1greenacre_m_primicerio_r_multivariate_analysis_of_ecological

Adding count variables to MDS maps

Performing nonmetric MDS on the same data gives a stress value of 13.5%, which is not a big improvement on the 16.3%, suggesting that the two resulting maps will not be as different as we found for the smaller data set of Jaccard indices. This is indeed the case, as shown by the quite similar maps in Exhibit 9.9.

In our experience, when there is a large number of samples (and by “large” we mean, as most statisticians do, 30 or more, as in this example), the metric and nonmetric approaches generally agree in their solutions. Where they disagree is in the quantification of the success of their results, with the stress measure always giving a more optimistic value because it does not measure the recovery of the proximities themselves, but their ordering in the map.

The maps in Exhibit 9.9 emanate originally from abundance data on five species, so the question now is how to include these species on the map. We shall consider alternative ways of doing this in future chapters, but for the moment let us use the same approach as in Exhibit 9.7 when the species were positioned at the averages of the samples that contained them. The difference here is that we have abundance counts for the species across the samples, so what we can do is to position each species at their weighted average across the samples. For example, species a has abundances of 0, 26, 0, 0, 13, etc., and a total abundance of 0 26 0 0

13 ... 404, so the position of a is at a weighted average position of the 30 species, with weights 26/404 0.064 on sample s2, 13/404 0.032 on sample s5, and

Dimension 1

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s17

Exhibit 9.10:

Nonmetric MDS solution (right hand map in Exhibit 9.9) with species a to e added by weighted

averaging of sample points, and sediment types C, S and G by averaging

60

0

–20

MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA

s8

s17

SUMMARY: Multidimensional scaling

so on. Exhibit 9.10 shows the species positions on the nonmetric MDS solution, showing, for example, that species a and b are relatively more abundant in the samples at lower left, while c is more associated with samples on the right. Similarly, even though the ordinal sediment types C (clay), S (sand) and G (gravel) have not been used in the mapping, they can be depicted at the averages of the subsets of samples corresponding to them. The samples thus appear to follow a trend from top right (more clay) to bottom left (more gravel).

1.Multidimensional scaling (MDS) is a method that attempts to make a spatial map of a matrix of proximities, either distances or dissimilarities defined between sample units, so that the interpoint distances in the map come as close as possible to the given proximities according to the chosen fit criterion.

2.The fit criterion in metric MDS involves approximating the actual proximity values by the mapped distances, for example by least-squares.

The multiple linear regression model postulates that the expected value of a response variable Y (i.e., the mean of Y) is a linear combination of several explanatory variables x1, x2, …, xp:

Algebra of multiple linear regression

Geometry of multiple linear regression

MULTIVARIATE ANALYSIS OF ECOLOGICAL DATA

For example, using the data of Exhibit 1.1, consider the regression of species labelled d on depth, pollution and temperature. The model is estimated as:

E(d) 6.271 0.148 depth 1.388 pollution 0.043 temperature (10.2)

Notice that, for the moment, we do not comment on whether this type of linear model of a count variable on three environmental variables would be sensible or not, because d is not an interval variable – we will return to this point later.

Since the coefficients in (10.2) depend on the units of the variables, we prefer to consider the regression using all variables in comparable units. Usually this is done by standardization of the variables, so that they are all in units of standard deviation. Let us denote these standardized variables (i.e., centred and normalized) with an asterisk, then the regression model becomes:

E(d*) 0.347 depth* 0.446 pollution* 0.002 temperature* (10.3)

The constant term now vanishes and the coefficients, called standardized regression coefficients, can be compared with one another. Thus it seems that pollution has the strongest influence on the average level of species d, reducing it by 0.446 of a standard deviation for every increase of one standard deviation of pollution. The effect of temperature is minimal and, in fact, is nonsignificant statistically (p 0.99), while depth and pollution are both significant (p 0.039 and p 0.010, respectively), so we drop temperature and consider just the regression on the other two variables, which maintains the value of the coefficients, but slightly smaller p -values: p 0.035 and p 0.008, respectively:

When referring to the multiple regression model, it is often said that a hyperplane is being fitted to the data. For a single explanatory variable this reduces to a straight line in the familiar case of simple linear regression. When there are two explanatory variables, as in (10.4), the model is a two-dimensional plane in three dimensions, the third dimension being the response variable d* – a view of this plane in three dimensions is given in Exhibit 10.1, with standardized depth* and pollution* forming the two horizontal dimensions and d* the vertical one. Notice how the plane is going down in the direction of pollution, but going up in the direction of depth, according to the regression coefficients (see the web site of the book which shows a video of this three-dimensional image). Notice too the lack of fit of the points to the plane – the value of R 2 for the regression is 0.442, which means that 44.2% of the variance of d is being