Ra Ev Yasm

Kenn Oswa

Sub-Saharan Africa

Yasmine Bekkouche ⋆

Kenneth Houngbedji

Oswald Koussihouede

Isabelle Chort

Denis Cogneau

Isabel Günther

Karen Macours

Karine Marazyan

Paul Glewwe

Place DOCU

Keywords: Education, Children, Climate JEL Classification: I21, Q54

Rainy days and learning outcomes: Evidence from sub-saharan Africa

About 59 million primary-aged children were not in school for the academic year ending in 2018. Worldwide, this number has stagnated since 2007, and its level is worrisome in Sub-Saharan Africa where six out of ten primary-aged children are not in school (see [START_REF] Unesco | New methodology shows that 258 million children, adolescents and youth are out of school[END_REF]. Moreover, learning outcomes in the region are estimated to be among the lowest in the world (see [START_REF] Angrist | Measuring human capital using global learning data[END_REF]. If low-level learning outcomes reduce incentives for schooling, efforts to improve school learning outcomes in Sub-Saharan Africa may prove effective at encouraging enrollment and attendance.

To contribute to the identification of public policy options that encourage demand for education in Sub-Saharan Africa, this paper studies the limits that climatic conditions can place on instruction time and learning outcomes. Throughout the region, academic calendars seldom vary across countries despite large variation in climatic conditions from one country to another. Likewise, within each country, most schools are usually subject to the same calendar regardless of the climatic differences between school locations. In that context, we argue that the number of school days provided by each school in an academic year may vary depending on rainfall at the local level during those days, which can have deleterious effects on learning.

Rainfall during school days may affect learning outcomes through two mechanisms which are of interest to this study. First, rainfalls induce flooding and hinder presence at school. Students attending schools in localities that experience more rainy days during the academic year may therefore receive less instruction time than their peers in localities with less rainfall.

Second, raindrops may disturb classroom acoustics and more generally create conditions that are less than ideal for the learning experience. However, this can be mitigated by implementing measures either to maintain school attendance for students and teachers or to compensate for learning time lost during rainy days. The net effect of rainfall activities on learning outcomes is thus theoretically ambiguous and warrants an empirical investigation to identify areas in which additional measures may be useful.

Since rainfall can also affect learning through other mechanisms, measuring the net effect of learning time lost due to climatic conditions can be challenging. For instance, the variation of rainfall patterns during school days across localities is likely determined in part by the differences in atmospheric conditions. In that case, the net effect of rainfall patterns on learning outcomes during the academic year may be explained by residential sorting. Families who chose to live in localities with more rainy days have different characteristics that affect their children's learning outcomes. Likewise, when the academic and agricultural calendars overlap, some students may spend time helping with agricultural activities. In that case, we would expect the opportunity costs of children attending schools in localities with more rainfall to be higher and students' involvement at school to be lower (see [START_REF] Nordman | Here comes the rain again: Productivity shocks, educational investments, and child work[END_REF], for a review of existing studies of this income-effect mechanism).

The remaining of the paper proceeds as follows: in Section 2 we will describe the research design and main data used to overcome the confounding mechanisms outlined above and explore the relationship between rainfall patterns during school days and student achievements.

Section 3 presents the main results of the study and explores the channels underlying the link between rainfall and academic performance. Section 4 offers concluding observations.

Data and Methods

Data

We were able to leverage several rounds of surveys conducted with the Programme for the Analysis of Education Systems (PASEC) across Sub-Saharan Africa. Between 2003 and2010, in a set number of countries, PASEC conducted a series of standardized tests in mathematics and reading one month after the beginning and one month before the end of the academic year on random samples of students in the second and fifth grades. Within each country, a sample of 140 to 180 schools was randomly selected from a sampling frame of active primary schools. Once a school was sampled, one class of second grade and one class of fifth grade were randomly selected, and up to 15 students were randomly drawn from each class to take the tests. Detailed questionnaires intended for the students who took part in the standardized tests, their teachers and the principals were also used to provide information on student characteristics along with their home and school environments. The resulting data set covered 1,643 schools, and we have identified and recovered the geographical extent of the 535 lowest administrative units for these schools. Class sizes varied from one school to another, and some schools may only have had one of the two grades. Hence, the number of second and fifth-grade classrooms was unbalanced, and we relied on information from 23,460 students in second grade from 1,579 classrooms and 22,312 students in fifth grade from 1,523 classrooms in Benin, Burkina Faso, Burundi, Cameroon, Chad, Congo, Gabon, Madagascar, Senegal and Togo (see Table A-1 and A-2 for details of the sample characteristics across the countries).

The academic calendars of the countries included in our study invariably start in October and end in June of the following calendar year. We used high-resolution map of daily rainfall produced by the Climate Hazards Group InfraRed Precipitation with Station data (CHIRPS) to count the number of weekdays with a rainy event detected within the boundaries of each school's administrative unit between the months of October and June of the relevant academic year. 1 We observed that rainfall patterns during weekdays in the academic year varied from zero rainy days in parts of Senegal to 177 rainy days in the tropical rainforest of the Republic of Congo. We also observed a wide variation of rainfall patterns across administrative units within a single country. In Cameroon, the number of rainy days measured during the academic year 2004-05 varied from 21 in the department of Logone-et-Chari to 153 in the department of Momo. As rain clouds travel through space, the number of rainy days may be higher in larger administrative units. Rainfall patterns may therefore be overestimated for schools located in large administrative units. To mitigate this measurement error, our analyses explicitly consider the area of the administrative units. To provide alternative estimates of the main results of the paper, we collected detailed Global Positioning System (GPS) coordinates for the schools surveyed in Benin and Senegal and exploited the variation of rainfall patterns at the exact school locations.

Methods

We considered the following specification to estimate the impact of rainfall during weekdays in the academic year:

Y e il = β 0 + β r R l + β a A l + µ c + ϵ il (1)
The primary variable of interest is Y e il , the normalized test score at the end of the academic year and the likelihood of grade repetition for student i attending school in the administrative unit (or locality) l. We sought to estimate the net effect of R l , the normalized number of school 1 CHIRPS' datasets combine observations of cold cloud derived from satellites, rain gauge measurements from stations and models of rainfall to produce detailed maps of daily precipitation for pixels of approximately 5 km × 5 km since 1981 (see Funk et al., 2015, for more details). In Benin, we compared the estimate of daily rainfall produced by CHIRPS to the measures of daily rainfall reported by the meteorological stations at the locations of each station (see Supplementary Materials S1). Our results show that the variation of the number of rainy days across stations reported by CHIRPS is highly correlated with the variation measured using the measurements produced by the meteorological stations (see Figure A-4). However, the levels of rainfall reported by both methods are poorly correlated. Our analyses use the measurements reported by CHIRPS to detect rainy events and do not consider the variations in rainfall intensity.

days with rainfall during the academic year. Since R l is measured throughout the locality l, students attending different schools in l share the same level of exposure to rainfall during the academic year. To attenuate the effects of the measurement errors related to using the number of rainy days across a locality to estimate the number of rainy weekdays at a school location, we included as a control variable A l , the area of the administrative unit of student i's school.

The survey fixed effects, µ c , help account for the difference in test items that were introduced to adapt the reading and mathematics tests to the context of each country. They also help use the variation of rainfall patterns across localities within the same country to identify the effect of rainy school days on the students' learning outcomes. Last, ϵ il represents the unobserved heterogeneity that affects test scores at the end of the academic year.

The parameter of interest β r , is the expected net variation of learning outcomes per standard deviation (s.d.) increase of rainy days in the administrative unit during the academic year.

