The textbook shows examples where splitting data into groups gives more insight. Today’s activity goes one step further: what if within each group there are subgroups that affect the distribution?

Specifically, we will explore the following questions.

• How informative is a direct comparison of the overall lung cancer incidence rates of two countries?
• Their populations are likely to differ by factors (specifically: age) that affect incidence rates.
• How can we take these differences into account in our comparison between countries?

Globally, lung cancer is one of the most common types of cancer with estimates of about 1.8 million cases or some 12.9% of all new cases of cancer in 2012 alone. The rates of lung cancer incidence may differ among countries because of factors such as levels of air pollution and smoking.

In this activity we will try to figure out if someone from, say, Viet Nam has a higher probability of getting lung cancer than someone from the UK. We will compare the following eight countries: Viet Nam, Singapore, the UK, Ethiopia, Austria, China, Georgia and the Philippines.

As part of your preparation for this activity, you made this bar chart.

Do you find the results shown by the bar chart surprising?

Let’s take the UK as an example.

uk <- lung_cancer[lung_cancer$Country == "UK", ]

uk$Incidence <- 100000 * uk$Cases / uk$Population

barplot(uk$Incidence,
        names.arg = uk$AgeClass,
        main = "UK Lung Cancer Incidence by Age Group",
        ylab = "Incidence rate per 100,000",
        col="violetred3",
        las = 2)
grid()

Here is the result of the code on the previous slide.

country <- "Singapore"
country_lc <- lung_cancer[lung_cancer$Country == country, ]
country_lc$Incidence <- 100000 * country_lc$Cases / country_lc$Population
barplot(country_lc$Incidence, names.arg = country_lc$AgeClass,
        main = paste(country,"Lung Cancer Incidence by Age Group"),
        ylab = "Incidence rate per 100,000", las = 2)

Here are all the bar plots combined into one figure so that we can compare them more easily.

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The bar plots show that

• incidence varies strongly between countries,
• the incidence rate tends to increase with age.

Can you think of reasons why this might be the case? Hans Rosling gave some hints in his video tutorial (https://www.youtube.com/watch?v=QBht72_PA-4).

Again let’s take the UK as an example. We assume that we have already done the subsetting as in the solution to our previous activity. Especially, we assume that the data frame uk is already in our environment.

uk_total_population <- sum(uk$Population)

barplot(uk$Population / uk_total_population,
        names.arg = uk$AgeClass,
        main = "UK Age Distribution",
        ylab = "Relative population",
        col="darkorchid3",
        las = 2)
grid()

country <- "Singapore"
country_lc <- lung_cancer[lung_cancer$Country == country, ]
country_total_population <- sum(country_lc$Population)
country_lc$Incidence <- 100000 * country_lc$Cases / country_lc$Population
barplot(country_lc$Population / country_total_population,
        names.arg = country_lc$AgeClass, main = paste(country,"Age Distribution"),
        ylab = "Relative population", las = 2)
grid()

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• Consider the implications of the results of activities 1 and 2. What could be an important factor in determining how the total incidence rates vary among these eight countries? Discuss.

• Given the foregoing discussion, would you amend the analyses in Challenge 1? How?

Age structure differs among countries, with a high percentage of older people in high income countries such as the UK and a very young population in countries like Ethiopia or even the Philippines.

You can compare the age structure of all countries in the world at https://www.populationpyramid.net/world/2012. In many countries, the population is aging while birth rates are falling. So it might be that some of the difference in total lung cancer incidence rates across countries is due to their different underlying demographic structure.

Ideally we would compute a “corrected” or “age-standardized” incidence rate that accounts for these age-specific differences in countries’ demographic structure. We’re going to use the UK as the “standard” population because it has the highest crude lung cancer incidence rate, at least for these 8 countries considered. Alternatively, if we had the data, we could use the 2012 World population for standardization.

Let’s compare two hypothetical countries \(A\) and \(B\).

Suppose there were only two age groups and

• the incidence rates were
  • low for the young group,
  • high for the old group, but
• identical for both age groups in country \(A\) and country \(B\).

