STA291 Statistical Methods Lecture 27

Inference for Regression

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 µ y is the population mean of y at a given value of x. We write µ y 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.

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 about µ y.  We can account for the difference between ŷ and µ y by adding the error residual, or ε : 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 β 0 and β 1 by making confidence intervals, as we’ve done for means and proportions. The Population and the Sample

Assumptions and Conditions In this order: 1.Linearity Assumption 2.Independence Assumption 3.Equal Variance Assumption 4.Normal Population Assumption

Summary of Assumptions and Conditions Assumptions and Conditions

Summary of Assumptions and Conditions 1.Make a scatterplot of the data to check for linearity. (Linearity Assumption) 2.Fit a regression and find the residuals, e, and predicted values ŷ. 3.Plot the residuals against time (if appropriate) and check for evidence of patterns (Independence Assumption). 4.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) 5.Make a histogram and Normal probability plot of the residuals (Normal Population Assumption) Assumptions and Conditions

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 s e ’s. 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 s x ’s. 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. The Standard Error of the Slope

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 ( b 1 ) with: Where s x is the ordinary standard deviation of the x ’s and

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. H 0 : β 1 = 0. This would mean that x and y are not linearly related. H a : β 1 ≠ 0. This would mean... A Test for the Regression Slope

CI for the Regression Slope When the assumptions and conditions are met, we can find a confidence interval for  1 from Where the critical value t* depends on the confidence level and has df = n – 2.

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 = Variable Coefficient SE(Coeff) t-ratio P-value Intercept < Day < What is the standard deviation of the residuals? What is the standard error of b 1 ? What are the hypotheses for the regression slope? At α = 0.05, what is the conclusion?

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 = Variable Coefficient SE(Coeff) t-ratio P-value Intercept < Day < What is the standard deviation of the residuals? s e = What is the standard error of ? SE( ) =

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 = Variable Coefficient SE(Coeff) t-ratio P-value Intercept < Day < 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.

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 = Variable Coefficient SE(Coeff) t-ratio P-value Intercept < Day < 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?

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 = Variable Coefficient SE(Coeff) t-ratio P-value Intercept < Day < 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.

 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.

 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.

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