Diploma in Statistics Introduction to Regression Lecture 3.11 Lecture 3.1 Multiple Regression (continued) Review Homework Review Analysis of Variance Review

Diploma in StatisticsIntroduction to Regression

Lecture 3.1 1

Lecture 3.1Multiple Regression (continued)

• Review Homework

• Review Analysis of Variance

• Review model fitting and testing procedure

• Case study: Predicting stamp sales for An Post

– Problem formulation

– Initial data analysis

– Fitting and checking

– Application


Lecture 3.1 2

Homework 2.2.1

Extend table of predictions of small medium and large jobs to include predictions based on the final fit. Compare and contrast.

Small Medium Large Original 155 447 969 Normal Revised 138 447 975 Final 140 445 965 Original 130 422 944 Rushed Revised 100 409 937 Final 102 407 927


Lecture 3.1 3

Homework 2.2.2

You have been asked to comment, as a statistical consultant, on a prediction formula for forecasting job completion times prepared by a former employee. The formula is, effectively, the one derived from the first fit discussed above. Write a report for management. Your report should refer to

(i) the practical usefulness of the employee's prediction formula, from a customer's perspective,

(ii) the significance of the exceptional cases from the customer's and management's perspectives, and

(iii) your recommended formula, with its relative advantages.


Lecture 3.1 4

Outline solution

(i) This formula is biased upwards for small jobs and downwards for large jobs.

Also, the prediction error associated with this prediction formula is ±75 hours, that is, ± 2 working weeks.

This means that we can predict the delivery time to be anywhere in a 4 week period. This is unlikely to be acceptable to our customers who have to meet exacting scheduling requirements of their own.


Lecture 3.1 5

Outline solution

(ii) There was one small job which took an excessively long time to complete. The causes for this need to be established with a view to preventing their recurrence.

The two longest jobs were subject to excessive variability, one taking an excessively long time and the other taking a remarkably short time. Again, the causes for these need to be established, with a view to reducing variability.

In the meantime, while the recommended prediction formula (see next) may be used with caution for long jobs, the prediction error is not valid for jobs longer than around 600 hours. Further experience with longer jobs is needed to establish a valid prediction formula.


Lecture 3.1 6

Outline solution

(iii) The prediction formula is

Jobtime = 44.2 – 0.0693 × Units + 9.83 × Ops + 0.108 × T_Ops

hours, less 38 hours for Rushed jobs,

± 15 hours.

This formula is unbiased and has a suitably small prediction interval width, likely to be acceptable to our customers.


Lecture 3.1 7

Homework 2.2.3

Make a table of the t values and corresponding s values for the three regressions

Compare, contrast and explain.


Lecture 3.1 8


• Review Homework







– Application


Lecture 3.1 9

Analysis of Variance

S = 7.41272 R-Sq = 99.8% R-Sq(adj) = 99.7%


Source DF SS MS F PRegression 4 299165 74791 1361.12 0.000Residual Error 12 659 55Total 16 299824

Residual Mean Square = s2: check!


Lecture 3.1 10


Regression Sum of Squares measuresexplained variation

Residual Sum of Squares measuresunexplained (chance) variation

Total Variation = Explained + Unexplained

Coefficient of Determination:

Check it!

%Total

ExplainedR2


Lecture 3.1 11


Regression Sum of Squares measuresexplained variation

Residual Sum of Squares measuresunexplained (chance) variation

Total Variation = Explained + Unexplained

F = MS(Reg) / MS(Res)

with 4 and 12 degrees of freedom.

Check it! Check F tables.


