Physics for Geologists, Second edition

142 Notes
each age group, for example, 5, 6,7 and 8, and within each 10cm of height from
1.2 to 1.5 m, you might draw up a table (called a contingency table) like this:
Height (m) Age (years)
The frequencies in each row are not all independent because when three are
known and we know the total, the fourth is determined and a degree of freedom
is lost. Likewise, the frequencies in each column are not all independent, and a
degree of freedom is lost. So the degrees of freedom for the table are not 4 x 4
but 3 x 3, or, more generally, where r is the number of rows, and c, the columns,
(r - l)(c - 1). Moroney (1956) is not very enlightening on this subject. When in
doubt, the best advice, as always, is to ask someone who knows these things.
As regards the assessment of significance in linear regression analysis, the pur-
pose is to establish if such a distribution of points could reasonably have arisen by
chance. The result is never that it could not have arisen by chance, but a probabil-
ity, P, that it arose by chance. The usual values chosen as levels of significance are
5 per cent, 1 per cent and 0.1 per cent. Perhaps the easiest to use is Student's t, or
the t-test, for the significance of the regression coefficient, r. This was used in the
examples. For this, you calculate
where N is the number of pairs in the analysis and N-2 are the degrees of freedom,
and enter this value of t into the tables of t with N - 2 degrees of freedom. From
the table we assess the probability P of having t as large or larger than some critical
number. If r is negative, this should be ignored in this test. Many calculators (and
spreadsheets, no doubt) include the statistical operations, and some include the
significance tests. For methods of assessing the significance of a regression analysis,
see Moroney (1956: 311f.), if you can find a copy, and Rosen (1992: 215f.).
It is worth repeating the warning that if a test of significance of a linear regression
analysis indicates that the probability P is very small of obtaining such a result by
chance, it does not necessarily confirm the analysis. It is very important to establish
linearity by plotting the logarithms of the data, or by computing the regression
coefficient of the logarithms. As mentioned in the text, any gentle curve or short
section of a curve, will appear to be significant.
Copyright 2002 by Richard E. Chapman