1.6. One way to understand missing data mechanisms is to generate hypothetical complete data and then create missing values by specific mechanisms. This is common in simulation studies of missing-data methods, since the deleted values can be retained and used to compare methods; missing-data methods are hard to assess on real data sets for the obvious reason that the missing values are rarely known. Consider 100 trivariate normal observations fð yi1, yi2, ui), i ¼ 1; ... ; 100g on ðY1, Y2, U) generated as follows: yi1 ¼ 1 þ zi1 yi2 ¼ 5 þ 2*zi1 þ zi2 ui ¼ a*ðyi1  1Þ þ b*ðyi2  5Þ þ zi3; where fðzi1, zi2, zi3Þ, i ¼ 1; ... ; 100g are independent standard normal (that is, mean 0, variance 1) deviates. Suppose the hypothetical complete data set consist of observations fðyi1, yi2), i ¼ 1; ... ; 100g, which have a bivariate normal distribution with means (1, 5), variances (1, 5) and correlation 2= ffiffiffi 5 p ¼ 0:89. The observed (incomplete) data are fyi1, yi2), i ¼ 1; ... ; 100g, where the values fyi1g of Y1 are all observed, but some values fyi2g of Y2 are missing. The latent variable U determines missingness of Y2 as follows: yi2 is missing if ui < 0: The missing indicator variable M2 for missingness of Y2 using this mechanism is thus: mi2 ¼ 1 if ui < 0 and mi2 ¼ 0 if ui  0. Since U as defined above has mean zero this mechanism should create missing values of Y2 for about 50% of the observations.

(a) Generate a data set of 100 observations using the above process with a ¼ b ¼ 0. Display the marginal distributions of Y1 and Y2 for complete and incomplete cases. Is this mechanism MCAR, MAR, or NMAR?

(b) Carry out a t-test comparing the means of Y1 for complete and incomplete cases. Is there evidence from this test that the data are not

(a) MCAR;

(b) MAR;

(c) NMAR?

(c) Repeat parts

(a) and

(b) with (i) a ¼ 2, b ¼ 0 and (ii) a ¼ 0, b ¼ 2.