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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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• 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

• 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

Equivalent to following two cohorts defined by exposure

• 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

• 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

• 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%

• 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

• 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

• 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

• 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)

Disease

Yes

No

Yes

a + b

b

a

Exposure

c + d

c

d

No

N = a+b+c+d

a + c

b + d

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

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

• 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

• 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

• 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)

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

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

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

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

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

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

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

• 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

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

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

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

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

• 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

• 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