You have asked, not a statistical question, but a policy question (or perhaps a question of definition). The essential question is, Are the several subjects equally important; or, Are the individual measures (or items?) equally important. (These are not the only possible ways of answering the question of weights, but they may be the simplest.)
Your example of a weighted arithmetic mean assigned equal importance to each item, and much less importance to subject B (who had three items'-worth of data) than to subject A (who had eight). If you think that subjects are equally important, you should weight them equally, producing the equally weighted mean (1 + 0.75)/2 = 0.875. This is commonly (if misleadingly?) called an "unweighted" mean. A related question that _is_ statistical has to do with how you chose to analyze the data: are you interested in the within-subject variances of the item scores? (This is what Rich Ulrich was alluding to in describing possible complications in the analysis.) Since both "unweighted" and weighted means produce a values that is more or less in the middle of the available data values, there may not be enough difference in the results to make the distinction worth making. (Your weighted mean of 0.818 and my equally-weighted mean of 0.875 are both between 0.75 and 1.00, for example.) My inclination would be to use "unweighted" means for each subject, and carry out the desired comparison (t test or ANOVA); then calculate the two weighted means (for group 1 and group 2), and see whether they are different enough from the unweighted means that the comparison might possibly turn out differently. Only if it looked as though the weighted means might have a different outcome would I bother to do the more elaborate analysis. Your initial decision (whether to assign equal weight to subjects or to items within subjects) is essentially a political one, as this anecdote illuminates: During the Philadelphia meetings which eventually produced the U.S. constitution about 1789, there was much debate as to whether the legislators should be elected in proportion to population or not. Delegates from the more populous colonies argued for proportionate representation; delegates from smaller colonies, concerned that their colonies' concerns would be overwhelmed by those of the larger colonies (and mindful that some matters of state would differ notably between more and less populous jurisdictions), argued for equal representation from each colony. As you can see, this is essentially your question, if in a different context: shall we assign equal weight to each person in the national population, or to each political unit in the nation? Eventually, of course, they came to what in some of our histories is called the Great Compromise: there would be two legislative houses, one (the House of Representatives) whose members would be elected in numbers proportional to the populations of the states, and one (the Senate) whose members were elected in equal numbers (precisely, two) from each state. On Thu, 10 Oct 2002, Jan Malte Wiener wrote: > hopefully a simple one -> > > let's assume i have data like this: > subject A: 1,0,1,1,0,1,1,1 -> mean=6/8=0.75 > subject B: 1,1,1 -> mean=3/3=1 > > weighted arithmetic mean of A+B-> (8*0.75 + 3*1)/11 = 0.82 > -> well that was easy, but what if i do have 20 subjects like this and i > want to compare their weighted arithmetic mean to the weighted mean of > another group of 20 subjects ?? i guess i need to weight every single > subject-mean before running any stat-test. and here is my problem: how > do i weight the individual subject means ?? ----------------------------------------------------------------------- Donald F. Burrill [EMAIL PROTECTED] 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 [Old address: 184 Nashua Road, Bedford, NH 03110 (603) 471-7128] . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
