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# Measures of Disease Association by ckwyK4to

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```									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
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
Cross-Sectional Study Design: A Prevalent Sample
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: Exposure=risk factor=predictor
2 x 2 table for association of disease and exposure
Disease
Yes              No

Yes
a              b             a+b

No         c               d            c+d

a+c              b+d      N = a+b+c+d
Note: data may not always come to you arranged as above.
STATA puts exposure across the top, disease on the side.
Prevalence ratio of disease in exposed and unexposed
Disease
Yes                 No

Yes           a              b                  a
a+b
PR =
c
No            c              d                 c+d
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
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 Pr>chi2 = 0.0783
STATA calls it a risk ratio by default
Prevalence ratio of disease in exposed and unexposed
Disease
Yes                 No

Yes           a              b                  a
a+b
PR =
c
No            c              d                 c+d

So a/a+b and c/c+d = probabilities of disease
and PR is ratio of two probabilities
Probability and Odds
• Odds another way to express probability of an event

• Odds =      # events
# non-events
• Probability =      # events
# events + # non-events
=      # events
# subjects
Probability and Odds
• Probability =    # events
# subjects
• Odds = # events
# subjects      = probability
# non-events     (1 – probability)
# subjects
• Odds = p / (1 - p)
[ratio of two probabilities]
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
Odds versus Probability
• Less intuitive than probability (probably
wouldn’t 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
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
Probability and odds in a 2 x 2 table
Disease
Yes                 No
What is p of disease
in exposed?
Yes            2             3        5
What are odds of
disease in exposed?

No             1             4        5   And the same for
the un-exposed?

3                7      10
Probability and odds ratios in a 2 x 2 table
Disease

PR = 2/5  1/5
Yes                 No

=2
Yes            2             3        5

0R = 2/3  1/4
= 2.67
No             1             4        5

3                7       10
Odds ratio of disease in exposed and unexposed
Disease
Yes             No                   a
a+b
Yes           a             b                      a
1-
a+b
OR =
c
No            c             d                   c+d
c
1-
c+d
Formula of p / 1-p in exposed / p / 1-p in unexposed
Odds ratio of disease in exposed and unexposed
a           a
a+b          a+b
a             b           a
1-
OR =      c         =             =        =
c           c           bc
c+d
c           c+d           d
1-
c+d           d
c+d
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
Odds ratio of exposure in diseased and not diseased
Disease
Yes             No                    a
a+c
Yes        a             b                       a
1-
a+c
OR =
b
No         c              d                   b+d
b
1-
b+d
Important characteristic of odds ratio
a      a
a+c     a+c
a        c      a
1-
ORexp =          =        =      =
b        b      b          bc
b+d     b+d
b               d
1-
b+d      d
b+d

OR for disease = OR for exposure
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
Measuring Association in a Cohort
Following two groups by exposure status within a cohort:
Equivalent to following two cohorts defined by exposure
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?
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
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
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
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
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
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 < 2.0 may still be valid
but are more likely to be the result of bias
– Second-hand smoke relative risk < 1.5
Example of Absolute vs. Relative
Measure of Risk
TB         No TB      Total
recurrence recurrence
Treated
> 6 mos        14          986      1000
Treated
< 3 mos        40          960      1000
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
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
Example of study reporting risk difference
Table 2. Survival and Functional
Outcomes from the Two Study Phases
Return of          Risk
Spontaneous     Difference
Study Phase      Circulation                     p-value
(95% CI)
Rapid
Defibrillation                      --
12.9%                           --
(N=1391)
Life Support       18.0%        5.1% (3.0-7.2)   <0.001
(N=4247)

Risk difference = 0.051; number needed to treat = 1/0.051 = 20

Stiel et al., NEJM, 2004
Risk Ratio
Diarrheal Disease
Yes       No          Total
Ate potato salad            54        16         70

Did not eat potato           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
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
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).
OR compared to Risk Ratio

If Risk Ratio = 1.0, OR = 1.0;
otherwise OR farther from 1.0

0                          1                     ∞
Stronger effect         Stronger effect

OR     Risk Ratio       Risk Ratio   OR
Risk ratio and Odds ratio
If Risk Ratio > 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
Risk ratio and Odds ratio
If Risk Ratio < 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
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 Pr>chi2 = 0.0783
Important property of odds ratio #2

• OR approximates Risk Ratio only if
disease incidence is low in both the
exposed and the unexposed group
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
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
“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
Important property of odds ratio #3

• Unlike Risk Ratio, OR is symmetrical:

OR of event = 1 / OR of non-event
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
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
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
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
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

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