Prediction concerning the response Y

Prediction concerning the response Y

Where does this topic fit in?

• Model formulation

• Model estimation

• Model evaluation

• Model use

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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height

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ight

Could we do better by taking into account a person’s 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?

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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Latitude

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S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Mortality = 389.189 - 5.97764 Latitude

Regression Plot

Example: Skin cancer mortality and latitude

1.150)40(9776.519.389ˆ y

“Point estimators”

is the best answer to each research question.

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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.

It is dangerous to “extrapolate” beyond scope of model.

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S = 2.67546 R-Sq = 66.8 % R-Sq(adj) = 63.5 %colonies = 16.0667 + 1.61576 conc

Regression Plot

It is dangerous to “extrapolate” beyond scope of model.

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S = 2.74819 R-Sq = 69.6 % R-Sq(adj) = 64.5 % - 0.276956 conc**2colonies = 15.0205 + 3.22113 conc

Regression Plot

A confidence interval for the population mean response μY

… when the predictor value is xh

Again, what are we estimating?

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(1-α)100% t-interval for mean response μY

Formula in notation:

Formula in words:

Sample estimate ± (t-multiplier × standard error)

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nMSEty

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hnh

Example: Skin cancer mortality and latitude

Predicted Values for New Observations

New 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 Observations

New Obs Lat

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Factors affecting the length of the confidence interval for μY

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nMSEty

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• 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 …?

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Var N StDevyhat(x=1) 5 0.320

… or here?

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Var N StDev yhat(x=1) 5 2.127

Does the estimate of μY vary more when xh = 1 or when xh = 5.5?

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y Var N StDev yhat(x=1) 5 2.127yhat(x=5.5) 5 0.512

Predicted Values for New Observations

New 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 center

Values of Predictors for New ObservationsNew Obs Latitude1 40.0 Mean of Lat = 39.5332 28.0

Example: Skin cancer mortality and latitude

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.

Prediction interval for a new response Ynew

Again, what are we predicting?

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(1-α)100% prediction interval for new response Ynew

Formula in notation:

Formula in words:

Sample prediction ± (t-multiplier × standard error)

2

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2,2

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xx

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nMSEty

i

hnh

Example: Skin cancer mortality and latitude

Predicted Values for New Observations

New 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 Observations

New Obs Lat

1 40.0

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.

What’s the difference in the two formulas?

Confidence interval for μY :

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nMSEty

i

hnh

Prediction interval for Ynew:

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Prediction of Ynew if the mean μY is known

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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?

And then reality sets in

• The mean μY is not known.

– Estimate it with the predicted response y

– The cost of using y to estimate μY is the

• The variance σ2 is not known.

variance of

y

– Estimate it with MSE.

Variance of the prediction

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which is estimated by:

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

hy

What’s the effect of the difference in the two formulas?

Confidence interval for μY :

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nMSEty

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hnh

Prediction interval for Ynew:

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nMSEty

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hnh

What’s the effect of the difference in the two formulas?

• A (1-α)100% confidence interval for μY at xh will always be narrower than a (1-α)100% prediction interval for Ynew at xh.

• The confidence interval’s standard error can approach 0, whereas the prediction interval’s standard error cannot get close to 0.

Confidence intervals and prediction intervals for response in Minitab

• Stat >> Regression >> 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.

Confidence intervals and prediction intervals for response in Minitab

Confidence intervals and prediction intervals for response in Minitab

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Predicted Values for New Observations

New 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 center

Values of Predictors for New ObservationsNew Obs Latitude1 40.0 Mean of Lat = 39.5332 28.0

Example: Skin cancer mortality and latitude

A plot of the confidence interval and prediction interval in Minitab

• Stat >> Regression >> 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.

A plot of the confidence interval and prediction interval in Minitab

A plot of the confidence interval and prediction interval in Minitab

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Mortality = 389.189 - 5.97764 LatitudeS = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %

Regression

95% CI

95% PI

Regression Plot