Prediction concerning the response YWhere does this topic fit in?Translating two research questions into two reasonable statistical answersCould we do better by taking into account a person’s height?One thing to estimate (μy) and one thing to predict (y)Two different research questionsExample: Skin cancer mortality and latitudeSlide 8“Point estimators”It is dangerous to “extrapolate” beyond scope of model.Slide 11A confidence interval for the population mean response μYAgain, what are we estimating?(1-α)100% t-interval for mean response μYSlide 15Factors affecting the length of the confidence interval for μYDoes the estimate of μY when xh = 1 vary more here …?… or here?Does the estimate of μY vary more when xh = 1 or when xh = 5.5?Slide 20When is it okay to use the confidence interval for μY formula?Prediction interval for a new response YnewAgain, what are we predicting?(1-α)100% prediction interval for new response YnewSlide 25When is it okay to use the prediction interval for Ynew formula?What’s the difference in the two formulas?Prediction of Ynew if the mean μY is knownAnd then reality sets inVariance of the predictionWhat’s the effect of the difference in the two formulas?Slide 32Confidence intervals and prediction intervals for response in MinitabSlide 34Slide 35Slide 36A plot of the confidence interval and prediction interval in MinitabSlide 38Slide 39Slide 40Prediction concerning the response YWhere does this topic fit in?•Model formulation•Model estimation•Model evaluation•Model useTranslating two research questions into two reasonable statistical answers•What is the mean weight, μ, of all American women, aged 18-24? –If we want to estimate μ, what would be a good estimate?•What is the weight, y, of a randomly selected American woman, aged 18-24?–If we want to predict y, what would be a good prediction?62 66 70 74110120130140150160170180190200210heightweightCould we do better by taking into account a person’s height?8.158yhw 1.65.266 One thing to estimate (μy) and one thing to predict (y)54321221814106High school gpaCollege entrance test score xYEY 10 iiixY10Two different research questions•What is the mean response μY when the predictor value is xh?•What value will a new observation Ynew be when the predictor value is xh?Example: Skin cancer mortality and latitude•What is the expected (mean) mortality rate for all locations at 40o N latitude?•What is the predicted mortality rate for 1 new randomly selected location at 40o N?504030200150100LatitudeMortalityS = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Mortality = 389.189 - 5.97764 LatitudeRegression PlotExample: Skin cancer mortality and latitude1.150)40(9776.519.389ˆy“Point estimators”is the best answer to each research question.hhxbby10ˆThat is, it is:• the best guess of the mean response at xh• the best guess of a new observation at xhBut, as always, to be confident in the answer to our research question, we should put an interval around our best guess.It is dangerous to “extrapolate” beyond scope of model.654321030252015conccoloniesS = 2.67546 R-Sq = 66.8 % R-Sq(adj) = 63.5 %colonies = 16.0667 + 1.61576 concRegression PlotIt is dangerous to “extrapolate” beyond scope of model.10 5 0302010conccoloniesS = 2.74819 R-Sq = 69.6 % R-Sq(adj) = 64.5 % - 0.276956 conc**2colonies = 15.0205 + 3.22113 concRegression PlotA confidence interval for the population mean response μY… when the predictor value is xhAgain, what are we estimating?54321221814106High school gpaCollege entrance test score xYEY 10 iiixY10(1-α)100% t-interval for mean response μYFormula in notation:Formula in words:Sample estimate ± (t-multiplier × standard error) 222,21ˆxxxxnMSEtyihnhExample: Skin cancer mortality and latitudePredicted Values for New ObservationsNew Obs Fit SE Fit 95.0% CI 95.0% PI1 150.08 2.75 (144.56, 155.61) (111.23,188.93) Values of Predictors for New ObservationsNew Obs Lat1 40.0Factors affecting the length of the confidence interval for μY 222,21ˆxxxxnMSEtyihnh• As the confidence level decreases, …• As MSE decreases, …• As the sample size increases, …• The more spread out the predictor values, …• The closer xh is to the sample mean, …Does the estimate of μY when xh = 1 vary more here …?1098765432125155xy Var N StDevyhat(x=1) 5 0.320… or here?109876543213020100xy Var N StDev yhat(x=1) 5 2.127Does the estimate of μY vary more when xh = 1 or when xh = 5.5?109876543213020100xy Var N StDev yhat(x=1) 5 2.127yhat(x=5.5) 5 0.512Predicted Values for New ObservationsNew Fit SE Fit 95.0% CI 95.0% PI1 150.08 2.75 (144.6,155.6) (111.2,188.93) 2 221.82 7.42 (206.9,236.8) (180.6,263.07)X X denotes a row with X values away from the centerValues of Predictors for New ObservationsNew Obs Latitude1 40.0 Mean of Lat = 39.5332 28.0 Example: Skin cancer mortality and latitudeWhen is it okay to use the confidence interval for μY formula?•When xh is a value within the scope of the model – xh does not have to be one of the actual x values in the data set.•When the “LINE” assumptions are met.–The formula works okay even if the error terms are only approximately normal.–If you have a large sample, the error terms can even deviate substantially from normality.Prediction interval for a new response YnewAgain, what are we predicting?54321221814106High school gpaCollege entrance test score xYEY 10 iiixY10(1-α)100% prediction interval for new response YnewFormula in notation:Formula in words:Sample prediction ± (t-multiplier × standard error) 222,211ˆxxxxnMSEtyihnhExample: Skin cancer mortality and latitudePredicted Values for New ObservationsNew Obs Fit SE Fit 95.0% CI 95.0% PI1 150.08 2.75 (144.56, 155.61) (111.23,188.93) Values of Predictors for New ObservationsNew Obs Lat1 40.0When is it okay to use the prediction interval for Ynew formula?•When xh is a value within the
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