Which of the following statements is true regarding the coefficient of correlation if coefficient of determination is equal to 1?

What Is the Coefficient of Determination?

The coefficient of determination is a statistical measurement that examines how differences in one variable can be explained by the difference in a second variable, when predicting the outcome of a given event. In other words, this coefficient, which is more commonly known as R-squared (or R2), assesses how strong the linear relationship is between two variables, and is heavily relied on by researchers when conducting trend analysis. To cite an example of its application, this coefficient may contemplate the following question: if a woman becomes pregnant on a certain day, what is the likelihood that she would deliver her baby on a particular date in the future? In this scenario, this metric aims to calculate the correlation between two related events: conception and birth.

R-Squared

Key Takeaways

• The coefficient of determination is a complex idea centered on the statistical analysis of models for data.
• The coefficient of determination is used to explain how much variability of one factor can be caused by its relationship to another factor.
• This coefficient is commonly known as R-squared (or R2), and is sometimes referred to as the "goodness of fit."
• This measure is represented as a value between 0.0 and 1.0, where a value of 1.0 indicates a perfect fit, and is thus a highly reliable model for future forecasts, while a value of 0.0 would indicate that the model fails to accurately model the data at all. 

Understanding the Coefficient of Determination

The coefficient of determination is a measurement used to explain how much variability of one factor can be caused by its relationship to another related factor. This correlation, known as the "goodness of fit," is represented as a value between 0.0 and 1.0. A value of 1.0 indicates a perfect fit, and is thus a highly reliable model for future forecasts, while a value of 0.0 would indicate that the calculation fails to accurately model the data at all. But a value of 0.20, for example, suggests that 20% of the dependent variable is predicted by the independent variable, while a value of 0.50 suggests that 50% of the dependent variable is predicted by the independent variable, and so forth.

Graphing the Coefficient of Determination

On a graph, the goodness of fit measures the distance between a fitted line and all of the data points that are scattered throughout the diagram. The tight set of data will have a regression line that's close to the points and have a high level of fit, meaning that the distance between the line and the data is small. Although a good fit has an R2 close to 1.0, this number alone cannot determine whether the data points or predictions are biased. It also doesn't tell analysts whether the coefficient of determination value is intrinsically good or bad. It is at the discretion of the user to evaluate the meaning of this correlation, and how it may be applied in the context of future trend analyses.

Which of the following statements is false? 1. When value of correlation coefficient is one, the two regression lines coincide2. The regression coefficients are independent of the change of origin and of scale3. The sign of the regression coefficients are always the same4. The square of the coefficient of correlation is called coefficient of determination  (adsbygoogle = window.adsbygoogle || []).push({});

1. 1
2. 2
3. 3
4. 4

Answer (Detailed Solution Below)

Option 2 : 2

Free

Teaching Aptitude Mock Test

10 Questions 20 Marks 12 Mins

Regression Coefficients

The regression coefficients are a statistical measure that is used to measure the average functional relationship between variables. In regression analysis, one variable is dependent and the other is independent. Also, it measures the degree of dependence of one variable on the other(s).

Some of the properties of regression coefficient:

1. It is generally denoted by ‘b’.
2. It is expressed in the form of an original unit of data.
3. If two variables are there say x and y, two values of the regression coefficient are obtained. One will be obtained when x is independent and y is dependent and the other when we consider y as independent and x as a dependent. The regression coefficient of y on x is represented by byx and x on y as bxy.
4. Both of the regression coefficients must have the same sign. If byx is positive, bxy will also be positive and it is true for vice versa.
5. If one regression coefficient is greater than unity, then others will be lesser than unity.
6. The geometric mean between the two regression coefficients is equal to the correlation coefficient
7. R=sqrt(byx*bxy)
8. Also, the arithmetic means (am) of both regression coefficients are equal to or greater than the coefficient of correlation.
9. (byx + bxy)/2= equal or greater than r.
10. The regression coefficients are independent of the change of the origin. But, they are not independent of the change of the scale. It means there will be no effect on the regression coefficients if any constant is subtracted from the value of x and y. If x and y are multiplied by any constant, then the regression coefficient will change.
11. The two lines of regression coincide i.e. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion.
12. The coefficient of determination is the square of the correlation (r) between predicted y scores and actual y scores; thus, it ranges from 0 to 1. With linear regression, the coefficient of determination is also equal to the square of the correlation between x and y scores.

Hence, from the above explanation, The regression coefficients are independent of the change of origin but they are not independent of the change of the scale. Therefore, Option 2 is false.

Last updated on Nov 25, 2022

University Grants Commission (Minimum Standards and Procedures for Award of Ph.D. Degree) Regulations, 2022 notified. As, per the new regulations, candidates with a 4 years Undergraduate degree with a minimum CGPA of 7.5 can enroll for PhD admissions. The UGC NET Final Result for merged cycles of December 2021 and June 2022 was released on 5th November 2022.Along with the results UGC has also released the UGC NET Cut-Off.  With tis, the exam for the merged cycles of Dec 2021 and June 2022 have conclude. The notification for December 2022 is expected to be out soon. The UGC NET CBT exam consists of two papers - Paper I and Paper II. Paper I consists of 50 questions and Paper II consists of 100 questions. By qualifying this exam, candidates will be deemed eligible for JRF and Assistant Professor posts in Universities and Institutes across the country.

What is the correlation coefficient if the coefficient determination is 1?

Which of the following is true regarding coefficient of correlation of determination is equal to 1?

Can coefficient of correlation and coefficient of determination be equal?

What is the relationship between correlation coefficient and coefficient of determination?