Beta-Dispersion

Beta-dispersion tests if the dispersion, variance, of two or more groups are significantly different or not. First, an item named “groups” must be made that contains lists of the location and number of isolates from that location used in the analysis. We will then check to make sure the lists in “groups” adds up to the number of isolated used in analysis.

First, make a list of the locations for each pathotype. Note that when you are using two or more locations you will need to make a list for each location with a length of isolates used.

groups <- factor(c(rep("Michigan_1", 11), rep("Michigan_2", 10)))

# this number shows how many isolates are in all "groups" lists combined
length(groups)

## [1] 21

# this shows the number of isolates within your data set, these numbers should
# match for downstream analyses to work!! 
length(unique(P_sojae_survey$Isolate))

## [1] 21

Next, beta-dispersion will be calculated using the Jaccard distance matrix made previously. An ANOVA is then performed to identify significance within the dataset. Post-hoc tests can be used to identify significant interactions between specific locations within the dataset. This can then be plotted to visualize how the beta-dispersion is different between groups.

 # calculates the beta-dispersion for each group, when comparing 2 or more
pathotype.disp <-
  betadisper(P_sojae_survey.matrix.jaccard, groups)

# tests if centroid distances are significantly different from each other
pathotype.disp.anova <- anova(pathotype.disp) 
pathotype.disp.anova

## Analysis of Variance Table
## 
## Response: Distances
##           Df   Sum Sq   Mean Sq F value Pr(>F)
## Groups     1 0.008375 0.0083752  0.9672 0.3377
## Residuals 19 0.164523 0.0086591

# test significance between each group
pathotype.disp.TukeyHSD <- TukeyHSD(pathotype.disp)
pathotype.disp.TukeyHSD

##   Tukey multiple comparisons of means
##     95% family-wise confidence level
## 
## Fit: aov(formula = distances ~ group, data = df)
## 
## $group
##                              diff        lwr        upr     p adj
## Michigan_2-Michigan_1 -0.03998626 -0.1250853 0.04511275 0.3377355

# plot showing the dispersion for each group
plot(pathotype.disp, hull = FALSE, ellipse = TRUE)

The ANOVA identified no significant differences between groups dispersion (p = 0.3377355). This means that the groups dispersion, or variance, is not significantly different from each other and the groups dispersion is likely homogeneous between groups. At this point, a Tukey HSD test is not warranted, but we use it as an example here. Again, a p-value of 0.3377355 is reported from the Tukey HSD tests and we reject the hypothesis that these groups may have different dispersion. We can plot the dispersion for each group using the plot() function. As expected, since we have identified no significant differences, the two groups dispersion overlap a great deal and are not distinct from each other. Again this shows that pathotype dispersion between the groups is homogeneous and not different in this instance.

If we were working with a data set that had groups with significantly different dispersion we would expect to see a significant ANOVA p-value (p < 0.05) as well as significance when using the Tukey HSD test. Lastly, the plotted dispersion will form distinct, separate, groups which can be observed.

Differences in beta-dispersion may indicate separate pathotype groups which should be further investigated with Permutation Based Analysis of Variance (PERMANOVA) and Analysis of Similarity (ANOSIM) analysis. Groups which have similar dispersion may still be significantly different in regards to their centroids, which will be tested using a PERMANOVA.

Permutation Based Analysis of Variance (PERMANOVA)

PERMANOVA tests if the centroids, similar to means, of each group are significantly different from each other. Likewise, an \(R^2\) statistic is calculated, showing the percentage of the variance explained by the groups.

pathotype.adonis <- adonis(P_sojae_survey.matrix.jaccard ~ groups)

