1greenacre_m_primicerio_r_multivariate_analysis_of_ecological

that is, the squared length of x is the sum of its three squared coordinates, hence the length is

dx = x12 + x22 + x32

It is also clear that placing a point Q in Exhibit 4.3 to depict another vector y and going through the motions to calculate the distance between x and y will lead to

Furthermore, we can carry on like this into four or more dimensions, in general

J dimensions, where J is the number of variables. Although we cannot draw the geometry any more, we can express the distance between two J -dimensional vectors x and y as:

j = 1

This well-known distance measure, which generalizes our notion of physical distance in twoor three-dimensional space to multidimensional space, is called the

Euclidean distance.

50

ds29,s30 = (51 − 99)2 + (6.0 − 1.9)2 + (3.0 − 2.9)2

= 2304 + 16.81 + 0.01 = 2320.82 = 48.17

The contribution of the first variable depth to this calculation is huge – one could say that the distance is practically just the absolute difference in the depth values (equal to 51-99 48) with only tiny additional contributions from pollution and temperature. This is the problem of standardization discussed in Chapter 3 – the three variables have completely different units of measurement and the larger depth values have larger inter-sample differences, so they will dominate in the calculation of Euclidean distances.

Some form of transformation of the data is necessary to balance out the contributions, and the conventional way to do this is to make all variables have the same variance of 1. At the same time we centre the variables at their means – this centring is not necessary for calculating distance, but it makes the variables all have mean zero and thus easier to compare. This transformation, commonly called standardization, is thus as follows:

The means and standard deviations (sd) of the three variables are:

leading to the table of standardized values given in Exhibit 4.4. These values are now on comparable standardized scales, in units of standard deviation with respect to the mean. For example, the standardized pollution value 0.693 for row s29 would signify 0.693 standard deviations above the mean, while 1.222 for row s30 would signify 1.222 standard deviations below the mean. The distance calculation thus aggregates squared differences in standard deviation units of each variable. As an example, the distance between the last two sites of the table in Exhibit 4.4 is:

51

For this particular pair of sites the difference in temperatures is still small but pollution now has a higher contribution than before. Depth still plays the largest role in this particular example, even after standardization, but this contribution is

52

53

MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA

where sj is the sample standard deviation of the j -th variable. Notice that we need not subtract the j -th mean from xj and yj because the means will just cancel out in the differencing. Now (4.7) can be rewritten in the following equivalent way:

where wj 1/s2j is the inverse of the j -th variance. We can think of wj as a weight attached to the j -th variable: in other words, we compute the usual squared differences between the variables on their original scales, as we did in the (unstandardized) Euclidean distance, but then multiply these squared differences by their corresponding weights. Notice in this case how the weight of a variable with high variance is low, while the weight of a variable with low variance is high, which is another way of thinking about the compensatory effect produced by standardization. The weights of the three variables in our example are (to 4 significant figures) 0.004101, 0.2181 and 12.64 respectively, showing how much the depth variable is downweighted and the temperature variable upweighted: depth has over 3000 times the variance of temperature, so each squared difference in (4.8) is downweighted relatively by that much. We call (4.8) weighted Euclidean distance.

Distances for count data So far we have looked at the distances between samples based on continuous data, now we consider distances on count data, for example the abundance data for the five species labelled a, b, c, d and e in Exhibit 1.1. First, notice that these five variables apparently do not have the problem of different measurement units that we had for the continuous environmental variables – all variables are counts. There are, however, different average frequencies of counts, and as we mentioned in Chapter 3, variances of count variables can be positively related to their means. The means and variances of these five variables are as follows:

Variable a with the highest mean also has the highest variance, while e with the lowest mean has the lowest variance. Only d is out of line with the others, having smaller variance than b and c but a higher mean. Because this variance mean relationship is a natural phenomenon for count variables, not one that is just par-

54

MEASURES OF DISTANCE BETWEEN SAMPLES: EUCLIDEAN

ticular to any given example, some form of compensation of the variances needs to be performed, as before. It is not common for count data to be standardized as Z-scores (i.e., with mean 0, variance 1), as was the case for continuous variables in (4.6). The most common ways of balancing the contributions of count variables to the distance measure are:

a power transformation: usually square root n1/2, where n is the count value, but also double square root (i.e., fourth root n1/4) when the variance increases faster than the mean (this situation is called overdispersion in the literature);

a Òshifted logÓ transformation: because of the many zeros in ecological count data,

a positive number, usually 1, has to be added to the data before log-transforming; that is, log(1 n);

chi-square distance: this is a weighted Euclidean distance of the form (4.8), which we shall discuss now.

