East Carolina University



Weighted Means and Unweighted Means One-Way ANOVAPsychologists almost always use weighted means when conducting a one-way ANOVA. Accordingly, groups that have more scores have a greater influence on the outcome of the analysis. I suspect this is the norm because ANOVA was designed for use with experimental research, and, when conducting an experiment, one will generally assign equal numbers of subjects to the groups, in which case the weighted means analysis will be identical to the unweighted means analysis. Furthermore, if attrition causes unequal sample sizes, the sample sizes should most often still be very close to equal, so weighting the groups will have little influence on the analysis. If attrition were cause great differences in sample sizes, then the researcher would have a lot more important things to worry about than whether or not to weight the groups.With nonexperimental data, however, the groups may have substantial differences in sample sizes, and these may or may not correspond to differences in group membership in the population of interest. For example, suppose we have randomly sampled voters. We classify them by political affiliation. To avoid offending anybody, I’ll refer to the three groups as Groups A, B, and C. Descriptive statistics are presented below. I randomly drew the scores from populations with absolute homogeneity of variance. We wish to compare these groups on a normally distributed score, a measure of political conservatism. The data used here are available in an Excel file.GroupMs2nA14.718.89260B15.627.14530C19.929.43310Notice that the groups differ considerably in sample size. If I am comfortable with the sample percentages (60%, 30%, and 10%) fairly representing the breakdown into groups in the population of interest, then it is a weighted means analysis that I would want to employ.Weighted Means AnalysisThe SStotal = 2399.The grand mean (GM) = .6(14.7) + .3(15.6) +.1(19.9) = 15.49.The weighted means SSamong = 60(14.7-15.49)2 + 30(15.6-15.49)2 + 10(19.9-15.49)2 = 232.29. Notice that each group’s deviation from the GM is weighted by its sample size.The SSerror = 2399 – 232.29 = 2166.7.p = .007Unweighted Means AnalysisImagine that I have good reason to believe that the three groups are actually equally represented in the population of interest. In fact, I got data from 100 subjects in each of the three groups. Unfortunately, rats got into my lab and chewed up a bunch of the data records. After losing much data to those dirty rats, I had greatly unequal sample sizes. I am comfortable with the notion that which data records were lost was unrelated to the scores. Accordingly, I decide to weight the groups equally.The harmonic mean sample size is The variance of the means is VAR(14.7, 15.6, 19.9) = 7.723. Notice that this is the same formula we used to compute the among groups mean square with equal sample sizes, except for the use of the harmonic mean sample size.SSamong = 2(154.47) = 308.94.Typically the groups are not equally weighted when estimating the error variance (which is assumed to be constant across populations), so the SSerror = same as from the weighted means analysis, 2166.7. To get this by hand you would simply calculate the sums of squares within each group and then sum them. Do note that this will result in the total sum of squares not being equal to the sum of the among groups sums of squares and the error sum of squares.Notice that the F and the 2 are larger than they were with the weighted means analysis. This is because the group which differed most from the grand mean (Group C) had the smallest sample size, and thus the least influence, in the weighted means analysis. When we went to unweighted (equally weighted) means, the influence of this group increased.Which Analysis Should I Use ?If you are comfortable with the assumption that the unequal sample sizes fairly reflect the composition of the population of interest, then the weighted means analysis is appropriate. If, for whatever reason, you feel that the groups should be equally weighted, use the unweighted means analysis – for example, if in experimental research one group had a much smaller sample size just because it was very expensive to provide the treatment for that group, you might well be justified in using an unweighted means analysis – but good luck finding a stat package that will do it for you. With experimental research, in most cases, the sample sizes will not differ enough for type of analysis to matter much.Differently Weighted Means Analysis.Suppose that the sample sizes were 10, 30, and 60, so that the group (C) with the greatest deviation from the GM also had the largest sample size, and thus greater influence on the solution.The grand mean (GM) = .1(14.7) + .3(15.6) +.6(19.9) = 18.09.The weighted means SSamong = 10(14.7-18.09)2 + 30(15.6-18.09)2 + 60(19.9-18.09)2 = 497.49. Notice that each group’s deviation from the GM is weighted by its sample size.The SSerror = 2399 – 497.49 = 1901.5.Notice that the F and the 2 are now much larger than they were before.Factorial DesignsWith factorial designs, the distinction between weighted and unweighted means becomes more important. Even with experimental data where subjects were randomly assigned to treatments, there may be unequal cell sizes because of random attrition. Unless the cell sizes are proportional, which is unlikely, these unequal cell sizes will result in the independent variables (factors) not being independent of each other. This correlation among the factors will result in there being proportions of the variance in the dependent variable (Y) which we cannot unambiguously say are due solely to one effect (main effect or interaction effect). We want to know what the results would have been if the factors were orthogonal. Since we experimentally manipulated the factors and randomly assigned subjects to treatments, the treatments really should be orthogonal. The mathematical solution most often (almost always) adopted here, these days, is to use Type III (Overall & Spiegel Method I) sums of squares, which do not allocate to any effect variance which cannot be unambiguously credited to that effect. The (now old-fashioned) unweighted means analysis was developed as an attempt to approximate (doing the analysis by hand) the more complicated Type III analysis that is now easily accomplished with computers and statistical software.Some statisticians have argued that if the cell sizes are proportional but unequal, then it does not matter whether you do the weighted or the unweighted means analysis. With proportional cells sizes, the factors will, in fact, be independent of one another, as demonstrated by conducting a contingency table analysis on the sample sizes – the resulting chi-square statistic will have a value of zero. However, if there are interactions, the results of the weighted means analysis will differ from those of the weighted means analysis.Keppel (1973, pp 346-350, briefly laid out how to conduct a weighted means analysis for the one-way layout.VassarStats has an online calculator that will do a one-way ANOVA for you, weighted or unweighted means. For the data used here, here is the unweighted means analysis computed using the VassarStats calculator:ReferenceKeppel, G. (1973). Design and analysis. Englewood Cliff, NJ: Prentice Hall.Copyright 2014, Karl L. Wuensch - All rights reserved.Fair Use of This Document ................
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