Explain the meaning and use of:
i)Partial regression coefficients
ii)Partial correlation
iii) Standardized/Beta coefficients.
i) Partial regression coefficients:
Partial regression coefficients, also known as partial slopes or partial regression coefficients, are used in regression analysis to quantify the relationship between a predictor variable and the response variable while holding other predictor variables constant. In multiple regression analysis, where there are multiple predictors, each predictor's partial regression coefficient represents the change in the response variable associated with a one-unit change in that predictor, while all other predictors remain constant.
To calculate the partial regression coefficient, you can use the following steps:
1. Build a multiple regression model using the predictor variables.
2. Calculate the correlation coefficients between each predictor variable and the response variable.
3. Determine the partial regression coefficient for a specific predictor variable by holding all other predictor variables constant and calculating the change in the response variable associated with a one-unit change in that predictor.
ii) Partial correlation:
Partial correlation is a statistical measure that quantifies the strength and direction of the linear relationship between two variables while controlling for the influence of one or more other variables. It helps to identify the unique relationship between two variables, excluding the effects of other variables.
To calculate the partial correlation coefficient, follow these steps:
1. Compute the correlation coefficients between each of the variables.
2. Calculate the partial correlation coefficient between two variables, say X and Y, while controlling for the influence of a third variable Z, by using the formula:
r_xy.z = (r_xy - (r_xz * r_yz)) / sqrt((1 - r_xz^2) * (1 - r_yz^2))
iii) Standardized/Beta coefficients:
Standardized coefficients, also known as beta coefficients, are used in multiple regression analysis to assess the strength and direction of the relationship between each predictor variable and the response variable. They represent the change in the response variable associated with a one-standard-deviation change in the predictor variable, while holding other predictors constant.
To obtain standardized coefficients in multiple regression analysis, follow these steps:
1. Build a multiple regression model using the predictor variables.
2. Standardize each predictor variable by subtracting its mean and dividing by its standard deviation.
3. Estimate the regression coefficients in the standardized model.
4. These estimated coefficients are the standardized coefficients, representing the change in the response variable associated with a one-standard-deviation change in each predictor, while controlling for the other predictors.
Standardized coefficients are useful to compare the relative importance of different predictors and to determine the strength of their effects on the response variable, regardless of the scales or units of the predictor variables.