4. Explain the meaning and use of:
i) Partial regression coefficients
ii) Partial correlation
iii) Standardized/Beta coefficients.
iv) Stepwise regression, Forward selection, and Backward elimination
i) Partial regression coefficients: In regression analysis, partial regression coefficients refer to the coefficients that measure the effect of one predictor variable on the dependent variable while holding other predictor variables constant. They indicate the unique contribution of a particular predictor variable to the variation in the dependent variable.
To calculate partial regression coefficients, you typically perform a multiple regression analysis. This involves including all predictor variables in the regression model simultaneously and estimating the coefficients for each variable while controlling for the others.
ii) Partial correlation: Partial correlation measures the strength and direction of the linear relationship between two variables, while controlling for the effects of other variables. It provides a way to assess the relationship between two variables after removing the influences of other variables.
To calculate the partial correlation coefficient, you need to run a partial correlation analysis. This involves computing the correlation between two variables while controlling for the influence of other variables by including them as covariates in the analysis.
iii) Standardized/Beta coefficients: Standardized coefficients, also known as beta coefficients, represent the change in the dependent variable (in standard deviation units) associated with a one standard deviation change in the predictor variable, while holding other variables constant. They allow for direct comparisons of the relative importance of different predictor variables in a regression model.
To obtain standardized coefficients, you can perform a regression analysis and request the standardized coefficients as part of the output.
iv) Stepwise regression, Forward selection, and Backward elimination: These are methods used for variable selection in regression analysis when dealing with a large number of predictor variables.
- Stepwise regression: This is an automated procedure that involves iteratively adding or removing predictor variables based on certain criteria (e.g., significance level, change in model fit) until an optimal subset of variables is obtained. It can be performed in a forward, backward, or bidirectional manner.
- Forward selection: This method starts with an empty model and adds predictor variables one by one based on predefined criteria (e.g., p-value, increase in adjusted R-squared) until no more variables meet the criteria.
- Backward elimination: This method starts with a model containing all predictor variables and removes variables one by one based on predefined criteria (e.g., p-value, decrease in adjusted R-squared) until no more variables meet the criteria.
Both forward selection and backward elimination can be manually performed or automated using statistical software. The goal of these methods is to identify the most predictive subset of variables for the regression model.