Indicator Value analysis with site group combinations
When the aim is to determine which species can be used as indicators
of certain site group an approach commonly used in ecology is the
Indicator Value (Dufrêne and Legendre
1997). These authors defined an Indicator Value (IndVal) index to
measure the association between a species and a site group. The method
of Dufrêne and Legendre (1997) calculates
the IndVal index between the species and each site group and then looks
for the group corresponding to the highest association value. Finally,
the statistical significance of this relationship is tested using a
permutation test. IndVal is the default index used to measure the
association between a species and a group of sites in
multipatt(). However, by default multipatt()
uses an extension of the original Indicator Value method, because the
function looks for indicator species of both individual site groups
and combinations of site groups, as explained in De Cáceres, Legendre, and Moretti (2010).
Indicator species analysis (with site group combinations) can be run
using:
indval <- multipatt(wetland, groups,
control = how(nperm=999))
As mentioned before, by default multipatt() uses the
IndVal index (func = "IndVal.g") as test statistic.
Actually, the square root of IndVal is returned by the
multipatt() function. The option
control = how(nperm=999) allows choosing the number of
random permutations required for the permutational test (this number
affects the precision of the p-value). Function how() from
the permute package allows defining more complex
permutational designs.
Displaying the results
When the indicator species analysis is completed, we can obtain the
list of indicator species for each site group (or site group
combination) using:
summary(indval)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 11
#> Number of species associated to 1 group: 7
#> Number of species associated to 2 groups: 4
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> stat p.value
#> Ludads 0.907 0.001 ***
#> Orysp. 0.823 0.002 **
#> Psespi 0.602 0.015 *
#>
#> Group 3 #sps. 4
#> stat p.value
#> Pancam 0.910 0.001 ***
#> Eupvac 0.724 0.004 **
#> Cynarc 0.602 0.007 **
#> Abemos 0.447 0.050 *
#>
#> Group 1+2 #sps. 1
#> stat p.value
#> Elesp. 0.741 0.005 **
#>
#> Group 2+3 #sps. 3
#> stat p.value
#> Melcor 0.876 0.001 ***
#> Phynod 0.715 0.010 **
#> Echell 0.651 0.020 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
In our wetland community data, ‘Ludads’ is strongly and significantly
associated with Group 1, whereas ‘Pancam’ would be a good indicator of
Group 3. In addition, there are some species whose patterns of abundance
are more associated with a combination of groups. For example, ‘Melcor’
is strongly associated with the combination of Groups 2 and 3.
It is important to stress that the indicator species analysis is
conducted for each species independently, although the results are often
summarized for all species. User should bear in mind possible problems
of multiple testing when making community-level statements such as
‘the number of indicator species is X’ (De Cáceres and Legendre 2009; Legendre and Legendre
2012).
Examining the indicator value components
If the association index used in multipatt() is
func = "IndVal" or func = "IndVal.g", one can
also inspect the indicator value components when displaying the results.
Indeed, the indicator value index is the product of two components,
called ‘A’ and ‘B’ (Dufrêne and Legendre 1997; De
Cáceres and Legendre 2009):
- Component ‘A’ is sample estimate of the probability that the
surveyed site belongs to the target site group given the fact that the
species has been found. This conditional probability is called the
specificity or positive predictive value of the
species as indicator of the site group.
- Component ‘B’ is sample estimate of the probability of finding the
species in sites belonging to the site group. This second conditional
probability is called the fidelity or sensitivity of
the species as indicator of the target site group.
