’ school outcomes.

2. Preceding Studies
Regardless of highly important policy implications, publications solely dealing with locally hired“contract teacher” effect is scarce (Atherton and Kingdon 2015, Bold et al., 2013, Muralidharan andSundaraman 2013, Duflo et al., 2012, Bourdon et al., 2010, Duthilleul 2005, and Vegas and De Laat2003). For the causal inference, randomized controlled experiments were conducted in two
countries and both indicate that PTA teachers’ effect on school outcomes is positive andsignificant3. On the other hand, in observational studies, Atherton and Kingdon (2010)
Duflo et al. (2012) has shown that the pupils taught by PTA teachers have significantly
increased test scores, compared to those taught by government hired teachers. Bold et al.
(2013) also shows the result of randomization controlled trials in Kenya that the test scores ofpupils taught by the PTA teachers have significantly increased when the program was
implemented by an International NGOs. Muralidharan and Sundaraman (2013) have showed thatthe randomly assigned contract teachers in India contribute the increase of test scores,-23

reports that by school fixed estimation method, contract teachers in North India are no lesseffective on school test scores than regular teachers. On the other hand, Vegas and De Laat
(2003) show that contract teacher performance is systematically underperformed, using
samples from Togo, while Bourdon, J. et al. (2010) shows that, by using observational data for Niger,

Togo and Mali with nonparametric quantile regression estimation, the average treatment effect as wellas the effect on a certain quantile are not consistently positive among three countries.

In this way, while the experimental result shows that the impact of PTA teacher deployment raises testscore, observational studies have the mixed result. The contribution that this paper makes on the past
literature is as follows: First, this study is to focus on the effect of “existing” PTA teachers on pupils’
school outcome, not to measure the marginal effect of “newly” hired or “randomly” deployed teacher.
It is worth noting that there is significant difference between newly deployed teacher by theexperiment and existing PTA teachers regularly paid by school communities. The nature of existing
PTA teachers is largely differed in the contract and payment scheme from the one of experiment
deployed teachers, which would greatly affect working motivation and effort level of teachers. As one
forth of total teachers are existing PTA teachers in Kenya, the nature of the contract and payment
scheme would greatly affect the working incentive of teachers. For this reason, the present study uses
the observational data.

For the use of observational data, obstacles exist in identification due to two types of unobserved
heterogeneity which affect the school outcomes. One is the unobserved heterogeneity among schools
and the other is the unobserved heterogeneity within a school. For the former heterogeneity, forexample, government prioritizes in allocating teachers to schools where pupils with higher test scores
are resident in and parents who are aware of school outcomes might self-select schools with less PTA

through reducing the PTR (pupil teacher ratio). The result is that the effect of PTR reduction
by contract teachers is not less effective to the test scores as reduction by regular teacher.
-3-teachers. The identification strategy of this paper must rely on the assumption that the selection into
schools with different PTA teacher ratio is random, conditional on the covariates. To sufficiently
remove the heterogeneity among schools with different PTA teacher ratio, this paper uses one of thelargest dataset existing in East African countries, which contains various educational inputs for schools
and household levels. The specification of propensity score does include the 23 covariates and 157
district level dummies so that the unobserved heterogeneity between different PTA teacher ratio
schools would be removed. By putting the conditional independence assumption that school average
test scores are independent from the PTA teacher ratio conditioned on the rich covariates, the paper
identify the treatment effect on test scores. As the propensity score matching method (Rosenbaum andRubin, 1983) is often utilized in the empirical works, the paper uses the extended propensity score
method by Hirano and Imbens (2004) for the case that treatment is continuous. Our treatment, PTA
teacher ratio does not take binary value but takes continuous values. In order to estimate thecontinuous treatment effect, the extended propensity score method is suggested (Imbens, 2000; Hirano
and Imbens, 2004; Imai and van Dyk, 2004). By using the framework of extended propensity score
method, this paper estimates the function of different treatment effect.

For the latter unobserved heterogeneity is within a school; for example, the assignment of PTA or TSC
teachers is not presumably random and pupils might be systematically sorted into particular type ofteachers without being observed. If TSC teachers are systematically assigned to higher score pupils,
then the effect of PTA teacher is estimated downward. Even in the case of randomized experiment,
when there is unobserved sorting process of pupils into different type of teachers (Bold et al., 2013), it
becomes serious obstacle to the identification, explained in detailed by Muralidharan and Sundaraman(2013). In order to avoid the unobserved heterogeneity within a school, the present study focuses on
the school level aggregated variable, the ratio of PTA teacher.

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The second contribution is that the present study uses observations from geographically broad areawith larger sample size. As for the experimental studies, two experiments were conducted in
Kenya. One is in 140 primary schools in Western province and the other is in 192 schools ineight provinces (Duflo et al., 2012 and Bold et al., 2013), while for India, the sample was takenfrom 200 primary schools in one state (Muralidharan and Sundaraman 2013). The present study
uses the nationally representing dataset containing 4,789 primary schools in all eight provinces4.

Broader geographical coverage is preferred, since the heterogeneity of region is not ignorable. From
this respect, the present paper contributes the past literature to confirm whether the estimated result isstill valid even considering the geographical diversity.

