More on Two-Variable Data Transforming Relationships Cautions about Correlation Regression Relations in Categorical Dat

1. Chapter 4 More on Two-Variable Data Transforming Relationships Cautions about Correlation & Regression Relations in Categorical Data

2. Carl Friedrich Gauss (1777-1855) Gauss�s contributions to the field of statistics include the method of least-squares and the normal distribution.

3. Remember from Chapter 3 Always plot your data. Correlation coefficient, r and coefficient of determination, r2 are only valid for LINEAR data. A residual plot will show if a LINEAR model is a good model to use for data.

4. Transformations If the residual plot shows a pattern, a linear model is NOT a good model to use. Try and estimate which type of nonlinear model to use, then transform the data to make it linear.

5. Types of Power Models Exponential Growth

6. Types of Power Models Power Even Exponent

7. Types of Power Models Square Root

8. Types of Power Models Inverse

9. Linear vs. Exponential Growth Linear Growth increases by a fixed amount in each equal time period. Exponential Growth increases by a fixed percentage of the previous total.

10. Transforming Exponential Data If the ratio of consecutive y-values remains the same, then your data is approximately exponential. To transform your data to be linear, transform the y-values to logarithmic y-values. Then plot (x, log y) to see if the transformed data is linear.

11. Exponential Growth Equation: y = a(b)x when b > 0, and where a is the y-intercept and b is the rate of change.

12. Review of Logarithms logbx = y if and only if by = x The rules for logarithms are log(AB) = log A + log B log(A/B) = log A - log B logXp = p log x

13. Transformation Table

14. 4.2 Caution about Correlation & Regression Correlation and regression describe only linear relationships. The correlation coefficient r, and the least-squares regression line are NOT resistant.

15. Extrapolation Extrapolation is the use of a regression line for prediction far outside the domain values of the explanatory variable x that you used to obtain the line or curve. Such predictions are often not accurate.

16. Lurking Variable Is a variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.

17. More on Lurking Variables A lurking variable can falsely suggest a strong relationship between x and y, or it can hide a relationship that is really there. Many lurking variables change systematically over time. Plot the residuals against time to check the lurking variable.

18. Using Averaged Data Correlations based on averages usually too high when applied to individuals.

19. The question of Causation In many studies of the relationship between two variables, the goal is to establish that changes in the explanatory variable CAUSE changes in the response variable.

20. Causation (Cause-and-Effect) Changes in x cause changes in y.

21. Common Response Changes in both x and y are caused by a lurking variable z.

22. Confounding The effect (if any) of x on y is confounded with the effect of a lurking variable z.

23. Remember Even a very strong association between two variables is not by itself good evidence that there is a cause-and-effect link between the variables.

24. Establishing the Cause-Effect Relationship The best way to establish a cause-effect relationship is to conduct a carefully designed experiment in which the effects of possible lurking variables are controlled.

25. Establishing the Cause-Effect Relationship when you cannot do an experiment The association is strong. The association is consistent. Higher doses are associated with stronger responses. The alleged cause precedes the effect in time. The alleged cause is plausible.