The Bangladeshi Child Mortality Miracle · 2019. 10. 25. · value was 143.6, and hence the target...

The Bangladeshi Child Mortality Miracle October 2014 Shruti Venkatraman

NORTH LONDON COLLEGIATE SCHOOL Mathematics Standard Level Internal Assessment Mathematics Standard Level Internal Assessment

Table of Contents

1. Introduction ....................................................................................................................................1

2. Data Collection and Initial Analysis .............................................................................................1

2.1 Overview.......................................................................................................................................................1

2.2 Has Bangladesh met the Millennium Development Goal?...........................................................................2

3. Modelling the Data Over Time .....................................................................................................3

The General Trend ..............................................................................................................................................3

Least Squares Regression Line ...........................................................................................................................4

Modelling the Data as a Polynomial Regression ................................................................................................8

Modelling the Data as an Exponential Regression .............................................................................................9

4. Conclusion.....................................................................................................................................11

5. Bibliography .................................................................................................................................12

5. 1 Works Cited ...............................................................................................................................................12

5.2 Works Consulted.........................................................................................................................................12

1. Introduction I chose this as the subject of my study because the Millennium Development Goals (MDGs)

are a recurring theme in Model United Nations (MUN) programs. I have participated and chaired at

many MUN conferences, and have always been interested in development issues, particularly those

relevant to health and medical infrastructure. The 4th Millennium Development Goal (MDG) is to

reduce child mortality by two thirds between 1990 and 2015.1 The significant and steady decrease

in child mortality rates in Bangladesh serves as an example of effective implementation of various

health and sanitation schemes. The Bangladeshi case is closely linked with political turbulence, and

demonstrates that these two factors go hand in hand in society.

In this study, I am aiming to work out in which year Bangladesh first met the MDG and to

determine what the best possible model for the child mortality data is using the least squares

method. Determining a model is important because it can inform government policy in terms of

resource management and allocation, both with the national context, and in the wider world. It will

also allow me to see if creating a mathematical model can give me further insights into this

situation. The data set itself is interesting because of the political and economic context, but I was

particularly interested in learning about how to determine how well a mathematical model fits a

data set, something that is not covered in the syllabus.

2. Data Collection and Initial Analysis 2. 1 Overview

To begin with, I wanted to find relevant data to investigate. Gap Minder is a program that

collates data from all countries to show the world’s most important trends. Using this program, I

was able to collate data (adapted from the United Nations Database) for the child mortality (number

of 0-5 year olds dying per 1000 born) in Bangladesh from 1951 to 2012.

I gave some thought about how to define the two variables, year and child mortality. I chose

to denote the year as x and I hence had to decide whether I would denote child mortality as a

function of n such as y(x), or as a sequence Yx or simply as a variable y. I chose not to define child

mortality as a function, as this implies causation between the two variables, which may not be true.

It would also not be appropriate to define it as a sequence because this would imply that child 1 "Bangladesh Exceeds MDG Target for Reducing Child Mortality." Dhaka Tribune. 24 Mar. 2014. Web. 07 June 2014

mortality has discrete values for each year, whereas child mortality is changes continuously

throughout each year. Hence I chose to define the variables as shown below.

• x is the year, and is the independent variable.

• y is child mortality (number of 0-5 year olds dying per 1000 born), and is the dependent

variable.

Raw Data 2 x y x y x y 1951 346.4 1972 220.3 1993 125.7 1952 335.3 1973 219.7 1994 119.8 1953 324.3 1974 218.7 1995 113.8 1954 313.8 1975 217.0 1996 108.1 1955 304.4 1976 214.4 1997 102.5 1956 294.9 1977 211.1 1998 97.2 1957 286.3 1978 207.1 1999 92.3 1958 278.1 1979 202.5 2000 87.7 1959 270.2 1980 197.8 2001 83.6 1960 262.4 1981 193.2 2002 79.7 1961 255.3 1982 188.6 2003 75.8 1962 248.4 1983 183.7 2004 71.8 1963 241.9 1984 178.7 2005 67.7 1964 235.9 1985 173.3 2006 63.4 1965 231.1 1986 167.5 2007 59.1 1966 227.1 1987 161.5 2008 54.9 1967 224.3 1988 155.5 2009 50.9 1968 222.3 1989 149.6 2010 47.2 1969 221.2 1990 143.6 2011 43.8 1970 220.6 1991 137.7 2012 40.9 1971 220.5 1992 131.7 2. 2 Has Bangladesh met the MDG?

We can use this data to see if Bangladesh has met and/or exceeded the MDG of reducing child

mortality by two thirds (~ 66.67 %) from 1950 to 2015.

