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Chapter 23 Inference for Regression BPS - 5th Ed. Chapter 23 1 Linear Regression (from Chapter 5) Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects with a given value of the explanatory variable. BPS - 5th Ed. Chapter 23 2 Case Study Crying and IQ Karelitz, S. et al., “Relation of crying activity in early infancy to speech and intellectual development at age three years,” Child Development, 35 (1964), pp. 769-777. Researchers explored the crying of infants four to ten days old and their IQ test scores at age three to determine if more crying was a sign of higher IQ BPS - 5th Ed. Chapter 23 3 Case Study Crying and IQ Data collection Data collected on 38 infants Snap of rubber band on foot caused infants to cry – recorded the number of peaks in the most active 20 seconds of crying (explanatory variable x) Measured IQ score at age three years using the Stanford-Binet IQ test (response variable y) BPS - 5th Ed. Chapter 23 4 Case Study Crying and IQ Data TABLE 23.1 BPS - 5th Ed. Chapter 23 5 Case Study Crying and IQ Data analysis Scatterplot of y vs. x shows a moderate positive linear relationship, with no extreme outliers or potential influential observations BPS - 5th Ed. Chapter 23 6 Case Study Crying and IQ Data analysis Correlation between crying and IQ is r = 0.455 (from formula in Chapter 4) Least-squares regression line for predicting IQ from crying is (as in Ch. 5) R2 = 0.207, so 21% of the variation in IQ scores is explained by crying intensity BPS - 5th Ed. Chapter 23 7 Inference We now want to extend our analysis to include inferences on various components involved in the regression analysis – slope – intercept – correlation – predictions BPS - 5th Ed. Chapter 23 8 Regression Model, Assumptions Conditions required for inference about regression (have n observations on an explanatory variable x and a response variable y) 1. for any fixed value of x, the response y varies according to a Normal distribution. Repeated responses y are independent of each other. 2. the mean response µy has a straight-line relationship with x: µy = + x . The slope and intercept are unknown parameters. 3. the standard deviation of y (call it ) is the same for all values of x. The value of is unknown. BPS - 5th Ed. Chapter 23 9 Regression Model, Assumptions regression model has three parameters: , , and the true regression line µy = + x says that the mean response µy moves along a straight line as x changes (we cannot observe the true regression line; instead we observe y for various values of x) observed values of y vary about their means µy according to a Normal distribution (if we take many y observations at a fixed value of x, the Normal pattern will appear for these y values) the BPS - 5th Ed. Chapter 23 10 Regression Model, Assumptions standard deviation is the same for all values of x, meaning the Normal distributions for y have the same spread at each value of x the BPS - 5th Ed. Chapter 23 11 Estimating Parameters: Slope and Intercept When using the least-squares regression line , the slope b is an unbiased estimator of the true slope , and the intercept a is an unbiased estimator of the true intercept BPS - 5th Ed. Chapter 23 12 Estimating Parameters: Standard Deviation standard deviation describes the variability of the response y about the true regression line a residual is the difference between an observed value of y and the value predicted by the leastsquares regression line: the standard deviation is estimated with a sample standard deviation of the residuals (this is a standard error since it is estimated from data) the BPS - 5th Ed. Chapter 23 13 Estimating Parameters: Standard Deviation The regression standard error is the square root of the sum of squared residuals divided by their degrees of freedom (n2): BPS - 5th Ed. Chapter 23 14 Case Study Crying and IQ yˆ a bx 91.27 1.493 x, b = 1.493 is an unbiased estimator of the true slope , and a = 91.27 is an unbiased estimator of the true intercept Since – because the slope b = 1.493, we estimate that on the average IQ is about 1.5 points higher for each added crying peak. The regression standard error is s = 17.50 – see pages 600-601 in the text for this calculation BPS - 5th Ed. Chapter 23 15 Hypothesis Tests for Slope (test for no linear relationship) The most common hypothesis to test regarding the slope is that it is zero: H 0: = 0 – says regression line is horizontal (the mean of y does not change with x) – no true linear relationship between x and y – the straight-line dependence on x is of no value for predicting y Standardize BPS - 5th Ed. b to get a t test statistic: Chapter 23 16 Hypothesis Tests for Slope the standard error of b is a multiple of the regression standard error: Test statistic for H0: = 0 : – follows t distribution with df = n2 BPS - 5th Ed. Chapter 23 17 Hypothesis Tests for Slope P-value: [for T ~ t(n2) distribution] Ha: > 0 : P-value = P(T t) Ha: < 0 : P-value = P(T t) Ha: 0 : P-value = 2P(T |t|) BPS - 5th Ed. Chapter 23 18 Case Study Crying and IQ Hypothesis Test for slope t = b / SEb = 1.4929 / 0.4870 = 3.07 P-value Significant linear relationship BPS - 5th Ed. Chapter 23 19 Test for Correlation The correlation between x and y is closely related to the slope (for both the population and the observed data) – in particular, the correlation is 0 exactly when the slope is 0 testing H0: = 0 is equivalent to testing that there is no correlation between x and y in the population