understanding of the applications proposed below. Note that we will use the expressions “measure”, “index” and “coefficient” as synonyms to refer to the quantities
used to compare pairs of objects or variables
3 .
3.2.1 Q Mode and R Mode
When pairs of objects are compared, the analysis is said to be in the Q mode. When
pairs of descriptors are compared the analysis is said to be in the R mode. This
distinction is important because the association measures in Q and R-mode analyses
are not the same.
In Q mode, the association measure is the dissimilarity or the similarity between
pairs of objects, e.g., Euclidean distance, Jaccard similarity. In R mode, one uses a
measure of dependence among variables, such as the covariance or correlation
coefficient.
3.2.2 Symmetrical or Asymmetrical Coefficients in Q Mode:
The Double-Zero Problem
Virtually all dissimilarity or similarity measures used in ecology are symmetric in
one sense: the value of the coefficient between objects n 1 and n 2 is the same as the
value of the coefficient between objects n 2 and n 1 . The same holds for dependence
measures in the R mode. The problem addressed here is different. It concerns the
treatment of double-zeros in the comparison of pairs of objects.
In certain cases, the zero value has the same meaning as any other value along the
scale of a descriptor. For instance, the absence (0 mg/L) of dissolved oxygen in the
deep layers of a lake is an ecologically meaningful information: the concentration is
below the measurement threshold and this poses severe constraints on aerobic life
forms, whatever the reason for this condition.
On the contrary, the zero value in a matrix of species abundances (or presenceabsence) is much more tricky to interpret. The presence of a species at two sites
generally implies that these sites provide a similar set of minimal conditions
allowing the species to survive; these conditions are the dimensions of the species’
ecological niche. The absence of a species from a relevé or site, however, may be due
to a variety of causes: the species’ niche may be occupied by a replacement species,
or the species may not have reached that site even though the conditions are
3 Although the term “coefficient” is sometimes narrowly defined as a multiplicative factor of a
variable in a mathematical expression, it has been applied for decades in multivariate data analysis
to the broader sense used in this book.
36
3 Association Measures and Matrices
used to compare pairs of objects or variables
3 .
3.2.1 Q Mode and R Mode
When pairs of objects are compared, the analysis is said to be in the Q mode. When
pairs of descriptors are compared the analysis is said to be in the R mode. This
distinction is important because the association measures in Q and R-mode analyses
are not the same.
In Q mode, the association measure is the dissimilarity or the similarity between
pairs of objects, e.g., Euclidean distance, Jaccard similarity. In R mode, one uses a
measure of dependence among variables, such as the covariance or correlation
coefficient.
3.2.2 Symmetrical or Asymmetrical Coefficients in Q Mode:
The Double-Zero Problem
Virtually all dissimilarity or similarity measures used in ecology are symmetric in
one sense: the value of the coefficient between objects n 1 and n 2 is the same as the
value of the coefficient between objects n 2 and n 1 . The same holds for dependence
measures in the R mode. The problem addressed here is different. It concerns the
treatment of double-zeros in the comparison of pairs of objects.
In certain cases, the zero value has the same meaning as any other value along the
scale of a descriptor. For instance, the absence (0 mg/L) of dissolved oxygen in the
deep layers of a lake is an ecologically meaningful information: the concentration is
below the measurement threshold and this poses severe constraints on aerobic life
forms, whatever the reason for this condition.
On the contrary, the zero value in a matrix of species abundances (or presenceabsence) is much more tricky to interpret. The presence of a species at two sites
generally implies that these sites provide a similar set of minimal conditions
allowing the species to survive; these conditions are the dimensions of the species’
ecological niche. The absence of a species from a relevé or site, however, may be due
to a variety of causes: the species’ niche may be occupied by a replacement species,
or the species may not have reached that site even though the conditions are
3 Although the term “coefficient” is sometimes narrowly defined as a multiplicative factor of a
variable in a mathematical expression, it has been applied for decades in multivariate data analysis
to the broader sense used in this book.
36
3 Association Measures and Matrices