the equator towards the pole? That question is most easily tested statistically if we
frame it in negative form, the so-called null hypothesisdenoted H 0 : mean body weight
of black bears does not change with latitude. If this hypothesis is falsified by data
showing that mean weights are not the same at all sampled latitudes, we reject the
null hypothesis in favor of an alternative hypothesisHa. Whereas a question can
generate only one null hypothesis, there may be a number of competing alternative
hypotheses. In the bear example the alternative to no change of weight with latitude
may be an increase with latitude, a decrease with latitude, a peak in the middle
latitudes, or a trough in the middle latitudes.
The procedures by which we test hypotheses make up the realm of statistical
analysis. They come quite late in the research sequence, which proceeds in the
following manner:
1 pose a research question (usually our best guess or prediction as to what is
going on);
2 convert that to a null hypothesis;
3 collect the data that will test the null hypothesis;
4 run the appropriate statistical test;
5 accept or reject the null hypothesis in the light of that testing;
6 convert the statistical conclusion to a biological conclusion.
Most statistical tests estimate the probability that a null hypothesis is false. A prob-
ability of say 10% is often interpreted loosely as meaning that there is only a 10%
chance of the null hypothesis being true. That is not quite right. Suppose our null
hypothesis states that there is no difference in bill length between the females of two
populations of a particular species. We draw a sample from each, perform the
appropriate statistical test for a difference, and find that the test statistic has a pro-
bability of (say) 10% for the sample sizes that we used. That 10% is the estimated
probability of drawing two samples as different or even more different in average bill
length as those that we drew if the populations from which they were drawn did not
differ in that estimated attribute.
If there really is no difference between the two populations in average bill length,
then the probability returned by the statistical test will be in the region of 50%. This
implies that the chance of drawing more disparate samples than those we actually drew
is the same as the chance of drawing less disparate samples than those we actually
drew. A probability greater than 50% means that the two samples are more similar than
we would expect from random sampling of identical populations. If that probability
approaches 95% or so, we should investigate whether the sampling procedure was biased.
Statistical tests deal in probabilities, not certainties. There is always a chance that we
are wrong. Such errors come in two forms, the Type 1 error(also known as an alpha
error) in which the null hypothesis is rejected even though true, and the Type 2
error(beta error) when the null hypothesis is accepted even though false. Following
Zar (1996), the relationship between the two kinds of error can be shown as a matrix:If H 0 is true If H 0 is false
If H 0 is rejected Type 1 error No error
If H 0 is accepted No error Type 2 errorObviously we are not keen to make either kind of error. The probability of com-
mitting a Type 1 error is simply the specified significance level. The probability of270 Chapter 16WECC16 18/08/2005 14:47 Page 270