Eigenvalue Multicollinearity Detection

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In this post, we learn about the role of eigenvalue multicollinearity detection. In the context of the detection of multicollinearity, eigenvalues are used to assess the degree of linear dependence among explanatory (regressors, independent) variables in a regression model. Therefore, by understanding the role of eigenvalue multicollinearity detection, one can take appropriate steps to improve the reliability and interpretability of the regression models.

Decomposition of Eigenvalues and Eigenvectors

The pair-wise correlation matrix of explanatory variables is decomposed into eigenvalues and eigenvectors. Whereas Eigenvalues represent the variance explained by each Principal Component and Eigenvectors represent the directions of maximum variance.

The Decomposition Process

Firstly, compute the correlation coefficients between each pair of variables in the dataset.

Secondly, find the Eigenvalues and Eigenvectors: solve the following equation for each eigenvalue ($\lambda$) and eigenvector ($vV)

$$A v = \lambda v$$

where $A$ is the correlation matrix, $v$ is the eigenvector, and $\lambda$ is the eigenvalue.

The above equation essentially means that multiplying the correlation matrix ($A$) by the eigenvector ($v$) results in a scaled version of the eigenvector, where the scaling factor is the eigenvalue. This can be solved using various numerical methods, such as the power method or QR algorithm.

Interpreting Eigenvalue Multicollinearity Detection

A set of eigenvalues of relatively equal magnitudes indicates little multicollinearity (Freund and Littell 2000: 99). A small number of large eigenvalues suggests that a small number of component variables describe most of the variability of the original observed variables ($X$). Because of the score constraint, a number of large eigenvalues implies that there will be some small eigenvalues or some small variances of component variables.

A zero eigenvalue means perfect multicollinearity among independent/explanatory variables and very small eigenvalues imply severe multicollinearity. Conventionally, an eigenvalue close to zero (less than 0.01) or condition number greater than 50 (30 for conservative persons) indicates significant multicollinearity. The condition index, calculated as the ratio of the largest eigenvalue to the smallest eigenvalue $\left(\frac{\lambda_{max}}{\lambda_{min}}\right)$, is a more sensitive measure of multicollinearity. A high condition index (often above 30) signals severe multicollinearity.

Eigenvalue Multicollinearity Detection

The proportion of variances tells how much percentage of the variance of parameter estimate (coefficient) is associated with each eigenvalue. A high proportion of variance of an independent variable coefficient reveals a strong association with the eigenvalue. If an eigenvalue is small enough and some independent variables show a high proportion of variation with respect to the eigenvalues then one may conclude that these independent variables have significant linear dependency (correlation).

Presence of Multicollinearity in Regression Model

Since Multicollinearity is a statistical phenomenon where two or more independent/explanatory variables in a regression model are highly correlated, the existence/presence of multicollinearity may result in

  • Unstable Coefficient Estimates: Estimates of regression coefficients become unstable in the presence of multicollinearity. A small change in the data can lead to large changes in the estimates of the regression coefficients.
  • Inflated Standard Errors: The standard errors of the regression coefficients inflated due to the presence of multicollinearity, making it difficult to assess the statistical significance of the coefficients.
  • Difficulty in Interpreting Coefficients: It becomes challenging to interpret the individual effects of the independent variables on the dependent variable when they are highly correlated.

How to Mitigate the Effects of Multicollinearity

If multicollinearity is detected, several strategies can be employed to mitigate the effects of multicollinearity. By examining the distribution of eigenvalues, researchers (statisticians and data analysts) can identify potential issues and take appropriate steps to address them, such as feature selection or regularization techniques.

  • Feature Selection: Remove redundant or highly correlated variables from the model.
  • Principal Component Regression (PCR): Transform the original variables into a smaller set of uncorrelated principal components.
  • Partial Least Squares Regression (PLSR): It is similar to PCR but also considers the relationship between the independent variables and the dependent variable.
  • Ridge Regression: Introduces a bias-variance trade-off to stabilize the coefficient estimates.
  • Lasso Regression: Shrinks some coefficients to zero, effectively performing feature selection.
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Correlation Regression MCQs 6

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The post is about a Quiz on Correlation Regression MCQs with Answers. There are 20 multiple-choice questions covering topics related to correlation and regression analysis, coefficient of determination, testing of correlation and regression coefficient, Interpretation of regression coefficients, and the method of least squares, etc. Let us start with Correlation Regression MCQs with answers.

