Measures of Disease Association

1
Measures of Disease Association

• Measuring occurrence of new outcome events can be
  an aim by itself, but usually we want to look at
  the relationship between an exposure (risk
  factor, predictor) and the outcome
• The type of measure showing an association
  between an exposure and an outcome event is
  linked to the study design

2
Main points to be covered

• Measures of association compare measures of
  disease between levels of a predictor variable
• Prevalence ratio versus risk ratio
• Probability and odds
• The 2 X 2 table
• Properties of the odds ratio
• Absolute risk versus relative risk
• Disease incidence and risk in a cohort study

3
Cross-Sectional Study Design A Prevalent Sample
4
Measures of Association in a Cross-Sectional Study

• Simplest case is to have a dichotomous outcome
  and dichotomous exposure variable
• Everyone in the sample is classified as diseased
  or not and having the exposure or not, making a 2
  x 2 table
• The proportions with disease are compared among
  those with and without the exposure
• NB Exposurerisk factorpredictor

5
2 x 2 table for association of disease and
exposure
Disease
Yes
No
Yes
a b
b
a
Exposure
c d
c
d
No
N abcd
a c
b d
Note data may not always come to you arranged as
above. STATA puts exposure across the top,
disease on the side.
6
Prevalence ratio of disease in exposed and
unexposed
Disease
Yes
No
a
a
Yes
b
a b
PR
Exposure
c
c
d
c d
No
7
Prevalence Ratio

• Text refers to Point Prevalence Rate Ratio in
  setting of cross-sectional studies
• We like to keep the concepts of rate and
  prevalence separate, and so prefer to use
  prevalence ratio

8
Prevalence ratio (STATA output)
Exposed
Unexposed Total ------------------------------
--------------------- Cases 14
388 402 Noncases
17 248
265 ----------------------------------------------
----- Total 31 636
667
Risk .4516129
.6100629 .6026987
Point estimate 95 Conf. Interval
----------------------------------------
----- Risk ratio .7402727
.4997794 1.096491
-----------------------------------------------
chi2(1) 3.10
Prgtchi2 0.0783
STATA calls it a risk ratio by default
9
Prevalence ratio of disease in exposed and
unexposed
Disease
Yes
No
a
a
Yes
b
a b
PR
Exposure
c
c
d
c d
No
So a/ab and c/cd probabilities of
disease and PR is ratio of two probabilities
10
Probability and Odds

• Odds another way to express probability of an
  event
• Odds events
• non-events
• Probability events
• events non-events
• events
  
• subjects

11
Probability and Odds

• Probability events
• subjects
• Odds events
• subjects
  probability
• non-events (1
  probability)
• subjects
• Odds p / (1 - p)
• ratio of two probabilities

12
Probability and Odds

• If event occurs 1 of 5 times, probability 0.2.
• Out of the 5 times, 1 time will be the event and
  4 times will be the non-event, odds 0.25
• To calculate probability given the odds
• probability odds / 1 odds

13
Odds versus Probability

• Less intuitive than probability (probably
  wouldnt say my odds of dying are 1/4)
• No less legitimate mathematically, just not so
  easily understood
• Used in epidemiology because the measure of
  association available in case-control design is
  the odds ratio
• Also important because the log odds of the
  outcome is given by the coefficient of a
  predictor in a logistic regression

14
Odds ratio

• As odds are just an alternative way of expressing
  the probability of an outcome, odds ratio (OR),
  is an alternative to the ratio of two
  probabilities (prevalence or risk ratios)
• Odds ratio ratio of two odds

15
Probability and odds in a 2 x 2 table
Disease
Yes
No
What is p of disease in exposed? What are odds
of disease in exposed? And the same for the
un-exposed?
2
Yes
3
5
Exposure
1
4
5
No
10
7
3
16
Probability and odds ratios in a 2 x 2 table
Disease
Yes
No
PR 2/5
1/5
2
2
3
Yes
5
0R 2/3
1/4
Exposure
2.67
1
4
5
No
10
7
3
17
Odds ratio of disease in exposed and unexposed
Disease
a
Yes
No
a b
a
b
a
Yes
1 -
a b
OR
Exposure
c
d
c
c d
No
c
1 -
c d
Formula of p / 1-p in exposed / p / 1-p in
unexposed
18
Odds ratio of disease in exposed and unexposed
a a b b a b c c d d c d
a
a b c d
a b
a
1 -
a b
ad bc

OR


c
c d
c
1 -
c d
19
Important Property of Odds Ratio 1

• The odds ratio of disease in the exposed and
  unexposed equals the odds ratio of exposure in
  the diseased and the not diseased
• Important in case-control design

20
Odds ratio of exposure in diseased and not
diseased
Disease
a
Yes
No
a c
a
b
a
Yes
1 -
a c
OR
Exposure
b
d
c
b d
No
b
1 -
b d
21
Important characteristic of odds ratio
a a c c a c b b d d b d
a
a c b d
a c
a
1 -
a c
ad bc



