MEASURES OF DISEASE ASSOCIATION

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MEASURES OF DISEASE ASSOCIATION

• Nigel Paneth

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MEASURES OF DISEASE ASSOCIATION

• The chances of something happening can be
  expressed as a risk or as an odds
• RISK the chances of something
  happening the chances of all things
  happening
• ODDS the chances of something
  happening the chances of it not happening

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• Thus a risk is a proportion, But an odds is
  a ratio.
•  
• An odds is a special type of ratio, one in which
  the numerator and denominator sum to one.

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• Example 1. Bookies are taking bets on the World
  Series. They are giving 31 odds on the Yankees.
  What does this mean?
• It means that they
  think that there it is three times as likely that
  the Yankees will not win the world series as that
  they will win.
• Expressed as a risk, the Yankees
  are expected to win one in four opportunities

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• Example 2. Among 100 people at baseline, 20
  develop influenza over a year.
• The risk is 1 in 5 (i.e. 20 among 100)
• The odds is 1 to 4 (i.e. 20 compared to 80)

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THE RELATIVE RISK(RISK OR RATE RATIO)

• The relative risk is a ratio of two risks.
• Assume that among the 100 people at risk, 50 are
  men and 50 women. If 15 men and 5 women develop
  influenza, then the relative risk of developing
  influenza in men, as compared with women, is
• Risk in men 15/50
• divided by
• Risk in women 5/50
• 15/50 5/50 3.0
• (Note that from the way the question was put, the
  two risks are cumulative incidence rates.)

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ODDS RATIO

• The odds ratio is a ratio of two odds
• The odds in men 15/35
  divided by The odds in women 5/35
• 15/35 5/45 3.9
• We conclude that the odds of men getting
  influenza over the year are 3.9 times as high as
  the odds of women getting influenza.
• Thought question note that the odds
  ratio in this example (3.9) is larger than the
  relative risk (3.0). Is this always the case?
  Is this important?

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MEASURES OF PUBLIC HEALTH IMPACT

• Four closely related measures are used
• Attributable risk
• Attributable (risk) fraction
• Population attributable risk
• Population attributable (risk) fraction
• Note all of these measures assume
  that the association between exposure and disease
  has already been shown to be causal.

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1. ATTRIBUTABLE RISK (AR)

• The incidence of disease in the exposed
  population whose disease can be attributed to the
  exposure.
• AR Ie - Iu

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2. ATTRIBUTABLE RISK FRACTION (ARF)

• The proportion of disease in the
  exposed population whose disease can be
  attributed to the exposure.
• ARF (Ie - Iu)/Ie

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3. POPULATION ATTRIBUTABLE RISK (PAR)

• The incidence of disease in the total population
  whose disease can be attributed to the exposure.
• PAR Ip - Iu

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4. POPULATION ATTRIBUTABLE RISK FRACTION (PARF)

• The proportion of disease in the total population
  whose disease can be attributed to the
  exposure.
• PARF (Ip - Iu)/Ip

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Note Ip can be linked to Ie and Iu if one knows
the proportions of the population who are exposed
(P) and unexposed (Q), (P and Q add to 1).  Ip
P (Ie) Q (Iu)
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EXAMPLE OF THESE MEASURES (data are invented)

• Red-meat eaters have a relative risk of 2.0 for
  colon cancer.
• If Iu 50/100,000/year, then
  Ie 100/100,00/year.
• If 25 of the population are red-meat eaters,
  what is Ip?
• Ip P (Ie) Q (Iu) , so
• Ip .25(100/100,000) .75 (50/100,000)
• Population incidence of colon cancer is thus
  62.5 /100,000/year

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INFERRING AN ATTRIBUTABLE RISK FRACTION FROM A
RELATIVE RISK

• Note that Ie Iu times the relative risk
  (RR) So substituting Iu x
  RR for Ie in the equation for attributable risk
  fraction
•  
• (Ie - Iu)/Ie

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• We get
• ARF RR (Iu) - Iu
• RR (Iu)
•  
• Dividing through by Iu gives
• ARF RR - 1
• RR

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• In other words, if we find a truly causal
  relative risk of 2.0 for a disease in relation to
  an exposure, we can assume that 50 of the
  disease in the exposed population is due to the
  exposure.
• Since the courts use a probability of 50 or
  greater as a threshold in liability cases, RR of
  2.0 has recently taken on great significance in
  lawsuits. It has been argued that when RR gt 2.0,
  it is more likely than not that the disease was
  due to the exposure in an exposed individual.
  What do you think of this legal reasoning?

