Mendelian Inheritance and Two Instances of Statistical Magic

The relationships between the genomes of relatives are governed by Mendelian inheritance. By exploiting this property when analyzing data from genetic studies, we demonstrate two instances of statistical magic, something usually considered impossible. The first case involves a linear regression model with three correlated explanatory variables and three coefficients of interest. The third explanatory variable is unobserved/missing. By ‘imputing’ a value for the third variable based on the observed values of the first two in a particular way, and performing the regression in a standard manner, (a) the fitted coefficients remain unbiased, and (b) the standard error matrix from the regression output is correct. (It is (b) in addition to (a) that is the most surprising). In the second case, we show that when there is bias in participating in a genetic study, the genetic component to that bias can be estimated from the genotypes of the participants alone, without using any other information about population and sample, e.g. we do not even need to know which population the sample was drawn from. The two methods had been applied to the data of the UK Biobank.