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In SPSS, I have created scatterplots of two continuous variables (X & Y) for two separate groups (G & P). Visual inspection comparing the scatterplots from the two groups suggests that distribution of the data points is similar; however, I would like to prove this. Is there a way to do this, preferably in SPSS? Would a one-way MANOVA be appropriate? If not, why would the one-way MANOVA not be appropriate?

Y versus X for groups P & G

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    $\begingroup$ MANOVA compares the mean of the variables X, Y between G, P taking into account correlation of X, Y. It is a different think to compare the distributions of X, Y between groups G, P. What exactly is your purpose? $\endgroup$ – Epaminondas Jun 14 '14 at 8:40
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    $\begingroup$ From just looking at the scatterplots of X & Y between G & P, it looks like the data points are similar. But, I would like to provide stronger support for this than just visual inspection. The scatterplot of Y versus X for groups P & G is pasted above. $\endgroup$ – dlj Jun 14 '14 at 8:48
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You can do a Wilcoxon test with the NPAR TESTS (Analyze > Nonparametric > Legacy Dialogs). You might also want to do a two-variable Q-Q plot, which is available in the SPSSINC QQPLOT2 extension command available via the SPSS Community website (www.ibm.com/developerworks/spssdevcentral)

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    $\begingroup$ Thank you for your input. I was taught that the Wilcoxon test can only be used with non-parametric data. In this case, groups P & G are not non-parametric (i.e.: they are separate groups). Also, according to the SPSS Community web site, QQPLOT2 creates Q-Q plots for two variables or two groups of cases for one variable. I am looking for a comparison between two groups for two variables. Would you have any other suggestions? $\endgroup$ – dlj Jun 15 '14 at 3:45
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    $\begingroup$ @dlj: You seem to misunderstand nonparametric analysis. It does not require knowledge or assumptions about population parameters, but it's fine if they exist. I don't think it would ever be meaningful to say data are nonparametric. $\endgroup$ – Nick Stauner Jul 21 '14 at 22:36
  • $\begingroup$ A simple powerful bivariate test for two sample location problems in experimental and observational studies (tbiomed.com/content/7/1/13) looks very promising. Also, stats.stackexchange.com/questions/55298/…, though it addresses this question for the case of multivariate normal distributions, may give you other useful pointers toward a strategy you can use. $\endgroup$ – rolando2 Jul 22 '14 at 2:59
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A simple powerful bivariate test for two sample location problems in experimental and observational studies looks very promising. Also, this equivalence test post, though it addresses this question for the case of multivariate normal distributions, may give you other useful pointers toward a strategy you can use.

Third, consider this really clumsy, simplistic, but maybe not so terrible strategy. It's easy to apply and makes no assumptions about the shape of either distribution. Draw a grid covering the area that is populated -- for this small sample, maybe a 3 x 3 grid. Determine how many P and how many G points sit in each square. Test for differences in group breakdowns using a 9 x 2 chi-square.

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The two-sample Kolmogorov–Smirnov test is a common approach for comparing two distributions. The original version applies to univariate distributions, but multivariate alternatives exist. Wikipedia:

A distribution free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997).[9] The test uses a statistic which is built using Rosenblatt's transformation and an algorithm is developed to compute it in the bivariate case.

The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to multivariate data. This is not straightforward because the maximum difference between two joint cumulative distribution functions is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of $\Pr(x < X \wedge y < Y)$ or $\Pr(X < x \wedge Y > y)$ or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.

One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In d dimensions, there are 2$^d$−1 such orderings [With your 2 dimensions, this is 3 orderings]...Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.

This test would be applied to the data that you used to produce the scatterplot. I'm not sure if it can be performed in SPSS without scripting it yourself, which is feasible. I think @Epaminondas' comment covers the difference between distribution tests and mean differences tests such as MANOVA.


Reference
9. Justel, A., Peña, D., & Zamar, R. (1997). A multivariate Kolmogorov–Smirnov test of goodness of fit. Statistics & Probability Letters, 35(3), 251–259. DOI:10.1016/S0167-7152(97)00020-5.

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