3. Characteristics of Teachers in Public Primary Schools in Kenya
Present data does not have variables of characteristics of teachers such as age, years of education andexperiences; however this section qualitatively describes the differences in general between the PTA
and TSC teachers.
i) The employment of TSC teachers
As for the hiring and firing authority for TSC teachers, it is highly concentrated in the institutions
called the Teacher Service Commission (TSC), delegated from the Ministry of Education Science andTechnology (MoEST). In order for teachers to be certified by the TSC, one needs to acquire academic
qualifications, ranging from the graduates of primary schools to Diploma. If non-graduate wishes to be
a TSC certified teacher, they need to acquire a relevant degree or qualification from a recognizedinstitution (UNESCO 2010). There are 25 public and 8 private colleges for training primary school
teachers. Once they become certified TSC teachers because of the protective laws and union, it isunlikely to get fired (Duflo et al., 2012, Glewwe et al., 2010). On the other hand, PTA teachers are
hired by the school community, parents and other school members such as headmasters. With scarce

In the estimation, we use the sample restricted for schools with positive value of PTA
teacher ratio. Hence, the total effective sample size is reduced to 3778 primary schools for 4th
grade pupils, 3728 schools for 3rd and 3706 schools for 2nd.
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qualitative information, it is often the case that PTA teachers’ contract is annually renewable based on
their performance.

ii) Monitoring on teachers
According to the laws and regulations of governmental monitoring policies, the work of government
teachers is principally monitored by the District of Education Office (DEO)5. However, as qualitative
studies explained, due to the large number of schools in one district, district education officers are lackof resources in mobilizing the monitoring school teachers (Omuga Thomas 2009, James Kariuki 2010,
John Ogamba et al., 2013). For example, it is articulated that the “comprehensive standard assessmentof all educational institutions be carried out by external Quality Assurance and Standards Officers(QASOs) at least once every term for primary schools” (James Kariuki, 2010)6. However, the table [1]shows merely 39% of the entire visits are confirmed by the headmaster during the term from January
to March 2012. The table [1] shows the number and percentage of schools that answers how many
months have passed since the previous Quality Assurance Officer visit7. In Kenya, there is a trimester
system. The first term starts from January to March, the second from May to July and the last term
starts from September to November. Although the regulation articulates the assurance officer must
visit schools at least once in the term, 61% visits were made before January 2012, including 30% of
missing information. When the previous term from September to November in 2011 included, it
becomes 53% of the total officers’ visit. The rest of nearly half of the visits were either done before
August of 2011, which is two terms before the surveyed or unknown about when the inspections
made.
5

In Kenya, country is divided into eight administrative regions (known as provinces) where each province
has a Provincial Director of Education Office. In each province, there are districts, and in total 158 districts
exist in the entire land and District of Education Office is (DEO) in charge of the monitoring of schools.
6 The department of Quality Assurance and Standards (DQAS) was established in 2004, andits’ mandate of this department is to ensure quality and standards in Kenya with the vision of “to provide
quality assurance feedback to all educational institutions in Kenya.” The Ministry of Education service
charter (2009) and the DOAS circular 2/1A/Vol.II/86 of 2009 states the frequency of the external quality
assurance (James Kariuki, 2010).
7
Out of 4,234 schools that 2012 dataset contains, the effective answers are from 2,968 schools and29.9% of schools have missing information.
-6-

Table [1]Not only has the monitoring from the government, but also from the school community on TSC
teachers become less since the introduction of the FPE (Free Primary Education) policy in 2003.
Qualitative studies explain that before the policy, school communities were in charge of collecting
school fees from parents and were entitled to decide how to spend the collected money. In that time,teachers were obliged to be accountable to payers, that is, parents and school community. Howeverafter introducing the FPE (Free Primary Education) policy, schools need to be more accountable to theDistrict of Education Office as well as to the Central Government which decides the allocation of thecapitation fund (Sasaoka et al. 2007). In this way, while the Free Primary Education has succeeded inassisting low income parents to send their children to school, however it happened to deprive theinterest of parents to manage schools or incentive to monitor schools (Sawamura 2004a).

iii) Wage payment scheme
In addition to the employment and monitoring teachers, the determination of wage level is clearly
different between two teachers. In developing countries, salaries and promotion of civil servants arelikely to be decided by educational qualifications and seniority, with less scope of performance(Chaudhury et al. 2006). In Kenya, the wage for TSC teachers is determined by education history, notby their performance (Sawamura 2004b). As for PTA teachers, although the wage level of PTA teacher
depends on schools, it is determined by the school local community who can directly verify theperformance of teachers. The amount of wage for PTA teachers is substantially lower than that of TSC
teachers. One report indicates that the average payment for PTA teachers is 4,151 Kenya Shillings,-7-

which is one fourth of the average salary 19,372 Kenya Shillings of the TSC teachers (Alice Muthoni,2010)8.

4. DataThis paper uses dataset of UWEZO Kenya, a part of Twaweza, “an independent East African initiative”(UWEZO East Africa 2012), which conducted one of the largest national assessments in three eastern
African countries for individual, school and local community. The annual national assessment survey
was conducted in three countries, namely Kenya, Uganda, and Tanzania, to assess the basic
competency of numeracy and literacy for children aged from six to sixteen. For the 2012 assessment inthree east African countries, 350,000 children in over 150,000 households were tested in ability oftasks at the Standard 2 level (UWEZO East Africa 2012). Specifically in Kenya, 2012 UWEZO annual
assessment covers 156 districts out of 158 (census) districts, which becomes 153,900 children assessedfrom 90,820 households, and 4,543 primary schools, that is, 20 percent of Kenya public primaryschools surveyed at the same time so that surveyed pupils and household information as well as school
information are merged at one dataset (UWEZO Kenya 2012).

Sampling methodology is based on the Census frame and the district is the main stratum9. Inside the
district, there are 30 villages randomly selected and at least one public primary school per village isincluded. Within a village, 20 households are randomly selected and for the selection, it does notmatter whether their children are attending the surveyed schools or not. All children are tested in theirhousehold environment and no tests are examined at their schools, hence children both attending and
not attending schools are included.

