IDmeasurer workflow examples

ANOVA F-value (F)

First quantification of individual identity in animal signals was done by using F-values from one-way ANOVA with individual identity as an independent variable and a trait as a dependent variable (F).

Differences between individuals are large and significant in the id3 dataset:

calcF(id0)
#>     traits   f pvals
#> 1 paramvec 0.5 0.873
calcF(id3)
#>     traits      f pvals
#> 1 paramvec 112.66     0

However, F-values reflect sample size of a dataset (number of observations for each individual) and, therefore, are not ideal for comparisons between studies with different number of observations per individual. Studies with many observations per individual will be more likely to report high individuality when using F-value as an individuality metric.

fewobs <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=0.5)
manyobs <- GenerateUnivariate(nindivs=10, nobs=50, betweenM=1000, individuality=0.5)

calcF(fewobs)
#>     traits    f pvals
#> 1 paramvec 1.44 0.184
calcF(manyobs)
#>     traits     f pvals
#> 1 paramvec 13.01     0

The dataset with many observations per individual manyobs has much larger F than the dataset with few observations per individual ‘fewobs’ despite the differences between individuals in both datasets are similar (Individuality = 0.5)

Because of this problem, alternative metrics were introduced.

Back to Contents

Potential of individual coding (PIC)

Potential of individual coding (PIC) is based on the coefficent of variation which itself is sample independent. PIC is the ratio of between and within individual coefficents of variation and it is robust to number of observarions per individual (as well as to number of individuals in dataset).

calcPIC(fewobs)
#> paramvec 
#> 1.047665
calcPIC(manyobs)
#> paramvec 
#> 1.110724

Using PIC is fine until we want to quantify individuality in a trait which has negative and positive values because in that case coefficients of variations do not make sense.

Note that individuality in the data remains the same and we only change the mean of the trait to be 0 (and thus negative and positive values are generated around this mean). However, PIC value is strongly affected.

postrait <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=0.5)
posnegtrait <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=0, individuality=0.5)

calcPIC(postrait)
#> paramvec 
#> 1.198084
calcPIC(posnegtrait)
#> paramvec 
#> 12.51082

Variants of PIC. In the literature, two variants of PIC were used depending on the way how between individual coefficient variation is calculated. Both ways can be called with functions calcPICbetweentot and calcPICbetweenmeans. The function calcPIC is equivalent to the calcPICbetweentot because values of calcPICbetweentot converge towards 1 with individuality approaching 0, and, hence, reflect the number of unique individual identity signatures in simalr way as 2^HS does.

calcPIC(id0)
#> paramvec 
#> 1.007073
calcPICbetweentot(id0)
#> paramvec 
#> 1.007073
calcPICbetweenmeans(id0)
#>  paramvec 
#> 0.2297909

Back to Contents

Beecher’s information statistic (HS)

Beecher’s information statistic (HS) uses ANOVA F-value but makes it sampling independent. HS is robust to number of observarions per individual as well as to number of individuals in dataset.

calcHS(fewobs)[2]
#> HS for all vars 
#>            0.03
calcHS(manyobs)[2]
#> HS for all vars 
#>            0.16

It can also handle traits with negative and positive values.

calcHS(postrait)[2]
#> HS for all vars 
#>            0.26
calcHS(posnegtrait)[2]
#> HS for all vars 
#>            0.13

This is very important advantage because it allows transforming trait variables. For example, it is possible to use Principal Component Analysis (PCA) on the raw traits and calculate HS of of the uncorrelated Principal components and sum them together to get the HS for an entire signal. Note that summing or averaging of PIC values is not a valid approach because the traits can be (and often are) correlated and, therefore, the resulting PIC values are overestimated.

Both HS and PIC are looking for the ratio of between and within individual variation and their values are closely related, but to see this, we have to convert HS into the number of potentially unique signatures. HS is expressed in bits of information and to convert it into number of signatures we have to use 2 as the base and and HS as an exponent. In that case, HS and PIC correlate almost perfectly, eventhough they are not numerically entirely equivalent.

id1 <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=1)
id2 <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=2)
id3 <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=3)
id4 <- GenerateUnivariate(nindivs=10, nobs=10, betweenM=1000, individuality=4)

PICs <- c(calcPIC(id1), calcPIC(id2), calcPIC(id3), calcPIC(id4))
HSs <- c(calcHS(id1)[2], calcHS(id2)[2], calcHS(id3)[2], calcHS(id4)[2])

plot((2^HSs) ~ PICs, xlim=c(0,5), ylim=c(0,5), pch=16)
abline(lm((2^HSs) ~ PICs))

Large overlap of trait values suggests low individuality

calcPIC(id4)
#> paramvec 
#>  3.82393
2^calcHS(id4)[2]
#> HS for all vars 
#>        3.837056

