15.1: Introduction

Professionals often want to know how two or more numeric variables are related. For example, is there a relationship between the heart rate during a submaximal treadmill test and the maximum oxygen consumption (VO2​ max) an athlete achieves? If there is a relationship, what is the relationship and how strong is it? In another example, your risk of chronic disease may be determined by your daily physical activity, your body mass index (BMI), your nutritional habits, and your genetic predisposition. The total calories burned during a workout is often determined by a baseline metabolic rate plus the intensity and duration of the exercise.

These examples may or may not be tied to a model, meaning that some theory suggested that a relationship exists. This link between a cause and an effect, often referred to as a model, is the foundation of the scientific method and is the core of how we determine what we believe about how the world works. Beginning with a theory and developing a model of the theoretical relationship should result in a prediction, what we have called a hypothesis earlier. Now the hypothesis concerns a full set of relationships. As an example, in Kinesiology, the model of muscle hypertrophy is based upon assumptions concerning physiological behavior: a desire to stimulate protein synthesis, the application of mechanical tension, and adequate recovery. These combined give us the dose-response curve for resistance training. From that, we have the prediction that as training volume increases, muscle cross-sectional area will increase up to a certain threshold. Public Health has models concerning the relationship between neighborhood walkability and the prevalence of obesity, or the impacts of community-based interventions on the long-term health of a population.

Models are not unique to Kinesiology, even within the health sciences. In Epidemiology, for example, there are models that predict the spread of a virus based upon assumptions of social contact and vaccination rates. There are models of biochemical behavior dealing with enzyme kinetics and metabolic pathways both for cellular respiration and systemic energy expenditure.

The so-called hard sciences are, of course, the source of the scientific method as they tried through the centuries to explain the confusing world around us. Some early models today make us laugh; the "miasma theory" of disease, for example. These early models are seen today as not much more than the foundational myths we developed to help us bring some sense of order to what seemed chaos.

The foundation of all model building is perhaps the arrogant statement that we know what caused the result we see. This is embodied in the simple mathematical statement of the functional form that y=f(x). The response, Y (such as blood pressure), is caused by the stimulus, X (such as sodium intake). Every model will eventually come to this final place and it will be here that the theory will live or die. Will the data support this hypothesis? If so then fine, we shall believe this version of the world until a better theory comes to replace it. This is the process by which we moved from believing exercise was dangerous for heart patients to understanding it is a vital part of cardiac rehabilitation.

The scientific method does not confirm a theory for all time: it does not prove “truth”. All theories are subject to review and may be overturned. These are lessons we learned as we first developed the concept of the hypothesis test earlier in this book. Here, as we begin this section, these concepts deserve review because the tool we will develop here is the cornerstone of the scientific method and the stakes are higher. Full theories will rise or fall because of this statistical tool; regression and its applications in biostatistics.

In this chapter we will begin with correlation, the investigation of relationships among variables that may or may not be founded on a cause-and-effect model. The variables simply move in the same, or opposite, direction. That is to say, they do not move randomly. Correlation provides a measure of the degree to which this is true. From there we develop a tool to measure cause-and-effect relationships: regression analysis. We will be able to formulate models and tests to determine if they are statistically sound. If they are found to be so, then we can use them to make predictions: if as a matter of policy we changed the value of this variable, what would happen to this other variable? If we implemented a sugary drink tax of 50 cents per bottle, how would that affect childhood obesity rates, sales of bottled water, or the prevalence of type 2 diabetes? The ability to provide answers to these types of questions is the value of regression as both a tool to help us understand our world and to make thoughtful public health decisions.