In a multiple regression analysis, if the variance of the observed variable Y is 1.57 and variance of the residue is 0.52, then R-square value is

Correct option is B

Introduction
· Multiple regression analysis is a statistical technique used to examine the relationship between a single dependent (observed) variable and two or more independent (predictor) variables.
· The total variance in the observed variable Y represents the total "information" or "spread" in the data that the model attempts to explain.
· The residual variance, also known as error variance, represents the portion of the data's variability that the regression model fails to account for.
Sol. 0.67
The $$R^2$$​value, known as the coefficient of determination, represents the proportion of the total variance in the dependent variable Y that is explained by the independent variables in the model. It is mathematically defined as:
$$R^2 = 1 - \frac{\text{Variance of the Residue}}{\text{Total Variance of } Y}$$​

By substituting the values provided in the question:
$$R^2 = 1 - \frac{0.52}{1.57}$$​ $$R^2 = 1 - 0.3312 = 0.6687$$​​
0.67, which indicates that the model has a reasonably good fit.

Information Booster
· The $$R^2$$​ value always ranges between 0 and 1 (or $$0\% to 100$$​), where $1$ indicates a perfect fit where all data points fall exactly on the regression line.
· An $$R^2$$​ of 0.67 means that approximately $67\%$ of the change in the dependent variable is explained by the factors included in the study, while the remaining 33% is due to unknown factors or inherent noise.
· In environmental modeling, the "residue" refers to the vertical distance between the actual data points and the predicted values on the regression plane.
· A higher $R^2$ value generally implies better predictive power, though it does not necessarily imply a "causal" relationship between the variables.
· This coefficient is a key output in software like SPSS or R when performing Biostatistical analyses for Environmental Science research.