Regression Modeling in Strength and Conditioning

Simple and multiple linear regression models assume linear relationships, but nonlinear models exist for non-linear relationships.
Gonzalez Bedillo and Sanchez Madina (2010) used an order polynomial relationship to model the nonlinear relationship between mean propulsive velocity and load.
The accuracy and shape of the relationship determine the order of the polynomial.
The goodness of fit (R-squared value) indicates the strength of the relationship; an R-squared of 98% indicates a strong fit.

Nonlinear regression shares assumptions with linear regression, notably the independence of observations.
Including all observations from a single participant violates this assumption due to the correlation between those observations.

Including all observations from a participant leads to autocorrelation error, inflating the perceived strength of the relationship.
Regression models are drawn to closely modeled participant scores, artificially strengthening the relationship.
Example: Gonzalez, Bodillo, and Sanchez Medina had 120 participants, but more observations due to multiple repetitions per load.
Including all repetitions artificially strengthens the model of the relationship between mean propulsive velocity and 1RM load.

Using minimal data points when extrapolating beyond known values is problematic.
Example: Using barbell velocity to predict 1RM with only two points. This is sometimes called the two-point method.
- One point represents velocity at 40% of 1RM, and another represents 80-90% of 1RM.
- A linear model is constructed using the slope and vertical intercept to estimate 1RM from velocity (X value).
Using only two points to construct the model is unreliable.
Furthermore, the actual 1RM velocity is an estimation from a population-wide assumption.
Predictions of 1RM from velocity are often inaccurate due to small numbers of observations used to build and make predictions.
Using five data points is common, but more data points leads to more accurate models.

Using linear models for curvilinear relationships is a problem.
The relationship between independent variable velocity and dependent variable force is curvilinear.

The force-velocity relationship is curvilinear, as shown by A.B. Hill's data from the 1930s.
However, linear models are often used to predict and assess the relationship between force and velocity, guiding training decisions.
The incorrect assumption of linearity causes issues with guidance and inferences drawn from these models.

Regression is useful for describing relationships between variables and predicting performance with sufficient data.
Caution is needed in strength and conditioning when applying models built on specific, small populations to the larger population.
Extrapolating models beyond their original sample or population is problematic due to small sample sizes and issues with underlying assumptions.
Be cautious about the use and application of regression modeling in strength and conditioning.