Example
Example 1: Apply Permutational MANOVA to Example 1 of MANOVA Basic Concepts based on the data in Figure 1.
Figure 1 – Data for Example 1
Figure 2 shows how to calculate the pseudo-F statistic for the data in Figure 1.
Figure 2 – Calculation of pseudo-F
Pairwise Distances
The entries in range H4:W19 contain the pairwise distance squared values based on the Euclidean distance formula. This range is filled in by inserting the following formula in cell H4, highlighting range H4:W19, and then pressing Ctrl-R and Ctrl-D.
=IF(H$3>=$G4,””,(INDEX($C$4:$E$19,$G4,1)-INDEX($C$4:$E$19,H$3,1))^2+(INDEX($C$4:$E$19,$G4,2)-INDEX($C$4:$E$19,H$3,2))^2+(INDEX($C$4:$E$19,$G4,3)-INDEX($C$4:$E$19,H$3,3))^2)
Cell L22 contains the formula =SUM(H4:W19) and cell L23 contains the formula =SUM(H4:K7,L8:O11,P12:S15,T16:W19). Cells M22, M23 and M24 contain the formulas =L22/I24, =L23/I22 and =M22-M23. Cells N22, N23 and N24 contain =I24-1, =I24-I23 and =I23-1. Cells O22, O23 and O24 contain the formulas =M22/N22, =M23/N23 and =M24/N24. Finally, the pseudo-F statistic in cell R22 is calculated by =O24/O23.
Iterations
By way of illustration, Figure 3 shows how to perform the Permutational MANOVA procedure using only two iterations.
Figure 3 – Permutational MANOVA
The first iteration of data, shown in range G4:I19, is obtained by randomly shuffling the order of the rows of the original data. This order is specified in column F using the array formula =SHUFFLE(A4:A19). The first row in the permutation corresponds to the 4th row of the original data (cell F4). Thus, range G4:I4 should have the values in range C7:E7. This is done by placing the array formula =INDEX(C4:C19,$F4:$F19) in range G4:G19 and pressing Ctrl-Shft-Enter, highlighting the range G4:I19 and pressing Ctrl-R. The pseudo-F statistic can then be calculated as for the original data or by placing the formula =PseudoF(G4:I19,4) in cell G21 (see Real Statistics Support for Permutational MANOVA for the definition of the PseudoF function).
The pseudo-F statistic for the original data is 3.387775. Since neither of the permutations results in a higher pseudo F value, the estimated p-value is 1/3 as shown in cell P4.
Results
Using 1,000 iterations, clearly a more credible value, we obtained a p-value = .022098 (cell P6). This can be obtained via the formula =PERMANOVA(C4:E19,4), where the PERMANOVA function is as described in Real Statistics Support for Permutational MANOVA.
Reference
Anderson, M. J. (2005) PERMANOVA
https://pdfs.semanticscholar.org/4d0c/430f6129b427e48fb407e59ac79ee29b4cae.pdf