To display the indicator value components ‘A’ and ‘B’ one simply
uses:
summary(indval, indvalcomp=TRUE)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 11
#> Number of species associated to 1 group: 7
#> Number of species associated to 2 groups: 4
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> A B stat p.value
#> Ludads 1.0000 0.8235 0.907 0.001 ***
#> Orysp. 0.6772 1.0000 0.823 0.002 **
#> Psespi 0.8811 0.4118 0.602 0.015 *
#>
#> Group 3 #sps. 4
#> A B stat p.value
#> Pancam 0.8278 1.0000 0.910 0.001 ***
#> Eupvac 0.6546 0.8000 0.724 0.004 **
#> Cynarc 0.7241 0.5000 0.602 0.007 **
#> Abemos 1.0000 0.2000 0.447 0.050 *
#>
#> Group 1+2 #sps. 1
#> A B stat p.value
#> Elesp. 1.0000 0.5484 0.741 0.005 **
#>
#> Group 2+3 #sps. 3
#> A B stat p.value
#> Melcor 0.8764 0.8750 0.876 0.001 ***
#> Phynod 0.8752 0.5833 0.715 0.010 **
#> Echell 0.9246 0.4583 0.651 0.020 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
This gives us additional information about why species can be used as
indicators. For example, ‘Ludads’ is a good indicator of Group 1 because
it occurs in sites belonging to this group only (i.e., A = 1.0000),
although not all sites belonging to Group 1 include the species (i.e., B
= 0.8235). In contrast, ‘Pancam’ can be used to indicate Group 3 because
it appears in all sites belonging to this group (i.e., B = 1.0000) and
it is largely (but not completely) restricted to it (i.e., A =
0.8278).
Inspecting the indicator species analysis results for all
species
In our previous calls to summary() only the species that
were significantly associated with site groups (or site group
combinations) were shown. One can display the result of the indicator
species analysis for all species, regardless of whether the
permutational test was significant or not. This is done by changing the
significance level in the summary:
summary(indval, alpha=1)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 1
#>
#> Total number of species: 33
#> Selected number of species: 29
#> Number of species associated to 1 group: 21
#> Number of species associated to 2 groups: 8
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 5
#> stat p.value
#> Ludads 0.907 0.001 ***
#> Orysp. 0.823 0.002 **
#> Psespi 0.602 0.015 *
#> Polatt 0.420 0.147
#> Casobt 0.243 1.000
#>
#> Group 2 #sps. 6
#> stat p.value
#> Aesind 0.445 0.245
#> Alyvag 0.335 0.449
#> Abefic 0.267 0.575
#> Poa2 0.267 0.583
#> Poa1 0.267 0.572
#> Helcri 0.267 0.572
#>
#> Group 3 #sps. 10
#> stat p.value
#> Pancam 0.910 0.001 ***
#> Eupvac 0.724 0.004 **
#> Cynarc 0.602 0.007 **
#> Abemos 0.447 0.050 *
#> Merhed 0.402 0.190
#> Ludoct 0.316 0.265
#> Passcr 0.316 0.254
#> Dendio 0.316 0.248
#> Physp. 0.316 0.227
#> Goopur 0.316 0.227
#>
#> Group 1+2 #sps. 2
#> stat p.value
#> Elesp. 0.741 0.005 **
#> Carhal 0.402 0.431
#>
#> Group 2+3 #sps. 6
#> stat p.value
#> Melcor 0.876 0.001 ***
#> Phynod 0.715 0.010 **
#> Echell 0.651 0.020 *
#> Echpas 0.584 0.262
#> Cyprot 0.500 0.068 .
#> Ipocop 0.354 0.324
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Parameter alpha is by default set to
alpha = 0.05, and hides all species association that are
not significant at this level. By setting alpha = 1 we say
we want to display the group to which each species is associated,
regardless of whether the association significant or not. However, note
that in our example we obtain the results of 29 (21+8) species. As there
are 33 species in the data set, there are still four species missing in
this summary. This happens because those species have their highest
IndVal value for the set of all sites. In other words, those species
occur in sites belonging to all groups. The association with the set of
all sites cannot be statistically tested, because there is no external