Variants of HS. In the literature, HS has been calculated by several different approaches. These approaches are implemented in functions: calcHSntot, calcHSngroups, calcHSnpergroup, and calcHSvarcomp. calcHS is equivalent to calcHSnpergroup because it matches the assumptions stated in the original article by Beecher (1989) (see ?calcHS for the reference). Unlike the other variants, calcHSvarcomp calculates HS by ucing variance components from mixed models. This approach allows to control for possible confounding factors and variables in the model, however, we found the values to be twice as large as the values of HS calculated with calcHS.

calcHS(id3)
#> HS for significant vars         HS for all vars 
#>                    1.77                    1.77
calcHSnpergroup(id3)
#> HS for significant vars         HS for all vars 
#>                    1.77                    1.77
calcHSntot(id3)
#> HS for significant vars         HS for all vars 
#>                    0.52                    0.52
calcHSngroups(id3)
#> HS for significant vars         HS for all vars 
#>                    1.77                    1.77
calcHSvarcomp(id3)
#> HS for all vars 
#>            3.54

Back to Contents

Univariate metrics with multivariate datasets

All the three functions can be used with multivariate datasets containing the identity codes in a first column and multiple traits in subsequent columns. In that case, metrics are calculated for each trait variable. For calculation of HS for separate variables the parameter sumHS must be set to FALSE.

calcF(CCformants)
#>   traits       f pvals
#> 1   s1f2  964.26     0
#> 2   s1f3 1153.80     0
#> 3   s1f4 1089.05     0
#> 4   s1f5  818.22     0
calcPIC(CCformants)
#>     s1f2     s1f3     s1f4     s1f5 
#> 10.81532 13.25913 13.89994 12.99410
calcHS(CCformants, sumHS=F)
#>   vars Pr   HS
#> 2 s1f2  0 3.30
#> 3 s1f3  0 3.43
#> 4 s1f4  0 3.39
#> 5 s1f5  0 3.19

Back to Contents

Discrimination score (DS)

Most of the studies subject similar data to the discriminant function analysis and calculate discrimination score - the proportion or percentage of the samples that were classified to the correct individual.

calcDS(id0)
#> [1] 0.16
calcDS(id5)
#> [1] 0.98

In case of id0, individuality is very low and it is hard to discriminate individuals. Hence, the value of DS should be close to value expected by chance classification (0.2 for 5 individuals). On the other hand, discrimination is easy and DS close to maximum value of 1 in case of id5.

However, if with many individuals in the sample, it is more likely to find very similar individuals that are easy to confuse. Hence, discrimination score will be lower for large datasets. This is shown here on two datasets with the same individuality (individuality = 5) but few (nindivs = 5) and many (nindivs = 100) individuals.

id5fewinds <- GenerateMultivariate(nindivs=5, nobs=10, nvar=2, covar=0.9, individuality=5)
id5manyinds <- GenerateMultivariate(nindivs=100, nobs=10, nvar=2, covar=0.9, individuality=5)
calcDS(id5fewinds)
#> [1] 0.76
calcDS(id5manyinds)
#> [1] 0.254

We might try to adjust DS with regard to chance value to see how many times DS exceeds classification by chance. However, again, we do not get similar DS values for both datasets as would be expected based on the fact they are both genereated with the same individuality parameter. In this case, the bias was only reversed to favour large datasets.

calcDS(id5fewinds)/0.2 # expected classification by chance for 5 individuals: 1 / 5 = 0.2
#> [1] 3.8
calcDS(id5manyinds)/0.01 # expected classification by chance for 5 individuals: 1 / 100 = 0.01
#> [1] 25.4

Tranforming the raw trait values into uncorrelated principal components with Principal Component Analysis has little impact on the DS values (HS, which is discussed later, requires PCA so this is shown here for comparison):

calcDS(calcPCA(id5fewinds))/0.2 # expected classification by chance for 5 individuals: 1 / 5 = 0.2
#> [1] 3.8
calcDS(calcPCA(id5manyinds))/0.01 # expected classification by chance for 5 individuals: 1 / 100 = 0.01
#> [1] 25.4

DS is also biased in respect to number of individuals in the sample.

id5fewobs <- GenerateMultivariate(nindivs=10, nobs=5, nvar=2, covar=0.9, individuality=1)
id5manyobs <- GenerateMultivariate(nindivs=10, nobs=100, nvar=2, covar=0.9, individuality=1)
calcDS(id5fewobs)
#> [1] 0.14
calcDS(id5manyobs)
#> [1] 0.275

Like in case of F-value in univariate case, if the values are biased it makes it difficult to compare the results of different studies with various sampling of individuals and observations per individual. Therefore, alternative metrics were suggested.