If \(A\) has a higher proportion of elderly, \(A\) will have a higher overall incidence rate.

But is it a “fair” comparison between \(A\) and \(B\)?

What would be the incidence rate in country \(A\) if, in all age groups, it had the same fraction of the population as country \(B\)? Let’s introduce some notation.

• \(c_{A, \text{age}}\): the number of cancer cases in \(A\) in a given age group.
• \(p_{B, \text{age}}\): the population in \(B\) in the same age group

The number of cases in this age group that would occur in country \(A\) if it had the population of country \(B\) is

\[ \tilde{c}_{A, \text{age}} = \frac{p_{B, \text{age}}} {p_{A, \text{age}}} \times c_{A, \text{age}}\ . \qquad \qquad \qquad (\text{I}) \]

In other words, if \(p_{B, \text{age}} = 2 \times p_{A, \text{age}}\), the number of cases in this age group that would occur in country \(A\) if it had the population of country \(B\) is twice the observed number

The “age-adjusted” incidence rate in country \(A\) is

\[ \frac{\sum_{\text{age}} \tilde{c}_{A, \text{age}}} {\sum_\text{age} p_{B, \text{age}}} \times 100\,000\ . \qquad \qquad \qquad (\text{II}) \]

The unadjusted incidence rate would have

• \(c_{A, \text{age}}\) in the numerator instead of \(\tilde{c}_{A, \text{age}}\),
• \(p_{A, \text{age}}\) in the denominator instead of \(p_{B, \text{age}}\).

# Preparation: Append population of UK in corresponding age group.
uk_population <- lung_cancer$Population[lung_cancer$Country == "UK"]
lung_cancer$Population_UK <- uk_population

# 1: How many cases would there be if the population had the age
#    structure of the UK?
lung_cancer$Cases_if_UK <-
  (lung_cancer$Population_UK / lung_cancer$Population) * lung_cancer$Cases

# 2: Total number of cases in each country if it had the population and
#    age structure of the UK.
adjusted <- aggregate(Cases_if_UK ~ Country, data = lung_cancer, sum)

# Preparation: Append population of UK in corresponding age group.
uk_population <- lung_cancer$Population[lung_cancer$Country == "UK"]
lung_cancer$Population_UK <- uk_population

# 1: How many cases would there be if the population had the age
#    structure of the UK?
lung_cancer$Cases_if_UK <-
  (lung_cancer$Population_UK / lung_cancer$Population) * lung_cancer$Cases

# 2: Total number of cases in each country if it had the population and
#    age structure of the UK.
adjusted <- aggregate(Cases_if_UK ~ Country, data = lung_cancer, sum)

# 3: Append a column with the age adjusted incidence rate.
adjusted$Incidence <- adjusted$Cases_if_UK / sum(uk_population) * 100000

barplot(adjusted$Incidence, names.arg = adjusted$Country,
        main = "Age-adjusted Lung Cancer Incidence", ylab = "Incidence rate per 100,000",
        col="tan2", las = 2)
grid()

Because most countries have a younger population than the UK, age-adjustment leads to higher incidence rates in most cases.

The age-adjusted incidence rate of China is strikingly high. China has a very high incidence in the oldest age group, which is (still) a relatively small fraction of China’s population.

Why do the corrected incidence rates provide a better answer to the main question of this activity?

“Does someone from one country (UK) have a higher chance of getting lung cancer than someone from another country (Vietnam)?”

The corrected or adjusted incidence rates can be compared among countries, having taken out any age-structure effects. If the adjusted rates are very different from the crude (i.e. overall) rates, the rank order of that country will change substantially.

For example, in Vietnam, the proportion of very young people, with almost no cases of lung cancer, is very high. By contrast, Singapore has more old people, a group with a much higher incidence rate of lung cancer. So the high crude rate in Singapore is strongly driven by the larger weight of the older age groups.

Therefore, in epidemiology, generally an age-adjusted rate is used to compare among countries or other groups. In general, the weighted average of the age-specific incidence rates is calculated based on the proportions of persons in the corresponding age groups of a standard population. The potential confounding effect of age is reduced when comparing age-adjusted rates computed using the same standard population.