Since the spatial variation of rainfall patterns is determined by meteorological processes, we assumed that the variation of R l across localities in the same country was exogenous to the unobserved heterogeneity ϵ il . To isolate the effects of rainy days on learning outcomes that are driven by a loss of instruction time, we documented how β r varies when we block alternative pathways through which the number of rainy days affects student learning outcomes. We focused on three such pathways. First, families who want to improve their children's learning outcomes might choose to live in administrative units with fewer rainy days during the academic year. In that case, end-of-year test scores would be lower (and, conversely, the likelihood of grade repetition would be greater) in rainy localities because of residential sorting and less investment in schools and students. Second, in response to residential sorting, student and school characteristics may differ across administrative units and confound the effects of the variation of rainy days across schools. Third, students in localities with more rainy days may contribute some of their time to helping with agricultural activities and invest less effort at school. We used Y b il , the normalized test score at the beginning of the academic year for student i attending school in locality l to help account for the effects of residential sorting that could mediate the impact of rainy days on end-of-year learning outcomes and considered the following specification:

Y e il = β 0 + β r R l + β y Y b il + β a A l + µ c + υ il . ( 2
)
Since start-of-year test scores are measured at the beginning of the year, we assumed that Y b il is exogenous to the unobserved heterogeneity υ il and we commented on the variations of the estimates of β r across (1) and (2).

To account for the mediating effects of student and school characteristics across localities with varying amounts of rainy days during the academic year, we used a vector of student and school characteristics.

Y e il = β 0 + β r R l + β y Y b il + β x X isl + β s S il + β a A l + µ c + ν il . (3)
The surveys do not document if students participated in income-generating activities between the start-of-year and end-of-year tests. However, each student stated at the beginning of the academic year whether they often contributed to farm and retail work. 2 We considered that students who had already contributed to income-generating activities would be more likely to divert efforts to income-generating activities when a locality has more rainy school days. If

that assumption holds and students in localities with more rainy days divert their efforts away from school, then we would expect lower end-of-year learning outcomes (a higher likelihood of grade repetition) per unit of R l for students who stated that they often contributed to farm and retail work. Conversely, the expected end-of-year learning outcomes per unit of R l for students who did not often contribute to income-generating activities would provide an estimate of the effect of rainy days that can be attributed to a loss of instruction time. With W il , a binary variable that is equal to one when student i in locality l declared they often contribute to farm 2 In the sample, 49% of second-grade and 57% of fifth-grade students stated at the beginning of the academic year that they had contributed to farm work. The corresponding estimates for students who had already contributed to retail work are 17% on second-grade and 22% of fifth-grade students. and retail work and zero otherwise, we estimated the following specification:

Y e il = β 0 + β r R l + β rw R l × W il + β y Y b il + β x X il + β w W il + β s S il + β a A l + µ c + η il (4)
We computed clustered standard errors that accounted for within-school correlation of outcomes since students in the same school share the same environment and estimated β r and β rw in (1), ( 2), ( 3) and (4).

Results

Main Results

The main findings of our work are presented in Table 1. We found that the end-of-year test scores of the second and fifth-grade students were on average lower in localities that experienced more rainy events during weekdays in the academic year. 3 When the competing mechanisms were accounted for, the results suggested that the expected end-of-year scores across reading and mathematics tests dropped by 0.09 s.d. for each s.d. increase of rainfall across localities for students in second grade compared to 0.04 for those in fifth grade.

A lesser effect of rainfall during school days on test scores in fifth grade is consistent with the increase by 1.96 percentage point per s.d. increase of rainy days of the expected likelihood of grade repetition across localities for students in second grade. This result indicates that the deleterious effect of rainfall on learning outcomes expands beyond lower test scores and prevents students in second grade from graduating. Consequently, students in fifth grade are selected and mostly comprised of those who benefit from an environment that is less vulnerable to disturbance induced by rainfall patterns. Furthermore, we found no evidence that more rainy days made these students more likely to repeat the grade at the end of the academic year.

The variation of the results across the different specifications was also consistent with our expectations and suggested that the effects of rainfall during school days on learning outcomes can be mediated through residential sorting (or a lower level of investment to support schooling) and the characteristics of students and schools in localities with more rainy days. We also found no evidence that the effect of rainfall on learning outcomes was mediated when effort 3 The results presented in Table 1 were estimated using the sample of students that were present at the start-ofyear and end-of-year tests. Students may be absent for a variety of reasons, but we found no evidence that they were more likely to be absent in localities that experienced more rainy days during weekdays in the academic year (see Table A-3).

was diverted to participating in income-generating activities. Indeed, the estimates of β rw provided no evidence that the deleterious effects of rainy days on learning outcomes were higher for students who were more likely to divert their effort to income-generating activities. Other results showed that the expected end-of-year test score per s.d. increase of rainy days was lower in reading than in mathematics (see Table A-4) and suggested that the amount of time students spend interacting with their teacher may have a higher impact on the learning process for mathematical skills covered in second and fifth grades.

Robustness

Even though we accounted for the area, the number of rainy days at school locations is overestimated when using data on rainfall during school days across administrative units. As a result, our methodological approach yields conservative estimates of the effect of rainy days on learning outcomes. Using rainfall during school days across administrative units is also worrisome when rainy days affect the actions of the administrative units in other ways that may affect student learning outcomes. This is the case when rainfall activities divert local resources away from primary schools toward flood mitigation or road maintenance. To test the extent to which we underestimated the effect of rainfall and to see whether we might explain away the effects of rainy days on learning outcomes with time-varying unobserved heterogeneity across administrative units, we collected GPS coordinates of the schools surveyed in Benin and Senegal and studied how learning outcomes varied with rainfall levels across locations within the same administrative unit.

For each school located using GPS coordinates, we computed the average level of daily precipitation in a buffer of 1 km around the school. Then, we counted the number of rainy school days at each school location and used the following specification to estimate their effect on learning outcomes:

Y e il = β 0 + β r R l + β y Y b il + β x X isl + β s S il + µ a + ϵ il . ( 5
)
The fixed effects, µ a , helped account for the unobserved heterogeneity that affects the learning outcomes of students from the same administrative unit during the academic year. The questionnaires used in Senegal did not allow us to measure how likely each student tested was to repeat the grade. Therefore, we restricted the estimation of the effect of the number of rainy school days on grade repetition on the students tested and surveyed in Benin.

The results are reported in

Heterogeneity

At the country level, we found that the expected end-of-year test scores were often lower per unit of s.d. increase of rainy days except in Congo and Madagascar (see Fig. 1). Across all the countries, the deleterious effects of rainfall during the academic year were also higher for students in second grade. We found no evidence of a gender-differentiated effect of rainfall during school days on student learning outcomes (see Table A-5). However, we found that the impact of rainfall during school days on test scores decreased with student age (see Table A-6).

This could suggest that older students are less likely to miss school on rainy days than their younger siblings. We also observed that the effects of rainfall during weekdays were more significant for students who had achieved higher scores at the start-of-year test (see Table A-7). This suggests that the disruption of instruction time induced by rainfall may affect all the students. In regression to the mean, with less instruction time, students who achieved higher test scores initially saw their score decrease the most by the end of the year.

Mechanisms

To test the mechanisms through which rainfall disturbs learning outcomes, we created a set of indicators of absenteeism and completion of the official curricula. These were derived using responses to the surveys administered to the teachers and the principals at the time of the endof-year test in certain countries. 4 Then, we used the specification of Eq. ( 3) to study how the number of rainy days affected the number of school days missed by each teacher, as reported by the principal; the number of students who were on average absent in each classroom; and the likelihood of completion of the official curricula in mathematics and reading by the teacher.

The results reported in Table 3 suggest that the expected number of school days missed by the teachers over a month increases by 0.06 s.d. per s.d. increase of rainy days. Teachers were also less likely to complete the curricula in reading and mathematics. We found no strong evidence that the average number of absent students increases with rainy school days.

Conclusion

Considering the context of a warming climate with varied rainfall patterns and intensity, this paper studies some of the hidden costs that rainy school days impose on learning outcomes at primary school. The results contribute to economic studies of the returns of instruction time.

