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:
Partial regression coefficients, also known as beta coefficients, are a statistical measure used in regression analysis to assess the relationship between an independent variable (predictor variable) and a dependent variable (outcome variable), while holding all other independent variables constant. These coefficients represent the change in the dependent variable for a one-unit change in the independent variable, considering the effects of the other independent variables.
To calculate partial regression coefficients, you can use multiple regression analysis, which involves fitting a linear regression model to the data. This approach enables you to estimate the influence of each independent variable on the dependent variable, accounting for the impact of all other variables included in the analysis.
ii) Partial correlation:
Partial correlation is a statistical technique used to identify and measure the strength of the relationship between two variables while controlling the influence of other variables. It helps to determine the unique association between two variables by excluding the common influence of the remaining variables in the analysis.
To compute partial correlations, you need to use multivariate analysis techniques such as multivariate regression or correlation matrices. By examining the partial correlation coefficient, which ranges from -1 to +1, you can determine the strength and direction of the relationship between two variables after removing the effects of other variables.
iii) Standardized/Beta coefficients:
Standardized coefficients, often referred to as beta coefficients, assess the relationship between an independent variable and a dependent variable in terms of standard deviations. Unlike the raw regression coefficients, which are measured in the original units of the variables, standardized coefficients allow for direct comparison of the magnitudes of the effects across different variables.
Standardized coefficients are obtained by converting the raw regression coefficients into z-scores, which are measures of the number of standard deviations a data point is from the mean. These coefficients enable you to assess the relative importance of different independent variables on the dependent variable. Additionally, they help in comparing the effects of variables when they are measured on different scales.
iv) Stepwise regression, Forward selection, and Backward elimination:
Stepwise regression, forward selection, and backward elimination are techniques used in regression analysis to select the most informative independent variables for the prediction of the dependent variable. These techniques iterate through subsets of variables and determine the predictive power of each variable and its contribution to the overall model fit.
- Stepwise regression: It is an automated process that starts with an empty model and progressively adds or removes variables based on predefined criteria, such as statistical significance, the goodness-of-fit statistic, or information criteria like AIC (Akaike Information Criterion) or BIC (Bayesian Information Criterion). Stepwise regression allows both forward selection and backward elimination to occur based on the selected criteria.
- Forward selection: This method starts with an initial model with no predictors and iteratively adds the independent variable that contributes the most to the model's improvement, as measured by a defined criterion (e.g., increase in adjusted R-squared). It continues adding variables until no further improvement is achieved.
- Backward elimination: In contrast to forward selection, backward elimination begins with a model that includes all independent variables and progressively removes the least useful variable, determined by a pre-defined criterion (e.g., p-value, decrease in adjusted R-squared). This process is repeated until no further improvement is observed.
Both forward selection and backward elimination can be used individually or as part of stepwise regression to select a subset of variables that contribute most significantly to the regression model. The choice of method depends on the specific research question, theoretical considerations, and statistical criteria.