Regression Inferential Methods

Height Weight Suppose you took many samples of the same size from this population & calculated the LSRL for each. Using the slope from each of these LSRLs – we can create a sampling distribution for the slope of the true LSRL. What is the standard deviation of the sampling distribution? What is the mean of the sampling distribution equal? What shape will this distribution have? b b b b b b b mb = b

What would you expect for other heights? Weight What would you expect for other heights? How much would an adult female weigh if she were 5 feet tall? This distribution is normally distributed. (we hope) She could weigh varying amounts – in other words, there is a distribution of weights for adult females who are 5 feet tall. What about the standard deviations of all these normal distributions? We want the standard deviations of all these normal distributions to be the same. Where would you expect the TRUE LSRL to be?

Regression Model The mean response my has a straight-line relationship with x: Where: slope b and intercept a are unknown parameters For any fixed value of x, the response y varies according to a normal distribution. Repeated responses of y are independent of each other. The standard deviation of y (sy) is the same for all values of x. (sy is also an unknown parameter)

What distribution does their weight have? Person # Ht Wt 1 64 130 10 175 15 150 19 125 21 145 40 186 47 121 60 137 63 143 68 120 70 112 78 108 83 160 Suppose we look at part of a population of adult women. These women are all 64 inches tall. What distribution does their weight have?

The slope b of the LSRL is an unbiased estimator of the true slope b. We use to estimate The slope b of the LSRL is an unbiased estimator of the true slope b. The intercept a of the LSRL is an unbiased estimator of the true intercept a. The standard error s is an unbiased estimator of the true standard deviation of y (sy). Note: df = n-2

Let’s review the regression model! x & y have a linear relationship with the true LSRL going through the my sy is the same for each x-value. For a given x-value, the responses (y) are normally distributed

Assumptions for inference on slope Linear: The true relationship is Linear - Check the scatter plot & residual plot Independent: The observations are Independent of each other - Check how collect/ 10% condition Normally: For any fixed value of x, the response y varies Normally about the true regression line. Check a histogram or boxplot of residuals Equal variance about regression line. The standard deviation of the response is constant. Check the scatter plot & residual plot Random: Data comes form a random sample or randomized experiment L I N E R

What is the slope of a horizontal line? Height Weight Suppose the LSRL has a horizontal line –would height be useful in predicting weight? A slope of zero – means that there is NO relationship between x & y!

Hypotheses Be sure to define b! H0: b = 0 1 Ha: b > 0 Ha: b < 0 This implies that there is no relationship between x & y Or that x should not be used to predict y What would the slope equal if there were a perfect relationship between x & y? H0: b = 0 Ha: b > 0 Ha: b < 0 Ha: b ≠ 0 1 Be sure to define b!

Because there are two unknowns a & b Formulas: Confidence Interval: df = n -2 Because there are two unknowns a & b

Formulas: Hypothesis test:

Inference for Linear Regression Example: Does Fidgeting Keep you Slim? In Chapter 3, we examined data from a study that investigated why some people don’t gain weight even when they overeat. Perhaps fidgeting and other “nonexercise activity” (NEA) explains why. Researchers deliberately overfed a random sample of 16 healthy young adults for 8 weeks. They measured fat gain (in kilograms) and change in energy use (in calories) from activity other than deliberate exercise for each subject. Here are the data: Inference for Linear Regression Construct and interpret a 90% confidence interval for the slope of the population regression line.

Inference for Linear Regression Example: Does Fidgeting Keep you Slim? State: We want to estimate the true slope β of the population regression line relating NEA change to fat gain at the 90% confidence level. Inference for Linear Regression Plan: If the conditions are met, we will use a t interval for the slope to estimate β. • Linear The scatterplot shows a clear linear pattern. Also, the residual plot shows a random scatter of points about the “residual = 0” line. • Independent Individual observations of fat gain should be independent if the study is carried out properly. Because researchers sampled without replacement, there have to be at least 10(16) = 160 healthy young adults in the population of interest. • Normal The histogram of the residuals is roughly symmetric and single-peaked, so there are no obvious departures from normality. • Equal variance It is hard to tell from so few points whether the scatter of points around the residual = 0 line is about the same at all x-values. • Random The subjects in this study were randomly selected to participate.

Inference for Linear Regression Do: We use the t distribution with 16 - 2 = 14 degrees of freedom to find the critical value. For a 90% confidence level, the critical value is t* = 1.761. So the 90% confidence interval for β is b ± t* SEb = −0.0034415 ± 1.761(0.0007414) = −0.0034415 ± 0.0013056 = (−0.004747,−0.002136) Conclude: We are 90% confident that the interval from -0.004747 to -0.002136 kg captures the actual slope of the population regression line relating NEA change to fat gain for healthy young adults.

