STA291

1. STA291 Statistical Methods Lecture 27

2. Inference for Regression • Does the cost of a movie depend on its length? • Now we want to know, how useful is this model?

3. The Population and the Sample The movie budget sample is based on 120 observations. But we know observations vary from sample to sample. So we imagine a true line that summarizes the relationship between x and y for the entire population, Where µy is the population mean of y at a given value of x. We write µy instead of y because the regression line assumes that the means of the y values for each value of x fall exactly on the line.

4. The Population and the Sample • For a given value x: • Most, if not all, of the y values obtained from a particular sample will not lie on the line. • The sampled y values will be distributed aboutµy. • We can account for the difference between ŷ and µy by adding the error residual, or ε:

5. The Population and the Sample • Regression Inference • Collect a sample and estimate the population β’s by finding a regression line (Chapter 6): • The residuals e = y – ŷ are the sample-based versions of ε. • Account for the uncertainties in β0 and β1 by making confidence intervals, as we’ve done for means and proportions.

6. Assumptions and Conditions • In this order: • Linearity Assumption • Independence Assumption • Equal Variance Assumption • Normal Population Assumption

7. Assumptions and Conditions Summary of Assumptions and Conditions

8. Assumptions and Conditions Summary of Assumptions and Conditions • Make a scatterplot of the data to check for linearity. (Linearity Assumption) • Fit a regression and find the residuals, e, and predicted values ŷ. • Plot the residuals against time (if appropriate) and check for evidence of patterns (Independence Assumption). • Make a scatterplot of the residuals against x or the predicted values. This plot should not exhibit a “fan” or “cone” shape. (Equal Variance Assumption) • Make a histogram and Normal probability plot of the residuals (Normal Population Assumption)

9. The Standard Error of the Slope For a sample, we expect b1 to be close, but not equal to the model slope β1. For similar samples, the standard error of the slope is a measure of the variability of b1 about the true slope β1. Spread around the line: se Spread of the x values: sx Sample size: n

10. The Standard Error of the Slope Which of these scatterplots would give the more consistent regression slope estimate if we were to sample repeatedly from the underlying population? Hint: Compare se’s.

11. The Standard Error of the Slope Which of these scatterplots would give the more consistent regression slope estimate if we were to sample repeatedly from the underlying population? Hint: Compare sx’s.

12. The Standard Error of the Slope Which of these scatterplots would give the more consistent regression slope estimate if we were to sample repeatedly from the underlying population? Hint: Compare n’s.

13. A Test for the Regression Slope When the conditions are met, the standardized estimated regression slope, Follows a t-distribution with df = n – 2. We estimate SE(b1) with: Where sx is the ordinary standard deviation of the x’s and

14. A Test for the Regression Slope • The usual null hypothesis about the slope is that it’s equal to 0. Why? • A slope of zero says that y doesn’t tend to change linearly when x changes. In other words, if the slope equals zero, there is no linear association between the two variables. • H0: β1 = 0. This would mean that x and y are not linearly related. • Ha: β1≠ 0. This would mean . . .

15. CI for the Regression Slope When the assumptions and conditions are met, we can find a confidence interval for b1 from Where the critical value t* depends on the confidence level and has df = n – 2.

16. 16.4 A Test for the Regression Slope • Example : Soap • A soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the weight (in grams) remaining. Conditions were met so a linear regression gave the following: • Dependent variable is: Weight • R squared = 99.5% s = 2.949 • Variable Coefficient SE(Coeff) t-ratio P-value • Intercept 123.141 1.382 89.1 <0.0001 • Day -5.57476 0.1068 -52.2 <0.0001 • What is the standard deviation of the residuals? • What is the standard error of b1? • What are the hypotheses for the regression slope? • At α = 0.05, what is the conclusion?

17. 16.4 A Test for the Regression Slope • Example : Soap • A soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the weight (in grams) remaining. Conditions were met so a linear regression gave the following: • Dependent variable is: Weight • R squared = 99.5% s = 2.949 • Variable Coefficient SE(Coeff) t-ratio P-value • Intercept 123.141 1.382 89.1 <0.0001 • Day -5.57476 0.1068 -52.2 <0.0001 • What is the standard deviation of the residuals? se= 2.949What is the standard error of ? SE( ) = 0.0168

18. 16.4 A Test for the Regression Slope • Example : Soap • A soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the weight (in grams) remaining. Conditions were met so a linear regression gave the following: • Dependent variable is: Weight • R squared = 99.5% s = 2.949 • Variable Coefficient SE(Coeff) t-ratio P-value • Intercept 123.141 1.382 89.1 <0.0001 • Day -5.57476 0.1068 -52.2 <0.0001 • What are the hypotheses for the • regression slope? • At α = 0.05, what is the conclusion? Since the p-value is small (<0.0001), reject the null hypothesis. There is strong evidence of a linear relationship between Weight and Day.

19. 16.4 A Test for the Regression Slope • Example : Soap • A soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the weight (in grams) remaining. Conditions were met so a linear regression gave the following: • Dependent variable is: Weight • R squared = 99.5% s = 2.949 • Variable Coefficient SE(Coeff) t-ratio P-value • Intercept 123.141 1.382 89.1 <0.0001 • Day -5.57476 0.1068 -52.2 <0.0001 • Find a 95% confidence interval for the slope? • Interpret the 95% confidence interval for the slope? • At α = 0.05, is the confidence interval consistent with the hypothesis test conclusion?

20. 16.4 A Test for the Regression Slope • Example : SoapA soap manufacturer tested a standard bar of soap to see how long it would last. A test subject showered with the soap each day for 15 days and recorded the weight (in grams) remaining. Conditions were met so a linear regression gave the following: • Dependent variable is: Weight • R squared = 99.5% s = 2.949 • Variable Coefficient SE(Coeff) t-ratio P-value • Intercept 123.141 1.382 89.1 <0.0001 • Day -5.57476 0.1068 -52.2 <0.0001 • Find a 95% confidence interval for the slope? • Interpret the 95% confidence interval for the slope? We can be 95% confident that weight of soap decreases by between 5.34 and 5.8 grams per day. • At α = 0.05, is the confidence interval consistent with the hypothesis test conclusion? Yes, the interval does not contain zero, so reject the null hypothesis.

21. Don’t fit a linear regression to data that aren’t straight. • Watch out for changing spread. • Watch out for non-Normal errors. Check the histogram and the Normal probability plot. • Watch out for extrapolation. It is always dangerous to predict for x-values that lie far away from the center of the data.

22. Watch out for high-influence points and unusual observations. • Watch out for one-tailed tests. Most software packages perform only two-tailed tests. Adjust your P-values accordingly.

23. Looking back • Know the Assumptions and conditions for inference about regression coefficients and how to check them, in this order: LIEN • Know the components of the standard error of the slope coefficient • Test statistic • CI Interpretation