There were three rounds of the national annual assessments in Kenya since UWEZO started to conduct
survey in 2009. The geographical coverage has been expanded since 2009 as table [2] shows indetailed. The first round was in 2009 (70 districts covered), the second in 2011 (124 districts covered)and the third round was conducted in 2012 (156 districts covered out of the 158). Per district, it is arotational panel with 10 villages newly replaced each year and 20 villages are remained. As for theschools, the remained 20 primary schools per district may or may not be changed for the following
8

It slightly varies from different sources. Duflo et al. (2012) reports the wage for PTA teachers as of 2004
is in the rage of 2000 Kenyan shilling, while that of the average civil service teacher is US$ 120 per monthplus benefits and allowance (Duflo et al., 2012).
The detailed sampling methodology is written in the report available from the website (Standard
Manual, Uwezo East Africa at Twaweza, Kenya).
9-8-

year. Hence, though 20 villages are overlapped for two consecutive years, those 20 primary schools
per district are not necessarily paneled. In this way, the number of overlapped effective schools fromyear 2011 to 2012 is small, in number 372 schools available in dataset10. Therefore, the present study
does not apply the panel data analysis; rather the pooled data of two consecutive years and our unit ofobservations is at school level11. The variables included in estimations are as follows;

Table [2]1) Independent Variables
Subjects of Swahili, English and Numeracy were scored at the time of household survey. Score of
Swahili and English is rated from 1 to 5 while Numeracy is rated from 1 to 812. Following variables
are selected as controlling covariates, based on past studies of education production literature.

2) Class size, Pupil Teacher Ratio (PTR) and PTA teacher ratio variables
Since PTA teachers are often hired by school community to compensate the shortage of deployed
government teachers (TSC teachers), without controlling the effect of shortage of teachers and pupilteacher ratio, PTA teacher ratio is only to capture the effect of large class size or large number of
pupils per one teacher. To control the effect, pupil teacher ratio (PTR) and class size are included. As
those above variables are skewed to the right, they are converted to logarithm and used in theestimation.

3) Household and Pupil Variables
Traditionally, the education production function literature assumes that the educational achievement isdetermined by the supply side such as school qualities or educational inputs and the demand side suchas household characteristics. As for the demand side, it is assumed that the household takes the utility
10This is the number of primary schools which are overlapped from 2011 to 2012 and pupils grade below 4
attend. Some schools do not have any children grade below 4, hence the number of overlapped schools becomes
smaller.
11
The detailed effective samples selection process is in table in Appendix A. In the dataset, those attending thesurveyed schools and grade lower than 4 are 35,642 for 2011 and 39,681 for 2012. Due to number of missing
values in covariates, the effective samples for 2011 and 2012 pooled dataset become 47,812. The subsample ofgrade below 4 is somehow arbitrary, hence specifications based on the subsample of pupils grade above 5 andbelow 8 are shown in appendix B. Furthermore, the estimate based on the subsample with missing information
drastically increases the sample size. The estimation with missing dummies yields qualitatively similar result andavailable from the author.
12
The level of score rate in each subject is as follows: As for Swahili and English, children can read 1. Nothing,
2. Letter, 3. Words, 4. Paragraph, and 5. Story. As for Numeracy, children can do 1. Nothing, 2. count 1-9, 3.
count 10-99, 4. count greater than 99, 5. calculate addition, 6. calculate subtraction, 7. calculate multiplication,and 8. calculate division.

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maximizing behavior in choosing the optimal investment amount in children subject to the budget
constraint. Hence, as proxies for wealth of a household, the asset variables were used, such as type ofhouse wall and possession of telephone. Also, the literature finds that there might be a resource
allocation issue about who to be invested most if the number of siblings is large. There is no variable
for the number of siblings; however as the proxy variable, total number of household members is used
to control the effect of resource competition.

In terms of unobserved ability of children, though not possible to totally control the effect, there arevariables to partially control the unobserved. The effect of parents’ education attainment is controlled
by years of education for mother and father, which can directly affect unobserved ability of children.
In order to control parents’ preference over children education, the variable whether parents pay tuition
or extra fee to school is included. As for pupil characteristics, variables included are age and theirschool grade. The older they are, it becomes easier to take high score, since the test level is set forStandard grade two level. Pupils are aged from three to sixteen and grade from one to fourteen.
Primary school pupils are from age five and grade one to grade eighth. Each age and grade category
dummy is included for the fixed effect. To control the physical health, the number of meal in one daythat pupil can take is also controlled.

4) School Input Variables
In the early study of education production function estimation, Heyneman and Loxley (1983)
explained that “the poorer the national setting in economic terms, the more powerful this (school
inputs and teacher quality) effect appears to be”. Since this “H-L effect” has been widely recognized
and supported by several empirical results, this paper also controls the supply side information,
including two variables for school inputs, in addition to PTR and class size. 1) Dummy variable if
school has learning supplement materials or not. 2) Dummy variable if school is fenced or not, since
the security in some areas is remained fragile after the large scale of post-election violence in 2007
December. While there are various types of educational input variables available in UWEZO Kenya
2012 and 2011 dataset, after most variables were tried to be included, however the coefficient of PTA
teacher ratio is not largely affected.

5) Village and District Variables
It is often pointed out that there exist large disparities in terms of educational input between regions in
Kenya. There are eight administrative regions in Kenya called Provinces and in each Province, there- 10 -

are countless tribes and ethnic minorities live. It is rationally assumed that the cultural intensity affects
the behavior of households and hence affects education attainment. In order to control thegeographical characteristics and ethnic diversity, 157 district dummy variables are added for the
district fixed effect. As for village characteristics where the school is located, the dummy variable ofexistence of paved roads and existence of electricity infrastructure are added.

5. Summary statistics and common support condition
Table [3.1] shows the summary statistics for the pupils in grade four before imposing the common
support condition. Due to the limited space, the summary statistics for pupils in grade three and two isin appendix. To implement the propensity score method, it requires that there are sufficient number ofschools which overlap the propensity score. It is because that propensity score matching relies on theassumption conditioned on the propensity score, the treatment is random to the outcome. Therefore,
even those schools with different treatment levels, the generalized propensity score of estimation
sample is sufficiently overlapped.

Following the literature (Flores et al. 2012, Kluve et al. 2012), we divide the treatment level intoquantile groups (25, 50, 75 and 100 percentile) and estimate the generalized propensity score using the
medium value of each quantile group. In general, in order to examine the overlapping condition for
quantile group “q”, we estimate the generalized propensity score 𝑅̂𝑖𝑞 using the representative treatment
value of the “q” group. In this paper, we use the medium value as the representative value of the group.
Table 3.1 shows each quantile medium and mean value of the treatment and number of observations.
As Flores et al. (2012) shows, the common support for quantile groups “q” is defined as follows. For
example, how the common support condition imposed in the first quantile group is that minimum
value of the estimated propensity score 𝑅̂𝑖𝑞 must be greater than the minimum value of the rest of
propensity score 𝑅̂𝑗𝑞 , while the maxim value of the estimated propensity score 𝑅̂𝑖𝑞 should be smaller𝑞than the maximum value of the rest of propensity score 𝑅̂𝑗 . Based on this definition, the effective sample

size is reduced as table 3.2. Figure 1 shows the distribution of generalized propensity score after
imposing the common support restriction and before13.
𝑞
𝑞
𝑞
𝑞
𝑞
𝐶𝑆𝑞 = {𝑖: 𝑅̂𝑖 ∈ [𝑚𝑎𝑥 {𝑚𝑖𝑛{𝑗:𝑄𝑗=𝑞} 𝑅̂𝑗 , 𝑚𝑖𝑛{𝑗:𝑄𝑗≠𝑞} 𝑅̂𝑗 } , 𝑚𝑖𝑛 {𝑚𝑎𝑥{𝑗:𝑄𝑗=𝑞} 𝑅̂𝑗 , 𝑚𝑎𝑥{𝑗:𝑄𝑗≠𝑞} 𝑅̂𝑗 }]}Table [3]For the brevity, the histograms for the grade 3 and 2 samples are not shown. The shape issimilar to the one of grade 4 school sample. It is available from author upon the request.
- 11 13

Table [3.1]
Table [3.2]Figure 1

6. Identification Strategy
6.1 Estimation of the propensity score
At first, the generalized propensity score is estimated by modeling the conditional distribution oftreatment variable. Instead of putting the specific assumption of modeling of treatment variable, thepaper uses more flexible modeling by the generalized linear models (McCullagh and Nelder, 1989).
The generalized linear models for treatment variable 𝑇𝑖 is unknown function “g” of the conditional
expectation of treatment variable 𝑇𝑖 given covariates 𝑋𝑖 is linear combination of covariates with
vector coefficients 𝛾.
𝑔[𝐸(𝑇𝑖 |𝑋𝑖 )] = 𝑋𝑖 𝛾
There are various modeling available by choosing the distribution function 𝐹 as well as the functional
form 𝑔. By comparing the model fitting, this paper uses the log-normal specification applied, basedon the Akaike information criteria. The estimated coefficients vector is shown in table [4.1].
𝑙𝑜𝑔(𝑇𝑖 |𝑋𝑖 ) ~ 𝑁{𝑋𝑖 𝛾, 𝜎 2 }As the definition of the generalized propensity score, the conditional density distribution of thetreatment variable given covariates is as follows:𝑟(𝑡, 𝑥) = 𝑓𝑇|𝑋 (𝑡|𝑋 = 𝑥)
Accordingly, the probability density function for the actual treatment level 𝑇𝑖 for the school i isestimated as follows.
𝑅̂𝑖 (𝑇𝑖 , 𝑋𝑖 ) =2(𝑇𝑖 − 𝑒 𝑋𝑖𝛾̂ )𝑒𝑥𝑝 {−}2𝜎̂ 2√2𝜋𝜎̂ 2
1𝑡

For the given treatment level “t”, we also define the random variable 𝑅̂𝑖 = 𝑟(𝑡, 𝑋𝑖 ).

6.2 Balancing property of the propensity score
The role of propensity score is to balance the difference between those schools in a certain treatment
level and those that are not in the same treatment level. To articulate in formula, given the certain
treatment value of “t”, the observed covariates are independent from whether it takes the value of “t”
or not, conditioned on the propensity score given the treatment level “t”.
𝑋𝑖 ⊥ 1{𝑇𝑖 = 𝑡}|𝑟(𝑡, 𝑋𝑖 )- 12 -

To confirm this balancing property of the propensity score satisfied, the literature has suggested
number of ways. Hirano and Imbens (2004) suggest ‘blocking on the score’ and divide the sample into
three different groups according to the distribution of treatment level. This paper uses the blocking on
the score method and treatment level is divided into three groups as follows.
1) {0 < 𝑃𝑇𝐴 𝑟𝑎𝑡𝑖𝑜 ≤ 16}2) {16 < 𝑃𝑇𝐴 𝑟𝑎𝑡𝑖𝑜 ≤ 26}3) {26 < 𝑃𝑇𝐴 𝑟𝑎𝑡𝑖𝑜 ≤ 100}
In each group, taking the mean value as the representative value of the group, the generalized𝑡propensity score is estimated 𝑅̂𝑖 = 𝑟(𝑡, 𝑋𝑖 ), given the mean level of the treatment level. Thisestimated propensity score is divided into five intervals (20, 40, 60, 80, 99 percentile) and eachinterval, it is confirmed whether the covariates are well balanced or not among the different groups.
For example, if the mean value of the treatment in 1) is 10 percent, using the value of 10 percent, thepropensity score is estimated and divided into 5 intervals. The covariates should be balanced in thesame interval, regardless of the fact the school belongs to the group 1) or to the other two groups.
More concretely, this blocking on the score method is to examine whether the mean of the covariates
are not statistically different each other as long as they are similar in generalized propensity scorevalue.