Existing studies report moderate to high returns of the length of the academic year on labor outcomes. In the United States of America (USA) and Germany, [START_REF] Card | Does School Quality Matter? Returns to Education and the Characteristics of Public Schools in the United States[END_REF] and [START_REF] Pischke | The impact of length of the school year on student performance and earnings: Evidence from the German short school years[END_REF] find evidence of persistent positive effects of the length of the academic year on labor market outcomes. In Argentina, Jaume and Willén (2019) find that teacher strikes in primary school have persistent long-run effects on earnings. Furthermore, a growing body of studies has documented the effects of instruction time on academic achievement (see [START_REF] Aucejo | Assessing the effect of school days and absences on test score performance[END_REF][START_REF] Barrios-Fernández | It's time to learn: School institutions and returns to instruction time[END_REF][START_REF] Cattaneo | The more, the better? the impact of instructional time on student performance[END_REF][START_REF] Goodman | In defense of snow days: students who stay home when school is in session are a much larger problem[END_REF][START_REF] Huebener | Increased instruction hours and the widening gap in student performance[END_REF][START_REF] Marcotte | Schooling and test scores: A mother-natural experiment[END_REF][START_REF] Marcotte | Unscheduled School Closings and Student Performance[END_REF][START_REF] Rivkin | Instruction Time, Classroom Quality, and Academic Achievement[END_REF][START_REF] Thompson | Is four less than five? Effects of four-day school weeks on student achievement in Oregon[END_REF] with potential deleterious effects of an increase in instruction time on student mental health (see [START_REF] Marcus | Increased instruction time and stress-related health problems among school children[END_REF].

Given its topical coverage, this paper expands the work of [START_REF] Marcotte | Schooling and test scores: A mother-natural experiment[END_REF]; [START_REF] Marcotte | Unscheduled School Closings and Student Performance[END_REF] and [START_REF] Goodman | Flaking Out: Student Absences and Snow Days as Disruptions of Instructional Time[END_REF] who studied the effect of snowfall on student' performance in the United States. In Maryland and Colorado, [START_REF] Marcotte | Schooling and test scores: A mother-natural experiment[END_REF] and [START_REF] Marcotte | Unscheduled School Closings and Student Performance[END_REF] find that students who took exams in years with snowfall performed significantly worse on the academic assessments than their peers in the same school who took the exams in other years. In Massachusetts, [START_REF] Goodman | In defense of snow days: students who stay home when school is in session are a much larger problem[END_REF] finds that school closings caused by extreme snowfall have little impact on student achievement whereas absences due to moderate snowfall cause a sharp reduction in math achievement. Considering that the number of school days lost to school closings was limited in Massachusetts, the results from Goodman (2015) suggest that teachers and schools can mitigate the adverse effects of the disruption of instructional time but have more issues with absences that affect different students at different times. These conclusions echo the results of [START_REF] Aucejo | Assessing the effect of school days and absences on test score performance[END_REF] who find that, in North Carolina, the increase in achievement from extending the school calendar is lower than the reduction in achievement due to absenteeism.

This paper finds that rainfall activities during school days have deleterious effects on student test scores and contribute to grade repetition by reducing the amount of instruction time for a sample of countries in Sub-Saharan Africa. Detailed analyses suggest that teacher absenteeism increases with rainfall, whereas the likelihood the official curricula will be completed decreases. These results echo the findings of learning losses associated to school closures during the COVID pandemic in low and middle income countries (see UNICEF et al., 2022). They also suggest that while learning poverty was highest pre-pandemic in Sub-Saharan Africa, schoolchildren have lower test scores during rainy school days when their teacher is absent.

The negative effects of rainfall are strongest for math achievement and penalize the youngest students the most. However, some countries are more resilient than others to the disruption of instruction time caused by rainfall during the academic year. This presents an opportunity for future research to study and identify the factors that contribute to making some countries more resilient than others to the loss of instruction time due to rain. Alternatively, it may be worth studying if regional calendars that reduce the number of rainy school days help reduce the inequality of instruction time across administrative units and improve enrollment and learning outcomes.

UNICEF, World Bank, UNESCO, USAID, FCDO, and Bill & Melinda Gates Foundation (2022). The state of global learning poverty: 2022 update. Technical report, UNICEF: United Nations Children's Fund. The table shows estimates of the expected end-of-year score in mathematics and reading per s.d. increase of rainy days during the school year. For a selected number of countries the surveys included complementary information that allowed us to identify students who were likely to repeat the grade. Hence, the column "Grade repetition" reports estimates of the expected likelihood of grade repetition per s.d. increase of rainy school days. For each outcome, column (1) reports estimates of β r obtained with the specification of Eq. ( 1), column (2) reports the estimates of β r from Eq. ( 2) and column (3) the estimates of β r from Eq. (3). Column (4) reports the estimates of β r and β rw from Eq. ( 4). The standard errors are clustered at the classroom level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Note: The table shows the estimates of the average effect of one standard deviation variation of rainy days during the school year in a 1 km radius around each school on the standardized end-of-year test scores (in reading, mathematics, and the average score for both subjects) and the likelihood that the student will repeat the grade according to the teacher. The sample used in this estimation is restricted to the students surveyed in Benin and Senegal where we were able to precisely locate each school using their GPS coordinates. The coefficients reported correspond to the estimates of β r in Eq. ( 3). The estimation of the average effect of rainy days on the likelihood of grade repetition is limited to the sample of students surveyed in Benin where teachers were asked this question. The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Grade 2

Grade 5

Fig. 1: Average effects of rainy days on average test scores across grades and countries

This figure shows how the expected end-of-year test scores across reading and mathematics per unit of s.d. increase of rainy days vary across country for students in second and fifth grade. Column p25 shows the number of rainy days at the local level for the 25% of locations with the lowest number of rainy days where a school was surveyed. Column p75 shows the minimum number of rainy days at the local level for the 25% locations with the most rainy days surveyed. Lastly, column max shows the maximum number of rainy days at the local level for each country. Note: The table presents descriptive statistics of the main variables considered in this study. The statistics reported under the heading "Second grade" refer to the sub-sample of students from the second grade. Under the heading "Fifth grade" we report the statistics from the sub-sample of fifth-grade students. a The infrastructure index at the locality level was constructed based on a list of infrastructures and services available near the school. The list includes roads, electrification, water supply networks, high schools, health-care centers, telephone lines, police stations and banks. b The infrastructure index at the school level is a composite index based on the availability of the following infrastructures within the school: kiosks, libraries, nurseries, canteens, toilets, playgrounds, athletics fields, electricity, and water supply networks. The table shows estimates of the expected end-of-year test score for each subject (reading or mathematics) per standard deviation increase of rainy days during the school year. For each outcome, column (1) reports the estimates of β r obtained with the specification of Eq. (1), column (2) reports the estimates of β r from Eq. ( 2) and column (3) the estimates of β r from Eq. ( 3). The standard errors are clustered at the classroom level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. The table shows estimates of the expected end-of-year test score (in reading, mathematics, and the average score of both subjects) and the likelihood that the student will repeat the grade per s.d. increase of rainy days during the school year and gender. Each column corresponds to an estimation where the end-of-year test score and the likelihood that the student will repeat the grade are regressed on the number of rainy school days at school locations interacted with a binary variable that is equal to one for girls and zero for boys while controlling for various characteristics of students and their schools as in Eq. ( 4). The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. The table shows the estimates of the average effect of one standard deviation variation of rainy days during the past summer break (July to mid-August) or during weekends of the school year in a 1 km radius around each school on the standardized end-of-year test scores (in reading, mathematics, and the average score of both subjects) and the likelihood that students will repeat the grade according to the teacher. The sample used in this estimation is restricted to the students surveyed in Benin and Senegal where we were able to precisely locate each school using their GPS coordinates. The coefficients reported correspond to the estimates of β r in Eq. ( 3). The estimation of the average effect of rainy days on the likelihood of grade repetition is limited to the sample of the students surveyed in Benin where teachers were asked this question. The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Note: The table shows how the estimates of the average effect of one standard deviation variation of rainy days during the school year on the standardized end-of-year tests scores (in reading, mathematics, and the average score of both subjects) and the likelihood that the student will repeat the grade vary depending on whether the student had already con-tributed to retail or farm work in the past. Each column corresponds to an estimation where student end-year test scores and the likelihood of grade repetition are regressed on the number of rainy days around school locations during the academic year interacted with a binary variable that is equal to one for students who participated to farm and retail work in the past and zero for those who did not. The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01. Matching the detailed maps of daily precipitation from CHIRPS with the coordinates of each school successfully located, we were able to estimate the number of rainy days for the pixel where each school is located. To account for rainfall variation across pixels for schools located near adjacent pixels, we associated the average level of precipitation estimated for a buffer of neighboring pixels to each school the average level of precipitation estimated for a buffer made of neighboring pixels. Hence, our approach appropriately reports on rainfall events occurring in the neighborhood of near each school during the academic year. To address concerns that spatial variation of rainfall produced by CHIRPS can fail to capture spatial variation of rainfall observed on the ground, we compared how the number of rainy days during the academic year 2019-20 at six (of the eight) meteorological stations throughout

