Correlation coefficient: show 1-r^2 is the ratio of 0th and 1st order models

  • Thread starter applestrudle
  • Start date
In summary, the correlation coefficient, represented as \( r \), indicates the strength and direction of a linear relationship between two variables. The metric \( 1 - r^2 \) demonstrates the proportion of variance in the dependent variable that is not explained by the independent variable in a first-order (linear) model, compared to a zero-order model, which assumes no relationship. This ratio highlights the effectiveness of the linear model relative to a constant mean model.
  • #1
applestrudle
64
0
Homework Statement
You have a linear model y = a+bx. Using the mean square error function for a zeroth order model (b=0 and a = <y>) and a first order model b=Covariance(x,y)/Variance(x) and a = <y> - b<x> show that E1/E0 = 1-r^2
Relevant Equations
MSE function E = <(y - a -bx)^2>
Correlation coefficient r = Covariance(x,y)/Standardev(x)Standarddev(y)
Standarddev = Square root of variance
The zeroth order model gives E0 = Var(y)

I've tried two methods:
Calculating 1-r^2 and trying to get E1/E0.
Calculating E1/E0 and trying to get 1-r^2.
 
Physics news on Phys.org
  • #2
applestrudle said:
I've tried two methods:
And what do you get?
 
  • #3
@applestrudle I'm a bit confused by your notation overall. You defined E as a function of x ( equiv y) then used E0, E1. Is E0:=E(0), E1:=E(1)?
 
  • #4
You haven't written down what ##r^2## is, which feels like an important piece.
 

FAQ: Correlation coefficient: show 1-r^2 is the ratio of 0th and 1st order models

What is the correlation coefficient?

The correlation coefficient, often denoted as "r," is a statistical measure that calculates the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.

What does r^2 represent in the context of the correlation coefficient?

In the context of the correlation coefficient, r^2 (the coefficient of determination) represents the proportion of the variance in the dependent variable that is predictable from the independent variable. It is a measure of how well the regression model explains the variability of the response data around its mean.

How is r^2 related to the 0th and 1st order models?

In regression analysis, the 0th order model refers to the mean of the dependent variable, while the 1st order model refers to the linear regression model. The r^2 value quantifies the improvement in predictive power when moving from the 0th order model to the 1st order model. Specifically, 1 - r^2 represents the ratio of the residual sum of squares (RSS) of the 1st order model to the total sum of squares (TSS) of the 0th order model.

How can we mathematically show that 1 - r^2 is the ratio of the 0th and 1st order models?

Mathematically, we can show this by considering the definitions of TSS and RSS. TSS is the total variation in the dependent variable, while RSS is the variation that remains after fitting the regression model. The r^2 value is given by (TSS - RSS) / TSS. Therefore, 1 - r^2 = RSS / TSS, which shows the ratio of unexplained variance (RSS) to the total variance (TSS).

Why is it important to understand the relationship between r, r^2, and the order models?

Understanding the relationship between r, r^2, and the order models is crucial for interpreting the effectiveness of a regression model. It helps in assessing how well the model captures the underlying data patterns and in making informed decisions about model selection and improvement. This understanding also aids in explaining the proportion of variance explained by the model, which is essential for evaluating its predictive power.

Similar threads

Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
Replies
7
Views
1K
Simplifying correlation coefficient
Replies
2
Views
1K
Differential Equations-Method of Undetermined Coefficients
Replies
7
Views
2K
Is the function v(r) = (e^(-mr))/r a solution to the Yukawa PDE equation?
Replies
1
Views
1K
How should I show that ## B ## is given by the solution of this?
Replies
2
Views
890
How should I show that this root is given approximately by this?
Replies
2
Views
297
Difficult Probability Density Function Question
Replies
2
Views
1K
Why smooth spherical waves with attenuation are only possible in 3-D
Replies
1
Views
1K
First Order Diffy Q Problem with Bernoulli/Integrating Factors
Replies
4
Views
1K
Doubt related to formation of a differential equation
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top