Table [4.2] shows that compared to the covariates without conditioning on the propensity score,
the balancing property is mostly satisfied for those conditioned. For example, when we compare thefirst treatment level (PTA teacher ratio is below 16 percent) with the rest of the schools, the mean of
all covariates except 5 out of 23 variables are significantly different. At the same time, when wecompare the last treatment interval group (PTA teacher ratio is over 26 percent), covariates except 7
are significantly different from zero. In this way, we find that without conditioning the propensity
score, mean of each covariate does differ between the treatment levels. However, once we conditionedon the propensity score, no statistical significant difference is observed between treatment levels.

7. Estimation result for the dose response function
After estimated the generalized propensity score 𝑅̂𝑖 (𝑇𝑖 , 𝑋𝑖 ) with the actual treatment value 𝑇𝑖 , theschool outcome is modeled as the linear function of treatment value as well as the estimatedpropensity score as follows. As emphasized by the Hirano and Imbens (2004), the estimated- 13 -

coefficient vectors α̂ does not have any interpretation of causal effect of treatment on outcome
variable.
𝐸[𝑌𝑖 |𝑇𝑖 , 𝑅𝑖 ] = 𝛼𝑜 + 𝛼1 𝑇𝑖 + 𝛼2 𝑇𝑖2 + 𝛼3 𝑅𝑖 + 𝛼4 𝑅𝑖2 + 𝛼5 𝑅𝑖 ∗ 𝑇𝑖
By storing the coefficient vectors, to estimate the dose response function at the particular level of thetreatment “t”, the conditional expectation is averaged over the generalized propensity score for the
fixed value of “t”, 𝑟(𝑡, 𝑋𝑖 ). Therefore, for the given treatment level “t”, the estimated averagepotential outcome is formulated as follows.
𝑁

1
2
2
̂
̂
̂ ̂𝑅
̂
𝐸[𝑌
∑(𝛼
̂𝑜 + 𝛼
̂𝑡
̂𝑡
+𝛼
̂𝑅
̂𝑅
𝑖 (𝑡)] =1 +𝛼
2
3 𝑖,𝑡 + 𝛼
4 𝑖,𝑡 + 𝛼
5 𝑖,𝑡 ∗ 𝑡)
𝑁𝑖=1

7.1 The Does response function (DRF) estimation
The DRF is estimated at each percent point of PTA teacher ratio for three subjects, Numeracy, English
and Swahili. The scale of Numeracy is ranged from 1 to 8 point, while scores of English and Swahili
are ranged from 1 to 5. Looking at the estimated DRF for pupils in Figure 2, 3, and 4, it is apparent
that the PTA teacher ratio is not only nonlinear to the school average test score for three subjects, buteach subject has different percent point that generates the lowest test score. For example, the score of
Numeracy for grade 4 pupils is lowest around 60 percent PTA teacher ratio, while, after reaching thelowest test score around 30 percent, English and Swahili subject start to raise the test score as PTA
teacher ratio increases. As for the grade three, the test score of Numeracy subject is highest around 10
percent and goes down to the lowest at the point of 46 to 52 percent. The Numeracy test score starts torecover above 55 percent of ratio. In the grade two pupils, it is more obvious that test score isnonlinearly affected by the PTA teacher ratio. As for the Numeracy, there are two peaks observed; thefirst peak is at 10 to 20 percent and the second peak is from 70 to 80 percent. Both scores in grade
three and grade two show that Swahili test score is overall higher than the English score and estimatedresponse function shape is similar, the score difference between English and Swahili is smaller ingrade two pupils.

7.2 Regional heterogeneity
As the past literature of randomized controlled trials has proved that the positive effect on test score ofpupils by PTA teachers in Western province in Kenya (Duflo et al. 2012), the present paper shows theheterogeneity of 7 regions in Kenya. The figures 5 to 11 show the estimated school average outcome- 14 -

response function of PTA teacher ratio at each percent point from 1 to 100 percent. The first obvious
evidence is that it is heterogeneous among regions in terms of the shapes. For example, the mean andmedium largely differ each other and the number of peaks differs among regions.

8. Interpretation and Policy Implication and further studies
The empirical results of dose response function are to show that treatment variable, that is, PTA
teacher ratio nonlinearly affects the school outcome of pupils. This is intuitively reasonable result thatthe schools with 90 percent of PTA teacher are not the same effect on the schools with 10 percent of
PTA teacher. This is the main contribution of this paper. Thought the existing literature of randomized
controlled trials has found the PTA teacher brings positive and statistical significant effect on schooloutcomes, this paper shows that the causal effect on school outcomes differs at each treatment percent.
While the estimation of the derivatives of dose response function and the statistical significance ofthem is required, the discussion should be opened to the level of the treatment.

- 15 -

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18

Table [1] The number and percentage of schools that answers how many months have passed since theprevious Quality Assurance Officer visit
Schoolyear

Number ofMonths

Month

Freq.