S1 Measuring Rainfall at School Location

Benin varies when we estimate it using CHIRPS or the observations reported by rain gauges installed at each station. Since rainfall measurements reported by CHIRPS are recorded for each 25 km 2 pixel, they are likely to include precipitations near the station that did not reach the station. Hence, we may expect that the number of rainy days reported by CHIRPS for the pixel or the buffer where the station is located might systematically be greater than the number of rainy days measured based on the rain gauge measurements available for the station. This expectation was vindicated by the data (see Fig. A-4). We also found that the three measures had a high level of internal consistence, and the ranking of stations depending on the number of rainy days did not show much variation from one method to another. The internal consistency (the Cronbach's alpha) between the number of rainy days reported by the stations and the number of rainy days reported by CHIRPS at the pixel level is 0.88 whereas the internal consistency between the number of rainy days reported by the stations and the number of rainy days reported at the buffer level by CHIRPS is estimated at 0.82.

  To measure daily precipitation in the neighborhood surrounding each school surveyed by PASEC, we used high-resolution maps of daily rainfall data collected by the Climate Hazards Group InfraRed Precipitation with Station data (CHIRPS). CHIRPS' data sets combine observations of cold cloud derived from satellites, rain gauge measurements from stations, and models of rainfall to produce detailed maps of daily precipitation for pixels with a size of 0.05°(approximately 5 km) since 1981 (see[START_REF] Funk | The climate hazards infrared precipitation with stations -a new environmental record for monitoring extremes[END_REF] more details). Combining the daily maps over a given academic year, it is possible to produce a detailed account of the number of rainy days during school days for each pixel of 25 km 2 (see Fig. A-3 for a visual representation of rainfall activities across Benin during school days in the academic year 2004-05 using daily maps produced by CHIRPS).

Fig. A- 3 :

 3 Fig. A-3: Rain activities during school days in the academic year 2004-05 in Benin. This figure describes the variation of the number of rainy days and total rainfall at local level across Benin for the academic year 2004-05.

  Fig. A-4: Number of rainy days during the academic year 2019-20 reported by CHIRPS and weather stations.This figure represents how the number of rainy days estimated using CHIRPS at the site of meteorological stations differ from the number of days recorded by the stations for the academic year 2019-20 for which we had comparable data sets.

  Table2and confirm the main results described earlier. We found that expected end-of-year test scores decreased by 0.21 s.d. for each s.d. increase of rainy days for students in second grade. Their likelihood to repeat their grade is also 11 percentage points higher. While these effects are stronger than reported in Table1-and confirm that specification (3) produces conservative estimates of the effects of rainy days -we found no noticeable effects on students in fifth grade.If rainy school days affect learning outcomes by reducing instruction time, we would expect that rainy days during weekends or summer break would have little to no effect on end-ofyear test scores or likelihood of grade repetition. Using the specification (5), we reproduced the results of Table2and found no evidence that rainy days on weekends or during the summer break impacted student learning outcomes (see TableA-8).

Table 3 .

 3

Table 1 :

 1 Effects of the number of rainy days on test score and likelihood of grade repetition

			End-of-year test score			Grade repetition
		(1)	(2)	(3)	(4)	(1)	(2)	(3)	(4)
	Panel A: Second-grade students
	Number of rainy days (β r )	-0.143*** -0.102*** -0.091*** -0.098***	3.326***	2.392***	1.960***	2.220**
		(0.02)	(0.02)	(0.02)	(0.02)	(0.72)	(0.71)	(0.73)	(0.93)
	× student worked (β rw )				0.023				-0.515
					(0.02)				(0.99)
	β r + β rw				-0.075***				1.705**
					(0.02)				(0.85)
	Number of students	20,140	20,140	20,140	20,140	14,224	14,224	14,224	14,224
	Panel B: Fifth-grade students
	Number of rainy days (β r )	-0.130*** -0.050*** -0.043*** -0.060***	2.086***	0.392	0.135	-0.638
		(0.02)	(0.01)	(0.01)	(0.02)	(0.79)	(0.79)	(0.81)	(1.06)
	× student worked (β rw )				0.037**				0.897
					(0.02)				(1.08)
	β r + β rw				-0.023				0.259
					(0.02)				(0.91)
	Number of students	19,584	19,584	19,584	19,584	13,262	13,262	13,262	13,262
	Area and country fixed effects	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
	Start-of-year test score	No	Yes	Yes	Yes	No	Yes	Yes	Yes
	Student and school characteristics	No	No	Yes	Yes	No	No	Yes	Yes
	Note:

Table 2 :

 2 Effects of the number of rainy days on test scores and likelihood of grade repetition using variation of rainfall patterns within administrative units

		End of year test score	Grade
		Reading	Math	Average Repetition
	Panel A: Second-grade students
	Number of rainy days	-0.109	-0.191** -0.213** 10.751**
		(0.09)	(0.09)	(0.10)	(4.60)
	Number of students	3,593	3,593	3,593	1,701
	Panel B: Fifth-grade students
	Number of rainy days	0.089	-0.008	0.021	8.546
		(0.07)	(0.07)	(0.06)	(5.60)
	Number of students	3,699	3,699	3,699	1,822
	Administrative unit fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes

Table 3 :

 3 Effects of the number of rainy days on absenteeism and the likelihood the curricula will be completed The table shows estimates of the effect of rainy school days on different variables associated with the amount of instruction time supplied at school. The column on teacher absenteeism reports the estimated s.d. change in the number of school days missed by each teacher per s.d. increase of rainy days. The column on student absenteeism reports the estimated s.d. change in the number of absent students per s.d. increase of rainy days. The last two columns under the headings "Curricula completed" report the expected percentage points change of the likelihood that a teacher will complete the reading and mathematics curricula per s.d. of rainy days. The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01.

		Absenteeism indicators Curricula completed
		Teachers	Students	Reading	Math
	Panel A: Second grade
	Number of rainy days	0.061**	-0.014	-2.038**	-2.951***
		(0.03)	(0.03)	(0.99)	(0.93)
	Number of classrooms	1,427	1,427	1,427	1,427
	Panel B: Fifth grade
	Number of rainy days	0.035	0.056	-1.871*	-2.430**
		(0.03)	(0.04)	(1.06)	(1.01)
	Number of classrooms	1,371	1,371	1,371	1,371
	Area and country fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes

Note:

Table A - 1 :

 A1 Summary of data used in this study

			Sample characteristics			Number of rainy days
	Country	Years	Number Number of students of Second Fifth	obs. min p25	p50 mean	p75 max	sd
			schools	grade	grade
	Benin	2004-05	145	2,034	2,098	61	53	70	79	79.6	88 121 14.1
	Burkina-Faso 2006-07	175	2,289	2,327	42	45	61	68	69.3	77 101 13.6
	Burundi	2008-09	180	2,694	2,625	89	42	48	50	50.7	53	65	4.4
	Cameroon	2004-05	174	2,427	2,361 112	21	87 113.5 101.5	129 153 36.4
	Chad	2009-10	169	2,454	1,988	43	22	45	63	62.4	77 100 19.7
	Congo	2006-07	140	2,052	1,958	32	95 141 150.5 149.7 161.5 177 17.1
	Gabon	2005-06	142	1,981	2,022	29	60	75	79	77.8	82	89	6.7
	Madagascar	2004-05	180	2,677	2,215	56	32	91 110.5 104.8 119.5 151 25.2
	Senegal	2006-07	159	2,201	2,140	38	0	38	46	45.9	53	81 16.1
	Togo	2009-10	179	2,651	2,578	33	57	81	99	94.4	109 124 19.0
	All	2004-10	1,643	23,460	22,312 535	0	53	77	82.4	109 177 34.5

Note: The table reports the characteristics of available PASEC surveys and spatial variability of rainfall across Sub-Saharan Africa. Column min reports the smallest number of days with rainfall measured at the lowest administrative unit during the academic year for each survey.