group for comparison. In order to know which species are those, one has
to inspect the object sign returned by
multipatt():
indval$sign
#> s.1 s.2 s.3 index stat p.value
#> Abefic 0 1 0 2 0.2672612 0.575
#> Merhed 0 0 1 3 0.4019185 0.190
#> Alyvag 0 1 0 2 0.3347953 0.449
#> Pancam 0 0 1 3 0.9098495 0.001
#> Abemos 0 0 1 3 0.4472136 0.050
#> Melcor 0 1 1 6 0.8757059 0.001
#> Ludoct 0 0 1 3 0.3162278 0.265
#> Eupvac 0 0 1 3 0.7236825 0.004
#> Echpas 0 1 1 6 0.5842649 0.262
#> Passcr 0 0 1 3 0.3162278 0.254
#> Poa2 0 1 0 2 0.2672612 0.583
#> Carhal 1 1 0 4 0.4016097 0.431
#> Dendio 0 0 1 3 0.3162278 0.248
#> Casobt 1 0 0 1 0.2425356 1.000
#> Aesind 0 1 0 2 0.4447093 0.245
#> Cyprot 0 1 1 6 0.5000000 0.068
#> Ipocop 0 1 1 6 0.3535534 0.324
#> Cynarc 0 0 1 3 0.6017217 0.007
#> Walind 1 1 1 7 0.4938648 NA
#> Sessp. 1 1 1 7 0.6984303 NA
#> Phynod 0 1 1 6 0.7145356 0.010
#> Echell 0 1 1 6 0.6509834 0.020
#> Helind 1 1 1 7 0.6984303 NA
#> Ipoaqu 1 1 1 7 0.4938648 NA
#> Orysp. 1 0 0 1 0.8229074 0.002
#> Elesp. 1 1 0 4 0.7405316 0.005
#> Psespi 1 0 0 1 0.6023402 0.015
#> Ludads 1 0 0 1 0.9074852 0.001
#> Polatt 1 0 0 1 0.4200840 0.147
#> Poa1 0 1 0 2 0.2672612 0.572
#> Helcri 0 1 0 2 0.2672612 0.572
#> Physp. 0 0 1 3 0.3162278 0.227
#> Goopur 0 0 1 3 0.3162278 0.227
After accessing the object indval$sign, we know that the
four species whose highest IndVal corresponded to the set of all sites
were ‘Valind’, ‘Sessp.’, ‘Helind’ and ‘Ipoaqu’, as indicated by the
missing values in the p.value column of the data frame. The
first columns of indicate (with ones and zeroes) which site groups were
included in the combination preferred by the species. Then, the column
indicates the index of the site group combination (see subsection
Excluding site group combinations in multipatt() below).
The remaining two columns are the association statistic and the p-value
of the permutational test.
Analyzing species ecological preferences with correlation
indices
Several other indices can be used to analyze the association between
a species and a group of sites (De Cáceres and
Legendre 2009). Diagnostic (or indicator) species are an
important tool in vegetation science, because these species can be used
to characterize and indicate specific plant community types. A statistic
commonly used to determine the association (also known as
fidelity, not to be confounded with the indicator value
component) between species and vegetation types is Pearson’s phi
coefficient of association (Chytrý et al.
2002). This coefficient is a measure of the correlation between
two binary vectors. It is possible to calculate the phi coefficient in
multipatt() after transforming our community data to
presence-absence:
wetlandpa <- ifelse(wetland>0,1,0)
phi <- multipatt(wetlandpa, groups, func = "r",
control = how(nperm=999))
What would be the association index if we had used abundance values
instead of presence and absences (i.e. wetland instead of
wetlandpa)? The abundance-based counterpart of the phi
coefficient is called the point biserial correlation
coefficient.
It is a good practice to correct the phi coefficient for the fact
that some groups have more sites than others (Tichý and Chytrý 2006). To do that, we need to
use func = "r.g" instead of func = "r":
phi <- multipatt(wetlandpa, groups, func = "r.g",
control = how(nperm=999))
Remember that the default association index of
multipatt() is func = "IndVal.g", which also
includes “.g”. In fact, the Indicator Value index defined by Dufrêne and Legendre (1997) already incorporated
a correction for unequal group sizes. It is possible to avoid this
correction by calling multipatt() with
func = "IndVal". However, in general we recommend using
either func = "IndVal.g" or func = "r.g" for
indicator species analysis.