Percent

Cum.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
17
18
19
20
21
22
23
25
26
27

March
February
January
DecemberNovember
October
SeptemberAugust
July
June
May
April
March
February
January
November
October
SeptemberAugust
July
June
May
March
February
January

13
974
662
10
219
207
182
18
62
87
64
12
54
82
153
7
19
15
2
11
5
6
5
10
30
30
29
1,266
4,234

0.31
23
15.64
0.24
5.17
4.89
4.3
0.43
1.46
2.05
1.51
0.28
1.28
1.94
3.61
0.17
0.45
0.35
0.05
0.26
0.12
0.14
0.12
0.24
0.71
0.71
0.68
29.9
100

0.31
23.31
38.95
39.18
44.36
49.24
53.54
53.97
55.43
57.49
59
59.28
60.56
62.49
66.11
66.27
66.72
67.08
67.12
67.38
67.5
67.64
67.76
68
68.71
69.41
70.1
100

Year 2012

Year 2011

Year 2010

Year 2009
Year 2008
Missing
Total

Schoolterm

Term 1

Term 3

Term 2

Term 1

Term 3

Term 2

Term 1

Table [2] The UWEZO Kenya National Assessment Scale

Annual Assessment
Survey Period

Round 1Round 2Round 3October, 2009

February –February –March, 2011

March, 2012

Districts reached (of census 158)

70

124

156

Households reached

40,386

72,10690,820Children assessed

68,945

134,243

153,900

Schools reached

2,030

3,574

4,543

Source: UWEZO Kenya (2009), (2010), (2012)19

Table [3] Descriptive statistics for Schools, village and pupils characteristics before imposing the common support condition
Grade 4

Obs

Mean

Variance

Std.

Min

Max

Dependent Variable

Math
EnglishSwahili

3778

6.82

1.67

1.29

1

8

3778

4.08

0.77

0.88

1

5

3778

4.19

0.78

0.88

1

5

3778

26.98

439.33

20.96

2

100

3778

3.50

0.33

0.58

0.69

6.86

3778

3.60

0.22

0.46

0.81

6.58

3778

0.69

0.21

0.46

0.00

1.00

3778

12.60

21.12

4.60

0.48

47.05

3778

13.14

12.29

3.51

0.66

43.34

3778

46.72

873.94

29.56

0.33

322.77

3778

48.85

444.65

21.09

0.53

285.33

3778

0.76

0.13

0.36

0

1

3778

0.54

0.25

0.50

0

1

3778

35.95

41.16

6.42

18

75

3778

10.64

2.03

1.43

6

16

3778

5.68

11.39

3.37

0

13

3778

6.66

13.03

3.61

0

17

3778

0.63

0.16

0.40

0

1

3778

0.01

0.00

0.06

0

1

3778

0.07

0.05

0.23

0

1

3778

0.10

0.07

0.27

0

1

3778

0.17

0.12

0.34

0

1

3778

2.72

0.22

0.47

1

3

3778

0.41

0.20

0.45

0

1

3778

6.84

3.95

1.99

1

24

3778

0.43

0.24

0.49

0

1

3778

0.13

0.11

0.34

0

1

Independent Variable (School level)

PTA teacher ratio
Log of pupil teacher ratio
Log of classsize
School fenced dummy (1 if school is fenced and 0 otherwise)
Log of pupil teacher ratio (Quadratic)
Log of classsize (Quadratic)
Log of pupil teacher ratio (Quadratic)
Log of classsize (Quadratic)
Independent Variable (Household level averaged at school)

Male household headYear dummy
Age of mother
Age of pupils
Years of education for motherYears of education for father
Asset (phone)
Quality of house wall (1 if material is Polythene)
Quality of house wall (1 if material is Iron sheet)
Quality of house wall (1 if material is Timber)
Quality of house wall (1 if material is Stone/bricks)
Number of meals that pupils take in a day
Payment to school extra tuition
Number of household members
Independent Variable (Village level)

Electricity dummy (1 if there is an electricity infrastructure in thevillage and 0 otherwise)
Tarmac road dummy (1 if there is a tarmac road and 0 otherwise)20

Table [3.1] Schools before Common support condition imposed and after.
Grade 4
Common support before
Observation
1st quantile947
2nd quantile1088
3rd quantile978
4th quantile
765
3778
Common support after
Observation
1st quantile947
2nd quantile1083
3rd quantile964
4th quantile
571
3565

mediun
10
19
28
50

mean
Min Max
9.453013
13
18.47059 14
22
28.1411 23
33
59.29804 35 100

mediun
10
19
28
44

mean
Min Max
9.453013
13
18.47461 14
22
28.09647 23
33
47.71629 35 100

Grade 3
Common support before
Observation mediun
1st quantile944
10
2nd quantile
1051
19
3rd quantile961
27
4th quantile
772
50
3728

mean
9.443856
18.50048
27.99688
59.28627

Common support after
Observation mediunmean
1st quantile942
10 9.440552
2nd quantile
1050
19.5 18.50286
3rd quantile949
27 27.9568
4th quantile579
44 47.51813
3520

21

Min

Max
2
14
23
35

Min

13
22
33
100

Max
3
14
23
35

13
22
33
100

Grade 2
Common support before
Observation
1st quantile1004
2nd quantile959
3rd quantile970
4th quantile773
3706
Common support after
Observation
1st quantile1002
2nd quantile957
3rd quantile958
4th quantile
567
3484

mediun
10
20
27
50

mean
9.959163
18.96038
28.02268
60.35058

Min

mediun
10
20
27
44

mean
9.956088
18.96029
27.97495
48.37919

Min

Max
3
15
23
35

14
22
33
100

Max
3
15
23
35

14
22
33
100

Table [3.2] Descriptive statistics for restricted schools satisfied with common support condition
Grade 4

Obs

Mean

Variance

Std.

