Prediction concerning the response Y

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Prediction concerning the response Y
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Where does this topic fit in?

• Model formulation
• Model estimation
• Model evaluation
• Model use

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Translating 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?

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Could we do better by taking into account a
persons height?
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One thing to estimate (µy) and one thing to
predict (y)
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Two 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?

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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?

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Example Skin cancer mortality and latitude
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Point estimators

• That is, it is
• the best guess of the mean response at xh
• the best guess of a new observation at xh

But, as always, to be confident in the answer to
our research question, we should put an interval
around our best guess.
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It is dangerous to extrapolate beyond scope of
model.
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It is dangerous to extrapolate beyond scope of
model.
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A confidence interval for the population mean
response µY

• when the predictor value is xh

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Again, what are we estimating?
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(1-a)100 t-interval for mean response µY
Formula in words
Sample estimate (t-multiplier standard error)
Formula in notation
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Example Skin cancer mortality and latitude
Predicted Values for New Observations New Obs
Fit SE Fit 95.0 CI 95.0 PI 1
150.08 2.75 (144.56, 155.61)
(111.23,188.93) Values of Predictors for New
Observations New Obs Lat 1 40.0
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Factors affecting the length of the confidence
interval for µY

• 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,

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Does the estimate of µY when xh 1 vary more
here ?
Var N StDev yhat(x1) 5 0.320
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or here?
Var N StDev yhat(x1) 5 2.127
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Does the estimate of µY vary more when xh 1 or
when xh 5.5?
Var N StDev yhat(x1) 5
2.127 yhat(x5.5) 5 0.512
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Example Skin cancer mortality and latitude
Predicted Values for New Observations New Fit
SE Fit 95.0 CI 95.0 PI 1 150.08 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 center Values of
Predictors for New Observations New Obs
Latitude 1 40.0 Mean of Lat
39.533 2 28.0

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When 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.

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Prediction interval for a new response Ynew
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Again, what are we predicting?
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(1-a)100 prediction interval for new response
Ynew
Formula in words
Sample prediction (t-multiplier standard
error)
Formula in notation
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Example Skin cancer mortality and latitude
Predicted Values for New Observations New Obs
Fit SE Fit 95.0 CI 95.0 PI 1
150.08 2.75 (144.56, 155.61)
(111.23,188.93) Values of Predictors for New
Observations New Obs Lat 1 40.0
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When is it okay to use the prediction interval
for Ynew 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 for the prediction interval depends
  strongly on the assumption that the error terms
  are normally distributed.

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Whats the difference in the two formulas?
Confidence interval for µY
Prediction interval for Ynew
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Prediction of Ynew if the mean µY is known
Suppose it were known that the mean skin cancer
mortality at xh 40o N is 150 deaths per
million (with variance 400)? What is the
predicted skin cancer mortality in Columbus, Ohio?
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And then reality sets in

• The mean µY is not known.

• Estimate it with the predicted response

• The cost of using

to estimate µY is the
variance of

• The variance s2 is not known.

• Estimate it with MSE.

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Variance of the prediction
The variation in the prediction of a new response
depends on two components
1. the variation due to estimating the mean µY
with
2. the variation in Y
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Whats the effect of the difference in the two
formulas?
Confidence interval for µY
Prediction interval for Ynew
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Whats the effect of the difference in the two
formulas?

• A (1-a)100 confidence interval for µY at xh will
  always be narrower than a (1-a)100 prediction
  interval for Ynew at xh.
• The confidence intervals standard error can
  approach 0, whereas the prediction intervals
  standard error cannot get close to 0.

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Confidence intervals and prediction intervals for
response in Minitab

• Stat gtgt Regression gtgt Regression
• Specify response and predictor(s).
• Select Options
• In Prediction intervals for new observations
  box, specify either the X value or a column name
  containing multiple X values.
• Specify confidence level (default is 95).
• Click on OK. Click on OK.
• Results appear in session window.

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Confidence intervals and prediction intervals for
response in Minitab
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Confidence intervals and prediction intervals for
response in Minitab
C6 40 28
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Example Skin cancer mortality and latitude
Predicted Values for New Observations New Fit
SE Fit 95.0 CI 95.0 PI 1 150.08 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 center Values of
Predictors for New Observations New Obs
Latitude 1 40.0 Mean of Lat
39.533 2 28.0

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A plot of the confidence interval and prediction
interval in Minitab

• Stat gtgt Regression gtgt Fitted line plot
• Specify predictor and response.
• Under Options
• Select Display confidence bands.
• Select Display prediction bands.
• Specify desired confidence level (95 default)
• Select OK. Select OK.

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A plot of the confidence interval and prediction
interval in Minitab
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A plot of the confidence interval and prediction
interval in Minitab
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