Percentage decrease from 1990 to 2012 =

€

100 × 143.6 − 40.0143.6

= 72.1% (to 3 s.f.)

Therefore Bangladesh has exceeded the MDG.

We can also determine the year in which Bangladesh met the MDG. In 1990, the child mortality

value was 143.6, and hence the target for decreasing child mortality by 2015 is 47.9 (to 3.s.f) deaths

of 0-5 year olds per 1000 born. This means that Bangladesh met the MDG in 2009-2010, during

which the child mortality value decreased from 50.9 to 47.2 deaths of 0-5 year olds per 1000 born.

2 "Under-‐five Mortality Rate (per 1,000 Live Births)." Gapminder. Web. 14 Sept. 2014. .

I thought it would be interesting to use the data prior to 1990 to see if the establishment of the MDG

in 1990 and the polio project in 1998 brought a great percentage decrease in child mortality.

Percentage decrease from 1951 to 1990 =

€

100 × 346.4 −143.6346.4

= 58.5% (to 3 s.f.)

This means that following the establishment of the MDG and the involvement of the WHO,

Bangladesh experienced a greater percentage decrease. This would support the notion that such

development schemes have been successful, at least within Bangladesh.

3. Modeling the Data Over Time 3. 1 The General Trend I began by plotting the raw data for child mortality on Excel to help me visualise the data. I have

labelled relevant data points that are linked to political or economic events.

The Indian partition took place in 1947, in which India, East Pakistan (now Bangladesh) and West

Pakistan were created (data was only available from 1951 onwards). This was followed by

Bangladesh’s independence from the Islamic Republic of Pakistan in 1971 to become an

independent state.3 Furthermore, Polio targets infants and young children almost exclusively, and a

reduction in cases following the World Health Organization’s (WHO) eradication project starting in

1988 has greatly aided development in Bangladesh.4 As part of this initiative, the WHO also

encouraged immunisation and vaccination.

3 "Bangladesh." Central Intelligence Agency. Central Intelligence Agency, Web. 14 Sept. 2014. 4 "Bangladesh Polio Free." The Daily Star., 27 Mar. 2014. Web. 14 Sept. 2014.

Independence

WHO polio and child health project

MDG set up

MDG met

0

50

100

150

200

250

300

350

400

1951 1961 1971 1981 1991 2001 2011

Child Mortality (number of 0-5 year olds

dying per 1000 born)

Year (x)

Child Mortality Over Time

The child mortality initially decreases at a rate that appears constant and this is surprising

given continued political turbulence and hindered economic and social growth in the aftermath of

the Indian partition. Following this, it appears to level off, from 1967 to 1975, around the time of

Bangladeshi independence. After the graph plateaus, child mortality continues to decrease until

2012. The WHO initiative starting in 1988 is reflected in child mortality falling at what appears to

be an increasing rate after this point.

3.2 Least Squares Regression Line

Theory: After understanding the general trend of the data, I wanted to find a mathematical model

that fits this data set. A least squares regression line is a formal version of a line of best fit that we

might draw by hand. Like a line of best fit, it can be used to make predictions of what the y value

will be for an x value. Rather than finding the regression line using Excel, I decided to calculate the

regression line by hand, so that I could truly understand the theory behind the least squares method.

A regression line in the form y = a + bx allows us to determine the effect of a change in x on

y, and this is shown by the slope, b. A change of one unit in x will change y by b, on average.5 This

is because if we replace x

by (x + 1), then y = a + b(x

+ 1) = a + bx +b. In this

case, the slope can be used

to calculate rates of

change. The least squares

method works by

considering the residual

for each data point, which

is the distance between the

point and the regression

line. For example, in the

example graph above, the value x = 5 would give a value of y = 2.3 according to the regression line.

However, the actual value is y = 3. Hence, the residual is 0.7, and this represents the error of the

regression line with regards to this value of x.

5 Baker, Samuel L. "Simple Regression Theory." (n.d.): n. pag. Arnold School of Public Health. University of South Carolina, 2010. Web. .

Because points lie both above and below the regression line, there are both positive and

negative residuals. Hence, to calculate error, the value of the residuals must be squared to prevent

the underestimation of total error. The best least squares regression line will be the one with the

least distance from the original data points.6 An example of a least squares regression line and

residuals is shown on the next page.

To determine if a regression fits the data well, the coefficient of determination (R2) can be

used. The R2 value represents the percentage of the change in y that can be explained by change in

x.7 If the distance between the points and the line small, this means that the model fits the data and

this reflected by a R2 value very close to 1. The formula for calculating R2 is shown below, where

SSreg is the sum of the squared residuals, and SStot is the sum of the squared difference between

each y value and the mean of y. 8

€

R2 =100 ×SSregSStot

The R2 value can be calculated directly on Excel. The closer the value is to 1, the better the fitted

model.