Min

Max

Dependent Variable

Math
EnglishSwahili

3565

6.82

1.68

1.29

1

8

3565

4.08

0.77

0.88

1

5

3565

4.19

0.77

0.88

1

5

3565

23.36

198.12

14.08

2

100

3565

3.41

0.17

0.41

0.69

4.69

3565

3.59

0.21

0.46

0.81

6.58

3565

0.69

0.21

0.46

0.00

1.00

3565

11.82

7.59

2.76

0.48

21.99

3565

13.08

12.08

3.48

0.66

43.34

3565

41.46

202.60

14.23

0.33

103.13

3565

48.51

429.55

20.73

0.53

285.33

3565

0.76

0.13

0.36

0

1

3565

0.56

0.25

0.50

0

1

3565

35.96

41.34

6.43

18

75

3565

10.63

2.02

1.42

6

16

3565

5.71

11.34

3.37

0

13

3565

6.70

12.94

3.60

0

17

3565

0.63

0.16

0.40

0

1

3565

0.01

0.00

0.06

0

1

3565

0.07

0.05

0.23

0

1

3565

0.10

0.08

0.27

0

1

3565

0.18

0.12

0.35

0

1

3565

2.72

0.22

0.47

1

3

3565

0.41

0.20

0.45

0

1

3565

6.84

3.97

1.99

1

24

3565

0.43

0.25

0.50

0

1

3565

0.13

0.12

0.34

0

1

Independent Variable (School level)

PTA teacher ratio
Log of pupil teacher ratio
Log of classsize
School fenced dummy (1 if school is fenced and 0 otherwise)
Log of pupil teacher ratio (Quadratic)
Log of classsize (Quadratic)
Log of pupil teacher ratio (Quadratic)
Log of classsize (Quadratic)
Independent Variable (Household level averaged at school)

Male household headYear dummy
Age of mother
Age of pupils
Years of education for motherYears of education for father
Asset (phone)
Quality of house wall (1 if material is Polythene)
Quality of house wall (1 if material is Iron sheet)
Quality of house wall (1 if material is Timber)
Quality of house wall (1 if material is Stone/bricks)
Number of meals that pupils take in a day
Payment to school extra tuition
Number of household members
Independent Variable (Village level)

Electricity dummy (1 if there is an electricity infrastructure in the village and 0
otherwise)
Tarmac road dummy (1 if there is a tarmac road and 0 otherwise)22

Figure 1 Histogram of generalized propensity score of grade 4 sample before imposing the common support condition and after

0

0

5

5

Percent

Percent

10

10

15

15

[ 1st quantile group]

0.01
the first quantile group

.02

.03

0

.01

the rest of the group.02

the first quantile group.03

the rest of the group

20

Percent

30

10

20

0

10
0

Percent

30

40

40

50

[ 2nd quantile group]0.01
the second quantile group

.02

.03
the rest of the group0.01
the second quantile group

23

.02

.03
the rest of the group

Percent

0

0

10

10

20

Percent

20

30

30

40

[ 3rd quantile group]0.01
the third quantile group

.02

.03

.005

the rest of the group

.01

.015

the third quantile group

.02

.025

.03

the rest of the group

15
5

10

Percent

20
10

0

0

Percent

30

20

40

[ 4th quantile group]

0.01
the fourth quantile group.02
the rest of the group

.03

0

.01
the fourth quantile group24.02
the rest of the group.03

Table [4.1] Generalized propensity score estimation result (157 district dummy is included)
Dependent Variable: PTA teacher ratio

Grade
four

Grade
three

Grade
two

Male household head

-0.0251(0.0230)

0.0136(0.0229)-0.00109(0.0224)

Year dummy-0.0262(0.0197)-0.00728(0.0198)-0.0225(0.0200)

Age of mother-0.00173 -0.000902 -0.00143(0.00127) (0.00129) (0.00127)

Age of pupils

0.004760.00494 0.0162***(0.00558) (0.00555) (0.00575)

Years of education for mother-0.00118 -0.00166 -0.00404(0.00372) (0.00388) (0.00386)

Years of education for father-0.00797** -0.00666* 0.00227

Asset (phone)-0.0438** -0.0427** -0.108***(0.00320) (0.00353) (0.00350)

Quality of house wall (1 if material is Polythene)(0.0209)(0.0212)(0.0211)0.279**

0.0126-0.143(0.115)(0.114)(0.0906)

Quality of house wall (1 if material is Iron sheet)

0.03860.0216-0.0894**(0.0363)(0.0383)(0.0391)

Quality of house wall (1 if material is Timber)

0.0938** -0.00133-0.0160(0.0428)(0.0465)(0.0429)

Quality of house wall (1 if material is Stone/bricks)-0.0432-0.163*** -0.0768**(0.0358)(0.0378)-0.00321

0.01660.0132(0.0183)(0.0180)(0.0181)

Payment to school extra tuition

0.0409**0.01050.0186(0.0194)(0.0206)(0.0211)

Number of household members-0.00423-0.00202

0.00156

Number of meals that pupils take in a day(0.0386)(0.00427) (0.00413) (0.00378)

Log of pupil teacher ratio-2.669*** -3.034*** -3.361***

Log of classsize(0.409)(0.372)0.712

1.541***(0.366)0.975(0.730)(0.447)(0.653)

School fenced dummy (1 if school is fenced and 0 otherwise)-0.115*** -0.0962*** -0.118***

Electricity dummy (1 if there is an electricity infrastructure in the village and 0
otherwise)
Tarmac road dummy (1 if there is a tarmac road and 0 otherwise)-0.0980*** -0.109*** -0.117***(0.0177) (0.0178) (0.0180)

Log of pupil teacher ratio (Quadratic)

0.829*** 0.917*** 0.991***(0.0922) (0.0836) (0.0842)

Log of classsize (Quadratic)-0.378** -0.621*** -0.411***(0.175)(0.110)(0.157)Log of pupil teacher ratio (Cubit)-0.0661***-0.0728***-0.0784***(0.00683) (0.00618) (0.00636)

Log of classsize (Cubic)

0.0401*** 0.0638*** 0.0407***(0.0135) (0.00899) (0.0123)

Constant

6.000*** 5.404*** 6.217***(0.890)(0.459)(0.780)(0.0177)

Observations
Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1-0.0369(0.0261)

3,778

25

(0.0178)-0.0367(0.0251)

