Understanding and interpreting a MANOVA using only between subjects differences
The below corresponds to the output using MANOVA>General Linear Model>Multivariate in SPSS.
A MANOVA is usually carried out to assess whether two or more characteristics (correlated variables) differ jointly between groups.
Zwick (1985) suggests carrying out a MANOVA after first ranking each characteristic (variable) separately across the groups. This nonparametric procedure she shows is more powerful when the assumptions underlying MANOVA using raw scores do not hold, namely a lack of multivariate normality (as seen by high univariate skews and kurtises) and unequal group covariance matrices (using Box's M test which is produced by default by SPSS).
Zwick also illustrates how to evaluate and understand what the MANOVA is testing with a worked example complete with raw data given in a table in her paper so you can try things out for yourself!
Essentially MANOVA uses two matrices, B and W, which are multivariate analogues of the between and within subject sums of square used by the univariate ANOVAs. The differences we use are similar to the MANOVA for repeated measures but instead of subtracting differences within subject we take differences between each subject score separately and a particular mean. The below describes in a little more detail how this is done.
We first need some definitions: Let's call B the matrix representing variability between groups. Suppose Di equals the difference: (variable i mean in group k - overall variable i mean) then for a pair of variables, i and j, the matrix, B, has i-jth element equal to (Sum over subjects of Di*Dj).
If we now define the difference, Ei, as (Variable i's value minus its group mean) then, for variables i and j, the i-jth element of W will have i,jth element: (Sum over the subjects of Ei*Ej).
You can also define a total sum of squares matrix, analogous to the total sum of squares in a univariate ANOVA, with i,jth element equal to the sum of the i-jth elements of B and E: ie Tij = Bij + Eij. If you read up about MANOVA you may find that the matrix B in the above is actually referred to as H.
Typically we use either B*inv(E) or B*inv(T) to obtain a ratio involving between group variation with within group variation as we do in the univariate ANOVA. There are various ways of converting these to a scalar quantity which we can convert to a F value. There does, however, appear to be a growing consensus among authors such as Zwick (1985), Field (2005, 2013), Howell (1997, 2002) and Olsen (1979) to quote the quantity, Pillai's trace (along with its F and p value), when seeing if the group variable means differ.
You can also partial out continuous variables (covariates) analogously to an ANCOVA by obtaining matrices containing terms such as Var(X'), Cov(X'Z') and Var(Z') where X' and Z' are the residuals from regressing covariate, Cov, on continuous outcomes X and Z. Respectively for X and Z these residuals equal (X - B1 Cov) and (Z - B1 Cov) where B1 and B2 are the respective regression coefficients of Cov with X and Cov with Z.
References
Field, A. (2005, 2013). Discovering statistics using SPSS. Second and Fourth Editions Sage:London. The fourth edition actually has a slightly different title of 'Discovering statistics using IBM SPSS statistics'.
Howell, D. C. (1997, 2002). Statistical methods for psychology (4th & 5th ed.). Belmont, CA: Duxbury Press.
Olsen, C. L. (1979). Practical considerations in choosing a MANOVA test statistic: A rejoinder to Stevens. Psychological Bulletin 86(6) 1350-1352.
Zwick, R. (1985). Nonparametric MANOVA. Psychological Bulletin 97(1) 148-152. This paper is available free via ScienceDirect to CBSU users.