ORexp
b
b d
b
1 -
b d
OR for disease OR for exposure
22
Measures of Association Using Disease Incidence

• With cross-sectional data we can calculate a
  ratio of the probability or of the odds of
  prevalent disease in two groups, but we cannot
  measure incidence
• A cohort study allows us to calculate the
  incidence of disease in two groups

23
Measuring Association in a
Cohort Following two groups by exposure status
within a cohort Equivalent to following two
cohorts defined by exposure
24
Analysis of Disease Incidence in a Cohort

• Measure occurrence of new disease separately in a
  sub-cohort of exposed and a sub-cohort of
  unexposed individuals
• Compare incidence in each sub-cohort
• How compare incidence in the sub-cohorts?

25
Relative Risk vs. Relative Rate

• Risk is based on proportion of persons with
  disease cumulative incidence
• Risk ratio ratio of 2 cumulative incidence
  estimates relative risk
• Rate is based on events per person-time
  incidence rate
• Rate ratio ratio of 2 incidence rates
  relative rate
• We prefer risk ratio, rate ratio, odds ratio

26
A Note on RR or Relative Risk

• Relative risk or RR is very common in the
  literature, but may represent a risk ratio, a
  rate ratio, a prevalence ratio, or even an odds
  ratio
• We will try to be explicit about the measure and
  distinguish the different types of ratios
• There can be substantial difference in the
  association of a risk factor with prevalent
  versus incident disease

27
Difference vs. Ratio Measures

• Two basic ways to compare measures
• difference subtract one from the other
• ratio form a ratio of one over the other
• Can take the difference of either an incidence or
  a prevalence measure but rare with prevalence
• Example using incidence cumulative incidence 26
  in exposed and 15 in unexposed,
• risk difference 26 - 15 11
• risk ratio 0.26 / 0.15 1.7

28
Summary of Measures of Association
Ratio Difference
Cross-sectional prevalence ratio prevalence difference
odds ratio odds difference
Cohort risk ratio risk difference
rate ratio rate difference
odds ratio odds difference
29
Why use difference vs. ratio?

• Risk difference gives an absolute measure of the
  association between exposure and disease
  occurrence
• public health implication is clearer with
  absolute measure how much disease might
  eliminating the exposure prevent?
• Risk ratio gives a relative measure
• relative measure gives better sense of strength
  of an association between exposure and disease
  for inferences about causes of disease

30
Relative Measures and Strength of Association
with a Risk Factor

• In practice many risk factors have a relative
  measure (prevalence, risk, rate, or odds ratio)
  in the range of 2 to 5
• Some very strong risk factors may have a relative
  measure in the range of 10 or more
• Asbestos and lung cancer
• Relative measures lt 2.0 may still be valid but
  are more likely to be the result of bias
• Second-hand smoke relative risk lt 1.5

31
Example of Absolute vs. Relative Measure of Risk
TB recurrence No TB recurrence Total
Treated gt 6 mos 14 986 1000
Treated lt 3 mos 40 960 1000
Risk ratio 0.04/0.014 2.9 Risk difference 0.04 0.014 2.6 Risk ratio 0.04/0.014 2.9 Risk difference 0.04 0.014 2.6 Risk ratio 0.04/0.014 2.9 Risk difference 0.04 0.014 2.6 Risk ratio 0.04/0.014 2.9 Risk difference 0.04 0.014 2.6
If incidence is very low, relative measure can be
large but difference measure small
32
Reciprocal of Absolute Difference ( 1/difference)

• Number needed to treat to prevent one case of
  disease
• Number needed to treat to harm one person
• Number needed to protect from exposure to prevent
  one case of disease
• TB rifampin example 1/0.026 38.5, means that
  you have to treat 38.5 persons for 6 mos vs. 3
  mos. to prevent one case of TB recurrence