Lecture 3.1 12

Analysis of VarianceSelected 5% critical values for the F distribution

with 1 numerator and 2 denominator degrees of freedom

1 2 3 4 5 6 7 8 10 12 24 ∞

1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 241.9 243.9 249.1 254.3 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 19.5 19.5 3 10.1 9.6 9.3 9.1 9.0 8.9 8.9 8.8 8.8 8.7 8.6 8.5 4 7.7 6.9 6.6 6.4 6.3 6.2 6.1 6.0 6.0 5.9 5.8 5.6 5 6.6 5.8 5.4 5.2 5.1 5.0 4.9 4.8 4.7 4.7 4.5 4.4 6 6.0 5.1 4.8 4.5 4.4 4.3 4.2 4.1 4.1 4.0 3.8 3.7 7 5.6 4.7 4.3 4.1 4.0 3.9 3.8 3.7 3.6 3.6 3.4 3.2 8 5.3 4.5 4.1 3.8 3.7 3.6 3.5 3.4 3.3 3.3 3.1 2.9 9 5.1 4.3 3.9 3.6 3.5 3.4 3.3 3.2 3.1 3.1 2.9 2.7

10 5.0 4.1 3.7 3.5 3.3 3.2 3.1 3.1 3.0 2.9 2.7 2.5 12 4.7 3.9 3.5 3.3 3.1 3.0 2.9 2.8 2.8 2.7 2.5 2.3 15 4.5 3.7 3.3 3.1 2.9 2.8 2.7 2.6 2.5 2.5 2.3 2.1 20 4.4 3.5 3.1 2.9 2.7 2.6 2.5 2.4 2.3 2.3 2.1 1.8 30 4.2 3.3 2.9 2.7 2.5 2.4 2.3 2.3 2.2 2.1 1.9 1.6 40 4.1 3.2 2.8 2.6 2.4 2.3 2.2 2.2 2.1 2.0 1.8 1.5

120 3.9 3.1 2.7 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.6 1.3


Lecture 3.1 13

Reduction in Prediction Error

No fit prediction error: sNo fit = sY = 202

1st fit prediction error: s1st fit = 37.5, less by factor of 5.4

2nd fit prediction error: s2nd fit = 13.8, less by factor of 2.7

3rd fit prediction error: s3rd fit = 7.4, less by factor of 1.9


Lecture 3.1 14


• Review Homework







– Application


Lecture 3.1 15

Step 1: Initial data analysis

• standard single variable summaries

– to determine extent of variation

– possible exceptional values;

• scatter plot matrix

– to view pair wise relationships between the response and the explanatory variables

and– to view pair wise relationships between the

explanatory variables themselves.


Lecture 3.1 16

Step 2: Least squares fit and interpretation

• calculate the best fitting regression coefficients

– check meaningfulness and statistical significance;

• calculate s

– check its usefulness for prediction

– its usefulness relative to alternative estimates of standard deviation.


Lecture 3.1 17

Step 3: Diagnostic analysis of residuals

• diagnostic plot

– check for exceptional residuals or patterns of residuals,

– possible explanations in terms of the fitted values;

• Normal plot

– check for exceptional residuals or non-linear patterns in the residuals


Lecture 3.1 18

Step 4: Iterate fit and check

• determine cases for deletion

– repeat steps 2 and 3 until checks are passed.