Online Multiple-Choice Questions about Correlation and Regression Analysis with Answers

1. The slope ($b_1$) represents

 
 
 
 

2. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is

 
 
 
 

3. Which one of the following situations is inconsistent?

 
 
 
 

4. If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a

 
 
 
 

5. If the coefficient of determination is 0.49, the correlation coefficient may be

 
 
 
 

6. The strength of the linear relationship between two numerical variables may be measured by the

 
 
 
 

7. The correlation coefficient

 
 
 
 

8. In a simple linear regression problem, $r$ and $\beta_1$

 
 
 
 

9. Which one of the following statements is true?

 
 
 
 

10. The $Y$ intercept ($b_0$) represents the

 
 
 
 

11. Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$

 
 
 
 

12. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is

 
 
 
 

13. Which of the following does the least squares method minimize?

 
 
 
 

14. The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by

 
 
 
 

15. The true correlation coefficient $\rho$ will be zero only if

 
 
 
 

16. Testing for the existence of correlation is equivalent to

 
 
 
 

17. If the correlation coefficient $r=1.00$ then

 
 
 
 

18. The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is

 
 
 
 

19. What do we mean when a simple linear regression model is “statistically” useful?

 
 
 
 

20. If the correlation coefficient ($r=1.00$) then

 
 
 
 

Online Correlation & Regression MCQs with Answers

Online Correlation Regression MCQs

  • The $Y$ intercept ($b_0$) represents the
  • The slope ($b_1$) represents
  • Which of the following does the least squares method minimize?
  • What do we mean when a simple linear regression model is “statistically” useful?
  • If the correlation coefficient $r=1.00$ then
  • If the correlation coefficient ($r=1.00$) then
  • Assuming a linear relationship between $X$ and $Y$ if the coefficient of correlation equals $-0.30$
  • Testing for the existence of correlation is equivalent to
  • The strength of the linear relationship between two numerical variables may be measured by the
  • In a simple linear regression problem, $r$ and $\beta_1$
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho=0$ against the one-sided alternative $H_1:\rho<0$ at a significance level of 0.193, the p-value is The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho = 0$ against the two-sided alternative $H_1:\rho\ne 0$. To test $H_0:\rho =0$ against the one-sided alternative $H_1:\rho >0$ at a significance level of 0.193, the p-value is
  • The sample correlation coefficient between $X$ and $Y$ is 0.375. It has been found that the p-value is 0.256 when testing $H_0:\rho=0$ against the one-sided alternative $H_1:\rho>0$. To test $H_0:\rho =04 against the two-sided alternative $H_1:\rho\ne 0$ at a significance level of 0.193, the p-value is
  • If you wanted to find out if alcohol consumption (measured in fluid oz.) and grade point average on a 4-point scale are linearly related, you would perform a
  • The correlation coefficient
  • If the coefficient of determination is 0.49, the correlation coefficient may be
  • The estimated regression line relating the market value of a person’s stock portfolio to his annual income is $Y=5000+0.10X$. This means that each additional rupee of income will increase the stock portfolio by
  • Which one of the following situations is inconsistent?
  • Which one of the following statements is true?
  • The true correlation coefficient $\rho$ will be zero only if
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MCQs Probability Quiz 10

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The post is about the Online MCQs Probability Quiz. There are 20 multiple-choice questions covering topics related to random experiments, random variables, expectations, rules of probability, events and types of events, and sample space. Let us start with the Probability Quiz.

Please go to MCQs Probability Quiz 10 to view the test

MCQs Probability Quiz 10

Online MCQs Probability Quiz with Answers

  • Consider a dice with the property that the probability of a face with $n$ dots showing up is proportional to $n$. What is the probability of the face showing 4 dots?
  • Let $X$ be a random variable with a probability distribution function $$f (x) = \begin{cases} 0.2 & \text{for  } |x|<1 \ 0.1 & \text{for } 1 < |x| < 4\ 0 & \text{otherwise} \end{cases}$$ The probability P (0.5 < x < 5) is ————-
  • Runs scored by batsmen in 5 one day matches are 50, 70, 82, 93, and 20. The standard deviation is ————-.
  • Find the median and mode of the messages received on 9 consecutive days 15, 11, 9, 5, 18, 4, 15, 13, 17.
  • $E (XY)=E (X)E (Y)$ if $x$ and $y$ are independent.
  • Mode is the value of $x$ where $f(x)$ is a maximum if $X$ is continuous.
  • A coin is tossed up 4 times. The probability that tails turn up in 3 cases is ————–.
  • If $E$ denotes the expectation the variance of a random variable $X$ is denoted as?
  • $X$ is a variate between 0 and 3. The value of $E(X^2)$ is ————-.
  • The random variables $X$ and $Y$ have variances of 0.2 and 0.5, respectively. Let $Z= 5X-2Y$. The variance of $Z$ is?
  • In a random experiment, observations of a random variable are classified as
  • A number of individuals arriving at the boarding counter at an airport is an example of
  • If $A$ and $B$ are independent, $P(A) = 0.45$ and $P(B) = 0.20$ then $P(A \cup B)$
  • If a fair dice is rolled twice, the probability of getting doublet is
  • If a fair coin is tossed 4 times, the probability of getting at least 2 heads is
  • If $P(B) \ne 0$ then $P(A|B) = $
  • The collection of all possible outcomes of an experiment is called
  • An event consisting of one sample point is called
  • An event consisting of more than one sample point is called
  • When the occurrence of an event does not affect the probability of occurrence of another event, it is called
https://itfeature.com probability quiz with answers

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