Back to Contents

Mutual information (MI)

Mutual information has been suggested as unbiased metric to complement DS. However, calcMI is subjected to the same kind of biases as DS, though these biases are reversed (like in case of DS adjusted for chance). For example, in case of number of individuals in dataset, DS will be higher for datasets with few individuals while MI will be higher for datasets with many individuals.

id5fewinds <- GenerateMultivariate(nindivs=5, nobs=10, nvar=2, covar=0.9, individuality=2)
id5manyinds <- GenerateMultivariate(nindivs=100, nobs=10, nvar=2, covar=0.9, individuality=2)
calcDS(id5fewinds)
#> [1] 0.82
calcDS(id5manyinds)
#> [1] 0.102
calcMI(id5fewinds)
#> [1] 1.566521
calcMI(id5manyinds)
#> [1] 3.50004

Back to Contents

Beechers information statistic (HS)

HS can be used for univariate as well as for multivariate signals. It calculates and sums partial HS values of each variable. The result prints HS summed over all trait variables in dataset or over traits that are significantly different between individuals only. Since non-significant variables contribute only a little to final value of HS we suggest to report HS for all variables.

calcHS(calcPCA(id5), sumHS=T)
#> HS for significant vars         HS for all vars 
#>                    4.24                    4.24

Note, that HS for multivariate signal needs to be calculated on uncorrelated variables. Here, we use the function calcPCA which uses Principal Component Analysis PCA to convert the raw trait variables into uncorrelated Principal components in the dataset. Including correlated variables overestimates the value of HS:

calcHS(id5, sumHS=T) # HS is overestimated if trait variables are correlated
#> HS for significant vars         HS for all vars 
#>                     4.8                     4.8

Also note, that for good estimate of HS values the number of trait variables should not exceed the number of individuals. For discussion of this issue, see the research article introducing this package by Linhart et al. (2019).

Back to Contents

Information capacity (HM)

HM has been suggested as variant of HS for particular type of signals where the variable traits are independent of each other. However, HS can be used for such signals as well.

id5 <- GenerateMultivariate(nindivs=10, nobs=10, nvar=2, covar=0, individuality=5)
calcHS(id5)
#> HS for significant vars         HS for all vars 
#>                    4.58                    4.58

HM is an adaptation of HS and is also expressed in bits (bits of individual identity information) but the value is surprisingly different.

calcHM(id5)
#> [1] 2.269866

It turns out that for cases in which trait variables are independent, HM calculates individual identity information per variable. In our case, we have two trait variables in signal and hance we need to multiply HM by 2 to get identity information in the entire signal. Now, the value is almost identical to HS calculated above.

calcHM(id5)*2
#> [1] 4.539732

We can try to generate datasets with different individualities to see how the HS and HM values match each other. We will also change number of variables to 5.

id1 <- GenerateMultivariate(nindivs=10, nobs=10, nvar=5, covar=0, individuality=1)
id3 <- GenerateMultivariate(nindivs=10, nobs=10, nvar=5, covar=0, individuality=3)
id5 <- GenerateMultivariate(nindivs=10, nobs=10, nvar=5, covar=0, individuality=5)
x <- c(calcHS(id1)[2],calcHS(id3)[2],calcHS(id5)[2])
y <- c(calcHM(id1)*5, calcHM(id3)*5, calcHM(id5)*5) # HM is multiplied by 5 because we have 5 trait variables.
plot(x,y)
abline(lm(y~x))

Problem arises, if the trait variables are correlated.

id5covar0 <- GenerateMultivariate(nindivs=50, nobs=10, nvar=5, covar=0, individuality=5) # uncorrelated traits
id5covar1 <- GenerateMultivariate(nindivs=50, nobs=10, nvar=5, covar=1, individuality=5) # perfectly correlated traits

In such cases, HS correctly reflects lower real dimensionality of the data and hence, lower total individual identity of a signal.

calcHS(calcPCA(id5covar0))
#> HS for significant vars         HS for all vars 
#>                   11.72                   11.72
calcHS(calcPCA(id5covar1))
#> HS for significant vars         HS for all vars 
#>                    3.54                    3.51

Lower dimensionality is also reflected in higher overlap of values between individuals and, hence, in lower discrimination of individuals.

calcDS(calcPCA(id5covar0))
#> [1] 0.986
calcDS(calcPCA(id5covar1))
#> [1] 0.274

Contrary, HM remains nearly unchanged and we would need to do separate analysis of dimensionality of data to estimate identity information of the entire signal. Simple multiplying HM by number of variables would overestimate individual identity information in case that traits are correlated.

calcHM(calcPCA(id5covar0))*5 # HM is multiplied by the number of variables to get individuality of entire signal
#> [1] 11.82591
calcHM(calcPCA(id5covar1))*5 # In this case, total individuality of entire signal is overestimated
#> [1] 11.62554

Back to Contents