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OCR for page 198

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

OCR for page 198

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
.