from which the data were drawn Therefore, BPS - 5th Ed. Chapter 23 20 Test for Correlation The t-score simplifies to: Degrees BPS - 5th Ed. of freedom: n-2 Chapter 23 21 Test for Correlation There does exist a test for correlation that does not require a regression analysis – Table E on page 695 of the text gives critical values and upper tail probabilities for the sample correlation r under the null hypothesis that the correlation is 0 in the population look up n and r in the table (if r is negative, look up its positive value), and read off the associated probability from the top margin of the table to obtain the P-value just as is done for the t table (Table C) BPS - 5th Ed. Chapter 23 22 Case Study Crying and IQ Test for H0: correlation = 0 Correlation between crying and IQ is r = 0.455 Sample size is n=38 From Table E: for Ha: correlation > 0 , the P-value is between .001 and .0025 (using n=40) – P-value for two-sided test is between .002 and .005 (matches two-sided P-value for test on slope) – one-sided P-value would be between .005 and .01 if we were very conservative and used n=30 BPS - 5th Ed. Chapter 23 23 Confidence Interval for Slope A level C confidence interval for the true slope is b t* SEb – t* is the critical value for the t distribution with df = n2 degrees of freedom that has area (1C)/2 to the right of it – recall, the standard error of b is a multiple of the regression standard error: BPS - 5th Ed. Chapter 23 24 Case Study Crying and IQ Confidence interval for slope b=1.4929, SEb = 0.4870, df = n2 = 382 = 36 (df = 36 is not in Table C, so use next smaller df = 30) For a 95% C.I., (1C)/2 = .025, and t* = 2.042 So a 95% C.I. for the true slope is: b t* SEb = 1.4929 2.042(0.4870) = 1.4929 0.9944 = 0.4985 to 2.4873 BPS - 5th Ed. Chapter 23 25 Inference about Prediction Once a regression line is fit to the data, it is useful to obtain a prediction of the response for a particular value of the explanatory variable ( x* ); this is done by substituting the value of x* into the equation of the line ( ) for x in order to calculate the predicted value We now present confidence intervals that describe how accurate this prediction is BPS - 5th Ed. Chapter 23 26 Inference about Prediction There are two types of predictions – predicting the mean response of all subjects with a certain value x* of the explanatory variable – predicting the individual response for one subject with a certain value x* of the explanatory variable Predicted values ( ) are the same for each case, but the margin of error is different BPS - 5th Ed. Chapter 23 27 Inference about Prediction To estimate the mean response µy, use an ordinary confidence interval for the parameter µy = + x* – µy is the mean of responses y when x = x* – 95% confidence interval: in repeated samples of n observations, 95% of the confidence intervals calculated (at x*) from these samples will contain the true value of µy at x* BPS - 5th Ed. Chapter 23 28 Inference about Prediction To estimate an individual response y, use a prediction interval – estimates a single random response y rather than a parameter like µy – 95% prediction interval: take an observation on y for each of the n values of x in the original data, then take one more observation y at x = x*; the prediction interval from the n observations will cover the one more y in 95% of all repetitions BPS - 5th Ed. Chapter 23 29 Inference about Prediction BPS - 5th Ed. Chapter 23 30 Inference about Prediction Both confidence interval and prediction interval have the same form. – both t* values have df = n2 – the standard errors (SE) differ for the two intervals (formulas on previous slide) BPS - 5th Ed. the prediction interval is wider than the confidence interval Chapter 23 31 Residual Plots x = number of beers y = blood alcohol Residuals: -2 731 -1 871 -0 91 0 5578 1 1 2 39 3 (4|1 = .041) 4 1 (close to Normal) Roughly linear relationship; spread is even across entire data range (‘random’ scatter about zero) BPS - 5th Ed. Chapter 23 32 Residual Plots ‘x’ = collection of explanatory variables, y = salary of player Standard deviation is not constant everywhere (more variation among players with higher salaries) BPS - 5th Ed. Chapter 23 33 Residual Plots x = number of years, y = logarithm of salary of player A clear curved pattern – relationship is not linear BPS - 5th Ed. Chapter 23 34 Checking Assumptions Independent observations – no repeated observations on the same individual True relationship is linear – look at scatterplot to check overall pattern – plot of residuals against x magnifies any unusual pattern (should see ‘random’ scatter about zero) BPS - 5th Ed. Chapter 23 35 Checking Assumptions standard deviation σ of the response at all x values Constant – scatterplot: spread of data points about the regression line should be similar over the entire range of the data – easier to see with a plot of residuals against x, with a horizontal line drawn at zero (should see ‘random’ scatter about zero) (or plot residuals against yˆ for linear regr.) BPS - 5th Ed. Chapter 23 36 Checking Assumptions Response y varies Normally about the true regression line – residuals estimate the deviations of the response from the true regression line, so they should follow a Normal distribution make histogram or stemplot of the residuals and check for clear skewness or other departures from Normality – numerous methods for carefully checking Normality exists (talk to a statistician!) BPS - 5th Ed. Chapter 23 37