For example, I might have a hypothesis oxygen limitation is an important constraint on body size aquatic ectotherms due to the low concentration of oxygen in water vs. air. Because elevated temepratures increase the metabolic rates of ectotherms and thus their demand for oxygen, I predict ectotherms that live in water waters will tend to be smaller. However because oxygen is less limiting in air, I don’t expect to see this relationship in terrestrial animals. To test my hypothesis, I might select 20 aquatic and 20 terrestrial species, and measure their body sizes in natural environments at locations ranging in temperature. For my dependent variable, I divide all body sizes by the mean body size of that species, so that values above 1 represent above average sizes, and below 1 represent below average sizes.
I could write down a model
y = b0 + b1 * environment + b2 * temperature + b3 * temperature * environment
where environment is a dummy variable that is zero for terrestrial environments and 1 for aquatic environments.
Problem Given my hypothesis, what are my expecations about the values of the parameters in this model? Which are relevant to hypothesis?
Because environment is a dummy variable that equals 0 for terrestrial animals, we can drop all terms with environment to get the equation for terrestial animals which is:
y = b0 + b2 * temperature
This b2 is the slope of the relationship between scaled body size and temperature. Our hypthesis is that oxygen limitation less limiting in air, so we don’t expect to see a large reduction in relative body size with increasing temperature. However, there could be other mechanisms that affect body size that our model does account for, so b2 may not nessicarly be 0. Our theory mostly pertains to the fact that we expect a much more dramitic decline in body size in aquatic environments due to oxygen limitation
Or even better I might have a set of alternative hypotheses about some pattern in nature. In these cases it is useful to write down each hypothesis you think is plasuable. Then think about what pattern you expect to see if each of the different patterns is true. For each of these models, you should then be able to write down a statistical model assocaite with each hypothesis.