Calculations:

The least squares regression line must be calculated in the form y = a +bx where a is the y intercept

and b is the slope. The formulas below are used. N is the number of values of x and y. 9

€

a =y −∑ b x∑N

€

b =N (x × y∑ ) − ( x)( y)∑∑

N x 2 − ( x)2∑∑

Using an excel spreadsheet for data handling, I was able to calculate all the values needed for the

equations of a and b. This is shown on the next page.

6 “S2 Bivariate Data Notes and Examples”. MEI Innovators in Mathematics Education 7 "Linear Regression." University of Leicester. Web. 14 Sept. 2014. 8 "Linear Regression." University of Leicester. Web. 14 Sept. 2014. 9 "Least Squares Regression-‐ Tutorial." Easy Calculations. N.p., n.d. Web. 4 Oct. 2014. .

Table Showing Calculation for the Least Squares Regression Line

Sum Values

These values can then be added together to give us the values we

need to substitute into the equations for a and b. This is shown on the

right.

I then substituted these values into the equations for a and b.

1. Calculating b, the slope

€

b =N (x × y∑ ) − ( x)( y)∑∑

N x 2 − ( x)2∑∑

€

=62(21753311) − (122853)(11023.8)62(243453075) − (122583)2

€

= −4.55 (to 3 s.f.)

x y xy x2 x y xy x2 x y xy x2 1951 346.4 675826.4 3806401 1972 220.3 434431.6 3888784 1993 125.7 250520.1 3972049 1952 335.3 654505.6 3810304 1973 219.7 433468.1 3892729 1994 119.8 238881.2 3976036 1953 324.3 633357.9 3814209 1974 218.7 431713.8 3896676 1995 113.8 227031.0 3980025 1954 313.8 613165.2 3818116 1975 217.0 428575.0 3900625 1996 108.1 215767.6 3984016 1955 304.4 595102.0 3822025 1976 214.4 423654.4 3904576 1997 102.5 204692.5 3988009 1956 294.9 576824.4 3825936 1977 211.1 417344.7 3908529 1998 97.2 194205.6 3992004 1957 286.3 560289.1 3829849 1978 207.1 409643.8 3912484 1999 92.3 184507.7 3996001 1958 278.1 544519.8 3833764 1979 202.5 400747.5 3916441 2000 87.7 175400.0 4000000 1959 270.2 529321.8 3837681 1980 197.8 391644.0 3920400 2001 83.6 167283.6 4004001 1960 262.4 514304.0 3841600 1981 193.2 382729.2 3924361 2002 79.7 159559.4 4008004 1961 255.3 500643.3 3845521 1982 188.6 373805.2 3928324 2003 75.8 151827.4 4012009 1962 248.4 487360.8 3849444 1983 183.7 364277.1 3932289 2004 71.8 143887.2 4016016 1963 241.9 474849.7 3853369 1984 178.7 354540.8 3936256 2005 67.7 135738.5 4020025 1964 235.9 463307.6 3857296 1985 173.3 344000.5 3940225 2006 63.4 127180.4 4024036 1965 231.1 454111.5 3861225 1986 167.5 332655.0 3944196 2007 59.1 118613.7 4028049 1966 227.1 446478.6 3865156 1987 161.5 320900.5 3948169 2008 54.9 110239.2 4032064 1967 224.3 441198.1 3869089 1988 155.5 309134.0 3952144 2009 50.9 102258.1 4036081 1968 222.3 437486.4 3873024 1989 149.6 297554.4 3956121 2010 47.2 94872.0 4040100 1969 221.2 435542.8 3876961 1990 143.6 285764.0 3960100 2011 43.8 88081.8 4044121 1970 220.6 434582.0 3880900 1991 137.7 274160.7 3964081 2012 40.9 82290.8 4048144 1971 220.5 434605.5 3884841 1992 131.7 262346.4 3968064

∑ x 122853

∑ y 11023.8

∑ xy 21753311

∑ x2 243453075

2. Calculating a, the y intercept

€

a =y −∑ b x∑N

€

=11023.8 + 4.55(122853)

62

€

= 9194.25 Hence, the equation for the least squares regression line is y = -4.55x + 9194.25

I then compared this to the least squares regression line that was fitted on Excel. This produced the

same equation.

