Lecture 19
Instructions for creating prediction intervals using weighted regression from the Minitab Program:
If you perform weighted regression and want to predict responses for new observations to obtain prediction intervals,
you must calculate the prediction intervals manually using the following steps.
Because of the way MSE is calculated in weighted regression, the prediction interval displayed in the output is incorrect.
If you are only interested in obtaining the fits, standard error of the fits,
or the confidence interval, you do not need to follow these steps; instead follow the steps in
To predict responses for new observations.
Preparing the data
- Create a column of 1s, the same length as the predictor and response columns.
- Create a column for each predictor containing the new observations.
The number of predictors columns for new observations must match the number of predictor columns in your original data.
- Create a column of weights for the new observations.
- Using Calc > Calculator, calculate the square roots of the weights for the new observations and store in a column called SqrtWeight.
- Using Calc > Calculator, multiply each predictor column containing the new observations by the SqrtWeight column and store in a column.
Performing weighted regression
- Choose Stat > Regression > Regression. In Response, enter the response column. In Predictors, enter the column of 1s as your first predictor. Then enter your original predictor columns.
- Click Options. In Weights, enter the column of weights for your original data. Then uncheck Fit intercept.
- In Prediction intervals for new observations, enter the SqrtWeight column you calculated in Step 4.
Then enter the predictor columns you created in Step 5, following the same order in which you entered the predictors in Step 6.
- Under Storage, check Prediction limits. Click OK in each dialog box.
Transforming prediction limits
- Using Calc > Calculator, divide the columns of stored prediction limits by the column SqrtWeight (from step 3). This transformation provides the correct prediction limits. It is important to note that if you also displayed or stored the fits, standard error of the fits, or confidence limits during this procedure, you must also divide them by the square roots of the weights to obtain the correct results.
Commands for plotting the prediction bands we just formed:
Plot 'PLIM1'*'experience' 'PLIM2'*'experience' 'salary'*'experience';
Symbol;
Type 0 0 6;
Color 1;
Size 1.0;
Connect;
Type 1 1 0;
Color 1;
Size 1;
Overlay.
Consider our (old) salary data from last time for weighted regression:
experience salary
7 26075
28 79370
23 65726
18 41983
19 62309
15 41154
24 53610
13 33697
2 22444
8 32562
20 43076
21 56000
18 58667
7 22210
2 20521
18 49727
11 33233
21 43628
4 16105
24 65644
20 63022
20 47780
15 38853
25 66537
25 67447
28 64785
26 61581
27 70678
20 51301
18 39346
1 24833
26 65929
20 41721
26 82641
28 99139
23 52624
17 50594
25 53272
26 65343
19 46216
16 54288
3 20844
12 32586
23 71235
20 36530
19 52745
27 67282
25 80931
12 32303
11 38371