Marginal and conditional relationships

In a linear model that predicts a response from several explanatory variables, the least squares coefficient associated with any explanatory variable describes its effect on the response if all other variables are held constant. This is also called the variable's conditional effect on the response.

This may be very different from the size and even the sign of the coefficient when a linear model is fitted with only that single explanatory variable. This simple linear model describes the marginal relationship between the response and that variable.

Example

In a model for predicting the percentage body fat of men, the best model (as determined by least squares) in a simple model with weight, is

Predicted body fat   =   -10.00  +  0.162 Weight

However if we add Abdomen circumference to the model, the best values for the coefficients are

Predicted body fat   =   -41.35  -  0.136 Weight  +  0.915 Abdomen