33
Example of study reporting risk difference
Table 2. Survival and Functional Outcomes from the Two Study Phases Table 2. Survival and Functional Outcomes from the Two Study Phases Table 2. Survival and Functional Outcomes from the Two Study Phases Table 2. Survival and Functional Outcomes from the Two Study Phases
Study Phase Return of Spontaneous Circulation Risk Difference (95 CI) p-value
Rapid Defibrillation (N1391) 12.9 -- --
Advanced Life Support (N4247) 18.0 5.1 (3.0-7.2) lt0.001
Risk difference 0.051 number needed to treat
1/0.051 20
Stiel et al., NEJM, 2004
34
Risk Ratio
Diarrheal Disease Yes No Diarrheal Disease Yes No Total
Ate potato salad 54 16 70
Did not eat potato salad 2 26 28
Total 56 42 98
Probability of disease, ate salad 54/70 0.77 Probability of disease, no salad 2/28 0.07 Risk ratio 0.77/0.07 11 Illustrates risk ratio in cohort with complete follow-up Probability of disease, ate salad 54/70 0.77 Probability of disease, no salad 2/28 0.07 Risk ratio 0.77/0.07 11 Illustrates risk ratio in cohort with complete follow-up Probability of disease, ate salad 54/70 0.77 Probability of disease, no salad 2/28 0.07 Risk ratio 0.77/0.07 11 Illustrates risk ratio in cohort with complete follow-up Probability of disease, ate salad 54/70 0.77 Probability of disease, no salad 2/28 0.07 Risk ratio 0.77/0.07 11 Illustrates risk ratio in cohort with complete follow-up
35
Risk Ratio in a Cohort with Censoring
Choose a time point for comparing two cumulative
incidences At 6 years, dead in low CD4 group
0.70 and in high CD4 group 0.26. Risk ratio
at 6 years 0.70/0.26 2.69
36
Comparing two K-M Curves
Risk ratio would be different for different
follow-up times. Entire curves are compared
using log rank test (or other similar tests).
37
OR compared to Risk Ratio
If Risk Ratio 1.0, OR 1.0 otherwise OR
farther from 1.0
0
1
8
Stronger effect Risk Ratio OR
Stronger effect OR Risk Ratio
38
Risk ratio and Odds ratio
If Risk Ratio gt 1, then OR farther from 1 than
Risk Ratio RR 0.4 2 0.2 OR
0.4 0.6 0.67 2.7 0.2
0.25 0.8
39
Risk ratio and Odds ratio
If Risk Ratio lt 1, then OR farther from 1 than
RR RR 0.2 0.67 0.3 OR
0.2 0.8 0.25 0.58 0.3
0.43 0.7
40
Odds ratio (STATA output)
Exposed Unexposed
Total ------------------------------------------
--------- Cases 14 388
402 Noncases 17
248 265 ---------------------
------------------------------ Total
31 636 667

Risk .4516129 .6100629
.6026987 Point estimate
95 Conf. Interval
--------------------------------------------- Risk
ratio .7402727 .4997794
1.096491 Odds ratio .5263796
.2583209 1.072801
-----------------------------------------------
chi2(1) 3.10
Prgtchi2 0.0783
41
Important property of odds ratio 2

• OR approximates Risk Ratio only if disease
  incidence is low in both the exposed and the
  unexposed group

42
Risk ratio and Odds ratio If risk of
disease is low in both exposed and unexposed, RR
and OR approximately equal. Text example
incidence of MI risk in high bp group is 0.018
and in low bp group is 0.003 Risk
Ratio 0.018/0.003 6.0 OR
0.01833/0.00301 6.09
43
Risk ratio and Odds ratio If risk of
disease is high in either or both exposed and
unexposed, Risk Ratio and OR differ Example, if
risk in exposed is 0.6 and 0.1 in unexposed
RR 0.6/0.1 6.0 OR 0.6/0.4
/ 0.1/0.9 13.5 OR approximates Risk Ratio only
if incidence is low in both exposed and
unexposed group
44
Bias in OR as estimate of RR

• Text refers to bias in OR as estimate of RR (OR
  RR x (1-incid.unexp)/(1-incid.exp))
• not bias in usual sense because both OR and RR
  are mathematically valid and use the same numbers
• Simply that OR cannot be thought of as a
  surrogate for the RR unless incidence is low

45
Important property of odds ratio 3

• Unlike Risk Ratio, OR is symmetrical
• OR of event 1 / OR of non-event

46
Symmetry of odds ratio versus non-symmetry of
risk ratio
OR of non-event is 1/OR of event RR of non-event
1/RR of event Example If cum. inc. in exp.
0.25 and cum. inc. in unexp. 0.07, then RR
(event) 0.25 / 0.07 3.6 RR (non-event)
0.75 / 0.93 0.8 Not reciprocal 1/3.6 0.28
0.8
47
Symmetry of OR
Example continued OR(event) 0.25
(1- 0.25) 4.43
0.07
(1- 0.07) OR(non-event) 0.07
(1- 0.07) 0.23
0.25 (1-
0.25) Reciprocal 1/4.43 0.23
48
Important property of odds ratio 4

• Coefficient of a predictor variable in logistic
  regression is the log odds of the outcome (e to
  the power of the coefficient OR)
• Logistic regression is the method of
  multivariable analysis used most often in
  cross-sectional and case-control studies

49
3 Useful Properties of Odds Ratios

• Odds ratio of disease equals odds ratio of
  exposure
• Important in case-control studies
• Odds ratio of non-event is the reciprocal of the
  odds ratio of the event (symmetrical)
• Regression coefficient in logistic regression
  equals the log of the odds ratio

50
Summary points

• Cross-sectional study gives a prevalence ratio
• Risk ratio should refer to incident disease
• Relative ratios show strength of association
• Risk difference gives absolute difference
  indicating number to treat/prevent exposure
• Properties of the OR important in case-control
  studies
• OR for disease OR for exposure
• Logistic regression coefficient gives OR