Introduction

Infant mortality rate (IMR) is a measurement of human infant deaths in a group younger than one year of age per 1000 births. IMR is an important indicator of the overall health and wellbeing of a country.
GDP per capita is the Gross Domestic Product at nominal values divided by the midyear population and considered as a measure of the standard of living in a country or the level of economic development.

The rationale of the investigation:

High infant mortality rates generally indicate human needs in medical care, nutrition, sanitation, etc. are unmet. Many studies suggest that higher income at country level is closely correlated with higher health status for that country’s population. It is also assumed that the indexes of IMR and GDP per capita have a negative relationship.

Problem Statement

The problem driving the investigation:

Understanding the relationship between two quantitative variables (IMR and GDP per capita) in order to allow making accurate predictions. It is interesting to know if the GDP per capita index of a country can be used to make predictions of infant mortality rate in that country.

The investigation problem will be solved by using linear regression model for modelling the relationship between the variables of interest. Correlation will be used to evaluate the strength and the direction of this relationship (if any). These methods assume that a predictor variable, GDP per carita, provides information about a dependent variable, infant mortality rate.

Data

The countries dataset used in this presentation is obtained from Kaggle.com https://www.kaggle.com/fernandol/countries-of-the-world
This dataset is open and has a Creative Commons Licence. It contains 227 observations of 20 variables that are self-explanatory.
countries dataset contains information on population, region, area size, infant mortality, population density, coast/area ratio, net migration, GDP, literacy, phones per 1000, arable(%), crops(%), climate, deathrate, birthrate, industry, agriculture, service and other(%) for 2010.
The variables of interest were selected in this step. These variables are:
- Infant mortality (per 1000 births)[numeric]: infant death number per 1000 births of the country, the range is from 2.29 to 191.19
- GDP ($ per capita)[integer]: total economic output of a country divided by the number of people and adjusted for inflation, the range is from 500 to 55100

countries <- read_csv("countries.csv")
countries <- countries %>% select(`Infant mortality (per 1000 births)`, `GDP ($ per capita)`)

Descriptive statistics

Summary statistics for variable Infant mortality (per 1000 births) and GDP ($ per capita)are as follow:

summary(countries$`Infant mortality (per 1000 births)`, na.rm =T)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##    2.29    8.15   21.00   35.51   55.70  191.19       3

sd(countries$`Infant mortality (per 1000 births)`, na.rm = T)

## [1] 35.3899

summary(countries$`GDP ($ per capita)`, na.rm=T)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##     500    1900    5550    9690   15700   55100       1

sd(countries$`GDP ($ per capita)`, na.rm = T)

## [1] 10049.14

Data preprocessing

missing values are scanned and removed from the dataset
using Z-score test, 4 outliers are identified and removed with subsetting

which(is.na(countries))

## [1]  48 222 224 451

countries <- na.omit(countries)
z.score<-countries$`Infant mortality (per 1000 births)` %>% scores(type = "z")
which(abs(z.score) > 3)

## [1]   1   6 183

z.scores<-countries$`GDP ($ per capita)` %>% scores(type = "z")
which(abs(z.scores) > 3)

## [1] 121

countries<- countries[-c(1,6,121,183),]

Data visualisation - part 1

Histogram of variable Infant mortality (per 1000 births) was produced and shows that it is right skewed
Transformation is implemented using logarithm to reduce the right skewness of the data

par(mfrow = c(1,2))
hist(countries$`Infant mortality (per 1000 births)`, main = 'Infant Mortality', xlab = 'Infant Mortality', col = "lightblue")
log(countries$`Infant mortality (per 1000 births)`) %>% hist(main = "log(Infant Mortality)", col = "lightblue")

Data visualisation - part 2

Histogram of variable GDP ($ per capita) was produced and shows that it is right skewed
Transformation is implemented using logarithm to reduce the right skewness of the data

par(mfrow=c(1,2))
hist(countries$`GDP ($ per capita)`, main = 'GDP', xlab = 'GDP($per capita)', col = "grey")
log(countries$`GDP ($ per capita)`) %>% hist(main = "log(GDP)", ylim = c(0,35), col = "grey")

Data visualisation - overview

a scatter plot of the transformed varaibles is as follow, to give an overview of the relationship between the two variables

plot(log(`Infant mortality (per 1000 births)`) ~ log(`GDP ($ per capita)`), data = countries)

Linear Regression - Overall Model

Hypotheses for the overall linear regression model:
- $H_0$: The countries data does not fit the linear regression model.
- $H_A$: The countries data fits the linear regression model.
- We test the overall model using the $F$-test.
Assumptions:
- Independence
- Linearity
- Normality of residuals
- Homoscedasticity
Decision Rules:
- Reject $H_0$: if $p$-value < 0.05
- Ohterwise, fail to reject $H_0$.
Conclusion:
- Test will be statistically significant if we reject $H_0$.
- Otherwise, the test is not statistically significant.

Linear Regression - Testing the Overall Model

The $p$-value for the $F$-test is really small, $F$(1, 218) = 610.9, $p$ < 0.001.
The model is statistically significant. We can intepret the coefficients.

model1 <- lm(log(`Infant mortality (per 1000 births)`) ~ log(`GDP ($ per capita)`), data = countries)
model1 %>% summary()

## 
## Call:
## lm(formula = log(`Infant mortality (per 1000 births)`) ~ log(`GDP ($ per capita)`), 
##     data = countries)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.64617 -0.32677 -0.01089  0.35710  1.62962 
## 
## Coefficients:
##                           Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                 9.5800     0.2677   35.79   <2e-16 ***
## log(`GDP ($ per capita)`)  -0.7637     0.0309  -24.72   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5409 on 218 degrees of freedom
## Multiple R-squared:  0.737,  Adjusted R-squared:  0.7358 
## F-statistic: 610.9 on 1 and 218 DF,  p-value: < 2.2e-16

Linear Regression - Interpreting the Coefficients and $R^2$

Interpreting the Intercept
- The best line fit: $log(Infant mortality (per 1000 births)) = 9.58 - 0.764 \times log(GDP (dollar per capita))$
- when $log(Infant mortality (per 1000 births)) = 0$, $log(GDP (dollar per capita)) = 9.58$
- however, $log(0) = -\infty$ doesn’t exist, therefore, the intercept doesn’t have ameaningful interpretation.
Interpreting the Slope
- As $log(Infant mortality (per 1000 births))$ increases by 1 unit, $log(GDP (dollar per capita))$ changes on average by -0.764.
$R^2$
- $R^2 = 0.74$
- 74% of the variability in log infant mortality can be explained by a linear relationship with log GDP.

Linear Regression - Testing Model Parameters

Hypotheses for linear regression modelparameters:
- Intercept:
  - $H_0: \alpha = 0$
  - $H_A: \alpha \neq 0$
- Slope:
  - $H_0: \beta = 0$
  - $H_A: \beta \neq 0$
Assumptions:
- Independence
- Linearity
- Normality of residuals
- Homoscedasticity
Decision Rules:
- Reject $H_0$: if $p$-value < 0.05, or if 95% CI of the parameter doesn’t capture $H_0: \alpha/\beta = 0$
- Ohterwise, fail to reject $H_0$.
Conclusion:
- Test will be statistically significant if we reject $H_0$.
- Otherwise, the test is not statistically significant.

Linear Regression - Testing Assumptions visualisation

par(mfrow=c(2,2))
model1 %>% plot(which = 1)
model1 %>% plot(which = 2)
model1 %>% plot(which = 3)
model1 %>% plot(which = 5)

Linear Regression - Testing Assumptions interpretation

Residuals vs. Fitted:
- Trend line is flat and variability on $y$ axis is constant across the range of values on the $x$ axis.
- This indicates linearity and homoscedasticity
Normal Q-Q:
- Residuals fall close to the line.
- This indicates normality
Scale-Location: +The red line is close to flat, and variance in the square root of the standardised residuals are consistent across predicted values.
- This indicates homoscedasticity
Residuals vs. Leverage:
- The bands are not visible and no values fall close to the upper and lower red bands.
- This indicates that there are no influential cases

Linear Regression - Strength and Direction of Linear Relationships

Correlation Coefficient: $r = -0.86$
Strong negative linear relationship between log infant and log GDP.

r <- cor(log(countries$`Infant mortality (per 1000 births)`), log(countries$`GDP ($ per capita)`), use = 'complete.obs')
r

## [1] -0.8584828

library(psychometric)
CIr(r, n = 220, level = .95)

## [1] -0.8897237 -0.8192381

detach('package:psychometric', unload = T)

Linear Regression - Interpretation

Simple linear regression summary:
- Linearity, normality of residuals, homoscedasticity, no influentilial cases were assumed.
- $r = -0.86, r^2 = 0.74$.
- $F(1, 218) = 610.9, p < 0.001$.
- $a = 9.58, p < 0.001$.
- $b = -0.764, p < 0.001$.
Decision:
- Overall model: Reject $H_0$.
- Intercept: Reject $H_0$.
- Slope: Reject $H_0$.
- $log(Infant mortality (per 1000 births)) = 9.58 - 0.764 \times log(GDP (dollar per capita))$
Conclusion:
- There was a statistically significant negative linear relationship between the infant mortality and the GDP of a country.

Discussion

Conclusion and findings:

Based on the investigation, there was a statistically significant negative linear relationship between GDP per capita and IMR index of a country. The estimated linear regression model could be used for further predictions.

Strengths:
- The dataset is relatively complete and did not have many missing values.
- We were able to generate a linear regression model that uses GDP as a predictor for infant mortality rate.
Limitations:
- The 2010 year data used for the investigation can be considered outdated.
- In order to get a more complete picture, other factors that affect infant mortality rate besides GDP need to be investigated.
Future investigations could use more recent data.

Assignment 3

Simple linear regression analysis

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