Indicator value and correlation indices usually produce similar
results. Indeed, if we display the results of the phi coefficient of
association we see that they are qualitatively similar to those of
IndVal:
summary(phi)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: r.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 10
#> Number of species associated to 1 group: 8
#> Number of species associated to 2 groups: 2
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> stat p.value
#> Ludads 0.870 0.001 ***
#> Orysp. 0.668 0.001 ***
#> Psespi 0.413 0.014 *
#>
#> Group 2 #sps. 1
#> stat p.value
#> Phynod 0.436 0.011 *
#>
#> Group 3 #sps. 4
#> stat p.value
#> Pancam 0.748 0.001 ***
#> Eupvac 0.537 0.005 **
#> Cynarc 0.492 0.014 *
#> Abemos 0.378 0.048 *
#>
#> Group 1+2 #sps. 1
#> stat p.value
#> Elesp. 0.538 0.002 **
#>
#> Group 2+3 #sps. 1
#> stat p.value
#> Melcor 0.612 0.001 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Nevertheless, there are some differences between indicator values and
correlation indices (De Cáceres, Font, and Oliva
2008; De Cáceres and Legendre 2009). Correlation indices are used
for determining the ecological preferences of species among a set of
alternative site groups or site group combinations. Indicator value
indices are used for assessing the predictive values of species as
indicators of the conditions prevailing in site groups, e.g. for field
determination of community types or ecological monitoring.
An advantage of the phi and point biserial coefficients is that they
can take negative values. When this happens, the value of the index is
expressing the fact that a species tends to ‘avoid’ particular
environmental conditions. We will find negative association values if we
inspect the strength of association in the results of
multipatt() when these coefficients are used:
round(head(phi$str),3)
#> 1 2 3 1+2 1+3 2+3
#> Abefic -0.110 0.221 -0.110 0.110 -0.221 0.110
#> Merhed -0.223 -0.047 0.270 -0.270 0.047 0.223
#> Alyvag -0.024 0.214 -0.190 0.190 -0.214 0.024
#> Pancam -0.585 -0.163 0.748 -0.748 0.163 0.585
#> Abemos -0.189 -0.189 0.378 -0.378 0.189 0.189
#> Melcor -0.612 0.142 0.470 -0.470 -0.142 0.612
In contrast, indicator values are always non-negative:
round(head(indval$str),3)
#> 1 2 3 1+2 1+3 2+3 1+2+3
#> Abefic 0.000 0.267 0.000 0.180 0.000 0.204 0.156
#> Merhed 0.000 0.117 0.402 0.079 0.245 0.354 0.271
#> Alyvag 0.113 0.335 0.000 0.311 0.089 0.256 0.271
#> Pancam 0.038 0.230 0.910 0.183 0.589 0.781 0.625
#> Abemos 0.000 0.000 0.447 0.000 0.272 0.289 0.221
#> Melcor 0.191 0.509 0.739 0.484 0.610 0.876 0.796
The fact that correlation indices take negative values means that
they can be used to detect negative associations (i.e. site
groups or site group combinations that are statistically avoided by the
species). Function multipatt() can be used for this purpose
without modification, if all groups combinations are explored. The
reason is that strength of correlation for a given site group
combination has always the same absolute value than the correlation for
the complementary combination (you can verify this in the table above
displaying the strength of association results for phi).
This entails that the site group combination that maximizes positive
association is the complement of the site group combination that
minimizes (i.e. maximizes the negative value) negative association. For
example, in species ‘Pancam’ the maximum positive phi correlation is for
Group 3 and the minimum negative phi correlation is for the combination
of Group 1 and Group 2 (i.e. the complement of Group 3). Hence, one
could use the complement of site group combinations indicated by
multipatt() for positive associations as results for
negative associations. Following the same example, species ‘Pancam’
would be negatively associated to the combination of Group 1 and Group 2
with the same p-value as obtained for its positive association with
Group 3.
Alternatively, one can allow multipatt() to explicitly
look for both negative and positive associations by
setting allow.negative = TRUE:
phi <- multipatt(wetlandpa, groups, func = "r.g",
control = how(nperm=999),
allow.negative = TRUE)
In this case, multipatt() will keep the association with
highest strength in absolute value. Since ties can occur (in absolute
value) between a group and its complementary, as explained above, the
function will in this case keep the association corresponding to group
combination of the smallest order.
summary(phi)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: r.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 9
#> Number of species associated to 1 group: 9
#> Number of species associated to 2 groups: 0
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 4
#> stat p.value
#> Ludads 0.870 0.001 ***
#> Orysp. 0.668 0.001 ***
#> Melcor -0.612 0.001 ***
#> Psespi 0.413 0.013 *
#>
#> Group 2 #sps. 1
#> stat p.value
#> Phynod 0.436 0.015 *
#>
#> Group 3 #sps. 4
#> stat p.value
#> Pancam 0.748 0.001 ***
#> Elesp. -0.538 0.005 **
#> Eupvac 0.537 0.005 **
#> Cynarc 0.492 0.012 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Note that in this case, species ‘Melcor’ and ‘Elesp’ appear as
negatively associated to single groups, whereas in the previous call
(not allowing negative associations) they were reported as positively
associated to the complementary group combination.