Lecture 3.1 19


• Review Homework







– Application


Lecture 3.1 20

The Stamp Sales Case Study

The problem

• January 1984, An Post established

• New business plan; sales forecasts required

• Historical sales data available


Lecture 3.1 21

Historical dataTable 1.4 Annual sales of stamps and metered mail, 1949 - 1983

Year Stamp Sales1

Meter Sales

Total Sales

Year Stamp Sales

Meter Sales

Total Sales

1949 245.2 42.0 287.2 1967 234.3 162.8 397.1 1950 224.4 48.6 273.0 1968 238.6 169.3 407.9 1951 241.3 52.1 293.4 1969 242.7 186.5 429.3 1952 251.3 60.9 312.3 1970 226.4 197.5 423.9 1953 236.7 65.8 302.5 1971 199.4 172.2 371.6 1954 231.6 69.1 300.7 1972 205.4 192.8 398.2 1955 235.8 75.1 310.8 1973 201.6 195.9 397.4 1956 253.0 90.4 343.4 1974 191.1 199.6 390.8 1957 262.6 98.1 360.7 1975 181.0 213.3 394.3 1958 265.4 104.6 370.0 1976 174.9 240.9 415.8 1959 266.0 107.5 373.4 1977 181.0 258.4 439.3 1960 278.4 112.4 390.8 1978 188.2 240.8 429.0 1961 277.7 116.9 394.6 1979 112.5 163.5 276.0 1962 235.9 105.0 340.9 1980 163.7 211.5 375.2 1963 230.0 105.2 335.2 1981 162.1 195.3 357.4 1964 234.8 121.3 356.1 1982 148.9 228.5 377.4 1965 228.8 149.0 377.8 1983 151.2 259.7 410.9 1966 230.1 153.7 383.8 1 Sales are recorded as millions of standard stamp equivalents, that is, total revenue in a year divided by

the price of a stamp for a standard sealed letter for internal delivery, and divided by 1,000,000.


Lecture 3.1 22

Trend projection?

Hire a consultant!

100

200

300

400

Data

1950 1960 1970 1980

Year

Stamp Sales

Meter Sales

Total Sales


Lecture 3.1 23

Factors influencing sales

• Economic growth

• Stamp prices

• Alternative product prices

measurement problems!


Lecture 3.1 24

Project: develop a sales forecasting system for An Post

Terms of reference

1. Identify and collect the relevant macro-economic data.

2. Establish a data base containing the data needed for model building;

3. Identify, estimate and check a dynamic regression model suitable for the purposes outlined below:


Lecture 3.1 25

(a) medium-term (one to five years) forecasting of aggregate demand for postal services;

(b) analysis of the effects of levels of general economic activity, postal prices and the prices of competing services, on aggregate demand for postal services;

(c) use as a benchmark for the analysis of the effects of demand stimulation activities.


Lecture 3.1 26

Project: develop a sales forecasting system for An Post

Terms of reference

1. Identify and collect the relevant macro-economic data.

2. Establish a data base containing the data needed for model building;

3. Identify, estimate and check a dynamic regression model suitable for the purposes outlined below:


Lecture 3.1 27

(a) medium-term (one to five years) forecasting of aggregate demand for postal services;

(b) analysis of the effects of levels of general economic activity, postal prices and the prices of competing services, on aggregate demand for postal services;

(c) use as a benchmark for the analysis of the effects of demand stimulation activities.


Lecture 3.1 28

Explanatory variables

• General economic activity:

– Gross National Product GNP

• Postal prices:

– Real Letter Price RLP

• Prices of competing services:

– Real Phone Charge RPC


Lecture 3.1 29

Definitions

• GNP measures the value of all goods and services produced by all residents of the state


Lecture 3.1 30

Definitions

Real Letter Price:

• the price of a standard sealed internal letter

divided by

• the Consumer Price Index (CPI);

measures relative change in the price of a stamp,

relative to changes in the prices of other goods and services


Lecture 3.1 31

Definitions

• Real Phone Charge:

the price of a local telephone call

divided by the

Consumer Price Index (CPI)


Lecture 3.1 32

Table 8.7 Annual postage stamp sales, GNP,real letter prices and real phone charges, 1949-1983

