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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-
         a+b          a+b           b           ad
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-
           a+c     a+c      c          ad
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)
  Advanced
  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
salad
   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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