6.2 - Numbers That Can Describe 2×2 Tables

Here we examine five measures that are often used to describe data collected in a 2 × 2 table.

  1. Risk = proportion with the undesirable trait = (number with trait/total)
  2. Relative Risk = Risk1 / Risk2
  3. Increased Risk = (Relative Risk - 1.0) × 100%
  4. Odds = (number or proportion with a trait/number or proportion without the trait)
  5. Odds Ratio= Odds1 / Odds2

The first three of these, involving "risk", are applied in situations describing an outcome variable that is undesirable (e.g. pertaining to outcomes that are diseases or injuries). The last two, involving "odds" are applied more generally.

Example 6.4 Section

Knee injury

A 2013 study in the journal Medicine & Science in Sports & Exercise examined the incidence of knee injuries for male and female high school athletes that occur in one of nine different sports as recorded in the new National Sports-Related Injury Surveillance System. The response variable was whether the athlete experienced a knee injury in a sports competition for each game they participated in. The data are summarized in Table 6.4 below.

Table 6.4. Experiencing Knee Injury in High School Sports by Gender
  Experienced Knee Injury?  
Sex Yes No Total number of games
Female 1,492 6,513,792 6,515,284
Male 3,624 10,655,200 10,658,824
Total 5,116 17,168,992 17,174,108

In this example, the undesirable trait (outcome) is experiencing a knee injury. So the calculated risk of injury for each gender is:

For Females: Risk = (number with trait)/total = 1492/6515284 = 0.000229 (0.0229%)
For Males: Risk = (number with trait)/total = 3624/10658824 = 0.000340 (0.0340%)

Risk interpretation: High School girls run a risk of knee injury in about 2.29 out of every 10,000 games they take part in while High School boys have a risk of about 3.40 knee injuries per 10,000 games.

Risk is just another name for a probability or proportion for an adverse outcome. Depending on the context, we might be speaking about the true population risk (probability) or about a sample-based proportion that provides an estimate of the probability. Risks are often reported in percentage terms or, for risks of rare events, in terms of cases per some large number of events (for example per 10,000 events as above).

Relative and increased risk

Risks between groups or between different situations are then compared using the relative risk and the increased risk. For example, we might compare the risk of knee injuries for males to the risk for females competing in high school sports.

Relative risk = Riskmales / Riskfemales = 3.40 / 2.29 ≈ 1.485

Relative risk interpretation: For each high school sporting event they take part in, a male athlete is 1.485 times more likely to experience a knee injury than a female athlete.

Increased risk = (Relative Risk - 1.0) × 100% = (1.485 - 1.0) × 100% = 48.5%

Increased risk interpretation: The risk of knee injury during a high school sporting event is 48.5% higher for male athletes than for female athletes.

Relative risks and increased risks are reported in the news all the time. However, these measures are almost always descriptive statistics arising from observational data so be careful to examine possible confounders before drawing conclusions. In this example, high school boys are seen to have a greater risk of knee injury than high school girls competing in sports competitions. However, there is a very different mix of the type of sports played and it turns out that girls have a higher risk if you condition on a particular sport played by both sexes like basketball, volleyball, or soccer. The data for the males is strongly affected by football which has the highest rate of knee injuries of any sport and is played exclusively in high school by boys.

Cautions Section

Some ways you can be misled...

 Watch the baseline:

In reading about the relative or increased risks, it is important to keep in mind the baseline risk when deciding if a risk is acceptable for your own life. For example, a recent report found that the risk of dying in a plane crash is about 2.7 times greater when flying on a commuter airline compared with flying on a major carrier. This high relative risk might dissuade you from ever flying on a commuter airline. But, for major U.S. carriers, the chance of being killed on a particular flight is about 1 in 20 million chances; so the risk is very small regardless.

 Think about whether the risk pertains to you:

As mentioned, the risk of knee injuries in high school sports depends a great deal on the particular sport. If you are taking part in swimming or diving competitions knee injuries occur in just one or two swim meets out of every 100,000. On the other hand, a boy playing football might injure his knee in about 1 or 2 of every 1000 games. That latter risk might be a deterrent to some when considering the risk during the 40 or 50 games that would be played over a high school player's four years in school. The lesson: risk calculations are often conditional on a specific population or situation. To evaluate your own risk it is important to consider how close the situation fits your own.

Example 6.5 Section

section of a sidewalk

An article in The Journal of Epidemiology and Community Health examined how well sidewalks are maintained in different neighborhoods in St. Louis, Missouri. A number of sidewalk segments were randomly selected from throughout the city and classified as to whether the sidewalk was too uneven for walking or whether it was satisfactory for walking. Some data from this study is provided in Table 6.5 below, which focuses on how the condition of the sidewalks relates to the poverty rate of the households in the neighborhood. Low poverty rate areas are those with less than 10% of the households in poverty, while high poverty rate neighborhoods would be those with more than 20% of the households in poverty.

Table 6.5. Sidewalk Condition and Poverty Rate in St. Louis Neighborhoods
Poverty Rate of neighborhood Uneven Sidewalks Even Sidewalks Total
High (over 20%) 98 418 516
Low (under 10%) 9 301 310
Total 107 719 826

You should be able to answer questions like the following about this table: 

  1. What percent of the sidewalk segments were too uneven for walking?
    Answer = 107 / 826 or about 13%
  2. What percent of the sidewalk segments in high poverty areas were too uneven for walking?
    Answer = 98 / 516 or about 19%
  3. What are the odds that a sidewalk would be too uneven for walking?
    Answer = 107 / 719 = 0.149 or about 1 to 6.7
  4. What is the ratio of the odds that a sidewalk segment is too uneven in a high poverty area to the odds of being too uneven in a low poverty area?
    Answer = 98 / 418 ÷ 9 / 301 = 98(301) / 418(9) ≈ 7.84
Odds Ratio

The last question leads us to the odds ratio.

Odds ratio interpretation: The odds of a sidewalk being uneven are about 7.84 times as big in a high poverty neighborhood compared with a low poverty area.

An interesting property of the odds ratio is that it comes out the same regardless of whether your condition on the explanatory variable or on the response variable (e.g. it will be the same for prospective studies that condition on the exposure and for retrospective studies that condition on the outcome). In the sidewalk study, the odds that an uneven sidewalk is in a high poverty area is 98/9 and the odds that an even sidewalk is in a high poverty area is 418/301. The odds ratio is then 98 / 9÷ 418 / 301 = 98(301) / 418(9) ≈ 7.84