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Lecture 13. Inference for regression

Lecture 13. Inference for regression. Objectives. Inference for regression (NHST Regression Inference Award)[B level award] The regression model Confidence interval for the regression slope β Testing the hypothesis of no linear relationship Inference for prediction

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Lecture 13. Inference for regression

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  1. Lecture 13. Inference for regression

  2. Objectives Inference for regression (NHST Regression Inference Award)[B level award] • The regression model • Confidence interval for the regression slope β • Testing the hypothesis of no linear relationship • Inference for prediction • Conditions for inference

  3. Most scatterplots are created from sample data. Common Questions Asked of Bivariate Relationships  Is the observed relationship statistically significant(not explained by chance events due to random sampling alone)?  What is the populationmean response my as a function of the explanatory variable x?my = a + bx

  4. The regression model The least-squares regression line ŷ = a + bxis a mathematical model of therelationship between 2 quantitative variables: “observed data = fit + residual” The regression line is the fit. For each data point in the sample, the residual is the difference (y - ŷ).

  5. At the population level, the model becomes yi = (a + bxi) + (ei)with residuals ei independent and Normally distributed N(0,s). The population mean response my is my = a + bx ŷunbiased estimate for mean responsemy aunbiased estimate for intercepta bunbiased estimate for slopeb

  6. For any fixed x, the responses y follow a Normal distribution with standard deviation s. The model has constant variance of Y (s is the same for all values of x). The regression standard error, s, for n sample data points is computed from the residuals (yi – ŷi): s is an unbiased estimate of the regression standard deviation s.

  7. Frog mating call Scientists recorded the call frequency of 20 gray tree frogs, Hyla chrysoscelis, as a function of field air temperature. MINITAB Regression Analysis: The regression equation is Frequency = - 6.19 + 2.33 Temperature Predictor Coef SE Coef T P Constant -6.190 8.243 -0.75 0.462 Temperature 2.3308 0.3468 6.72 0.000 S = 2.82159 R-Sq = 71.5% R-Sq(adj) = 69.9% s: aka “MSE”, regression standard error, unbiased estimate of s

  8. Confidence interval for the slope β Estimating the regression parameter b for the slope is a case of one-sample inference with σunknown. Hence we rely on t distributions. The standard error of the slope b is: (s is the regression standard error) Thus, a level C confidence interval for the slope b is: estimate ± t*SEestimate b± t* SEb t* is t critical for t(df = n – 2) density curve with C% between –t* and +t*

  9. Frog mating call Regression Analysis: The regression equation is Frequency = - 6.19 + 2.33 Temperature Predictor Coef SE Coef T P Constant -6.190 8.243 -0.75 0.462 Temperature 2.3308 0.3468 6.72 0.000 S = 2.82159 R-Sq = 71.5% R-Sq(adj) = 69.9% n = 20, df = 18 t* = 2.101 standard error of the slope SEb 95% CI for the slope b is: b± t*SEb = 2.3308 ± (2.101)(0.3468) = 2.3308 ± 07286, or 1.6022 to 3.0594 We are 95% confident that a 1 degree Celsius increase in temperature results in an increase in mating call frequency between 1.60 and 3.06 notes/seconds.

  10. Testing the hypothesis of no relationship To test for a significant relationship, use a sampling distribution with the parameter for the slope bis equal to zero, using a one-sample t test. The standard error of the slope b is: We test the statistical hypotheses H0: b= 0 versus a one-sided or two-sided Ha. We computet = b / SEb which has the t (n – 2) distribution to find the P-value of the test.

  11. Frog mating call Regression Analysis: The regression equation is Frequency = - 6.19 + 2.33 Temperature Predictor Coef SE Coef T P Constant -6.190 8.243 -0.75 0.462 Temperature 2.3308 0.3468 6.72 0.000 S = 2.82159 R-Sq = 71.5% R-Sq(adj) = 69.9% Two-sided P-value for H0: β = 0 We test H0: β = 0 versus Ha: β ≠ 0 • t = b / SEb= 2.3308 / 0.3468 = 6.721, with df = n – 2 = 18, • Table C: t > 3.922  P < 0.001 (two-sided test), highly significant. •  There is a significant relationship between temperature and call frequency.