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INFERRING A POPULATION ATTRIBUTABLE RISK FRACTION
FROM A RELATIVE RISK  (this is a little heavier
going)

• Remember that
• PARF (Ip - Iu)/Ip
• and that Ip P(Ie) Q(Iu)
• and that Ie Iu x RR

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Therefore, the equation for PARF can be rewritten
in terms of RR
P(Ie) Q(Iu) Iu
PARF
P(Ie) Q(Iu)

• Replacing Ie with Iu x RR, we get

P(Iu)RR Q(Iu) Iu
PARF
P(Iu)RR Q(Iu)
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Going From A Relative Risk To An Attributable
Risk Fraction Contd

• Iu can be factored out and cancelled 

Iu (P x RR Q - 1)
X
PARF
Iu (P x RR Q)
X
If we now replace Q with 1-P (since P Q 1)
P x RR 1 - P - 1
PARF
P x RR 1 - P
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• or P (RR - 1) P (RR - 1) 1
• In other words, if we find a truly causal
  relative risk of 2.0 for a disease in relation to
  an exposure, and if 50 of the population has the
  exposure, then 33 of the disease in the
  population is due to the exposure.
• (Again, always assuming that we are discussing a
  exposure whose causal role has been established).

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EXAMPLE OF HOW FAILURE TO UNDERSTAND WHAT AN ODDS
RATIO MEANS CAN LEAD TO TROUBLE
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Schulman et al The Effect of Race and Sex on
Physicians' Recommendations for Cardiac
Catheterization. N Eng J Med 1999 340 619-625

• To study doctors recommendations for managing
  chest pain, the study used actors to portray
  patients with particular characteristics in
  scripted interviews about their symptoms.
• 720 primary care physicians viewed a recorded
  interview and were given other data about a
  hypothetical patient. He or she then made
  recommendations about that patient's care.
• The study used multivariate logistic-regression
  analysis to assess the effects of the race and
  sex of the patients on treatment recommendations

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The number of White and Black patients who
doctors thought should be referred for cardiac
catheterization based on their symptoms
Referred Not referred
White 326 (90.6) 34 360
Black 305 (84.7) 55 360
331 89 720
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Risk ratio and odds ratio in this table

• Relative risk or risk ratio for Blacks is
• 305 divided by 326
• 360 360
• or 0.93
• Odds ratio for Blacks is
• 305 x 35
• 326 x 55
• or 0.58

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THIS IS HOW THE AUTHORS DESCRIBED THEIR FINDINGS

• Logistic-regression analysis indicated that
  blacks (odds ratio, 0.60 95 percent confidence
  interval, 0.4 to 0.9 P0.02) were less likely to
  be referred for cardiac catheterization than
  whites.

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HEART BIAS STUDY WAS MISINTERPRETED (AP 8/15/99)

• The editors of the NEJM say they take
  responsibility for media reports which greatly
  exaggerated conclusions in a study about possible
  gender and sex bias in heart care. The study,
  published in the journal on Feb 25, reported what
  happened when doctors viewed taped interviews of
  actors describing their identical symptoms and
  asked what treatment they would recommend. It
  found that in cases of equally sick patients,
  doctors were less likely to refer blacks and
  women than they were white and men to have
  cardiac catheterization, a test used to diagnose
  heart disease. Several news organizations,
  including the AP, interpreted the study to show
  that doctors were 40 less likely to order the
  tests for women and blacks than for men and
  whites

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• However, a follow up published in the
  Journal recently concluded that the likelihood of
  women and blacks being referred for the tests was
  actually 7 percent less than for men and whites.
• The follow up, written by Dr. Lisa M. Schwartz
  and others from the VA Outcomes Group in White
  River Junction, Vt., said the misunderstanding
  resulted from the original study's use of an
  "odds ratio" to report the differences rather
  than a more commonly used "risk ratio."
• The researchers calculated the odds in favor of
  blacks being offered the test and of whites being
  offered the test. Then they calculated the ratio
  of these two figures. The ratio of blacks' odds
  to whites' odds worked out to 0.6, as did the
  ratio of women's odds to men's. The media
  interpreted this to mean that women and blacks
  were 40 percent less likely to be offered
  catheterization. But the true difference is much
  smaller.

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• A table published with the study shows that
  actually 85 percent of women and blacks were
  referred for catheterization as were 91 percent
  of men and whites. This means that the risk ratio
  was .93. In other words, the probability of
  referral was 7 percent lower for blacks and women
  than for whites and men.
• The journal editors said they "take
  responsibility for the media's
  overinterpretation" of the study's findings and
  said they should not have allowed the use of odds
  ratios in the study's summary.