Year Stamp Sales

GNP RLP1 RPC2 Year Stamp Sales

GNP RLP1 RPC2

1949 245.2 552.6 1.047 0.419 1967 234.3 848.4 1.090 0.654 1950 224.4 557.0 1.031 0.413 1968 238.6 919.4 1.040 0.624 1951 241.3 564.3 0.957 0.383 1969 242.7 970.7 1.164 0.582 1952 251.3 580.1 0.880 0.352 1970 226.4 1002.6 1.206 0.714 1953 236.7 598.1 0.946 0.501 1971 199.4 1037.3 1.570 0.655 1954 231.6 603.9 0.998 0.499 1972 205.4 1112.5 1.453 0.603 1955 235.8 616.0 0.974 0.487 1973 201.6 1154.5 1.464 0.541 1956 253.0 608.3 0.934 0.622 1974 191.1 1201.7 1.526 0.557 1957 262.6 611.8 0.897 0.598 1975 181.0 1223.5 1.616 0.544 1958 265.4 600.9 0.859 0.572 1976 174.9 1229.7 1.764 0.626 1959 266.0 626.6 0.859 0.572 1977 181.0 1316.2 1.677 0.551 1960 278.4 658.9 0.855 0.570 1978 188.2 1388.4 1.598 0.639 1961 277.7 690.9 0.832 0.554 1979 112.5 1422.8 1.526 0.564 1962 235.9 716.0 0.997 0.532 1980 163.7 1462.2 1.607 0.577 1963 230.0 749.6 1.039 0.519 1981 162.1 1492.6 1.835 0.580 1964 234.8 780.7 1.113 0.730 1982 148.9 1473.2 2.114 0.601 1965 228.8 800.7 1.158 0.695 1983 151.2 1462.6 1.993 0.651 1966 230.1 806.8 1.124 0.675

1 The Real Letter Price (RLP) is the price of a standard sealed internal letter divided by the Consumer Price Index.

2 The Real Phone Charge (RPC) is the price of a local telephone call divided by the Consumer Price Index.


Lecture 3.1 33

Prediction model?Multiple linear regression

How are Stamps Sale (Y) related to

• Gross National Product (GNP = X1 ),

• Real Letter Price (RLP = X2 ),

• Real Phone Charge (RPC = X3 ) ?

Try

332211 XXXY


Lecture 3.1 34

Example

Best prediction equation:

• Predicted Sales = 343 – .0577 GNP – 53.2 RLP

• To calculate the predicted sales for any year, find the values of GNP and RLP for that year and substitute them in the equation.

Application

Evaluate the effect on sales of the industrial action in 1979.

Actual sales (1979): 112.5GNP(1979): 1,422.8RLP(1979): 1.526


Lecture 3.1 35

Application

Evaluate the effect on sales of the industrial action in 1979.

Actual sales (1979): 112.5GNP(1979): 1,422.8RLP(1979): 1.526

"Predicted" Sales:

343 – .0577 GNP – 53.2 RLP

= 343 – .0577 × 1422.8 – 53.2 × 1.526

= 179.7

Effect = 112.5 – 179.7 = – 67.2


Lecture 3.1 36


• Review Homework







– Application


Lecture 3.1 37

Step 1: Initial data analysis, dotplots


Lecture 3.1 38

Initial data analysis, time plots

1950 1960 1970 1980

Year

100

150

200

250

300

Stamp Sales

1950 1960 1970 1980

Year

600

800

1000

1200

1400

GNP

1950 1960 1970 1980

Year

1.0

1.2

1.4

1.6

1.8

2.0

RLP

1950 1960 1970 1980

Year

0.4

0.5

0.6

0.7

RPC


Lecture 3.1 39

Initial data analysis, scatterplot matrix


Lecture 3.1 40



Lecture 3.1 41



Lecture 3.1 42


• Review Homework







– Application


Lecture 3.1 43

The regression equation isStamp Sales = 300 - 0.0603 GNP - 54.6 RLP + 73.5 RPC

Predictor Coef SE Coef T PConstant 300.26 19.34 15.52 0.000GNP -0.06033 0.02406 -2.51 0.018RLP -54.57 21.15 -2.58 0.015RPC 73.52 32.83 2.24 0.032

S = 15.2996 R-Sq = 86.6%R-Sq(adj) = 85.3%


Source DF SS MS F PRegression 3 46807 15602 66.65 0.000Residual Error 31 7256 234Total 34 54063

Step 2: Regression Analysis, First Fit


Lecture 3.1 44

Exercise

Explain the Degrees of Freedom

Check the calculation of:

MS(Regression)

MS(Error)

s

R2

F

T

Check the statistical significance of the coefficients


Lecture 3.1 45

Step3: Diagnostic Analysis


Lecture 3.1 46

Step 4: Iterate the analysis, 1979 deleted


S = 9.22460

Exercise:Compare s to previous value.

