MEASURES OF DISEASE ASSOCIATION Nigel Paneth MEASURES OF

MEASURES OF DISEASE ASSOCIATION Nigel Paneth

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

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.

Example 1. Bookies are taking bets on the World Series. They are giving 3: 1 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

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)

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. )

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?

MEASURES OF PUBLIC HEALTH IMPACT Four closely related measures are used: 1. Attributable risk 2. Attributable (risk) fraction 3. Population attributable risk 4. Population attributable (risk) fraction Note: all of these measures assume that the association between exposure and disease has already been shown to be causal.

1. ATTRIBUTABLE RISK (AR) The incidence of disease in the exposed population whose disease can be attributed to the exposure. AR = Ie - Iu

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

3. POPULATION ATTRIBUTABLE RISK (PAR) The incidence of disease in the total population whose disease can be attributed to the exposure. PAR = Ip - Iu

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

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)

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

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

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

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 > 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?

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

Therefore, the equation for PARF can be rewritten in terms of RR: PARF = P(Ie) + Q(Iu) – Iu P(Ie)+ Q(Iu) Replacing Ie with Iu x RR, we get: PARF = P(Iu)RR + Q(Iu) – Iu P(Iu)RR + Q(Iu)

Going From A Relative Risk To An Attributable Risk Fraction Cont’d Iu can be factored out and cancelled: XIu (P x RR + Q - 1) PARF = XIu (P x RR + Q) If we now replace Q with 1 -P (since P + Q = 1): PARF = P x RR + 1 - P - 1 P x RR + 1 - P

or: 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).

EXAMPLE OF HOW FAILURE TO UNDERSTAND WHAT AN ODDS RATIO MEANS CAN LEAD TO TROUBLE

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 logisticregression analysis to assess the effects of the race and sex of the patients on treatment recommendations

The number of White and Black patients who doctors thought should be referred for cardiac catheterization based on their symptoms White Black Referred Not referred 326 (90. 6%) 305 (84. 7%) 331 34 360 55 360 89 720

Risk ratio and odds ratio in this table • Relative risk or risk ratio for Blacks is: 305 360 divided by 326 360 or 0. 93 • Odds ratio for Blacks is: 305 x 35 326 x 55 or 0. 58

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; P=0. 02) were less likely to be referred for cardiac catheterization than whites.

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

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.

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.