Table A

 A

		-2: Descriptive statistics of key variables
			Second grade			Fifth grade
		Obs Min Mean s.d. Max	Obs Min Mean s.d. Max
	Reading score
	-start-of-year	23,460	0	20.4 22.4	100 22,312	0	24.2 16.9	100
	-end-of-year	20,140	0	32.5 24.4	100 19,584	0	25.9 17.7	100
	Mathematics score
	-start-of-year	23,460	0	29.6 28.0	100 22,312	0	31.1 21.2	100
	-end-of-year	20,140	0	32.6 24.1	100 19,584	0	28.6 18.7	100
	Average score
	-start-of-year	23,460	0	25.0 24.0	100 22,312	0	27.6 17.9 97.4
	-end-of-year	20,140	0	32.5 23.2	100 19,584	0	27.2 17.0	100
	Student attrition	23,460	0	0.14 0.35	1 22,312	0	0.12 0.33	1
	Student will repeat the grade	14,224	0	23.8 42.6	100 13,262	0	28.7 45.3	100
	Student has dropped out	14,224	0	2.86 16.7	100 13,262	0	2.84 16.6	100
	Student has repeated a grade	23,460	0	0.46 0.50	1 22,312	0	0.55 0.50	1
	Age in years	23,460	4	8.27 1.69	18 22,312	4	11.9 1.82	21
	Student is a girl	23,460	0	0.48 0.50	1 22,312	0	0.46 0.50	1
	Number of school literate parents
	-none	23,460	0	0.37 0.48	1 22,312	0	0.39 0.49	1
	-one	23,460	0	0.30 0.46	1 22,312	0	0.32 0.46	1
	-two	23,460	0	0.33 0.47	1 22,312	0	0.29 0.46	1
	Child work
	-agricultural activities	23,333	0	0.49 0.50	1 22,162	0	0.57 0.50	1
	-retail and services	23,248	0	0.17 0.38	1 22,087	0	0.22 0.41	1
	Teacher absence in days	1,599	0	2.99 5.85	153	1,538	0	2.74 5.64	153
	Number of absent students	1,427	0	5.04 7.65	98	1,371	0	4.00 7.57	98
	Curricula was completed in:
	-Reading	1,599	0	10.4 30.5	100	1,538	0	8.65 28.1	100
	-Mathematics	1,599	0	10.3 30.4	100	1,538	0	9.69 29.6	100
	School and classroom organization
	-single-shift	1,599	0	0.57 0.50	1	1,538	0	0.62 0.48	1
	-multi-grade	1,599	0 0.096 0.29	1	1,538	0	0.10 0.30	1
	-double-shifting	1,599	0	0.33 0.47	1	1,538	0	0.28 0.45	1
	Infrastructure index
	-locality a	1,599	0	0.53 0.42	2	1,538	0	0.54 0.41	2
	-school b	1,599 -0.44	0.35 0.46	2	1,538 -0.44	0.35 0.46	2
	Number of rainy days	527	0	82.6 34.7	177	517	0	82.9 34.6	177

Table A - 4 :

 A4 Effects of the number of rainy days on end-of-year test scores in reading and mathematics

			Reading			Math
		(1)	(2)	(3)	(1)	(2)	(3)
	Panel A: Second-grade students
	Number of rainy days	-0.139*** -0.089*** -0.078*** -0.105*** -0.100*** -0.088***
		(0.02)	(0.02)	(0.02)	(0.02)	(0.02)	(0.02)
	Number of students	20,140	20,140	20,140	20,140	20,140	20,140
	Panel B: Fifth-grade students
	Number of rainy days	-0.115*** -0.049*** -0.039*** -0.148*** -0.081*** -0.074***
		(0.02)	(0.02)	(0.01)	(0.02)	(0.02)	(0.02)
	Number of students	19,584	19,584	19,584	19,584	19,584	19,584
	Area and country fixed effects	Yes	Yes	Yes	Yes	Yes	Yes
	Start-of-year test score	No	Yes	Yes	No	Yes	Yes
	Student and school characteristics	No	No	Yes	No	No	Yes
	Note:

Table A - 5 :

 A5 Effect of the number of rainy school days by student gender

		End of year test score	Grade
		Reading	Math	Average Repetition
	Panel A: Second-grade students
	Number of rainy days	-0.080*** -0.077*** -0.087***	1.720**
		(0.02)	(0.02)	(0.02)	(0.80)
	× student is a girl	0.005	-0.023*	-0.009	0.522
		(0.01)	(0.01)	(0.01)	(0.69)
	Number of students	20,140	20,140	20,140	14,224
	Panel B: Fifth-grade students
	Number of rainy days	-0.034**	-0.070*** -0.039***	0.129
		(0.02)	(0.02)	(0.01)	(0.88)
	× student is a girl	-0.011	-0.008	-0.009	0.014
		(0.01)	(0.01)	(0.01)	(0.82)
	Number of students	19,584	19,584	19,584	13,262
	Area and country fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes
	Note:

Table A - 6 :

 A6 Effect of the number of rainy days by student age The table shows estimates of the expected end-of-year test scores (in reading, mathematics, and the average score of both subjects) and the likelihood that the student will repeat the grade per s.d. increase of rainy days during school year and age of the student. Each column corresponds to an estimation where the end-of-year test scores and the likelihood of grade repetition are each regressed on the number of rainy days around school locations during the academic year interacted with age (in years) of the student while controlling for various characteristics of students and their schools as in Eq. (4). The standard errors are clustered at the school level and are reported in parentheses. Significance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01.

		End of year test score	Grade
		Reading	Math	Average Repetition
	Panel A: Second-grade students
	Number of rainy days	-0.237*** -0.230*** -0.244***	1.511
		(0.06)	(0.06)	(0.06)	(2.18)
	× age in years	0.019***	0.017***	0.018***	0.055
		(0.01)	(0.01)	(0.01)	(0.25)
	Number of students	20,140	20,140	20,140	14,224
	Panel B: Fifth-grade students
	Number of rainy days	-0.261*** -0.244*** -0.190*** -3.146
		(0.07)	(0.07)	(0.06)	(3.13)
	× age in years	0.019***	0.014**	0.012***	0.279
		(0.01)	(0.01)	(0.00)	(0.27)
	Number of students	19,584	19,584	19,584	13,262
	Area and country fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes
	Note:

Table A

 A The table shows estimates of the expected end-of-year score in reading, mathematics, and the average score of both subjects per s.d. increase of rainy school days by level of start-of-year test scores. The column "Grade repetition" reports the estimates of the expected likelihood of grade repetition per s.d. increase of rainy school days by levels of start-of-year test scores. Each column corresponds to an estimation where the end-of-year test score and the likelihood of grade repetition are regressed on the number of rainy school days at school locations interacted with binary variables that identify different quartiles of start-of-year test scores while controlling for various characteristics of students and their schools as in Eq. (4).

	-7: Effect of the number of rainy school days on end-of-year learning out-
	comes by initial level of academic performance
		End-of-year test score	Grade
		Reading	Math	Average Repetition
	Panel A: Second-grade students
	Number of rainy days	-0.048**	-0.051**	-0.060**	2.306*
		(0.02)	(0.02)	(0.02)	(1.31)
	× 2nd quartile start-of-year score -0.005	-0.012	0.004	-1.057
		(0.02)	(0.02)	(0.02)	(1.34)
	× 3rd quartile start-of-year score	-0.035	-0.063**	-0.041	0.211
		(0.03)	(0.02)	(0.03)	(1.42)
	× 4th quartile start-of-year score	-0.071**	-0.046	-0.071** -0.369
		(0.03)	(0.03)	(0.03)	(1.41)
	Number of students	20,140	20,140	20,140	14,224
	Panel B: Fifth-grade students
	Number of rainy days	-0.017	-0.088*** -0.044**	0.849
		(0.02)	(0.02)	(0.02)	(1.42)
	× 2nd quartile start-of-year score -0.001	0.041*	0.019	-0.603
		(0.02)	(0.02)	(0.02)	(1.46)
	× 3rd quartile start-of-year score	-0.013	-0.020	0.001	-0.287
		(0.02)	(0.03)	(0.02)	(1.60)
	× 4th quartile start-of-year score	-0.115*** -0.001	-0.067** -0.941
		(0.03)	(0.03)	(0.03)	(1.77)
	Number of students	19,584	19,584	19,584	13,262
	Area and country fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes
	Note: The standard errors are clustered at the school level and are reported in parentheses. Signifi-
	cance levels are denoted as follows: * p<0.10, ** p<0.05, *** p<0.01.

