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1. Does correlation equal causation? Does the strength of correlation depend on the direction? What is the meaning of a zero correlation? Explain your answers.

2. What is the difference between an independent and a dependent variable? Does a regression model imply causation? Explain why or why not.

3. What terms describe the fit of a regression equation to the data? What is the importance of the coefficient of determination (r2)? How do you identify outliers in your data? How do they affect your regression equation?

4. What are the requirements that must be met for a regression analysis? What happens if these requirements are violated? Why is analysis of residuals important?

|(1) In correlation analysis, we calculate the correlation coefficient which is a measure of the degree of covariablity between X and Y. Correlation is merely a |

|tool to ascertain the degree of relationship between X and Y. We cannot assign cause and effect relationship between X and Y. Correlation is neither the same as |

|causation nor does it show causation. It is regression that deals with causation. |

|[For example, the average income of people in a city may have been increasing over the last 10 years. Similarly, there is evidence that the number of plant species|

|in nature is decreasing with time. These two variables have a negative correlation, but there is no (straightforward) causal connection between them.] |

|The strength of the correlation does not depend upon direction. |

|A correlation coefficient of ± 1 ( a perfect positive/negative correlation between X and Y. |

|A zero coefficient ( no correlation. Higher the value of r, stronger is the correlation between X and Y. |

| |

|(2) A variable (x) which is the cause and which can be manipulated is called the independent variable (or explanatory variable) and the variable (y) which is the |

|effect is called the dependent variable (or response variable). |

|Generally speaking, the two variables are not interchangeable because one is the cause and the other is the effect. But in some specific and rare cases, the two |

|may be used interchangeably - for example, happiness at the workplace and happiness at home. |

|Yes, regression implies causation. Coefficient of determination, R^2 is a measure of the cause and effect relationship between the two variables. It describes how |

|well the regression model fits with the actual data. A higher value of R^2 means the model is a good fit. If R^2 = 0.85 for a model, it means 85% of the variation |

|in y is explained by the variation in x. |

| |

|(3) The β coefficients and R^2 together characterize a regression model. How well the data fit the regression equation is measured by the coefficient of |

|determination (R^2). A high value of R^2 implies the model is successful in representing the correlation between the variables. A low value of R^2 implies a poor |

|fit. Moreover, statistical significance of the β coefficients can be tested using a hypothesis tests such as t- test. |

| Outliers are observations that lie outside (beyond) most of the data. There may be low outliers or high outliers. It is very important to remember that a |

|regression model is very sensitive to outliers. We can identify outliers on a scatter graph as points that are far off from the trend line. If we can somehow |

|eliminate them, the capability of the model to be used for forecasting improves. |

| |

|(4) Meaningful regression analysis requires the following: |

|(a) The values of the independent variable should be themselves independent (no relation should exist between them). |

|(b) There should be a good rational or experimental basis for identifying the independent variables and the resultant dependent variable. This is required because |

|regression models can be built both ways- x vs y and y vs x, and they will be different. |

|(c) Good (sufficient) sample size, with the pairs of units sampled randomly |

|(d) The random error terms e are independent and, for any value of x, have a normal distribution with μ = 0 and variance σ^2 |

|If these requirements are not satisfied, the model will not be robust. |

|The analysis of residuals plays an important role in validating the regression model. If the error term in the regression model satisfies the assumptions for |

|regression, then the model is considered valid. |

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