Dear Constantine: hope the following comments help. Rod Little. On Tue, 6 Mar 2001, Constantine Daskalakis wrote: > Hi all. > One minor technical question and a more substantial issue regarding > mixed-effects models. > > More than once in the past year I've come across a statement regarding > testing MAR in SPSS, such as > "we tested whether the data were missing at random using SPSS". > > I am fairly certain that testing MAR (vs. non-ignorable) is not possible > logically or statistically, at least not without external info. What is > possible (and relatively straight-forward) is testing MCAR vs. MAR. Please > correct me if I have it wrong! > > So, can anyone explain what, if any, test SPSS performs (it's not a package > I'm familiar with)? Is it a MCAR test? You are right. I am not very familiar with SPSS but I believe this is a test of MCAR based on the following reference: Little, R.J.A. (1988). A test of missing completely at random for multivariate data with missing values. Journal of the American Statistical Association , 83, 1198-1202. > > Now, to the more important issue that I would like to ask about. > > It is generally stated that mixed-effects regression is a full-information > likelihood method, thus valid under ignorable missingness. In other words, > the estimates and standard errors (e.g., as obtained from SAS) are valid > even in the case of unbalanced data (i.e., different number of observations > per subject) -- provided that's MAR. True for the estimates, but a fact that perhaps should be more widely known is that the standard errors assume MCAR, not just MAR. This is because the information matrix is assume to be block diagonal between the mean and covariance parameters, which is not generally true under MAR. It would be better if SAS provided se's under MAR, which is perfectly feasible. Molenberghs, Bijnens and Shaw have a book chapter that addresses this issue but I don't have the full reference. > > But consider this argument. Suppose that we have a study of N=100 > subjects, with up to 4 timepoints for each. Suppose that everyone has the > first 3 measurements but that the 4th measurement is missing for some > subjects. Further suppose that whether Y_4 is missing or not depends on > the value of Y_3 (which is always observed -- i.e., MAR for > Y_4). Specifically, suppose that the probability of Y_4 missing increases > with increasing values of Y_3. > > With the mixed model, among the N subjects, some are going to have steeper > slopes of Y (increasing) over time and some flatter ones. The former will > then have higher probability of Y_4 missing than the latter. Thus, > subjects with steeper slopes will be more likely to have fewer datapoints > than subjects with flatter slopes. But that means that the former will > have somewhat lower "weight" in the estimation of the fixed effect than the > latter, and therefore the estimate will be biased (toward the null in this > case). > > Can you comment on the possible incompatibility of the two lines of thinking? I like your intuitive approach, but I too have managed to confuse myself with such arguments! Full ML with unbalanced data involves the LS slopes _and_ intercepts, and is not quite as simple as weighting the LS slopes for each individual by their precision. The latter method is not full ML (since data are being discarded) and can lead to bias under MAR. The fact is that ignorable ML works under MAR provide the model is right ... The simulations and discussion in the following article are quite germane to your question. Little, R.J.A. and Raghunathan, T. E. (1999). On Summary-Measures Analysis of the Linear Mixed-Effects Model for Repeated Measures When Data are Not Missing Completely at Random. Statistics in Medicine, 18, 2465-2478. > > Thanks in advance, > cd > > > > ____________________________________________________________ > > Constantine Daskalakis, ScD > Assistant Professor, > Biostatistics Section, Division of Clinical Pharmacology, > Thomas Jefferson University, > 132 South 10th Street, Philadelphia, PA 19107 > Tel: 215-955-5695 > Fax: 215-955-5681 > Email: [EMAIL PROTECTED] > ____________________________________________________________ > ___________________________________________________________________________________ Roderick Little Chair, Department of Biostatistics (734) 936-1003 U-M School of Public Health Fax: (734) 763-2215 M4208 SPH II [EMAIL PROTECTED] 1420 Washington Hgts http://www.sph.umich.edu/~rlittle/ Ann Arbor, MI 48109-2029
