I have performed an Analysis of Variance in SPSS and asked for Descriptive Statistics and Estimated Marginal Means. But the standard errors for the Estimated Marginal Means are all the same. When I look at the standard deviations for each group shown in the Descriptives table, they are all different. Can this be right?
Resolving The Problem
Both are correct, because the models are different. The standard errors in the Descriptives table (or from EXAMINE) are calculated separately for each group, from the variation about that group's mean. No information about the cases in the other groups is used.
The UNIANOVA model uses all the cases to compute a single estimate of the standard error. The model is that each group has its own mean, but that the variation about that mean is the same for all the groups. This assumption that the variation about the group mean is the same for all groups is called Homogeneity of Variance, and Levene's test may be used to determine if the assumption has been violated. The model standard error is the square root of the Mean Square Error found in the ANOVA table.
For each mean, the model standard error gets multiplied by a number, which in a one-way ANOVA is the reciprocal of the square root of the number of cases in each group. For example, if there are a hundred observations in each group, the Mean Square Error is divided by 10. The means will all have the same standard error only if all the groups have an equal number of cases. This will happen only when the ANOVA design is balanced. (This mechanism for calculating the standard error for estimated marginal means also applies to the MIXED command, although random and repeated effects may lead to nonequal standard errors across factor levels.)
The same machinery used to calculate this number for the estimated marginal means is also used to find the standard error of any contrast, though the number used to multiply the Mean Square Error must be calculated using linear algebra. If COMPARE had been added to /EMMEANS, the differences between each pair of group means would have an estimate of the standard error, which would be larger than the error for either mean by itself. Likewise if the parameter estimates had been requested, they would have error estimates based on the single estimate of Mean Square Error.
The standard errors are used to construct the t-tests, from which the significance of the contrast is obtained. By contrast, one could mean the estimate of the parameter, an estimated marginal mean, the difference of two means, or any user-specified linear combination.
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16 April 2020