# 5.4. Principal components to reduce the number of explanatory variables

With an increasing number of explanatory
variables (related or not-related, similar or dissimilar units) in one
experiment, multivariate statistics may be of interest. Multivariate statistics
are widely used in ecology (Leps and Smilauer, 2003), but less often
in bee research. Multivariate statistics can be used to reduce the number of
response variables without losing information in the response variables (van Dooremalen and Ellers, 2010), or to reduce
the number of explanatory variables (especially valuable if they are
correlated). A Principle Component Analysis (PCA) can be used to examine, for
example, morphometric or physiological variables (such as protein content of
different bee body parts or several volatile compounds in the head space of bee
brood cells). The PCA is usually used to obtain only the first principal
component that forms one new PC variable (the axis explaining most variation in
your variables). The correlations between the original variables and the new PC
variable will show the relative variation explained by the original variables
compared to each other and their reciprocal correlation. The new PC variable
can then be used to investigate effects of different treatments (and/or
covariates) using statistics as explained above in section 5. For an example in
springtails see van Dooremalen* et al.* (2011), or in
host-parasite interactions see Nash*
et al.* (2008). Note that the
new PC variables are uncorrelated with each other, which improves their
statistical properties. Unfortunately, it is also easy to lose track of what
they represent or how to interpret them. However, by reducing dimensionality
and dealing with uncorrelated variables one can transform a data set with a
great many explanatory and response variables into one with only a few of each,
and ones which capture most of the variability (i.e. the underlying processes) in
the data set. Related procedures are factor analysis, partial least squares,
and PC regression.