Table A - 8 :

 A8 Effects of the number of rainy days during weekends and the previous summer break on test scores and likelihood of grade repetition

		End of year test score	Grade
		Reading Math Average Repetition
	Panel A: Second-grade students
	Number of rainy days in 1km buffer during:
	-holidays	0.165	0.184	0.153	-39.579*
		(0.13)	(0.13) (0.13)	(20.89)
	-weekends	0.062	-0.075 -0.017	1.918
		(0.06)	(0.06) (0.06)	(8.41)
	Number of students	3,593	3,593	3,593	1,701
	Panel B: Fifth-grade students
	Number of rainy days in 1km buffer during:
	-holidays	-0.169	-0.018 -0.108	-13.307
		(0.11)	(0.15) (0.11)	(21.09)
	-weekends	0.029	0.018	0.022	12.386
		(0.05)	(0.05) (0.04)	(8.03)
	Number of students	3,699	3,699	3,699	1,822
	Administrative unit fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes

Note:

Table A

 A

	-9: Effect of the number of rainy days by student participation in retail or
	farm work
		End of year test score	Grade
		Reading	Math	Average Repetition
	Panel A: Second-grade students
	Number of rainy days	-0.080*** -0.081*** -0.098***	2.220**
		(0.02)	(0.02)	(0.02)	(0.93)
	× student worked	0.002	0.040**	0.023	-0.515
		(0.02)	(0.02)	(0.02)	(0.99)
	Number of students	20,140	20,140	20,140	14,224
	Panel B: Fifth-grade students
	Number of rainy days	-0.044**	-0.084*** -0.060*** -0.638
		(0.02)	(0.02)	(0.02)	(1.06)
	× student worked	0.040**	0.031	0.037**	0.897
		(0.02)	(0.02)	(0.02)	(1.08)
	Number of students	19,584	19,584	19,584	13,262
	Area and country fixed effects	Yes	Yes	Yes	Yes
	Start-of-year test score	Yes	Yes	Yes	Yes
	Student and school characteristics	Yes	Yes	Yes	Yes

The questions on absenteeism were not included in the survey administered to teachers and principals in Cameroon. Consequently, we excluded Cameroon from the sample used to produce the estimates reported in

, Moussa Hamani Ounteni and Alexis Le Nestour for assistance in recovering Map-Benin, as well as Saliou Abdou and Parfait Menou for excellent assistance and fieldwork when mapping primary schools in Benin. Funding from the French National Institute for Sustainable Development, the UKRI GCRF Accelerating Achievement for Africa's Adolescents Hub and the J-PAL's Post-Primary Education (PPE) Initiative is gratefully acknowledged. The study benefited from excellent research assistance by Edouard Pignede and Claire Kremer, scoping work on education in Benin undertaken by Kamon Yebie, as well as comments and insights from Luc Behaghel,

data and information on past surveys carried out at the PASEC, TAHMO for access to data from their weather stations, Open Street

Supplementary tables

The table shows estimates of the effect of one standard deviation variation of rainy weekdays on the likelihood that students will take the end-of-year tests. This allows us to check if presence at the end-of-year tests is affected by rainfall patterns during school days. We provide different estimates of the effect on the sample of students in second and fifth grades in separate columns using the specifications outlined in Eq. ( 1), ( 2), ( 3) and (4). Column (1) reports the estimates obtained with the specification of Eq. ( 1), column (2) corresponds to the estimates from Eq. ( 2) and column (3) the estimates from Eq. ( 3).


Julien Bourgeois
email: julien.bourgeois@univ-fcomte.fr

Adrien Henriet
email: adrien.henriet@univ-fcomte.fr

Jerome Lacan
email: jerome.lacan@ensica.fr

Tanguy Pérennou
email: tanguy.perennou@ensica.fr

Using a trace-based simulation tool to test the efficiency of error codes applied to wireless multimedia streaming

Keywords: Multimedia, FEC, error codes, simulation, wireless, streaming

This article presents an original method to validate the efficiency of the short length maximum separable distance forward error codes when using broadcasting mode in 802.11b/g multimedia streaming. Our method is based on real traces on which the forward error code will be mapped. Then, the quality of the streaming is evaluated by calculating the Peak Signal-to-Noise Ratio (PSNR).

Introduction

The development of wireless networks bandwidth is changing the way to access to multimedia contents. WiFi, also known as 802.11b/g, is the most popular local wireless network. WiFi can communicate according to two modes: point to point and broadcast. Point to point communications offer reliability by ensuring retransmission of lost packets. The lost packets retransmission is managed by the MAC layer of 802.11b/g. When using broadcasting mode, the MAC layer does not manage retransmission of lost packets. Therefore, in a video streaming session, errors will occur and video quality will decrease. In this article, we propose to evaluate the benefit of using adaptive Forward Error Code (FEC) by calculating the Peak Signal-to-Noise Ratio (PSNR) of the transmitted videos. This experiment based on real traces ensures a result very close to the reality. In the following section , we present conventional error control mechanisms. Section , discusses the efficiency of shortlength MDS FEC. This efficiency is calculated using real experiments that generate traces. Simulations use these traces to obtain results. Section concludes the article and describes future work to be done.

Conventional Error Control Mechanisms

In order to ensure packets transmission reliability, two essential types of mechanisms can be distinguished [START_REF] Lin | Error Control Coding: Fundamentals and Applications[END_REF] :

• reactive mechanisms (ARQ: Automatic Repeat request): when a packet is lost, the transmitter retransmits it. The packet loss indication can be carried out by the emission of positive or negative acknowledgments.

• proactive mechanisms (FEC: Forward Error Correction): the transmitter sends redundancy packets a priori, making it possible for the receiver to recover certain lost packets thanks to the redundancy packets. These packets are usually calculated by using block error correcting codes. These codes generate n -k redundancy packets starting from a set of k information packets (to be protected) so that the receiver can recover the k information packets as soon as it has received a sufficient number of packets among all the sent packets.

It should be pointed out that these two mechanisms are often combined by using acknowledgments to adjust the quantity of redundancy of FEC codes. Such a combination is often called Hybrid ARQ of type I (packets already received by the receiver are not combined with redundancy packets received after the retransmission request) or of type II (all packets that relate to the same block of k packets are used by the decoder).

Being relatively flexible and easy to implement, reactive mechanisms are the most used in packet transmissions (in TCP for example). Proactive mechanisms are particularly used in multicast transmissions (where a redundancy packet allows to compensate for the loss of a different packet for each of the receivers). In addition, real-time transmissions which do not allow delays due to retransmissions, often use proactive mechanisms.

In the context considered here, i.e. real-time multicast video transmissions, the use of FEC is clearly the best solution.

Two main families of FEC can be distinguished: the Maximum Separable Distance (MDS) codes and the Low-Density Parity-Check (LDPC) ones. The MDS codes, often based on Reed-Solomon codes, have the property that the information packets can be recovered as soon as any k packets among the n sent ones are received. However, for long lengths (n > 500), these codes, which have quadratic encoding and decoding complexities, become too costly in CPU. On the other hand, LDPC codes have linear complexity and then can be used with long lengths [START_REF] Luby | Efficient erasure correcting codes[END_REF]7]. However, they need more received packets (between 5 and 20% [START_REF] Plank | A practical analysis of lowdensity parity-check erasure codes for wide-area storage applications[END_REF]) than MDS ones to recover the information packets, especially for short codes.

