Chapter 4, Problem 14AP

(a)

To determine

Determine the health disparity.

Explanation of Solution

The health disparity in Town E can be calculated, since the cancer rate is 50% for nobles and 40% of peasants, as follows:

$\begin{array}{c}\text{Health disparity}=\frac{{\text{Cancer rate}}_{\text{N}}-{\text{Cancer rate}}_{\text{P}}}{{\text{Cancer rate}}_{\text{P}}}\\ =\frac{50\%-40\%}{40\%}\\ =25\%\end{array}$

Hence, in Town E chance of nobles contracting cancer is 25% more than that for peasants.

The health disparity in Town W can be calculated, since the cancer rate is 10% for nobles and 8% of peasants, as follows:

$\begin{array}{c}\text{Health disparity}=\frac{{\text{Cancer rate}}_{\text{N}}-{\text{Cancer rate}}_{\text{P}}}{{\text{Cancer rate}}_{\text{P}}}\\ =\frac{10\%-8\%}{8\%}\\ =25\%\end{array}$

Hence, in Town W chance of nobles contracting cancer is 25% more than that for peasants.

Therefore, in both towns, the chances for nobles contracting cancer is 25% more than that for peasants

(b)

To determine

Determine the nationwide cancer rate for nobles as well as peasants.

Explanation of Solution

The number of nobles having in cancer in Town E and Town W can be calculated as follows:

$\begin{array}{c}{\text{Total number of cancer patients}}_{\text{Nobles}}=\left({\text{Population}}_{\text{E}}\times {\text{Cancer rate}}_{\text{E}}\right)\\ +\left({\text{Population}}_{\text{W}}\times {\text{Cancer rate}}_{\text{W}}\right)\\ =\left(100\times 0.5\right)+\left(900\times 0.1\right)\\ =140\end{array}$

Thus, the total number of nobles having in cancer in both towns is 140.

The total number of nobles in both Towns E and W is $1,000(100+900)$.  Thus, the nation wise cancer rate can be calculated as follows:

$\begin{array}{c}{\text{Cancer rate}}_{\text{N}}=\frac{\text{Number of cancer rates}}{\text{Total population}}\times 100\\ =\frac{140}{1,000}\times 100\\ =14\%\end{array}$

Thus, the nationwide cancer rate for nobles is 14%.

The number of peasants having cancer in Town E and Town W can be calculated as follows:

$\begin{array}{c}{\text{Total number of cancer patients}}_{\text{peasants}}=\left({\text{Population}}_{\text{E}}\times {\text{Cancer rate}}_{\text{E}}\right)\\ +\left({\text{Population}}_{\text{W}}\times {\text{Cancer rate}}_{\text{W}}\right)\\ =\left(1000\times 0.4\right)+\left(500\times 0.08\right)\\ =440\end{array}$

Thus, the total number of peasants having cancer in both towns is 440.

The total number of nobles in both Towns E and W is $1,500(1,000+500)$.  Thus, the nationwide cancer rate can be calculated as follows:

$\begin{array}{c}{\text{Cancer rate}}_{\text{P}}=\frac{\text{Number of cancer rates}}{\text{Total population}}\times 100\\ =\frac{440}{1,500}\times 100\\ =29.33\%\end{array}$

Thus, the nationwide cancer rate for peasants is 29.33%.

Therefore, the nationwide cancer rate is higher for peasants.

(c)

To determine

The Simpson’s paradox.

Explanation of Solution

As Part (a) explains that the region wise health disparity is higher for nobles, while the nationwide disparity is higher for peasants. This effect is called ‘Simpson’s paradox’. This Simpson’s paradox exists because the number of Peasants is higher than the number of nobles. Therefore, a small percentage of peasants having cancer translates to larger absolute number. That is, the absolute number of peasants is 1,500, while the absolute number of nobles is 1,000. Even in geographical perspective, the absolute number of peasants is very high (1000 peasants) for Town E than Town W (500 peasants). Therefore, even though the rate of getting cancer in Town E is smaller for peasants, it turns out that 400 peasants have cancer and only 50 nobles have cancer in Town E. Hence, the health disparity is mainly due to the socioeconomic status.  

(d)

To determine

Determine the three conditions for the situation of Simpson’s paradox. 

Explanation of Solution

The three conditions for the situation of Simpson’s paradox are listed below (using Table 4.7):

• In Town E, ‘a’ amount of nobles live in which ‘w’ suffer with cancer, and ‘b’ amount of peasants live in which only ‘x’ suffer from cancer. This means the cancer rate of nobles should be greater than the cancer rate of peasants.
• The above explained condition must be same for Town ‘W’. This means the rate of people who suffer with the cancer should be higher for nobles than the peasants.
• The third and final condition is that the nationwide disparity for peasants should be higher than nobles. Hence, the total population (b and d) of peasants together should be higher than the total population of nobles (a and c).   

(e)

To determine

Prove that Simpson’s paradox will not occur if the population of nobles and peasants are same in both towns.  

Explanation of Solution

As explained, Simpson’s paradox occurs if the rate of cancer is high among nobles in individual towns, while it is high among peasants in nationwide. This happens mainly due to difference in population between nobles and peasants. Assume that the situation of the same population of the nobles and peasants in both towns, then what happen is that the cancer rate will be also less in both the cases. As a result, the nationwide result will also change, that is, the nobles will be more in number of suffering cancer.  Hence, Simpson’s paradox will not occur if the population and peasants are same in both towns.  

(f)

To determine

Determine whether Simpson’s paradox may invalidate due to some of the evidence for health disparities.

Explanation of Solution

According to the efficient producer hypothesis, better educated individuals are more efficient producers of health than less well-educated individuals. As a result, health disparity will exist between well-educated people and less educated people. In this case, nobles can be consider as the well-educated people and peasants can be consider as less-educated people. As per the efficient producer hypothesis, nobles should have better health than peasants. However, in this case, nobles suffer more from cancer in individual towns. Hence, Simpson’s paradox, might extent invalidate some of the evidence for the health disparities.