3,728(0.0183)-0.0136(0.0267)

3,706

Table [4.2] The Balancing Property of covariates for estimation sample
PTA ratio<=16Adjusted
Not Adjusted
0 <PTA ratio <=16Male household head

Adjusted
Not Adjusted
26 <PTA ratio[-0.01266-0.358 ]-0.0239[-1.91]*-0.0024[ -0.144 ]-0.01[-0.77]0.0263[ 0.728 ]0.0347[2.71]***

[

0.07
1.704

]

0.0046[0.26][

0.02786
0.706 ]

0.0195[1.09]-0.1[ -2.324 ]-0.0238[-1.34]

[

0.024
0.038

]

-0.0846[-0.38]-0.3334[ -0.612 ]-0.3145[-1.35]0.2533[ 0.552 ]0.3951[1.72]*[

0.0903
0.692

]

0.225
[4.54]***[

0.048960.486 ]

0.0868[1.69]*-0.207[ -1.598 ]-0.3195[-6.33]***[-0.1876-0.616 ]-0.753[-6.44]***-0.1615[ -0.716 ]-0.356[-2.93]***0.5595[ 1.926 ]*1.1335[9.56]***[-0.24426-0.648 ]-0.8031[-6.43]***-0.2212[ -0.852 ]-0.404[-3.11]***0.5926[ 1.972 ]*1.2326[9.73]***[-0.0279-0.866 ]-0.0925[-6.72]***[0.010880.29 ]-0.0046[-0.32]0.0367[ 1.064 ]0.101[7.20]***[-0.0038-0.376 ]0.0009[0.43][

0.00168
0.434 ]

0.0021[0.97]

0.0007[ 0.218 ]-0.003[-1.39][-0.00568-0.258 ]-0.0082[-1.01][0.006420.344 ]

0.0073[0.88]-0.003[ -0.138 ]0.0014[0.17][-0.00536-0.284 ]-0.0313[-3.27]***[-0.0169-0.78 ]

-0.0164[-1.66]*0.0398[ 1.318 ]0.0486[4.99]***[-0.03102-1.116 ]-0.0742[-6.18]***[0.01350.534 ]

0.0012[0.10]

0.0323[ 1.066 ]0.0763[6.22]***[-0.01958-0.49 ]

-0.0544[-3.33]***-0.0089[ -0.314 ]-0.022[-1.30]0.0484[ 1.206 ]0.0782[4.70]***[

0.021420.61]-0.0002[-0.01]-0.0108[ -0.342 ]-0.0189[-1.16]-0.012[ -0.292 ]0.0187[1.16][

0.10960.744]

0.2274[3.27]***-0.0258[ -0.156 ]-0.0223[-0.31]-0.155[ -0.814 ]-0.2156[-3.04]***[-0.03436-0.972 ]0.039[2.72]***-0.0333[ -0.808 ]-0.021[-1.41]0.0502[ 1.426 ]-0.0202[-1.38][0.04821.304]

0.0481[2.98]***[-0.0121-0.22 ]

-0.0119[-0.71]-0.07[ -1.372 ]-0.0386[-2.34]**[0.033480.81]-0.0311[-1.93]*[-0.0098-0.27 ]

-0.0247[-1.49]-0.002[ -0.056 ]0.0566[3.45]***[-0.06998-1.854 ]-0.1966[-11.58]***[0.03278
0.818 ]

0.0078[0.44]

0.063[ 1.652 ]*0.1977[11.38]***[-0.02464-0.914 ]-0.0677[-5.71]***[0.0046
0.156 ]

0.0002[0.02]

0.0365[ 1.148 ]0.0705[5.82]***[-0.28448-1.21 ]

0.2423[2.52]**-0.2324[ -0.816 ]-0.13[-1.31]0.3756[ 1.614 ]-0.1263[-1.28][0.342741.21]

0.3331[2.75]***-0.0809[ -0.178 ]-0.0655[-0.52]-0.532[ -1.316 ]-0.2839[-2.29]**[-1.72664-1.428 ]1.1265[2.27]**-1.2063[ -0.786 ]-0.597[-1.16]2.0978[ 1.756 ]*-0.594[-1.17][1.891221.104 ]

1.7456[2.42]**-0.361[ -0.112 ]-0.2191[-0.29]-3.16[ -1.246 ]-1.609[-2.18]**Year dummy
Age of mother
Age of pupils
Years of education for motherYears of education for father
Asset (phone)
Quality of house wall (1 if material is Polythene)
Quality of house wall (1 if material is Iron sheet)
Quality of house wall (1 if material is Timber)
Quality of house wall (1 if material is Stone/bricks)
Number of meals that pupils take in a day
Payment to school extra tuition
Number of household members
Log of pupil teacher ratio
Log of classsize
School fenced dummy (1 if school is fenced and 0 otherwise)Electricity dummy (1 if there is an electricity infrastructure in the village and 0
otherwise)
Tarmac road dummy (1 if there is a tarmac road and 0 otherwise)

Adjusted
Not Adjusted
16 <PTA ratio<=26

Log of pupil teacher ratio (Quadratic)
Log of classsize (Quadratic)
Log of pupil teacher ratio (Cubit)
Log of classsize (Cubic)- 26 -

Figure 2, 3, 4 the estimated dose response function for grade four, three and two pupils

27

Figure 5 Central Province, Grade 4 pupil school average

Figure 6 Coast Province, Grade 4 pupil school average

Figure 7 North Eastern Province, Grade 4 pupil school average

28

Figure 8 Eastern Province, Grade 4 pupil school average

Figure 9 Western Province, Grade 4 pupil school average

29

Figure 10 Rift Valley Province, Grade 4 pupil school average

Figure 11 Nyanza Province, Grade 4 pupil school average

30