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I am using the mantel.test() function in the "ape" package in R to compare the similarity of two matrices. The function returns a p-value and Z-statistics. While the p-value is understandable, but I am not sure what the significance of the Z-statistics is. i.e Does positive high Z-statistic value means that both matrices are similar, or the similarity can only be described if the p-value is <0.001 and is independent of Z-statistic?

http://svitsrv25.epfl.ch/R-doc/library/ape/html/mantel.test.html

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For the Mantel test, the z score is the sum of the products of the corresponding elements of the matrices...so don't confuse it with the standard z-score based on the normal distribution.

The problem with the z-score (sensu Mantel) is that the value depends on the magnitude of your measured variables (kind of like covariance) and is, thus, hard to interpret. So one usually uses a normalized Mantel cofficient, rM, [calculated in the same way as Pearson's product moment correlation coefficient, r,...and interpreted in the same way (on a scale from -1 to 1...albeit the magnitude tends to be downward biased relative to a standard correlation coefficient)]. A positive rM would indicated positive correspondence and a negative rM would indicate a negative correspondence, but be sure to look at the P-value to determine if the value is greater (or less) than would be expected based on the randomized matrices.

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  • $\begingroup$ passagesoftware.net/webhelp/Mantel_test.htm $\endgroup$
    – coreydevinanderson
    Feb 2, 2017 at 21:23
  • $\begingroup$ Thanks alot for the response. Can you kindly elaborate on two parts. Firstly, if the Z statistics is positive and p less than 0.001, then does it imply that both are similar?. Second, how can I compute normalized Z statistics? $\endgroup$
    – show_stopper
    Feb 3, 2017 at 20:48
  • $\begingroup$ As long as the distances are greater than zero, the Z statistic will be a always be a positive number! $\endgroup$
    – coreydevinanderson
    Feb 5, 2017 at 2:08
  • $\begingroup$ Thanks. Apologies for asking this again as it is still unclear. Given two matrices, how can I tell if they are similar or not using the Mantel test? $\endgroup$
    – show_stopper
    Feb 6, 2017 at 2:45
  • $\begingroup$ As I already stated (above), you need to check the normalized Mantel coefficient (r), which is just the Pearson product moment correlation coefficient. If you don't want to unfold the matrices and calculate it yourself, then find software (such as PASSaGE 2) that will report it for you. If r is greater than zero, then there is evidence of positive correspondence; however, make sure you check the P-value to see if it is greater than you would expect for the randomized data. As stated, Z will always be positive and it is only interpretable relative to the randomized Z values. $\endgroup$
    – coreydevinanderson
    Feb 6, 2017 at 16:58

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