nonlinearly with age (Figures 14b and 14c): early growth is dominantly in height and later growth is dominantly in weight, leading indirectly to the pattern of Figure 14a.
Time series in particular, and nonindependent sampling in general, jeopardize interpolation and especially extrapolation. Nonlinearities are also a hazard, and we shall explore their impacts more fully in the subsequent section. First, however, let us assume the ideal correlation situation -- independent sampling and a linear relationship. How can we confidently and quantitatively describe the correlation between two variables?
Correlation Statistics
The type of test appropriate for identifying significant correlations depends on the kind of measurement scale. For classification data, such as male and female responses to an economic or psychological study, a test known as the contingency coefficient searches for deviations of observed from expected frequencies. For ranked, or ordinal, data where relative position along a continuum is known, the rank correlation coefficient is appropriate. Most scientific measurement scales include not just relative position but also measurable distance along the scale, and such data can be analyzed with the correlation coefficient or rank correlation coefficient. This section focuses on analysis of these continuous-scale data, not of classification data.
Suppose that we suspect that variable Y is linearly related to variable X. We need not assume existence of a direct causal relationship between the two variables. We do need to make the three following assumptions: first, that errors are present only in the Yi; second, that these errors in the Yi are random and independent of the value of Xi; and third, that the relationship between X and Y (if present) is linear. Scientists routinely violate the first assumption without causing too many problems, but of course one cannot justify a blunder by claiming that others are just as guilty. The second assumption is rarely a problem and even more rarely recognized as such. The third assumption, that of a linear relationship, is often a problem; fortunately one can detect violations of this assumption and cope with them.
The hypothesized linear relationship between Xi and Yi is of the form: Y = mX+b, where m is the slope and b is the Y intercept (the value of Y when X equals zero). Given N pairs of measurements (Xi,Yi) and the assumptions above, then the slope and intercept can be calculated by linear regression, from:
