Lenth’s method is a statistical method used to determine the relative importance of each predictor variable in a multiple regression analysis. This method involves calculating the relative efficiency of each predictor variable, which is a measure of how well the predictor variable explains the variance in the response variable.

The first step in Lenth’s method is to fit a multiple regression model using all of the predictor variables. This model is then used to calculate the residual sum of squares (RSS), which is a measure of the amount of variance in the response variable that is not explained by the predictor variables.

Next, the relative efficiency of each predictor variable is calculated by fitting a separate regression model for each predictor variable and calculating the corresponding RSS. The relative efficiency of each predictor variable is then calculated as the ratio of the RSS from the full model to the RSS from the model with only the predictor variable in question.

For example, suppose we have a multiple regression model with three predictor variables: X1, X2, and X3. The full model has an RSS of 500, and the models with only X1, X2, and X3 as predictors have RSS values of 400, 300, and 200, respectively. The relative efficiencies of X1, X2, and X3 are then calculated as 500/400 = 1.25, 500/300 = 1.67, and 500/200 = 2.5, respectively.

This analysis reveals that X3 is the most important predictor variable in the model, since it has the highest relative efficiency. In contrast, X1 is the least important predictor variable, since it has the lowest relative efficiency.

Lenth’s method is useful for identifying the relative importance of predictor variables in a multiple regression model. This can help researchers to focus on the most important predictor variables and to interpret the results of the regression analysis more accurately.

For example, suppose a researcher is interested in predicting the weight of a chicken based on its age, diet, and amount of exercise. The researcher fits a multiple regression model with these three predictor variables and uses Lenth’s method to determine their relative importance. The analysis reveals that age is the most important predictor variable, with a relative efficiency of 1.8, followed by diet (1.5), and exercise (1.2). This indicates that age is the primary factor influencing the weight of the chicken, with diet and exercise having a lesser impact.

Another example of Lenth’s method is in studying the factors that influence the success of a marketing campaign. A researcher fits a multiple regression model with predictor variables such as advertising spend, target audience, and product type. Lenth’s method reveals that advertising spend is the most important predictor variable, with a relative efficiency of 2.5, followed by target audience (1.7) and product type (1.3). This indicates that the success of the marketing campaign is primarily influenced by the amount of money spent on advertising, with target audience and product type having a lesser impact.