6.3 - Simpson's Paradox

Example 6.6 Section

President Johnson signing the Civil Rights Act of 1964 - Martin Luther King Junior in the background

The Civil Rights Act of 1964 outlawed discriminatory practices based on race in voting rights, segregation in schools, workplace rules, and at facilities that serve the public. After a long filibuster by some southern Senators, the final bill was approved in a bipartisan vote in June of 1964.

The following tables show the results of the votes in both the Senate and the House broken down by political party and region of the country.

Table 6.6: House and Senate Votes
House Votes
The House Vote Democrats Republicans
Yes No Yes No
Northern 145 9 138 24
Southern 7 87 0 10
Total 152 96 138 34
Senate Votes
The Senate Vote Democrats Republicans
Yes No Yes No
Northern 45 1 27 5
Southern 1 20 0 1
Total 46 21 27 6

Try it! - By Party Section

Use Table 6.6 above to answer the following questions (the answers are all shown in Table 6.7 - but try not to look there until you try them on your own).

  1. What percent of Democrats in the House voted for the bill?
    61%
  2. What percent of Republicans in the House voted for the bill?
    80%
  3. What percent of Democrats in the Senate voted for the bill?
    69%
  4. What percent of Republicans in the Senate voted for the bill?
    82%
  5. In each chamber of Congress, what party voted proportionately more for the Civil Rights Bill of 1964?
    Republicans

Table 6.7: House and Senate Votes with Percentages
House Votes with Percentages
The House Vote Democrats Republicans
Yes No Yes No
Northern 145 (94%) 9 138 (85%) 24
Southern 7 (7%) 87 0 (0%) 10
Total 152 (61%) 96 138 (80%) 34
Senate Votes with Percentages
The Senate Vote Democrats Republicans
Yes No Yes No
Northern 45 (98%) 1 27 (84%) 5
Southern 1 (5%) 20 0 (0%) 1
Total 46 (69%) 21 27 (82%) 6

Try it! - By Region Section

Now use Table 6.6 to consider the following questions where the region represented (north or south) is taken into account (again the answers are all shown in Table 6.7 - but try not to look there until you try them on your own).

  1. What percent of Northern Senate Democrats voted for the bill? How about Southern Senate Democrats?
    Northern Democrats: 98%
    Southern Democrats: 5%
  2. What percent of Northern Senate Republicans voted for the bill? How about Southern Senate Republicans?
    Northern Republicans: 84%
    Southern Republicans: 0%

  3. What percent of Northern House Democrats voted for the bill? How about Southern House Democrats?

    Northern Democrats: 94%
    Southern Democrats: 7%
  4. What percent of Northern House Republicans voted for the bill? How about Southern House Republicans?
    Northern Republicans: 85%
    Southern Republicans: 0%
  5. In each chamber of Congress, and in each region of the country, what party voted proportionately more for the Civil Rights Bill of 1964?
    Chamber: Senate
    Region: Northern
    Party: Republicans

Table 6.7: House and Senate Votes with Percentages
House Votes with Percentages
The House Vote Democrats Republicans
Yes No Yes No
Northern 145 (94%) 9 138 (85%) 24
Southern 7 (7%) 87 0 (0%) 10
Total 152 (61%) 96 138 (80%) 34
Senate Votes with Percentages
The Senate Vote Democrats Republicans
Yes No Yes No
Northern 45 (98%) 1 27 (84%) 5
Southern 1 (5%) 20 0 (0%) 1
Total 46 (69%) 21 27 (82%) 6

Simpson’s Paradox: Section

Answering the above questions leads to an interesting finding. For each chamber of Congress Democrats have a higher percentage than Republicans voting for the bill in both the North and the South (the only two possibilities), and yet they had a lower percentage overall. Look again at the numbers in Table 6.7 and you will see how that happened. Back in 1964, there was a huge imbalance in representation by region -most Southern Senators and Representatives were Democrats at the time and the negative votes were almost all associated with that region.

The relationship between Party and vote on the civil rights bill was highly affected by a third variable - the region represented. This was an example of Simpson's Paradox.

An observed association between two variables can change or even reverse direction when there is another variable that interacts strongly with both variables.

Example 6.7: Smoking and Survival Section

A person smoking a cigarette

1314 women took part in a study of heart disease and smoking that was conducted in 1972-1974 in Newcastle, United Kingdom. A follow-up study of the same subjects was recently conducted nearly thirty years later. Of the 582 women who were smokers in the original study, 76.2% were still alive in the follow-up study. Of the 732 non-smokers, 68.6% were still alive twenty years later. Does this show a beneficial effect of smoking? What might have caused this counter-intuitive result?

Answer: To find the confounding factor that might be driving this paradoxical result, think about the variable most associated with dying over a two or three-decade period? Of course, the key variable associated with your survival over a 20-30 year period is your current age. The smokers were much younger women than the non-smokers at the beginning of the study (there aren't very many old smokers). For example, if you look just at those who were over 64 years old in the original study, 14% of the 49 smokers were still alive compared with 17% of the 193 non-smokers. This is another example of Simpson's Paradox (the perils of aggregation across a potential confounding factor) - a disproportionate number of nonsmokers were over the age of 64 at the beginning of the study.