Compare coefficient estimates to previous values.


Lecture 3.1 47

Compare fits


Lecture 3.1 48

1980

Diagnostic plots, 1979 deleted

after1970

up to1970


Lecture 3.1 49

19851980197519701965196019551950

300

250

200

150

100

Year

Sta

mp

Sal

es

Next step:

Model recent data, 1971-83 (\1979)


Lecture 3.1 50

Regression 1971-1983 \ 1979


S = 5.8924


Lecture 3.1 51

210200190180170160150140

3

2

1

0

-1

-2

-3

-4

Fitted Value

De

lete

d R

esi

du

alVersus Fits

(response is Stamp Sales)


Lecture 3.1 52

4

3

2

1

0

-1

-2

-3

-4210-1-2

De

lete

d R

esi

du

al

Score

N 12

AD 0.543

P-Value 0.128

Normal Probability Plot(response is Stamp Sales)


Lecture 3.1 53

Regression with 1980, RPC deleted

Predictor Coef SE Coef T PConstant 339.58 10.62 31.96 0.000GNP -0.03158 0.01329 -2.38 0.045RLP -70.155 9.660 -7.26 0.000

S = 3.92988 R-Sq = 96.8%


Lecture 3.1 54


• Review Homework







– Application


Lecture 3.1 55

Exercise

Calculate the predicted stamp sales for 1984 and 1985. Assume no change in nominal stamp price.

Compare with the actual outcomes:

1984 1985

Sales 163.6 172.1

GNP 1487.5 1466.6

RLP 1.835 1.741

Comment on the prediction errors.


Lecture 3.1 56

Exercise

Predicted Sales = 300 – .0312 GNP – 70.155 RLP

To calculate the predicted sales for any year, find the values of GNP and RLP for that year and substitute them in the equation.

Problem: how to get GNP and RLP for future years?

Answer: use "official" predictions.


Lecture 3.1 57

Central Bank predictions for 1984, 1985

1984 1985GNP: + 1.5% + 1.5%Inflation: + 8.6% + 5.5%

NB: no change in nominal stamp price in 1984 or 1985

GNP(83) = 1462.6;predicted GNP(84) = 1462.6 × 1.015 = 1484.5

RLP(83) = 1.993;

assuming no change in nominal stamp price,

predicted RLP(84) = 1.993 / 1.086 = 1.835


Lecture 3.1 58

Prediction for 1984

GNP(84) = 1484.5

RLP(84) = 1.835

Predicted Sales = 340 – .0316 × GNP – 70.155 × RLP

= 340 – .0316 × 1484.5 – 70.155 × 1.835

= 164.4

Actual outcome: 163.6

Prediction for 1985? Homework 3.1.1


Lecture 3.1 59

Homework 3.1.2

Carry out the analysis of stamp sales data prior to 1970, leading to the prediction formula

Sales = 371 – 176 RLP + 84 RPC,

s = 5.5.

Compare early and recent prediction formulas, including prediction errors.

Ref:SA pp. 282-4


Lecture 3.1 60

Reading

SA § 1.6, §8.7

Alternative readings

Hamilton, Ch 2, up to p. 53,

Ch 3, pp. 65-72, 74-75, 77-80,

Ch 4, pp. 109-117

Documents

Diploma in Statistics Introduction to Regression Lecture 3.11 Lecture 3.1 Multiple Regression (continued) Review Homework Review Analysis of Variance Review