SUMMARY OF COMMONLY USED FORMULAE

SUMMARY OF COMMONLY USED FORMULAE 1. Prevalence Rate = # of individuals with a health condition at a given point in time # individuals in the relevant population Prevalence and Incidence Cumulative Incidence Rate = # of new cases of a health outcome during a given period of time # at risk of developing the outcome at the start of the period Incidence Density = # new cases of a health outcome during a given period of time Total person tim e of observatio n 2. Statistical Estimation Accuracy = Bias + Reliability Means, proportions, and rates are all averages: E(X ) = µ and X= ∑X i =1 n i n , E(p ) = π and p= ∑X i =1 n i n , E(r ) = λ and r= ∑X i =1 n i n 271 The standard deviation is a measure of the average deviation of individual observations around their mean: Standard Deviation of X = S = ∑ (X n i =1 I − X) 2 n −1 The standard error is a measure of the average deviation of summary statistics (means, proportions, rates) around their mean assuming infinite repeated sampling: ∑ (X n i Normal : s.e.(X ) = i =1 − X) 2 S n = n −1 n The calculation of standard errors for proportions and rates depends on whether they are in their decimal form, (0<=p<=1, 0<=r<=1), or in their integer form (percents, or per 1,000, 10,000, 100,000 etc.): Binomial : s.e.(p ) = p(1 − p ) n where p takes on values from 0 to 1 or %(100 − % ) Binomial : s.e.(%) = n where % takes on values from 1 to 100 Poisson : s.e.(r ) = r n where r takes on values from 0 to 1 or rate Poisson : s.e.(rate) = × multiplier n where rate takes on values >= 1, and multiplier = 1,000, 10,000, 100,000 Just as summary statistics (means, proportions, and rates) are all averages and analogous to one another, so too, their standard errors are analogous under certain conditions: 272 ∑ (X n i =1 i −X ) 2 n −1 n ≅ p(1 − p ) r ≅ n n In particular, when an event is very rare (a proportion is very small), the formulas for the binomial and Poisson standard errors are close approximations of one another: p(1 − p ) p(1) r ≅ ≅ n n n 3. Difference Measures: Measures of Association Means : X1 − X 2 , Proportion s : p1 − p 2 , Rates : r1 − r2 Risk Factor Yes No Outcome Yes No a b c d m1 m2 n1 n2 N a c − = p1 − p 2 or r1 − r2 n1 n 2 Attributable Risk = p1 − p2 Population Attributable Risk = Attributable Risk × Prevalence of Risk n p 0 − p 2 = (p1 − p 2 )× 1 N p 2 − p1 Preventive Fraction = = 1 − Relative Risk p2 Ratio Measures: a a a + b = n1 = r1 or p1 RR and RP = c c r 2 p2 c+d n2 Crude: a ad OR = b = c bc d 273 Rothman - Boice Summary Relative Risk : # strata = i =1 # strata ∑ i =1 a i n 2i Ni ∑ c i n1i Ni Adjusted: Mantel - Haenszel Summary Odds Ratio : # strata = i =1 # strata ∑ i =1 a i di Ni ∑ b i ci Ni 4. General form of test statistics: Test Statistics Test Statistic = Observed Associatio n - Expected Associatio n Standard Error of the Associatio n Test for the Difference Between Two Independent Means: t= X1 − X 2 − 0 ( 2 n 1 − 1 S1 + (n 2 − 1)S 2 2 n1 + n 2 − 2 )  1 1  + n  1 n2     where S is the standard deviation of the observed values Test for the Difference Between Two Independent Proportions or Rates χ2 = ∑ i =a d (O i − E i )2 Ei where O i = the observed value in each cell and E i = row total × column total N or 274 z= (p1 − p 2 ) − 0 1 1  p0 (1 − p0 ) +  n n  2  1 and z= r1 − r2 − 0  1 1   r0  + n   1 n2  For the z tests, p1 and r1 = a c m , p 2 and r2 = , and p0 and r0 = 1 n1 n2 N Test for Difference Between a Proportion or Rate and a Standard z= Standard(1 − Standard ) n1 p1 − Standard or z = r1 − Standard Standard n1 Test for the Relative Risk and Relative Prevalence: r ln  1 r  2  −0       z= 1 b  1 d  × + × a n  c n 1  2   Test for the Odds Ratio a×d ln  −0  b×c z= 1 1 1 1 + + + a b c d 5. General form of confidence intervals: Confidence Intervals CI = Estimate ± Critical Value × Standard Error of the Estimate CI = Association ± Critical Value × Standard Error of the Association Confidence intervals around single estimates: 275 CI(X ) = X ± 1.96 CI(p ) = p ± 1.96 S n p(1 − p ) n r n CI(r ) = r ± 1.96 CI(d ) = d ± 1.96 d where d is a count of rare health events Confidence intervals around measures of association: Difference Measures Normal : CI = X1 − X 2 ± 1.96 2 S1 S2 + 2 n1 n 2 Binomial : CI = p1 − p 2 ± 1.96 p1 (1 − p1 ) p2 (1 − p 2 ) + n1 n2 r1 r2 + n1 n 2 Poisson : CI = r1 − r2 ± 1.96 Ratio Measures  r  ln  1  r   2   ± 1.96   1 b  1 d  ×  + × a n  c n 1   2          CI RR and RP = e CI OR = e   a ×d  1 1 1 1  ln  ± 1.96 + + +   b ×c  a b c d   276