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If the relatove frequency distributions of the variable associated with the disorder in affected and unaffected individuals are Gaussian (normal), the detection rate for a 5% false positive rate (DR5) is given by the area under the Gaussian curve (Φ ) for affected individuals, corresponding to a cut off level of (D- 1.645)/W, which can be looked up in statistical tables. Here 1.645 is the number of standard deviations (SDs) from the mean corresponding to the fifth centile for unaffected individuals, SDa and SDu are the SDs for affected and unaffected individuals respectivly, W is the ratio of the two (W=SDa/SDu), and D is the distance between the mean for affected individuals (Ma) and the mean for unaffected individuals (Mu) in SD units, so that D=(Ma-Mu)/SDu.
DR5=Φ [[(D- 1.645)/W]] (1)
The relative odds of having or developing a disease for people in the top fifth of the distribution of unaffected individuals compared with people in the bottom fifth (ROQ1-5) is equal to the ratio of the proportion of affected individuals in the top fifth to the proportion in the bottom fifth, that is:
ROQ1-5=Φ [[(D- 0.842)/W]]/[[Φ (- 0.842- D)/W]] (2)
where 0.842 corresponds to the cut off values (in SD units) which define the upper and lower fifths. Again, the two areas specified by Φ can be looked up in tables or determined using statistical software.
Having defined DR5 and ROQ1-5 in equations (1) and (2), equation (1) can be rearranged in terms of D and the expression for D can then be substituted in equation (2); the relation between them is then given by:
where Φ-1 (DR5) is the notation used to specify the cut off level required to obtain a given detection rate for a 5% false positive rate. Again this can be determined using statistical tables of the Gaussian distribution or statistical software.
The statistical basis of the terms given above can be found in standard statistical textbooks such as Armitage and Berry and Chatfield and Collins. The equations are simplified if the SD of the variable in affected individuals is the same as that in unaffected individuals, so that W=SDa/SDu=1. Using these equations we show selected DR5 values for ROQ1-5 in the table[t1], and in figs 3[f3] and 4[f4] we illustrate the relation between the two measures of risk for specific examples.
- Armitage P, Berry B. Statistical methods in medical research. 2nd ed. Oxford: Blackwell Scientific Publications, 1991.
- Chatfield C, Collins AJ. Introduction to multivariate analysis. London: Chapman and Hall, 1980.
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