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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 ?

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Prediction concerning the response Y

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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?

Two different research questions

• What is the mean responseμY when the predictor value is xh?

• What value will anew observationYnew 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?

Example: Skin cancer mortality and latitude

is the best answer to each research question.

“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.

A confidence interval for the population mean response μY

… when the predictor value is xh

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

Formula in words:

Sample estimate ± (t-multiplier × standard error)

Formula in notation:

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

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, …

Does the estimate of μY when xh = 1 vary more here …?

Var N StDev

yhat(x=1) 5 0.320

… or here?

Var N StDev

yhat(x=1) 5 2.127

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

Var N StDev

yhat(x=1) 5 2.127

yhat(x=5.5) 5 0.512

Example: Skin cancer mortality and latitude

Predicted Values for New Observations

New Fit SE Fit95.0% 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

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

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

Formula in words:

Sample prediction ± (t-multiplier × standard error)

Formula in notation:

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

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 :

Prediction interval for Ynew:

Prediction of Ynewif 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?

And then reality sets in

• The mean μY is not known.

• Estimate it with the predicted response

• The cost of using

to estimateμY is the

variance of

• The variance σ2 is not known.

• Estimate it with MSE.

which is estimated by:

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 μYwith

2. the variation in Y

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

Confidence interval for μY :

Prediction interval for Ynew:

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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Example: Skin cancer mortality and latitude

Predicted Values for New Observations

New Fit SE Fit95.0% CI95.0% 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

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.