Unlike with indicator value coefficients, the set of all sites can
never be considered with the phi or point biserial coefficients, because
these coefficients always require a set of sites for comparison, besides
the target site group or site group combination of interest.
Excluding site group combinations in multipatt()
There are two situations where site group combinations may be
excluded from the analysis:
- When conducting indicator species analysis, it may happen that some
combinations of site groups are difficult to interpret
ecologically. In those cases, we may decide to exclude those
combinations from the analysis, so our species may appear associated to
other (more interpretable) ecological conditions.
- The combinations to be explored will need to be restricted if
the initial number of groups is large (e.g. more than
15), because the number of possible combinations grows exponentially and
calculations will become computationally unfeasible if combinations are
not restricted.
There are three ways to restrict the site group combinations to be
considered in multipatt(), as detailed in the following
subsections.
Indicator species analysis without site groups combinations
The original Indicator Value method of Dufrêne
and Legendre (1997) did not consider combinations of site groups.
In other words, the only site group combinations permitted in the
original method were singletons. When using multipatt() it
is possible to avoid considering site group combinations, as in the
original method, by using duleg = TRUE:
indvalori <- multipatt(wetland, groups, duleg = TRUE,
control = how(nperm=999))
summary(indvalori)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 8
#> Number of species associated to 1 group: 8
#> Number of species associated to 2 groups: 0
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> stat p.value
#> Ludads 0.907 0.001 ***
#> Orysp. 0.823 0.001 ***
#> Psespi 0.602 0.014 *
#>
#> Group 2 #sps. 1
#> stat p.value
#> Phynod 0.676 0.007 **
#>
#> Group 3 #sps. 4
#> stat p.value
#> Pancam 0.910 0.001 ***
#> Melcor 0.739 0.001 ***
#> Eupvac 0.724 0.002 **
#> Cynarc 0.602 0.005 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Restricting the order of site groups combinations
The second way to exclude site group combinations from a
multipatt() analysis is to indicate the maximum order of
the combination to be considered. Using the option
max.order we can restrict site group combinations to be,
for example, singletons (max.order = 1, which is equal to
duleg=TRUE), singletons and pairs
(max.order = 2), or singletons, pairs and triplets
(max.order = 3). In the follow example, only singletons and
pairs are considered:
indvalrest <- multipatt(wetland, groups, max.order = 2,
control = how(nperm=999))
summary(indvalrest)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 10
#> Number of species associated to 1 group: 6
#> Number of species associated to 2 groups: 4
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> stat p.value
#> Ludads 0.907 0.001 ***
#> Orysp. 0.823 0.004 **
#> Psespi 0.602 0.012 *
#>
#> Group 3 #sps. 3
#> stat p.value
#> Pancam 0.910 0.001 ***
#> Eupvac 0.724 0.003 **
#> Cynarc 0.602 0.012 *
#>
#> Group 1+2 #sps. 1
#> stat p.value
#> Elesp. 0.741 0.006 **
#>
#> Group 2+3 #sps. 3
#> stat p.value
#> Melcor 0.876 0.001 ***
#> Phynod 0.715 0.011 *
#> Echell 0.651 0.017 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
In this case the output looks like a the output of an unrestricted
multipatt() execution, because the only combination that is
excluded is the set of all sites, which cannot be tested for
significance and thus never appears in the summary.
Limiting the maximum order of combinations is the commonest way to
restrict combinations to avoid computational problems derived from a
large number of site groups.
Specifying the site groups combinations to be considered
There is a third, more flexible, way of restricting site group
combinations. The input parameter vector restcomb allows
specifying the combinations of site groups that are permitted in
multipatt(). In order to learn how to use parameter
restcomb, we must first understand that inside
multipatt() site groups and site group combinations are
referred to with integers. Site group combinations are numbered starting
with single groups and then increasing the order of combinations. For
example, if there are three site groups, the first three integers ‘1’ to
‘3’ identify those groups. Then, ‘4’ identifies the combination of Group
1 and Group 2, ‘5’ identifies the combination of Group 1 and Group 3,
and ‘6’ identifies the combination of Group 2 and Group 3. Finally, ‘7’
identifies the combination of all three groups.
