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Appendix D Equation Relating to the Effect of Error in Assessing Childhood Milk-consumption The approximation for the inflation in sample sizes ~ -M ~MR2 due to errors in milk consumption is derived by noting that if the dose estimate is linear both in milk intake, Im, and also in other variables, W. which are independent of milk consumption, then the variance of the close distribution can be divided into two portions. For example if dose, D, is approximately equal to D = a + bW + CIm Then the variance of the dose distribution is approximately equal to var( D ) = b 2 var( W ) + c 2 var( Im ) Now if milk consumption is estimated using a questionnaire to give value Qm then which has correlation R with true milk consumption, then in order to estimate a linear dose response function in dose with no dose error attenuation bias, we replace D in the response mode} with E(D ~ Qm ~ = a + bell ~ Qm ~ + CE(lm ~ Qm (Carroll and others, 1995~. The variance of this quantity determines the sample size required to detect a nonzero dose response, as described in chapter X on statistical power. Assuming 198
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Appendix D 199 also that the non-milk variables are independent of the questionnaire estimate of milk consumption, we have var(E(D ~ Qm JO = b2var6W ~ Qm ~ + c2var<~m ~ Qm = b2 var(W) + c2R 2 var(]m ~ . Letting M=c2Var(Im)/Var(D)be the fraction ofthe variance of dose that depends upon milk and 1-M the remainder, the we have var(E(D ~ Qm' = var(D)~-M +MR2) . Thus if we do not know true milk consumption, but only Qm, in order to have the same power to detect a linear dose response function we must (from the equation for the noncentrality parameter) have N2 var(D ~ Qm = N var(D) where N is the sample size needed with no errors in estimating milk consumption, and N2 is the sample size required when Qm is used to estimate milk. Therefore the inflation in sample size needed is N2 = I-M +MR2 .