A strong constraint on FEC used for real-time transmissions is that they must have a short length. Indeed, the decoder must necessarily wait the reception of the last packets to begin the decoding. Then it appears that the FEC codes which must be considered here are short-length MDS codes. Next Sections present an evaluation of these codes.

Evaluating the short-length MDS FEC efficiency

To evaluate the efficiency of FEC codes, and extremely important point is to consider a realistic representation of the channel in terms of packet loss rate and losses distribution.

Even if some modeling of the 802.11 channel losses were already proposed (see e.g. [8] ), we preferred consider real traces to evaluate the codes.

Experiments

A platform of wireless tests was installed in order to collect IP broadcast packets loss measurements. It relies on an 802.11b infrastructure mode network, which comprises one access point, 10 nodes with an 802.11b interface, and one node with an Ethernet interface, directly connected to the access point. Various types of 802.11b interfaces (PCI, USB and PCMCIA) and material architectures (PC, laptop PC, Pocket PC) were used.

Methodology

First, a serie of 5 experiments was carried out during which all the nodes remained motionless. The nodes were placed at various places, with a distance of 0 to 4 walls with the access point. In a second series of 6 experiments, 4 of the nodes were mobile: each one was carried by an operator walking without a specified trajectory while always remaining in the range of the access point.

In each experiment, the fixed node pacman transmits a multicast MPEG1/RTP video-only stream, which consisted of 300 to 1400 bytes packets with a mean bandwidth of 500kbps. Each experiment is approximately 90 seconds long. The video stream is produced by the Java Media Studio application and is broadcasted in multicast on the wireless network. The transmitter is fixed in all of the experiments and remains close to the access point, with a minimal attenuation of the signal on the corresponding link. The resulting MAC packets are automatically re-emitted by the access point towards 9 nodes over the wireless network, and 1 node over the Ethernet. Thus, each experiment made it possible to record the sequence numbers of the RTP packets actually received by each receiver. In 802.11b, there is no acknowledgement nor retransmission in the MAC layer for broadcasted packets [1]. Hence, a received UDP packet was transmitted only once from the transmitter towards the access point and then towards the receivers. By making the assumption that no packet is lost on the Ethernet, the Ethernet receiver serveur-mitv allows the measurement of MAC losses over the uplink between the transmitter and the access point, the wireless receivers allowing to measure the additional MAC losses over the downlink.

Results

Figure 1. Traces of packets loss with fixed stations Figure 2. Traces of packets loss with mobiles stations

The 5 experiments, where all the nodes were fixed, produced similar traces, just as the 6 experiments with mobile nodes. We can see here that the mobility isn't a significant factor who influate in packet lost. Figures 1 and2 gives the traces of one of the experiments with fixed stations and one with mobile stations (pda, miura, montardo and harfang).

The x-axis corresponds to the sequence of transmitted packets and the y-axis illustrates the burstiness of packet losses for each station. The receiving stations are listed on the left. The percentage given after the station name is the packet loss rate observed for this station in this experiment. Each line indicates if packet #x was received (lowest position) or lost (higher positions) by the corresponding station. Loss bursts are represented by their length along the y-axis. To get an idea of the length of the bursts, consider that station loss lines are 10 losses apart. In experiment #1 for example, serveur-mitv does not lose any packet, which results in a straight line. In contrast netlook experiences a big burst of 32 packets starting at packet #1645 and a few other less important bursts: this results in an almost straight line with a high peak for the 32-packet long burst and a few smaller peaks corresponding to bursts of 1, 2, 3 or 5 packets. Finally, pda experiences a lot more loss bursts (255 bursts have a length greater than 1, generally 2, 3, 4, or 5 packets long), which results in a very irregular line. The ten receiving stations are classified on the y-axis from the nearest to the most distant with respect to the access point, except for meta-b which is close to the access point but receives few packets due to hardware limitations. The experiments show that practically no data is lost on the uplink, and that many losses are observed on the downlink. They are related to the used material, the position of the fixed receiver or to mobility. Some simultaneous bursts attest the occurrence of important disturbances on the downlink, which could not be diagnosed. The peaks of very important amplitude recorded on meta-b reveal a disconnection phase due to a hardware configuration problem. More generally, a considerable quantity of packets are lost in bursts, which can be temporally correlated for a given receiver and/or spatially correlated for several receivers. The peaks observed for each individual receiver of Figure 2 denote temporal correlation. Spatial correlation can be observed too when peaks occur for the same packets, as in e.g. experiment #1 for packets #2168 to #2088, which are simultaneously lost by all stations except serveur-mitv and netlook. An analysis of these correlations can be found in [START_REF] Perennou | évaluation de mécanismes de contrôle d'erreur pour des transmissions multipoints sur des réseaux de mobiles[END_REF].

Simulation driven by traces

To evaluate the efficiency of the FEC codes applied to a video sequence, one solution is to count the ratio between wellreceived and losts packets. If this solution can show the theoritical efficiency of the FEC codes, it does not show the real efficiency of the FEC codes applied to a video sequence. The idea is to calculate the PSNR of the received video. Indeed of the methodology, and the type of trace, we can't know the size of each packet, and so, can't have a more realistic simulation based not only on packet, but on data. This isn't a limitation, because of the structure is the same, and be relatively near a real situation.

Methodology

As described in figure 3, our simulation tool takes two inputs: a movie and the experiment trace file described in section .

The movie is the Foreman, a reference sequence for testing multimedia platform. The aim of our tool is to compute the PSNR of the movie when packets are lost according to the trace file. This PSNR represents the available quality for the client for each decoded frame of the stream. Indeed, during multimedia streaming the lost packets will not have the same effect on the video quality. Some packets will seriously affect the visual quality of the video whereas others will not. This difference depends on various parameters like the kind of lost packet, the moment in the group of pictures (GOP) the packet is lost, etc. In fact, if the last P-frame of a GOP is lost, the quality will not be affected because, just after this image, an intra image will be decoded. As the last P-frame of a GOP and the first I-frame of the following GOP do not have any temporal dependances, only one frame will be damaged. An another example is when a packet is lost on an P-frame just before a camera movement. The resulting image will be damaged but the camera movement will delete this error.

That is why it is necessary to calculate the PSNR and not just to count the lost packets to evaluate the resulting video quality.

Our simulation tool can compute the PSNR with or without the short-length MDS FEC. In fact, it can apply any kind of processing on the packets. Our simulation tool is composed of two main boxes: the Simulation Box and the PSNR Box. The Simulation Box is composed of mplayer an open-source movie player and the Damager which is able to alter the packets transmitted by mplayer. The Damager does not randomly alter the packets, it takes as input the trace file generated in section . When a packet is generated in mplayer, the Damager alters it or not according to the trace file. It can also take an error code information into account. Here, we use our short-length MDS FEC as a plugin for the Damager. This plugin modifies the trace file by correcting the lost packets when it is possible. The PSNR Box receives only images, not packets. MPlayer receives packets and generates Portable Network Graphic (PNG) images. These images are then gathered by the PSNR Box which converts them into a raw format and then computes the PSNR. The output of our simulation tool is a text file which contains the PSNR for each generated image.

Results

The three figures 4, 5 and 6 show the PSNR of the movie Foreman visualized respectively by the client "coucouyaya", "meta-b" and "harfang" in the mobile environment. Strong or light FECs represent the minimum size of the redundancy information in the code. The solid curve represents the PSNR of video sequence without any errors. The dotted curve shows the PSNR calculated when strong FECs are applied to the packets whereas the dashed curve presents the PSNR calculated when light FECs are applied to the packets. The dotted-dashed curve shows the PSNR of the transmission without error codes. "Coucouyaya" receives the packets with few errors, that is why the three curves, PSNR with strong FECs, light FECs and without FECs are very close. The interesting thing to see is that the use of strong codes or light codes when there are no long bursts of errors is nearly equivalent. "Meta-b" and "harfang" have nearly the same error profile: lots of errors, and long bursts of errors. For these station, it is clear that the FECs enhance the visual quality of the video.

Figure 3. The simulation tool

But, when the bursts are too long, even the strong FECs can not correct the errors and retreiving the original content of the packets. This, is the second interesting fact taught by these experiments.

Thanks to these results, we can say that even strong FECs can not cope with long bursts as the stream will be interrupted.