The numbers composing the vector passed to restcomb
indicate the site groups and site group combinations that we want
multipatt() to considered as valid options. For example, if
we do not want to consider the combination of Group 1 and Group 2, we
will exclude combination ‘4’ from vector restcomb:
indvalrest <- multipatt(wetland, groups, restcomb = c(1,2,3,5,6),
control = how(nperm=999))
summary(indvalrest)
#>
#> Multilevel pattern analysis
#> ---------------------------
#>
#> Association function: IndVal.g
#> Significance level (alpha): 0.05
#>
#> Total number of species: 33
#> Selected number of species: 9
#> Number of species associated to 1 group: 6
#> Number of species associated to 2 groups: 3
#>
#> List of species associated to each combination:
#>
#> Group 1 #sps. 3
#> stat p.value
#> Ludads 0.907 0.001 ***
#> Orysp. 0.823 0.001 ***
#> Psespi 0.602 0.023 *
#>
#> Group 3 #sps. 3
#> stat p.value
#> Pancam 0.910 0.001 ***
#> Eupvac 0.724 0.004 **
#> Cynarc 0.602 0.006 **
#>
#> Group 2+3 #sps. 3
#> stat p.value
#> Melcor 0.876 0.001 ***
#> Phynod 0.715 0.003 **
#> Echell 0.651 0.014 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
If we compare these last results with those including all possible
site group combinations, we will realize that species ‘Elesp.’ was
formerly an indicator of Group 1 and Group 2, and now it does not appear
in the list of indicator species. If fact, if we examine the results
more closely we see that the highest IndVal for ‘Elesp’ is achieved for
group 1, but this relationship is not significant:
indvalrest$sign
#> s.1 s.2 s.3 index stat p.value
#> Abefic 0 1 0 2 0.2672612 0.593
#> Merhed 0 0 1 3 0.4019185 0.187
#> Alyvag 0 1 0 2 0.3347953 0.413
#> Pancam 0 0 1 3 0.9098495 0.001
#> Abemos 0 0 1 3 0.4472136 0.062
#> Melcor 0 1 1 5 0.8757059 0.001
#> Ludoct 0 0 1 3 0.3162278 0.250
#> Eupvac 0 0 1 3 0.7236825 0.004
#> Echpas 0 1 1 5 0.5842649 0.191
#> Passcr 0 0 1 3 0.3162278 0.235
#> Poa2 0 1 0 2 0.2672612 0.571
#> Carhal 1 0 0 1 0.3313667 0.713
#> Dendio 0 0 1 3 0.3162278 0.247
#> Casobt 1 0 0 1 0.2425356 1.000
#> Aesind 0 1 0 2 0.4447093 0.235
#> Cyprot 0 1 1 5 0.5000000 0.075
#> Ipocop 0 1 1 5 0.3535534 0.347
#> Cynarc 0 0 1 3 0.6017217 0.006
#> Walind 1 0 1 4 0.4406873 0.708
#> Sessp. 0 1 1 5 0.5901665 0.728
#> Phynod 0 1 1 5 0.7145356 0.003
#> Echell 0 1 1 5 0.6509834 0.014
#> Helind 0 1 1 5 0.5720540 0.831
#> Ipoaqu 1 0 1 4 0.4053049 0.900
#> Orysp. 1 0 0 1 0.8229074 0.001
#> Elesp. 1 0 0 1 0.5534178 0.673
#> Psespi 1 0 0 1 0.6023402 0.023
#> Ludads 1 0 0 1 0.9074852 0.001
#> Polatt 1 0 0 1 0.4200840 0.155
#> Poa1 0 1 0 2 0.2672612 0.579
#> Helcri 0 1 0 2 0.2672612 0.579
#> Physp. 0 0 1 3 0.3162278 0.237
#> Goopur 0 0 1 3 0.3162278 0.237
Restricting site group combinations is also possible with the phi and
point biserial coefficients.