The R2 value is 0.981, which is relatively close to 1. This signifies that there is a strong correlation

between child mortality and the year. However, this does not necessarily suggest a causal

relationship. There are evident flaws to this model. Most importantly, in terms of predictions, this

equation will cross the x-axis and have negative y values. However, we know this is impossible, and

it also impossible for child mortality to reach zero. In the United Kingdom, which has one of the

most advanced healthcare systems in the world, the child mortality rate is 5 0-5 year olds out of

1000 born.10 Child mortality rates such as the UK’s are what developing countries are aiming to

reach. Furthermore, the data doesn’t exactly form a straight line, hence I chose to explore other

modelling options.

10 "Mortality Rate, Under-‐5 (per 1,000 Live Births)." The World Bank. N.p., n.d. Web. 04 Oct. 2014. .

y = -‐4.5503x + 9194.2 R² = 0.98124

0

50

100

150

200

250

300

350

400

1951 1961 1971 1981 1991 2001 2011


dying per 1000 born) (y)

Year (x)

Child Mortality Over Time Using Least Squares Regression Line

3. 3 Modelling the data as a polynomial regression

I thought that the shape of the data seemed similar to the graph of f(x) = -x3. I decided to model the

data on Excel using a polynomial regression, as it is complicated to use the least squares method. I

initially tried to model the data using a polynomial regression to the third order.

While this produced a higher R2 value than the least squares regression line, it doesn’t appear to be

a close fit. I then tried modelling the data using polynomial regressions of a higher order. When I

did this, I found that a polynomial regression to the 6th order fit the data very closely.

This regression has a higher R2 value than the cubic and the least squares regression lines. This

means it is more accurate. However, we encounter the same problem, because the y values do

eventually become negative. I therefore decided to try and a find a model that tends to zero and

does not intersect with the x-axis.

y = -‐0.0011x3 + 6.8137x2 -‐ 13501x + 9E+06 R² = 0.98521

0

50

100

150

200

250

300

350

400

1951 1961 1971 1981 1991 2001 2011 Child Mortality (number of 0-5 year

olds dying per 1000 born) (y)

Year (x)

Child Mortality Over Time Using a Cubic Regression

y = -‐2E-‐07x6 + 0.0022x5 -‐ 10.99x4 + 29015x3 -‐ 4E+07x2 + 3E+10x -‐ 1E+13 R² = 0.99928

0 50 100 150 200 250 300 350 400

1951 1961 1971 1981 1991 2001 2011

Child Mortality (number of 0-5 year

olds dying per 1000 born) (y)

Year (x)

Child Mortality Over Time Using a Polynomial Regression to the 6th Order

3.4 Modelling the data as an exponential regression

In order to create a model

that is useful for predicting

future trends, the y values

have to tend to zero, and

not cross the x-axis.

Immediately, I thought of

the graph of e-x. Evidently,

all the data would not

match this regression. A

graph of e-x is shown to the

right. Another reason I chose to look into further this is because typically all natural systems are

exponential. For example, birth rates are exponentials. This is because the rate of change is

proportional to the number of births, and a higher birth rate leads to even higher birth rate. Bearing

this mind, we can expect that lower child mortality will lead to even lower child mortality. Hence I

decided to model just the data from 1990 onwards, to see if there is an exponential regression that

will allow us to make accurate predictions. I used Excel to do this.

This seems to fit the last 12 data points very well. This regression also has a very high R2 value. It

is important to note that equation for this regression is y = 550e-‐0.056x, which would mean that

y = 5E+50e-‐0.056x R² = 0.99258

0

20

40

60

80

100

120

140

160

1985 1990 1995 2000 2005 2010 2015 Child Mortality (number of 0-5 year olds


Year (x)

Child Mortality From 1990

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 0.5 1 1.5 2 2.5 3

y

x

Graph of e-‐x

the y intercept (the child mortality at year 0) would be 550, and this is evidently impossible.

Hence the equation is only applicable to this segment of data. To determine if this is a useful

model to use for predictions based on this segment of the data, I increased the domain to see if it the

forecasts seemed reasonable.

By extending the domain up to 2062, 50 years after the last data point, we can see that child

mortality is tending to zero. This model is more useful because we can make forecasts on further

decreases of child mortality rates. This can be used to inform government policy. While this

regression does not take into account all the data, it fits the final data points after the WHO polio

eradication project extremely well. This is not necessarily a limitation, as the national circumstances

are changing constantly. If the years to come are consistent with the past 12 years, then we can

make forecasts that will be useful, despite a certain level of uncertainty and error.

