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Main points to be covered. Measures of association compare measures of disease occurrence between levels of a predictor variable (eg, exposed/unexposed) Disease incidence and risk in a cohort study Absolute risk vs. relative risk Properties of the 2 X 2 table: Relative risk vs. odds ratio.

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Main points to be covered l.jpg
Main points to be covered

  • Measures of association compare measures of disease occurrence between levels of a predictor variable (eg, exposed/unexposed)

  • Disease incidence and risk in a cohort study

  • Absolute risk vs. relative risk

  • Properties of the 2 X 2 table: Relative risk vs. odds ratio


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Measuring Association in a Cohort Study

  • Simplest case is to have a dichotomous exposure

  • Everyone in cohort is classified as exposed or unexposed

  • Incidence of the outcome is measured in the two groups of exposed and unexposed

  • Two incidences are compared


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Following two groups by exposure status within a cohort:

Equivalent to following two cohorts defined by exposure



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Difference vs. Ratio

  • Two basic ways to compare two incidence measures:

    • difference: subtract one from the other

    • ratio: form a ratio of one over the other

  • Example: if cumulative incidence is 26% in exposed and 15% in unexposed,

    • risk difference = 26% - 15% = 11%

    • risk ratio = relative risk = 26/15 = 1.7


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Why use difference vs. ratio?

  • Risk difference gives an absolute measure of the association of exposure on 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 etiologic inferences


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Example of Absolute vs. Relative Measure of Risk

  • If disease incidence is very low, can have very strong association on relative measure but absolute difference is small

  • Example: incidence in unexposed is 0.3% and incidence in exposed is 1.5%

    • relative risk = 1.5/0.3 = 5.0

    • risk difference = 1.5 - 0.3 = 1.2%


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Relative Risk

  • Relative risk sometimes used to mean either the ratio of two cumulative incidences (incidence proportions) or the ratio of two incidence rates

  • Text distinguishes risk and rate and so distinguishes relative risk from relative rate


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Relative Risk vs. Relative Rate

  • Risk is based on proportion of persons with disease = cumulative incidence

  • Relative risk = ratio of 2 cumulative incidence estimates

  • Rate is based on events per person-time = incidence rate

  • Relative rate = ratio of 2 incidence rates


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Relative Risk in Cohort

  • Relative risk = ratio of two cumulative incidences in a cohort

  • To simplify the presentation, in text ratio of two cumulative incidences assumes no censoring (and no confounding)

  • Allows presentation of relative risk and odds ratio in the setting of a 2 X 2 table


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Relative Risk in Cohort

  • Ratio of two cumulative incidences by Kaplan-Meier method in a cohort with censoring is still a relative risk

  • Standard error of RR is calculated differently if there is censoring

  • Significance testing of RR cannot be done with one 2 x 2 table when censoring present

    • most common statistic for testing difference between two K-M incidences is log rank test

      (series of 2 x 2 tables weighted by sample size)


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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 = a+b+c+d

a + c

b + d


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2 x 2 table translated into a cohort with

no losses to follow-up

Disease

No disease

(a + b)

a

b

Exposed

time

Disease

No disease

c

d

(c + d)

Unexposed

Relative risk = disease proportion in exposed / disease proportion

in unexposed


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Relative risk of disease in exposed and unexposed

Disease

Yes

No

a

b

a

Yes

a + b

RR =

Exposure

c

d

c

c + d

No


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Odds versus Probability

  • Odds based on probability; expresses probability (p) as ratio: odds = p / (1 - p)

    • odds is always > p because divided by < 1

  • For example, if probability of dying = 1/5, then odds of dying = 1/5 / 4/5 = 1/4

  • Thinking of odds as 2 outcomes, the numerator is the # of times of one outcome and the denominator the # of times of the other

  • P = odds / (1 + odds), so 1/4 / 1 + 1/4 = 1/5


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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 primarily because the log of the ratio of two odds is given by the coefficients in logistic regression equations


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Odds ratio of disease in a cohort

  • Since odds = p / 1- p, odds of disease in exposed = cumulative incidence in exposed / 1 - cumulative incidence in exposed

  • And odds in unexposed = cumulative incidence in unexposed / 1 - cumulative incidence in unexposed

  • Ratio of two odds is the odds ratio (OR)


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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


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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

ad

bc

is called the cross-productof a 2 x 2 table

Better to calculate two odds than cross-product


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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


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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


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Relative risk and Odds ratio

If incidence in exposed and unexposed

is the same, RR = 1 and OR = 1

Odds is always > probability because

odds is p divided by (1 - p) = < 1

If RR = 1, OR will be farther from 1 than RR

For example:

RR=0.4/0.2=2 then OR=0.67/0.25=2.7 and

RR=0.2/0.3=0.7 then OR=0.25/0.43=0.6


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Relative risk and Odds ratio

If risk of disease is low in both exposed and

unexposed, RR and OR approximately =

Text example: incidence of MI risk in high bp

group is 0.018 and in low bp group is 0.003:

RR = 0.018/0.003 = 6.0

OR = 0.01833/0.00301 = 6.09


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Relative risk and Odds ratio

If risk of disease is high in either or both

exposed and unexposed, RR 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 RR only if incidence is

low in both exposed and unexposed group


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“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


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Symmetry of OR versus non-symmetry of RR

OR of non-event is 1/OR of event

RR of non-event = 1/RR of event

Example:

If cum. inc. in exp. = 0.2529 and

cum. inc. in unexp. = 0.0705, then

RR (event)= 0.2529 / 0.0705 = 3.59

RR(non-event)= 0.0705 / 0.2529 = 0.8

Not reciprocal: 1/3.59 = 0.279 = 0.8


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Symmetry of OR versus non-symmetry of RR

Example continued:

OR(event)= 0.2529/1- 0.2529 / 0.0705/

1- 0.0705 = 4.46

OR(non-event)= 0.0705/1- 0.0705 /

0.2529/1- 0.2529 = 0.22

Reciprocal: 1/4.46 = 0.22


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Confidence interval for RR

Calculated on log scale (See Appendix A.3)

Text example: RR=6.0 and log 6.0= 1.792

SE(log RR) =  b/a(a+b) + d/c(c+d) = 0.197

95% CI (RR) = exp {log RR + [1.96 x

 b/a(a+b) + d/c(c+d)]}

1.96 x 0.197 = 0.386

95% CI = exp [1.792 + 0.386] =

exp(1.406) and exp(2.178) =

95% CI = 4.08 to 8.83


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RRHypothesis Testing

H0: RR = 1

Equivalent to testing that proportion

with outcome in exposed equals

proportion with outcome in unexposed

Statistic: Chi-square or Fisher’s exact


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Prevalence Ratios

  • Text refers to Point Prevalence Rate Ratio, but avoiding rate with prevalence, just use prevalence ratio (PR)

  • Analogous to incidence ratio: prevalence in exposed (+) divided by prevalence in unexposed (-)

  • Analogous ratio exists for odds ratio called prevalence odds ratio

    • prevalenceexp/1-prevalenceexp /

      prevalenceunexp/1- prevalenceunexp


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Summary points

  • Cohort with no loss to follow-up can be displayed as a 2 x 2 table

  • Risk difference gives absolute difference; risk ratio gives relative difference

  • Both RR and OR calculated from 2 x 2

  • OR farther from 1.0 than RR

  • OR approximates RR if incidence low in both exposed and unexposed


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