SSE can never be
a. larger than SST
b. smaller than SST
c. equal to 1
d. equal to zero
The Correct Answer and Explanation is :
The correct answer is (b) smaller than SST.
Explanation:
In the context of regression analysis, particularly in the method of least squares, we encounter three important quantities: SST (Total Sum of Squares), SSE (Sum of Squared Errors), and SSR (Sum of Squares for Regression). These quantities help us evaluate the fit of a regression model.
- SST (Total Sum of Squares): This represents the total variation in the observed data, and it is calculated as the sum of the squared differences between each observed value and the overall mean of the dependent variable.
- SSE (Sum of Squared Errors): This represents the unexplained variation in the dependent variable after fitting the regression model. It is calculated as the sum of the squared differences between the observed values and the predicted values (the residuals).
- SSR (Sum of Squares for Regression): This represents the variation explained by the regression model, calculated as the sum of the squared differences between the predicted values and the overall mean of the dependent variable.
The relationship between these sums of squares is described by the equation:
[
SST = SSR + SSE
]
This equation indicates that the total variation (SST) is the sum of the explained variation (SSR) and the unexplained variation (SSE).
Key Points:
- SSE cannot be larger than SST because SST represents the total variation in the data, and SSE represents the portion of that variation that the model fails to explain. Therefore, SSE ≤ SST.
- SSE can never be smaller than SST because it is a part of SST (as shown in the equation above), and it always represents some portion of the total variation, meaning SSE > 0 unless the model perfectly predicts all values, in which case SSE = 0.
- SSE cannot be equal to 1 or 0 in all cases; it depends on the fit of the model. If the model perfectly fits the data, SSE equals zero. If the model fits poorly, SSE will be larger.
Thus, the correct answer is that SSE is never smaller than SST.