4. Conclusion In this study, I investigated child mortality data from Bangladesh from 1951 to 2012. I

began by collating the data, and doing percentage decrease calculations to determine the overall

decrease in child mortality and in which year Bangladesh met the MDG of reducing child mortality

by two thirds. I plotted the data to get a general idea of the trend, and I linked certain data points to

important regional events. I then calculated the least squares regression line by hand and verified

this with the least squares regression line produced by Excel. I then also modelled the data using

y = 5E+50e-‐0.056x R² = 0.99258

0

20

40

60

80

100

120

140

160

1980 1990 2000 2010 2020 2030 2040 2050 2060 2070



Year (x)

Child Mortality From 1990

polynomial regressions, and modelled the last segment of data (from 1990 onwards) using an

exponential regression. I used R2 values to determine whether or not a model fitted the data well,

and to make comparisons. Each regression had its limitations, showing me how complex it can be

to model a natural system accurately.

Modelling data using regressions has many applications. We can use it to establish whether

there is a relationship between two variables, even when this relationship is not linear, and this can

help to make predictions about future changes. With regards to child mortality, by analysing the

Bangladeshi child mortality model, other countries facing similar developmental hurdles, can map

their relative growth and mould solutions to be effective in their national context. By plotting child

mortality rates against time, we can draw links between economic and political events and child

mortality. This can help when constructing child health and medical infrastructure policies. Most

importantly, the model can be used to predict future decreases in child mortality, assuming that all

external factors remain constant. All of this can help to provide the basis for improving

infrastructure, sanitation, and the physician to patient ratio.

I thoroughly enjoyed being able to apply a variety of mathematical skills to a real life

situation, ranging from basic percentage decrease calculations to more difficult calculations of the

least squares regression line, something which is out of the realm of the syllabus. With regards to

fitting the data to different regressions, I learnt how to evaluate the usefulness of different models.

Furthermore, having used the R2 value in statistics in other subjects such as Chemistry, I enjoyed

learning about what this value signifies and how it is calculated. It was equally fascinating to link

characteristics of the data to the national context. Following Bangladesh’s independence in 1971,

child mortality decreases steadily until 2012, leading to the country meeting the MDG. This can be

attributed to international aid to Bangladesh shortly after its birth as a nation, as well as the decrease

in the incidence of polio following the World Health Organization’s (WHO) eradication program11,

and an increase in the availability of baby immunizations12. This project has allowed me to apply

mathematics learnt in the classroom to a current global discussion, thereby broadening my

understanding of mathematics as a dynamic subject.

11 "Bangladesh Polio Free." The Daily Star. N.p., 27 Mar. 2014. Web. 14 Sept. 2014. 12 Breiman, Robert F., Peter Kim Streatfield, Maureen Phelan, Naima Shifa, Mamunur Rashid, and Mohammed Yunus. "Effect of Infant Immunisation on Childhood Mortality in Rural Bangladesh: Analysis of Health and Demographic Surveillance Data." The Lancet 364.9452 (2004): 2204-211. Web. 7 June 2014.

5. Bibliography

Works Cited

Baker, Samuel L. "Simple Regression Theory." Arnold School of Public Health. University of South Carolina, 2010. Web. .

"Bangladesh." Central Intelligence Agency. Web. 04 June 2014.

. "Bangladesh Exceeds MDG Target for Reducing Child Mortality." Dhaka Tribune. 24 Mar. 2014.

Web. 07 June 2014. .

"Bangladesh Polio Free." The Daily Star. 27 Mar. 2014. Web. 14 Sept. 2014.

. "Least Squares Regression-‐ Tutorial." Easy Calculations. Web. 4 Oct. 2014.

.

"Mortality Rate, Under-‐5 (per 1,000 Live Births)." The World Bank. Web. 04 Oct. 2014.

. "Under-five Mortality Rate (per 1,000 Live Births)." Gapminder. Web. 14 Sept. 2014.

.

Works Consulted

"Bangladesh: Children and Women Suffer Severe Malnutrition." UN Office for the Coordination of

Humanitarian Affairs, 19 Nov. 2008. Web. 07 June 2014.

. "Bangladesh and Development." The Economist. 03 Nov. 2012. Web. 06 June 2014.

.

Breiman, Robert F., Peter Kim Streatfield, Maureen Phelan, Naima Shifa, Mamunur Rashid, and

Mohammed Yunus. "Effect of Infant Immunisation on Childhood Mortality in Rural Bangladesh: Analysis of Health and Demographic Surveillance Data." The Lancet 364.9452 (2004): 2204-211. Web. 7 June 2014. .

"Ending Preventable Child Deaths before 2035: Bangladesh Call to Action." UNICEF. 21 July

2013. Web. 07 June 2014. .

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The Bangladeshi Child Mortality Miracle · 2019. 10. 25. · value was 143.6, and hence the target...

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