Relative risk: 7 exam questions solved step by step

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What this lesson covers

  • Relative risk compares the risk in one group with the risk in another.
  • 'Twice as likely' and 'twice more likely' do not mean the same thing.
Key idea

7 questions, one comparison

  • 1. Risk = P(outcome | group).
  • 2. RR = risk ÷ reference risk.
  • 3. Say "times as likely".
  • 4. Does the claim need a ratio?
Your turn

Question 1 · Achieved, in context

  • 500 Year 13 students were surveyed.
  • Find the risk of poor sleep for
  • students who drink energy drinks.
GroupPoor sleepNotTotal
Energy drink6684150
No energy drink70280350
Total136364500
Worked example

Q1: one risk

  • Given the group.
  • Divide by its row total.
  1. risk = P(poor | drink)given the group
  2. P(poor | drink) = 66 ÷ 150
  3. = 0.44
  4. 66 ÷ 136 = 0.4853that is P(drink | poor)
  5. 44% of drinkers sleep poorly
Your turn

Question 2 · Achieved

  • The same survey.
  • Find the relative risk of poor sleep,
  • drinkers compared with non-drinkers,
  • and say it in a sentence.
GroupPoor sleepNotTotal
Energy drink6684150
No energy drink70280350
Total136364500
Worked example

Q2: drinkers v non-drinkers

  • Reference: no drink.
  • Say "times as likely".
  1. P(poor | no drink) = 70 ÷ 350
  2. = 0.2
  3. RR = 0.44 ÷ 0.2reference: no drink
  4. = 2.2
  5. drinkers: 2.2 times as likely
  6. (2.2 − 1) × 100 = 120percent more
  7. 120% more likely
  8. "2.2 times more likely"ambiguous
Your turn

Question 3 · Merit

  • A website claims energy drinks
  • "raise the risk of poor sleep by 24%".
  • Use the survey to judge the claim.
  • Careful: 24 of what?
GroupPoor sleepNotTotal
Energy drink6684150
No energy drink70280350
Total136364500
Worked example

Q3: points or percent?

  • Subtract: points.
  • Divide: percent.
  1. 44% − 20% = 24subtract
  2. a rise of 24 percentage points
  3. RR = 0.44 ÷ 0.2 = 2.2divide
  4. (2.2 − 1) × 100 = 120%relative rise
  5. "24% higher"mixes the two
  6. a survey: no proof of cause"raise" claims cause
  7. 24 points, or 120% higher
Your turn

Question 4 · Merit, in context

  • A clinic followed 800 patients: 25%
  • smoked. 18% of smokers and 6% of
  • non-smokers got a chest infection.
  • Build the table and find the RR.
Worked example

Q4: counts, then the ratio

  • Percent of which group?
  • Reference: non-smokers.
  1. smokers: 0.25 × 800 = 200
  2. non-smokers: 800 − 200 = 600
  3. infected smokers: 0.18 × 200 = 36
  4. infected others: 0.06 × 600 = 36
  5. RR = 0.18 ÷ 0.06reference: non-smokers
  6. = 3
  7. smokers: 3 times as likely
  8. (3 − 1) × 100 = 200% more
Your turn

Question 5 · Merit

  • The clinic table, completed.
  • Claim: "most of the chest infections
  • were in smokers".
  • Is the claim supported?
GroupInfectionNoneTotal
Smoker36164200
Non-smoker36564600
Total72728800
Worked example

Q5: does it need a ratio?

  • "Most" means over half.
  • Given infection: new total.
  1. infections: 36 + 36 = 72
  2. P(smoker | infection)infected only
  3. = 36 ÷ 72 = 0.5
  4. 0.5 is not over 0.5half, not most
  5. RR = 3answers another claim
  6. not supported: exactly half
Your turn

Question 6 · Excellence

  • A randomised trial of a hay fever spray.
  • Claim: "the spray halves the risk
  • of symptoms".
  • Evaluate the claim with numbers.
GroupSymptomsTotal
Spray45300
Placebo100400
Worked example

Q6: spray v placebo

  • Reference: placebo.
  • Below 1: lower risk.
  1. P(symptoms | spray) = 45 ÷ 300
  2. = 0.15
  3. P(symptoms | placebo) = 100 ÷ 400
  4. = 0.25
  5. RR = 0.15 ÷ 0.25reference: placebo
  6. = 0.6
  7. spray: 0.6 times as likely
Worked example

Q6: does it halve?

  • Halving: RR = 0.5.
  • Randomised: cause is fair.
  1. RR = 0.6
  2. halves: RR would be 0.5
  3. 0.6 > 0.5not halved
  4. 1 − 0.6 = 0.440% lower
  5. 25% − 15% = 10percentage points
  6. randomised: spray caused it
  7. 40% lower risk, not halved
Your turn

Question 7 · Excellence

  • 1200 drivers surveyed; 300 use a phone
  • while driving. 165 had a near miss.
  • Phone users: 2.5 times as likely.
  • How many phone users had a near miss?
Worked example

Q7: form the equation

  • Unknown reference risk: r.
  • Every near miss counted.
  1. 165 × 2.5 ÷ 3.5 = 117.9groups differ in size
  2. no phone: 1200 − 300 = 900
  3. P(miss | no phone) = rreference
  4. P(miss | phone) = 2.5rRR = 2.5
  5. 300 × 2.5r + 900r = 165all near misses
  6. 750r + 900r = 165
  7. 1650r = 165
  8. r = 0.1
Worked example

Q7: the counts

  • Risk times group size.
  • Check both facts.
  1. r = 0.1
  2. P(miss | phone) = 2.5 × 0.1
  3. = 0.25
  4. phone: 300 × 0.25 = 75
  5. no phone: 900 × 0.1 = 90
  6. 75 + 90 = 165
  7. 0.25 ÷ 0.1 = 2.5
  8. (2.5 − 1) × 100 = 150% more
  9. 75 phone users had a near miss
Summary

The method that solved all seven

  • Name the group, divide, say it as "times as likely", then judge the claim.
StepHow
RiskP(outcome | group)
RRrisk ÷ reference risk
Say it"times as likely"
Percent(RR − 1) × 100
Pointssubtract the risks
"Most"one probability, no ratio

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