Strong FECs delay the interruption but do not avoid it. The second fact is that when there are no long bursts, light FECs are sufficient. As the strong codes need a lot more bandwidth than the light ones, the idea is to preferably use them. But, the problem of long bursts remains. Solutions will be investigated later.

Conlusion and perpectives

We have presented our method to test the efficiency of the short length maximum separable distance forward error codes. Our simulation tool is able to calculate the quality of any streamed video by using trace files of previous transmission. The results show that the FECs can not cope with long bursts of error. They can delay the error but at last, the video quality will dramatically decrease. Future works will have to deal with this problem. The idea we will have to experiment is the following. Light FECs are kept to deal with short bursts of error and an RTP mixer [START_REF] Bourgeois | Video transmission adaptation on mobile devices[END_REF] capable of regenerating an I-frame is adding between the server and the clients to overcome the long bursts of error. When a interruption occurs, the RTP mixer will transform a P-Frame into a I-Frame, and then the video can continue thanks to this new GOP. With this mixed solution (reactive and proactive), the bigger part of the errors will be treated without adding too much traffic on the wireless channel.

[1] ANSI/IEEE Std 802.11. Wireless LAN MAC and PHY Speci-

  with high FEC PSNR with low FEC PSNR without FEC

Figure 4 Figure 6 .

 46 Figure 4. PSNR computation for coucouyaya


Schwab Camille
email: camille.schwab@aphp.fr

PhD Student Hindlet Patrick

Pharmacy student Nazoiri Charifa

PhD Korb-Savoldelli Virginie

Dr Camille Schwab

EXTERNAL VALIDATION AND COMPARISON OF EARLY READMISSION'S RISK SCORES IN THE ELDERLY: PROSPECTIVE OBSERVATIONAL STUDY READMISSION SCORES: EXTERNAL VALIDATION & COMPARISON

Keywords:

We conducted a prospective observational study to externally validate and statistically compare eight clinical risk scores in order to identify elderly at high risk

Introduction

Unplanned readmissions in older patients are a prominent and current issue for healthcare professionals and regulators. They can indeed be very deleterious for patients, leading to an increased risk of dependence and functional or psychosocial decline [START_REF] Arendts | The impact of early emergency department allied health intervention on admission rates in older people: a non-randomized clinical study[END_REF] and for hospitals as these readmissions are costly [START_REF] Zook | Repeated hospitalization for the same disease: a multiplier of national health costs[END_REF] . Moreover, the reduction of readmissions was introduced into guidelines by health system organizations as quality indicators [START_REF]Comment réduire le risque de réhospitalisations évitables des personnes âgées[END_REF][START_REF] Fischer | Is the Readmission Rate a Valid Quality Indicator? A Review of the Evidence[END_REF] Several clinical care interventions have been shown to reduce the risk of early unplanned readmission [START_REF] Hansen | Interventions to reduce 30-day rehospitalization: a systematic review[END_REF] but they may be difficult to implement and costly in terms of time, money and caregiving if delivered to all patients. Thus, it is necessary to predict patient readmission risk to target the patients who will most likely benefit from these interventions. Clinical risk scores can help physicians and pharmacists select these patients.

Eight clinical risk scores identifying inpatients at risk of unplanned readmission have been published to date (ISAR, TRST, Nomogram, LACE, VIP, SST, HOSPITAL, 80+ score [START_REF] Schwab | Risk scores identifying elderly inpatients at risk of 30-day unplanned readmission and accident and emergency department visit: a systematic review[END_REF][START_REF] Donzé | International Validity of the HOSPITAL Score to Predict 30-Day Potentially Avoidable Hospital Readmissions[END_REF][START_REF] Alassaad | A tool for prediction of risk of rehospitalisation and mortality in the hospitalised elderly: secondary analysis of clinical trial data[END_REF][START_REF] Ben-Chetrit | A simplified scoring tool for prediction of readmission in elderly patients hospitalized in internal medicine departments[END_REF] ).

They were externally validated in diverse populations and with different readmission time lapses.

In order to have a reliable comparison of these eight scores, we aimed to externally validate them in a homogeneous older population and with a unique definition of early unplanned readmission (i.e. emergency department (ED) visit or hospitalization 30 days after discharge).

Methods

An observational prospective cohort study was conducted in seven medical wards of two French university hospitals (February-August 2018). This study was approved by an ethic committee (Comité de Protection des Personnes Ouest 6, number 1086 RNI).

All patients aged 65 and older, discharged home (or nursing home) after hospitalization and reachable by telephone the 31st day after discharge, were included.

For each included patient, the eight scores were calculated from the electronic medical records (EMR). On the 31st day after discharge, medical records were checked and/or a phone call was made to ascertain unplanned readmission (ED visit or hospitalization).

Scores contain 3 to 8 questions divided into five categories: demographics, functional impairment, hospitalization, medication, comorbidities.

The discrimination of each score was assessed with the area under the receiver operating characteristic curve (AUC ROC) and metrological characteristics were calculated.

Results

Among 503 included patients, 134 (26.64%) had an early unplanned readmission. All the AUC ROCs of the scores were between 0.56 and 0.65. The VIP score had the lowest AUC ROC (0.56, 95%IC: 0.51-0.61) and LACE had the highest (0.65 95%CI: 0.60-0.70). The metrological characteristics of each score are presented in table I. The ISAR score had the highest sensitivity (76.12%) but the lowest specificity (40.38%). The SST, the VIP and the HOSPITAL scores had the highest specificity but the lowest sensitivity. Furthermore, the SST had also the highest positive predictive value and accuracy.

The HOSPITAL score and LACE index could not be calculated before discharge, because one of the items is the length of stay.

The ISAR, TRST and VIP scores might not be automatable since they call for a reading of the EMR, or even a face to face interview.

Discussion

In this observational prospective study, we externally validated eight clinical risk scores for 30day unplanned readmissions in an older population. These eight scores had all fair discrimination (AUC ROC greater than 0.5, but lower than 0.7), which is consistent with the literature [START_REF] Schwab | Risk scores identifying elderly inpatients at risk of 30-day unplanned readmission and accident and emergency department visit: a systematic review[END_REF][START_REF] Braes | Screening for risk of unplanned readmission in older patients admitted to hospital: predictive accuracy of three instruments[END_REF] .

In order to implement an intervention, an early and ideally automatable identification of patients is essential. Three scores (80+, Nomogram and SST) meet these requirements but have very scattered metrological characteristics. The SST has the highest specificity and, therefore, the best ability to correctly rank patients who will not be readmitted as they are at low risk for readmission.

As it seems more convenient to choose a specific test and intervene with patients classified as low risk of readmission but who will be readmitted (i.e. not missing false negative) rather than a sensitive test, the SST, which has the highest specificity, might be preferable. Moreover, it has good negative predictive values (76.52%) and low positive predictive values (50%).

The limitations of our study were that we included only certain medical wards and not surgical specialties, we included only persons discharged home (no rehabilitation) and 5.45% of the participants were lost to follow up.

Table : Metrological characteristics of the scores

 :

	Score	Cut-off	Specificity	Sensitivity	npv*	ppv*	Accuracy
		(≥)	(%)	(%)	(%)	(%)	(%)
	ISAR	3	40.38	76.12	82.32	31.68	49.90
	80+ Score	4	43.63	71.64	80.90	31.58	51.09
	LACE	11	50.68	69.40	82.02	33.82	55.67
	Nomogram 100	50.95	64.18	79.66	32.21	54.47
	TRST	3	72.09	38.81	76.44	33.55	63.22
	VIP	2	87.26	22.39	75.68	38.96	69.98
	HOSPITAL 7	90.79	11.19	73.79	30.61	69.58
	SST	3	91.87	22.39	76.52	50	73.36
	*npv = negative predictive value, ppv = positive predictive value

funding from agencies in the public, commercial or not-for-profit sectors.

Competing interests: None declared

Conclusion and implications

In this external validation study, we showed that the SST score is a useful tool to predict the risk of early unplanned readmission of the elderly patients. The perspectives of this study are to integrate the score into medical software in order to assess its ability to be automatically calculated, thus facilitating its implementation.