Simple Linear Regression: Prediction. Instructor: G. William Schwert

Transcription

1 APS 425 Fall 2015 Simple Linear Regression: Prediction Instructor: G. William Schwert Ciba-Geigy Ritalin Experiment Ritalin is tested to see if it helps with Central Auditory Processing Disorder (CAPD) Similar symptoms to ADD/ADHD Experiment: Randomly select 64 children All receive auditory test 32 (control group) receive no drug (or placebo?) 32 (treatment group) receive varying doses of Ritalin All children are tested a second time (c) Prof. G. William Schwert,

2 Ciba-Geigy Ritalin Experiment DOSAGE i = amount of Ritalin received by child i Measured as Mg of Ritalin per Kg of body weight IMPROVE i = child s 2 nd test score 1 st test score Dataset A425_ritalin.wf1 also contains: AGE of child in months Gender (FEMALE = 1, for girls) Predictions (c) Prof. G. William Schwert,

3 Predictions Predictive model: IMPROVE i = DOSAGE i = the estimate of E[IMPROVE i DOSAGE i, b 0, b 1 ] What is your estimate of the average IMPROVE score for all children who receive a dosage of 0.35 mg/kg? IMPROVE i = DOSAGE i = This question asks about the average or expected value for all children who get a DOSAGE of 0.35mg/kg. Predictions A given child has been administered a DOSAGE of 0.35mg/kg. What value do you predict for the child s IMPROVE score? ^ IMPROVE i = = This question asks you to predict the value for an individual child who gets a DOSAGE of 0.35mg/kg (c) Prof. G. William Schwert,

4 Predictions The prediction of the value for an individual child = the expected value for the population (of all children with DOSAGE = 0.35mg/kg) (see previous two slides) However, standard errors are different! Let s derive them next Std Error of Predictions Standard error for predicting an individual value The linear model is: Y i = + X i + e i Our prediction is: ^ Y i = b 0 + b 1 X i Three sources of error: Error in estimating Error in estimating Error in estimating e i (c) Prof. G. William Schwert,

5 Std Error of Predictions Standard error of Y i : SDP = [ s 2 + s 2 / n + s 2 (X i X) 2 / x i 2 ] ½ _ Error term Intercept Slope uncertainty uncertainty uncertainty where s 2 ^ = e 2 i / (n-2) is the residual variance and s 2 / x 2 i is the variance of b 1 Prediction Intervals A 100(1 )% confidence interval for Y i is: [Y i - t /2 SDP, Y i + t /2 SDP] where Pr{t n-2 > t /2 }= /2 In the Ritalin case with one child getting a DOSAGE of 0.35mg/kg, we have SDP = , so a 95% confidence interval for IMPROVE i is [ (12.056), (12.056)] = [ 19.61, 28.59] (c) Prof. G. William Schwert,

6 Using Eviews to Get Prediction Intervals Redefine the workfile range so that you can generate an out-ofsample prediction Using Eviews to Get Prediction Intervals Double-click dosage to open the spreadsheet Then click Edit+/- Move to observation 65 (which will say NA ) Type in.35 into the workspace bar to enter this value for the 65 th observation (c) Prof. G. William Schwert,

7 Using Eviews to Get Prediction Intervals Upper limit Point estimate Using the regression equation predicting IMPROVE as a function of DOSAGE, click FORECAST Then specify the forecast sample as the value of DOSAGE = 0.35 you just entered Lower limit Prediction Intervals for Improvement from DOSAGE Upper 95% PI Predicted IMPROVE Lower 95% PI DOSAGE = DOSAGE (c) Prof. G. William Schwert,

8 Std Error of Prediction for the Average Standard error for predicting the expected or average value A group of children have been administered a DOSAGE of 0.35mg/kg. What value do you predict for the average IMPROVE score of these children on the test? ^ IMPROVE i = = Std Error of Prediction for the Average Now let s consider the standard error The linear model is: Y i = + X i + e i We are predicting: E[Y 0 ] = + X 0 Our prediction is: Y i = b + b X i Two sources of uncertainty: Error in estimating Error in estimating (c) Prof. G. William Schwert,

9 Std Error of Prediction for the Average Standard error of E(Y i ): SEP = [s 2 / n + s 2 (X i X) 2 / x i 2 ] ½ _ Intercept uncertainty Slope uncertainty A 100(1 )% confidence interval for E(Y i ) is: [Y i - t /2 SEP, Y i + t /2 SEP] where Pr{t n-2 > t /2 }= /2 Std Error of Prediction for the Average In the Ritalin case with one child getting a DOSAGE of 0.35mg/kg, we have SEP = 1.585, so a 95% confidence interval for E(IMPROVE i DOSAGE = 0.35, b 0, b 1 ) is [ (1.585), (1.585)] = [1.319, 7.657] (c) Prof. G. William Schwert,

10 Prediction Intervals for Expected Improvement from DOSAGE Upper 95% PI Predicted IMPROVE Lower 95% PI Note: prediction interval for expected improvement are much narrower and curvature is more apparent DOSAGE Predictions and Eviews for DOSAGE = 0.35 (SEP) 2 = s 2 / n + (SE(b 1 )) 2 (X i X) 2 = / 64 + (5.723) 2 ( ) 2 = => SEP = (SEP) 2 + (SE of Regression) 2 = (SDP) 2 [ ] 1/2 = Note that SEP and SDP depend on X i = 0.35 _ (c) Prof. G. William Schwert,

11 Predictions and Eviews The range of DOSAGE values in the data is 0 to 0.71 Given this sample, the predictive model is: ^ IMPROVE i = = We have no support from the data whether this relation extends outside of the sample range (e.g., to dosages > 0.71 mg/kg) To predict outside the sample range is called extrapolation Extrapolation is ill-advised and subject to much criticism Predictions of IMPROVE from Eviews Generate predicted values, predict, and the standard deviation of the prediction, sdp (c) Prof. G. William Schwert,

12 Prediction Interval for IMPROVE from Eviews Generate upper and lower limits for 95% prediction interval, predup and preddown Prediction Interval for IMPROVE from Eviews Create a group of dosage, predict, predup, and preddown (c) Prof. G. William Schwert,

13 Prediction Interval for IMPROVE from Eviews Graph XY line One X against all Y s Make sure that the predictor variable, DOSAGE, is the first one in the set Prediction Interval for IMPROVE from Eviews Note that the 95% prediction interval for IMPROVE covers a wide range of outcomes for individual students No assurance that any one child will improve if given Ritalin (c) Prof. G. William Schwert,

14 Standard Error of the Regression Line for IMPROVE from Eviews We will calculate the standard error of the regression line (SEP) First, save the standard error of the regression, SER Next, derive SEP from SDP and SER Standard Errors Around the Regression Line for IMPROVE Generate upper and lower limits for 95% regression line, regrup and regrdown (c) Prof. G. William Schwert,

15 Standard Errors Around the Regression Line for IMPROVE Create a group of dosage, predict, regrup, and regrdown Standard Errors Around the Regression Line for IMPROVE Graph simple scatter Make sure that the predictor variable, DOSAGE, is the first one in the set (c) Prof. G. William Schwert,

16 Standard Errors Around the Regression Line for IMPROVE Note that the 95% confidence interval for the regression line is much narrower For the dosages used in this experiment the entire interval covers positive improvement Links Ritalin Data Return to APS 425 Home Page (c) Prof. G. William Schwert,