## 'adonis' will be deprecated: use 'adonis2' instead

pathotype.adonis

## $aov.tab
## Permutation: free
## Number of permutations: 999
## 
## Terms added sequentially (first to last)
## 
##           Df SumsOfSqs  MeanSqs F.Model      R2 Pr(>F)
## groups     1   0.11358 0.113583  1.6229 0.07869  0.175
## Residuals 19   1.32976 0.069988         0.92131       
## Total     20   1.44335                  1.00000       
## 
## $call
## adonis(formula = P_sojae_survey.matrix.jaccard ~ groups)
## 
## $coefficients
## NULL
## 
## $coef.sites
##                     [,1]        [,2]        [,3]       [,4]       [,5]
## (Intercept)  0.356206294  0.47247525  0.45504258 0.36111093 0.33087421
## groups1     -0.007721445 -0.02921472 -0.04813257 0.01795243 0.01040219
##                    [,6]       [,7]        [,8]       [,9]       [,10]
## (Intercept)  0.33955905 0.27518277  0.35412523 0.28774687  0.31463605
## groups1     -0.05774087 0.01311983 -0.04013468 0.03179116 -0.03474624
##                  [,11]      [,12]      [,13]      [,14]      [,15]      [,16]
## (Intercept) 0.33437434 0.38567493 0.32215777 0.32867133 0.30119623 0.29928127
## groups1     0.02883821 0.03019708 0.03363795 0.02097902 0.03420322 0.05551671
##                  [,17]      [,18]      [,19]      [,20]       [,21]
## (Intercept) 0.32090891 0.29928127 0.28518807 0.28518807 0.412384837
## groups1     0.05108374 0.05551671 0.05197906 0.05197906 0.008538684
## 
## $f.perms
##                  [,1]
##    [1,]  0.8193056960
##    [2,]  1.3038466377
##    [3,]  1.0862746899
##    [4,]  1.4279771623
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##  [955,]  1.1678331815
##  [956,]  1.1756405727
##  [957,]  2.2378565740
##  [958,]  0.1773011916
##  [959,]  0.8301576873
##  [960,]  0.3461615953
##  [961,]  0.5466808821
##  [962,]  0.4259064883
##  [963,] -0.0309491085
##  [964,]  1.0812243088
##  [965,]  0.2398193442
##  [966,]  0.4744263492
##  [967,]  0.6263446816
##  [968,]  1.2196121587
##  [969,]  1.6450009388
##  [970,]  0.5582230310
##  [971,]  2.0902385740
##  [972,]  1.1446680450
##  [973,]  0.7094555971
##  [974,]  0.9477943655
##  [975,]  0.7743675319
##  [976,]  0.1697948876
##  [977,]  1.7822122791
##  [978,]  0.6882726888
##  [979,]  2.1238557175
##  [980,]  3.3792336033
##  [981,]  0.5926037262
##  [982,]  1.9392021885
##  [983,]  1.5648920737
##  [984,]  0.0306015992
##  [985,]  0.8890504188
##  [986,]  0.7559606319
##  [987,]  1.1899166392
##  [988,]  0.7692286415
##  [989,]  1.9831670820
##  [990,]  0.1688320470
##  [991,]  0.5770212875
##  [992,]  0.6537947872
##  [993,]  0.0045088870
##  [994,]  4.2600817744
##  [995,]  2.3360372060
##  [996,]  1.9978138476
##  [997,]  0.5891620053
##  [998,]  2.0311077682
##  [999,]  0.1735987483
## 
## $model.matrix
##    (Intercept) groups1
## 1            1       1
## 2            1       1
## 3            1       1
## 4            1       1
## 5            1       1
## 6            1       1
## 7            1       1
## 8            1       1
## 9            1       1
## 10           1       1
## 11           1       1
## 12           1      -1
## 13           1      -1
## 14           1      -1
## 15           1      -1
## 16           1      -1
## 17           1      -1
## 18           1      -1
## 19           1      -1
## 20           1      -1
## 21           1      -1
## 
## $terms
## P_sojae_survey.matrix.jaccard ~ groups
## attr(,"variables")
## list(P_sojae_survey.matrix.jaccard, groups)
## attr(,"factors")
##                               groups
## P_sojae_survey.matrix.jaccard      0
## groups                             1
## attr(,"term.labels")
## [1] "groups"
## attr(,"order")
## [1] 1
## attr(,"intercept")
## [1] 1
## attr(,"response")
## [1] 1
## attr(,".Environment")
## <environment: R_GlobalEnv>
## 
## attr(,"class")
## [1] "adonis"

The PERMANOVA identified no significant differences between the groups centroids, or means (p = 0.175). In addition to identifying significance between group centroids, the PERMANOVA also calculates how much of the variance can be explained by the specified groups (see the \(R^2\) column in the PERMANOVA output). In this case, the \(R^2\) is 0.175, so 17.5% of the variance is explained by the groups used in analysis. Based on the PERMANOVA results we can conclude that these two groups are not different from each other and likely have similar pathotypes to each other.

Analysis of Similarity (ANOSIM)

ANOSIM statistic (R) ranges from between -1 and 1. Positive numbers suggest that there is more similarity within groups than there is between groups. Values close to zero indicate no difference between groups (i.e., similarities are the same between groups).

pathotype.anosim <- anosim(P_sojae_survey.matrix.jaccard, groups)

pathotype.anosim

## 
## Call:
## anosim(x = P_sojae_survey.matrix.jaccard, grouping = groups) 
## Dissimilarity: jaccard 
## 
## ANOSIM statistic R: 0.06882 
##       Significance: 0.136 
## 
## Permutation: free
## Number of permutations: 999

ANOSIM statistic (R) was 0.0688182, so there are more similarities between groups than there are within groups. This is evidence that the groups are not different from one another. Likewise the significance is >0.05 so there is no significant difference between groups’ similarities.