In medical and biological research we are often interested in making predictions or in building a model to predict a continuous outcome based on a continuous predictor type variable or possible biomarker. After constructing the scatter plot and visually looking at the paired data points we may conclude that the relationship is linear and that this seems plausible and then go about using least squares regression to derive a fitted line or equation of the form, Y = a + bX. Before we can rely on this strategy as a research tool, how do we assess whether the relationship in indeed linear? Other than residual plots, is there a useful statistical test or assessment to verify this?
Dear David,
Thank you for your interesting question. I just wanted to clarify that simple linear regression is used to explain part of the variation between two variables, not all of it. It this therefore a question of whether it is valid to use a linear model to explain part of the variation.
If an underlying relationship doesn't appear to be exactly linear, perhaps a different nonlinear function would better explain it? It is probably only in a scientific context that one is trying to prove a relationship is exactly linear.
A linear relationship is often an oversimplification of the association between two variables. It is chosen mostly because it lends itself to easy analysis and interpretation. Residual analysis is one of the best test for model adequacy.
A more likely but non-linear model might also be fitted and compared with the linear model on the basis of criteria like R squared, AIC, BIC, mean square error, etc. Analysis of variance is another way of checking for model adequacy. A model is always comparative and not absolute.
In your fields, I think you must recognize that most such relationships are highly nonlinear. Thus, the scatter approach, least-squares measures "crutch," and bit-fit attempts should be fine in the vast majority of cases in constructing a reasonable model. If you suspect or want a linear model to simplify your work, the only thing I can presently, without taking more time, is to decrease your measurement and plotting errors to the absolute minimums; if you do so, the best fit curve may look very linear, perhaps. But keep taking data, and I'll bet things will look nonlinear again. :-)
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