Worksheet Week 2

Self-Assessment Questions³

Give an example for a two-sample test for a mean.
Give an example for a two-sample test for a proportion.
Why do we calculate the t-score as $t =\frac{\text{Estimate of parameter - null hypothesis value of parameter}}{\text{Standard error of estimate}}$ ?
What is the difference between a t-score and a z-score?
What are the strengths and weaknesses of two-sample tests?

Please stop here and don’t go beyond this point until we have compared notes on your answers.

Two-Sample Tests in R

Data Preparation

We are working with the World Development Indicators again. Data are taken from World Bank (2024), Boix et al. (2018), and Marshall & Gurr (2020).
To save you rooting in your files, here is the codebook (you are welcome):

Table 2: WDI Codebook
variable	label
Country Name	Country Name
Country Code	Country Code
year	year
democracy	0 = Autocracy, 1 = Dictatorship (Boix et al., 2018)
gdppc	GDP per capita (constant 2010 US$)
gdpgrowth	Absolute growth of per capita GDP to previous year (constant 2010 US Dollars)
enrl_gross	School enrollment, primary (% gross)
enrl_net	School enrollment, primary (% net)
agri	Employment in agriculture (% of total employment) (modeled ILO estimate)
slums	Population living in slums (% of urban population)
telephone	Fixed telephone subscriptions (per 100 people)
internet	Individuals using the Internet (% of population)
tax	Tax revenue (% of GDP)
electricity	Access to electricity (% of population)
mobile	Mobile cellular subscriptions (per 100 people)
service	Services, value added (% of GDP)
oil	Oil rents (% of GDP)
natural	Total natural resources rents (% of GDP)
literacy	Literacy rate, adult total (% of people ages 15 and above)
prim_compl	Primary completion rate, total (% of relevant age group)
infant	Mortality rate, infant (per 1,000 live births)
hosp	Hospital beds (per 1,000 people)
tub	Incidence of tuberculosis (per 100,000 people)
health_ex	Current health expenditure (% of GDP)
ineq	Income share held by lowest 10%
unemploy	Unemployment, total (% of total labor force) (modeled ILO estimate)
lifeexp	Life expectancy at birth, total (years)
urban	Urban population (% of total population)
polity5	Combined Polity V score

Load the data set
The Polity V Score (variable polity5) codes regimes from -10 (indicating perfect autocracy) to +10 (indicating perfect democracy). With the tidyvserse, create a new variable called democracy which codes all countries with a Polity V score lower than +1 as dictatorships, and all countries with a Polity V score from +1 to +10 as democracies.
Apply the same procedure to gdp, cutting it at its median into two levels, Developing and Developed, creating a new variable called gdpcat.
Last up is the variable gdpgrowth. Create a new variable called growth which divides countries into “slow-growing” and “fast-growing” countries, using the mean as the cut-off point.

Guided Example – Two-Sample Test for a Proportion

Let us find out whether a higher proportion of developed countries is democratic than developing countries.
State the null hypothesis and the directional alternative hypothesis for this research question.
In order to test this hypothesis, we need to first create a cross-tabulation:

table(wdi$gdpcat, wdi$politybin)
            
             Dictatorship Democracy
  Developing           21        36
  Developed            19        71

We now take the number of observations which are classed as democracies per development status.
We also calculate the row totals, as this gives us the total number of developing and developed countries, respectively.
Then we are ready to use the prop.test() command, by first specifying the number of countries which are democracies, then the total number of developing and developed countries, then advising R that a correction for small sample sizes is not necessary in our case.
Our hypothesis is directional, because we expect a higher proportion of developed countries to be democratic than developing countries. The status Developing is the lower category, and we thus expect this proportion to be smaller, or “less”. We add this to the test function as option alternative = "less".

prop.test(c(36,71),c(57,90), correct=F, alternative = "less")

    2-sample test for equality of proportions without continuity correction

data:  c(36, 71) out of c(57, 90)
X-squared = 4.3602, df = 1, p-value = 0.01839
alternative hypothesis: less
95 percent confidence interval:
 -1.00000000 -0.03061662
sample estimates:
   prop 1    prop 2 
0.6315789 0.7888889

Which proportion of developing and developed countries are democratic?
Do we verify or falsify our hypothesis at a 95% confidence level?

How would the R code change, if we investigated whether a higher proportion of developing countries is autocratic than developed countries?

Exercise – Two-Sample Test for a Proportion

Is a higher proportion of fast-growing countries democratic than slow-growing countries? Use a 95% confidence level.

Guided Example – Two-Sample Test for a Mean

Now we are interested whether people live longer in developed countries than in developing countries?
For this, we use the variable life
Once again, state the Null- and the Alternative Hypothesis
The t-test assumes an equal variance in both samples, and so we need to test whether this is the case. We will do this with the so-called Levene Test, where:
- H$_{0}$: The variance among the groups is equal.
- H$_{\text{A}}$: The variance among the groups is not equal.

This is essentially another two-sample test in which we ascertain whether the difference between the variances of the two groups is different from zero (H$_{0}$)

For the Levene test you need the car package, where “car” stands for “Companion to Applied Regression”:

install.packages("car")

Now we call the package and perform the test:

library(car)

leveneTest(wdi$lifeexp ~ wdi$gdpcat)
Levene's Test for Homogeneity of Variance (center = median)
       Df F value  Pr(>F)  
group   1  3.7671 0.05395 .
      167                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The result is insignificant, and we therefore accept the null hypothesis. This means that the variance in the two samples is equal.
Now we can perform the t-test. We once again specify alternative="less" as an option, due to the same reasoning as before.

t.test(lifeexp ~ gdpcat, data=wdi, var.equal = TRUE, alternative="less")

    Two Sample t-test

data:  lifeexp by gdpcat
t = -12.01, df = 167, p-value < 2.2e-16
alternative hypothesis: true difference in means between group Developing and group Developed is less than 0
95 percent confidence interval:
     -Inf -9.53028
sample estimates:
mean in group Developing  mean in group Developed 
                64.65938                 75.71177

Can we conclude at a 95% confidence level, that people live longer in developed countries than in developing countries?

What is the precise p-value for the hypothesis that the true difference in means between group Developing and group Developed is greater than 0?

Exercise – Two-Sample Test for a Mean

Do people live longer under democracies than under dictatorahips (use the politybin variable)? Use a 95% confidence level.

Homework for Week 3

There is no separate reading for the Week 3 seminar
Work through this week’s flashcards to familiarise yourself with the relevant R functions.
Find an example for each NEW function and apply it in R to ensure it works
Complete the Week 2 Moodle Quiz
Complete Exercise 4.55 in Agresti (2018)
Work through the Section “Introduction to Matrices”.

Solutions

You can find the Solutions in the Downloads Section.

Some of the content of this worksheet is taken from Reiche (forthcoming).↩︎