Inference for Linear Regression Performing a Significance Test for the Slope Suppose the conditions for inference are met. To test the hypothesis H0 : β = hypothesized value, compute the test statistic Find the P-value by calculating the probability of getting a t statistic this large or larger in the direction specified by the alternative hypothesis Ha. Use the t distribution with df = n - 2. t Test for the Slope of a Least-Squares Regression Line Inference for Linear Regression When the conditions for inference are met, we can use the slope b of the sample regression line to construct a confidence interval for the slope β of the population (true) regression line. We can also perform a significance test to determine whether a specified value of β is plausible. The null hypothesis has the general form H0: β = hypothesized value. To do a test, standardize b to get the test statistic: To find the P-value, use a t distribution with n - 2 degrees of freedom. Here are the details for the t test for the slope.

Inference for Linear Regression Example: Crying and IQ Infants who cry easily may be more easily stimulated than others. This may be a sign of higher IQ. Child development researchers explored the relationship between the crying of infants 4 to 10 days old and their later IQ test scores. A snap of a rubber band on the sole of the foot caused the infants to cry. The researchers recorded the crying and measured its intensity by the number of peaks in the most active 20 seconds. They later measured the children’s IQ at age three years using the Stanford-Binet IQ test. A scatterplot and Minitab output for the data from a random sample of 38 infants is below. Inference for Linear Regression Do these data provide convincing evidence that there is a positive linear relationship between crying counts and IQ in the population of infants?

Inference for Linear Regression Example: Crying and IQ State: We want to perform a test of H0 : β = 0 Ha : β > 0 where β is the true slope of the population regression line relating crying count to IQ score. No significance level was given, so we’ll use α = 0.05. Inference for Linear Regression Plan: If the conditions are met, we will perform a t test for the slope β. • Linear The scatterplot suggests a moderately weak positive linear relationship between crying peaks and IQ. The residual plot shows a random scatter of points about the residual = 0 line. • Independent Later IQ scores of individual infants should be independent. Due to sampling without replacement, there have to be at least 10(38) = 380 infants in the population from which these children were selected. • Normal The Normal probability plot of the residuals shows a slight curvature, which suggests that the responses may not be Normally distributed about the line at each x-value. With such a large sample size (n = 38), however, the t procedures are robust against departures from Normality. • Equal variance The residual plot shows a fairly equal amount of scatter around the horizontal line at 0 for all x-values. • Random We are told that these 38 infants were randomly selected.

Inference for Linear Regression Example: Crying and IQ Do: With no obvious violations of the conditions, we proceed to inference. The test statistic and P-value can be found in the Minitab output. Inference for Linear Regression The Minitab output gives P = 0.004 as the P-value for a two-sided test. The P-value for the one-sided test is half of this, P = 0.002. Conclude: The P-value, 0.002, is less than our α = 0.05 significance level, so we have enough evidence to reject H0 and conclude that there is a positive linear relationship between intensity of crying and IQ score in the population of infants.

Body fat = -27.376 + 0.250 weight r = 0.697 r2 = 0.485 Example: It is difficult to accurately determine a person’s body fat percentage without immersing him or her in water. Researchers hoping to find ways to make a good estimate immersed 20 male subjects, and then measured their weights. Find the LSRL, correlation coefficient, and coefficient of determination. Body fat = -27.376 + 0.250 weight r = 0.697 r2 = 0.485

b) Explain the meaning of slope in the context of the problem. For each increase of 1 pound in weight, there is an approximate increase in .25 percent body fat. c) Explain the meaning of the coefficient of determination in context. Approximately 48.5% of the variation in body fat can be explained by the regression of body fat on weight.

a = -27.376 b = 0.25 s = 7.049 d) Estimate a, b, and s. e) Create a scatter plot and residual plot for the data. Weight Body fat Weight Residuals

f) Is there sufficient evidence that weight can be used to predict body fat? Assumptions: Scatterplot and residual plot shows Linear association. Have an Independent SRS of male subjects; 20 is less than 10% of pop Since the boxplot of residual is approximately symmetrical, the responses are approximately Normally distributed. Since the points are evenly spaced across the LSRL on the scatterplot, sy is approximately Equal for all values of weight H0: b = 0 Where b is the true slope of the LSRL of weight Ha: b ≠ 0 & body fat Since the p-value < a, I reject H0. There is sufficient evidence to suggest that weight can be used to predict body fat.

Be sure to show all graphs! g) Give a 95% confidence interval for the true slope of the LSRL. Assumptions: Scatter plot and residual plot show LINEAR association Have an INDEPENDENT SRS of male subjects Since the boxplot of residualS is approximately symmetrical, the responses are approximately NORMALLY distributed. Since the points are evenly spaced across the LSRL on the scatterplot, sy is approximately EQUAL for all values of weight We are 95% confident that the interval from 0.12 and 0.38 lbs captures the actual slope of the population LSRL relating weight and body fat. Be sure to show all graphs!

What does “s” represent (in context)? h) Here is the computer-generated result from the data: Sample size: 20 R-square = 48.5% s = 7.0491323 df? What does “s” represent (in context)? Parameter Estimate Std. Err. Intercept -27.376263 11.547428 Weight 0.24987414 0.060653996 Correlation coeficient? Be sure to write as decimal first! What does this number represent? What do these numbers represent?