  12. Testing for lack of correlation The regression slope b and the correlation coefficient r are related and b = 0 r = 0. Conversely, the population parameter for the slope β is related to the population correlation coefficient ρ, and when β= 0 ρ = 0. Thus, testing the hypothesis H0: β = 0 is the same as testing the hypothesis of no correlation between x and y in the population from which our data were drawn.

  13. Hand length and body height for a random sample of 21 adult men Regression Analysis: Hand length (mm) vs Height (m) The regression equation is Hand length (mm) = 126 + 36.4 Height (m) Predictor Coef SE Coef T P Constant 125.51 39.77 3.16 0.005 Height (m) 36.41 21.91 1.66 0.113 S = 6.74178 R-Sq = 12.7% R-Sq(adj) = 8.1%  No significant relationship

  14. Objectives (PSLS Chapter 23) Inference for regression (NHST Regression Inference Award)[B level award]Note Change • The regression model • Confidence interval for the regression slope β • Testing the hypothesis of no linear relationship • Inference for prediction • Conditions for inference

  15. Two Types of Inference for prediction 1. One use of regression is for predictionwithin range: ŷ = a + bx. But this prediction depends on the particular sample drawn. We need statistical inference to generalize our conclusions. 2. To estimate an individual responsey for a given value of x, we use a prediction interval. If we randomly sampled many times, there would be many different values of yobtained for a particular x following N(0,σ) around the mean response µy.

  16. Confidence interval for µy Predicting the population mean value of y, µy, for any value of x within the range (domain) of data tested. Using inference, we calculate a level Cconfidence intervalfor the population mean μy of all responses y when x takes the value x*: This interval is centered on ŷ, the unbiased estimate of μy.The true value of the population mean μy at a givenvalue of x, will indeed be within our confidence interval in C% of all intervals computedfrom many different random samples.

  17. A level Cprediction interval for a single observation on y when x takes the value x* is: ŷ ± t*SEŷ A level Cconfidence interval for the mean response μy at a given value x* of x is: ŷ ± t*SEm^ Use t* for a t distribution with df = n – 2

  18. Confidence interval for the mean call frequency at 22 degrees C: µy within 43.3 and 46.9 notes/sec Prediction interval for individual call frequencies at 22 degrees C: ŷ within 38.9 and 51.3 notes/sec Minitab Predicted Values for New Observations: Temperature=22 NewObs Fit SE Fit 95% CI 95% PI 1 45.088 0.863 (43.273, 46.902) (38.888, 51.287)

  19. Blood alcohol content (bac) as a function of alcohol consumed (number of beers) for a random sample of 16 adults. What do you predict for 5 beers consumed?What are typical values of BAC when 5 beers are consumed? Give a range.

  20. Conditions for inference • The observations are independent • The relationship is indeed linear (in the parameters) • The standard deviation of y, σ, is the same for all values of x • The response y varies Normallyaround its mean

  21. Using residual plots to check for regression validity The residuals (y – ŷ) give useful information about the contribution of individual data points to the overall pattern of scatter. We view the residuals in a residual plot: If residuals are scattered randomly around 0 with uniform variation, it indicates that the data fit a linear model, have Normally distributed residuals for each value of x, and constant standard deviation σ.

  22. Residuals are randomly scattered  good! Curved pattern  the shape of the relationship is not modeled correctly. Change in variability across plotσ not equal for all values of x.

  23. Frog mating call The data are a random sample of frogs. The relationship is clearly linear. The residuals are roughly Normally distributed. The spread of the residuals around 0 is fairly homogenous along all values of x.

  24. Objectives (PSLS Chapter 23) Inference for regression (NHST Regression Inference Award)[B level award]Note Change • The regression model • Confidence interval for the regression slope β • Testing the hypothesis of no linear relationship • Inference for prediction • Conditions for inference

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