14.2: Simple Coefficients of Determination and Correlation Flashcards
What does the coefficient of determination (r²) represent in simple linear regression?
The coefficient of determination, denoted as r², measures the proportion of the total variation in the dependent variable that is explained by the independent variable in the regression model.
How is total variation calculated in the context of regression analysis?
Total variation is calculated as the sum of squared differences between each observed value and the mean of the dependent variable, denoted as Σ(yi - ȳ)².
What is the difference between explained and unexplained variation in regression analysis?
Explained variation is the portion of total variation that is explained by the regression line, calculated as Σ(ŷi - ȳ)².
Unexplained variation, or SSE, is the portion of total variation not explained by the model, calculated as Σ(yi - ŷi)².
What does a high r² value suggest about a regression model?
A high r² value suggests that a large proportion of the variability in the dependent variable is accounted for by the variability in the independent variable.
The closer the r² value is to 1, the more effective the model is at predicting the dependent variable.
Are there any limitations to using r² as a measure of a model’s effectiveness?
Yes, r² does not indicate whether a regression model is an appropriate fit for the data, nor does it indicate the direction or strength of any relationship.
Additionally, a high r² does not guarantee that the model will predict accurately for individual cases.
How can the coefficient of determination inform the practical usefulness of a regression model?
The coefficient of determination helps in assessing the practical usefulness of a model.
If r² is close to 1, the model might be considered useful for predicting the dependent variable.
However, a value of r² that is not close to 1 indicates that the model might not make accurate predictions for y.
What is the simple correlation coefficient (r) in regression analysis?
The simple correlation coefficient, r, measures the strength and direction of the linear relationship between two variables.
It is the square root of the coefficient of determination (r-squared), and it ranges from -1 to 1.
How do you calculate and interpret the simple correlation coefficient (r)?
The simple correlation coefficient r is calculated as either the positive or negative square root of r-squared, depending on whether the slope of the regression line is positive or negative.
A positive r indicates a positive correlation, while a negative r indicates a negative correlation.
What does it mean if the correlation coefficient (r) is positive or negative?
A positive r value indicates that as one variable increases, the other also increases, showing a positive correlation.
A negative r value indicates that as one variable increases, the other decreases, showing a negative correlation.
What do the values of r = 1 and r = -1 represent?
The value r = 1 represents a perfect positive linear relationship, where the variables move in sync;
r = -1 represents a perfect negative linear relationship, where the variables move in opposite directions.
Is the correlation coefficient (r) the same as the slope (b1) of the regression line?
No, the correlation coefficient r is not the slope b1 of the regression line.
The slope b1 quantifies how much the dependent variable changes for each unit change in the independent variable, whereas r measures the strength of the linear relationship between the variables.
Can a high correlation coefficient imply a cause-and-effect relationship?
No, a high correlation coefficient does not imply a causal relationship between the variables.
It indicates that the variables tend to move together in a linear fashion but does not establish causation.