The chi-square distance is special because it is at the heart of correspond- Chi-square distance ence analysis, used extensively in ecological research. The first premise of this

distance function is that it is calculated on relative counts,1 and not on the original ones, and the second is that it standardizes by the mean and not by the variance.

In our example, the count data are first converted into relative counts by dividing the rows by their row totals so that each row contains relative proportions across the species, which add up to 1. These sets of proportions are called proÞles, site profiles in this example – see Exhibit 4.6.

The extra row at the end of Exhibit 4.6 gives the set of proportions called the average proÞle. These are the proportions calculated on the set of column totals, which are equal to 404, 262, 252, 327 and 89 respectively, with grand total 1334. Hence, 404/1334 0.303, 262/1334 0.196, etc. Chi-square distances are then calculated between the profiles, in a weighted Euclidean fashion, using the inverse of the average proportions as weights. Suppose cj denotes the j -th element of the average profile, that is the abundance proportion of the j -th species in

the whole data set. Then the chi-square2 distance, denoted by , between two sites with profiles x x1 x2 ··· xJ and y y1 y2 ··· yJ is defined as:

1A definition of chi-square distance on raw counts is referred to in the bibliographical appendix.

2From the definition of this distance function it would have been better to call it the chi distance function, because it is not squared, as in the chi-square statistic! But the “chi-square” epithet persists in the literature, so when we talk of its square we say the “squared chi-square distance”.

55

Exhibit 4.7 shows part of the 30 30 triangular matrix of chi-square distances. Once again, this is a large matrix with more numbers (435) than the original table of counts (150), and we shall see the benefit of calculating these distances

56

57

MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA

Then the number of matches is 2 and the number of mismatches is 3, hence the distance between the two samples is 3 divided by 5 (the number of variables), that is 0.6. This is called the simple matching coefÞcient. Sometimes this coefficient is expressed in terms of similarity, not dissimilarity, in which case the similarity would be equal to 0.4, the relative number of matches – so one should check which way it is being defined. Here we stick to distances, in other words dissimilarities or mismatches. Note that this coefficient is directly proportional to the squared Euclidean distance calculated between these data in dummy variable form, where each category defines a zero-one variable:

The squared Euclidean distance sums the squared differences between these two vectors: if there is an agreement (there are two matches in this example) there is zero sum of squared differences, but if there is a discrepancy there are two differences, 1 and 1, which give a sum of squares of 2. So the sum of squared differences here is 6, and if this is expressed relative to the maximum discrepancy that can be achieved, namely 10 when there are no matches in the 5 variables, then this gives exactly the same value 0.6 as before.

There are several variations on the theme of the matching coefficient, and one of them is the chi-square distance for multivariate categorical data, which introduces a weighting of each category inverse to its mean value, as for profile data based on counts. Suppose that there are J categories in total (in the above example

J 15) and that the total occurrences of each category are denoted by n1,…,nJ , with total n j nj (since the totals for each variable equal the sample size, n will be the sample size times the number of variables). Then define cj as follows: cj nj /n and use 1/cj as weights in a weighted Euclidean distance between the samples coded in dummy variable form. The idea here is, as before, that mismatches on a rare category should have a higher weight in the distance calculation than that of a frequent category. Just like the chi-square distance function is at the heart of correspondence analysis of abundance data, so this form of the chi-square for multivariate categorical data is at the heart of multiple correspondence analysis. We do not treat multiple correspondence analysis specifically in this book,

58

59