Biology Reference
In-Depth Information
sexually dimorphic in jaw shape. Instead, we want to generalize from that sample to
T. alpinus. That concern for generality motivates statistical analysis because we would not
need statistical tests if we cared only about the particular organisms that we've observed.
We could easily determine if their means differ in whatever variables interest us
we'd
just measure those individuals, calculate the mean of each sample and look at the num-
bers. It is precisely because we want to make inferences about the populations from which
the samples were drawn that we need statistical methods of inference.
This chapter presents an introduction to formulating and testing hypotheses. We will
focus on two simple hypotheses: (1) adult T. alpinus vary in shape because they vary in size;
(2) adult T. alpinus are sexually dimorphic in shape. In the next chapter, we will test the more
complex hypothesis that adult T. alpinus are sexually dimorphic in shape, controlling for
size, i.e. they differ in shape when compared at the same size as well as other complex
hypotheses. In this chapter, we also restrict ourselves to balanced designs, meaning that our
sample sizes are equal in all groups which, in the case of an analysis of sexual dimorphism, a
balanced design means that we have equal numbers of males and females. In the next chap-
ter, we consider unbalanced designs, i.e. the case in which groups differ in sample sizes.
In general outline, the first step in any statistical analysis is to turn the biological
hypothesis into a formal statistical model. Then the coefficients of that model are esti-
mated, and the model is tested for its statistical significance. To explain these steps, we
will consider our first example, the biological hypothesis that chipmunk jaw shapes vary
because of variation in size. An important distinction between that biological hypothesis
and our statistical model is that our mathematical model says nothing about causality.
Instead, the model says that we can predict one variable (shape) from another (size). Based
on the good fit of our model to the data, we might conclude that size predicts shape and,
in light of that, we might be tempted to conclude that size explains shape. However, even
if the model fits well, size might not be a cause of shape for at least two reasons. First, size
is not a process. In the context of developmental biology, we can explain size in terms of
the proliferation of cells that add tissue to a structure. Because growth rates vary over the
organism, cell proliferation (in conjunction with cell death, cell differentiation, deposition
of an extracellular matrix, etc.) produces changes in shape. In this context, saying that size
“explains” shape does not mean that size itself causes shape; rather, it means that we are
using “size” as shorthand for all those developmental processes that jointly alter size and
shape. Second, even if size predicts shape, we cannot infer that it actually causes shape
because we have not manipulated size and determined that those manipulations affect
shape. If the model fits the data, what we have demonstrated is that the relationship
between size and shape is predicted by a particular mathematical model.
We begin with the formulation of the model for the simple bivariate case in which we
have one dependent variable (Y) and one independent variable (X), each of which is measured
on N individuals. The model is the equation of a straight line, hence the term “linear
regression”. We are fitting the equation of a straight line to the data to find the coefficients
that best predict shape from values of the independent variable (e.g. size). More specifi-
cally, we are trying to find the best estimates of the coefficients m and b of the equation:
Y
i
mX
i
b
1
ε
(8.1)
5
1
i
