diploma in statistics introduction to regression lecture 2.21 introduction to regression lecture 2.2...

Diploma in StatisticsIntroduction to Regression

Lecture 2.2 1

Introduction to RegressionLecture 2.2

1. Review of Lecture 2.1

– Homework– Multiple regression– Job times case study

2. Job times continued

– residual analysis– model fitting and testing

3. Model fitting and testing procedure

4. t-tests

5. Analysis of Variance


Lecture 2.2 2

Update: Accessing data files

• Access the data in mstuart's get folder:– in ISS Public Access labs, click Start, then

Network Shortcuts, open Get– on your own computer with TCD network

access, navigate to Ntserver-usr / get– once in get, type ms, open mstuart, Diploma

Reg, Excel Data,

or• Access the data on the Diploma web page at

https://www.scss.tcd.ie:453/courses/dipstats/Local/ST7002_0809.php

• Open the relevant Excel file and copy the data




Lecture 2.2 3

Homework 2.1.1The shelf life of packaged foods depends on many factors. Dry cereal (such as corn flakes) is considered to be a moisture-sensitive product, with the shelf life determined primarily by moisture. In a study of the shelf life of one brand of cereal, packets of cereal were stored in controlled conditions (23°C and 50% relative humidity) for a range of times, and moisture content was measured. The results were as follows.

Draw a scatter diagram. Comment. What action is suggested? Why?

Storage Time

0 3 6 8 10 13 16 20 24 27 30 34 37 41

Moisture Content

2.8 3.0 3.1 3.2 3.4 3.4 3.5 3.1 3.8 4.0 4.1 4.3 4.4 4.9


Lecture 2.2 4

Draw a scatter diagram. Comment. What action is suggested? Why?

2 exceptional cases; delete and investigate

Storage Time

Mois

ture

Conte

nt

403020100

5.0

4.5

4.0

3.5

3.0

Scatterplot of Moisture Content vs Storage Time


Lecture 2.2 5

Following appropriate action, the following regression was computed.

The regression equation isMoisture = 2.86 + 0.0417 Storage

Predictor Coef SE Coef T PConstant 2.86122 0.02488 115.01 0.000Storage 0.041660 0.001177 35.40 0.000

S = 0.0493475

Calculate a 95% confidence interval for the daily change in moisture content; show details.

)04401.0,03931.0(00235.004166.0)ˆ(SE2ˆ


Lecture 2.2 6

Was the action you suggested on studying the scatter diagram in part (a) justified? Explain.

Predict the moisture content of a packet of cereal stored under these conditions for 5 weeks; calculate a prediction interval.

What would be the effect on your interval of not taking the action you suggested on studying the scatter diagram? Why?

Taste tests indicate that this brand of cereal is unacceptably soggy when the moisture content exceeds 4. Based on your prediction interval, do you think that a box of cereal that has been on the shelf for 5 weeks will be acceptable? Explain.

What about 4 weeks? 3 weeks? What is acceptable?


Lecture 2.2 7







4. t-tests



Lecture 2.2 8

Example 5A production prediction problem

Erie Metal Products: The problem

Metal products fabrication:

customers order varying quantities of products of varying complexity;

customers demand accurate and precise order delivery times.


Lecture 2.2 9

Table 8.1 Times, in hours, to complete jobs with varying numbers of units, numbers of operations per unit and priority status (normal or rushed)

Order Jobtime Units Operations Normal (0)

number (hours) per unit or Rushed (1)? 1 153 100 6 0

2 192 35 11 0 3 162 127 7 1 4 240 64 12 0 5 339 600 5 1 6 185 14 16 1 7 235 96 11 1 8 506 257 13 0 9 260 21 9 1

10 161 39 8 0 11 835 426 14 0 12 586 843 6 0 13 444 391 8 0 14 240 84 13 1 15 303 235 9 1 16 775 520 12 0 17 136 76 8 1 18 271 139 11 1 19 385 165 14 1 20 451 304 10 0

Erie Metal Products: The data


Lecture 2.2 10

The multiple linear regression model

Jobtime =

Units × Units

Ops × Ops

T_Ops × T_Ops

Rushed × Rushed


Lecture 2.2 11

Model parameters

The regression coefficients:

Units, Ops, T_Ops, Rushed

The "uncertainty" parameter:

standard deviation of


Lecture 2.2 12

Regression of Jobtime on other variables

Predictor Coef SE Coef T PConstant 77.24 44.76 1.73 0.105Units -0.1507 0.1121 -1.34 0.199Ops 7.152 4.305 1.66 0.117T_Ops 0.11460 0.01322 8.67 0.000Rushed -24.94 19.11 -1.31 0.211

S = 37.4612


Lecture 2.2 13

Homework

Predict job times for

small (U=100, O=5),

medium (U=300, O=10) and

large (U=500, O=15) jobs,

both normal and rushed.

Present the results in tabular form.


Lecture 2.2 14

Homework Solution


Lecture 2.2 15

Are these predictions useful?

What is S?

What is 2S?

When will my order arrive?

NEXT

Diagnostics; analysis of residuals


Lecture 2.2 16







4. t-tests



Lecture 2.2 17

Checking model fit

Assumptions:

explanatory variables are adequate

error term ():

variation is Normal

variation is stable

Check via residuals

Response = Fit + Residual


Lecture 2.2 18

Regression diagnostics

• The diagnostic plot, 'deleted' residuals vs fitted values

– checking for homogeneity of error

• The Normal residual plot,

– checking the Normal model


Lecture 2.2 19

Residuals

Job 9, a rushed job with 21 units and 9 operations per unit, took 260 hours to complete.

Prediction

Jobtime = 77 – 0.15 × 21 + 7.1 × 9 + 0.11 × 189 – 25

= 135,

Residual = 260 – 135 = 125


Lecture 2.2 20

Deleted residuals

Job 9, a rushed job with 21 units and 9 operations per unit, took 260 hours to complete.

Deleted prediction, regression with case 9 deleted:

Jobtime = 42 – 0.08 × 21 + 10 × 9 + 0.11 × 189 - 38

= 113,

DeletedResidual = 260 – 113 = 147

Standardised deleted residual ≈ DR / s = 147 / 14

= 10.5


Lecture 2.2 21

Deleted residuals

• Residual

– observed – fitted

• Standardised Residual

– using an estimate of based on current data

• Standardised Deleted Residual

– calculated from data with suspect case deleted

– estimated from data with suspect case deleted


Lecture 2.2 22

The Diagnostic Plot

200 300 400 500 600 700 800

Fitted values

-2

0

2

4

6

8

10

Deletedresiduals


Lecture 2.2 23

Scatterplot of artificial datawith a highly exceptional case

NB: exceptionally large Y value corresponds to small X value


Lecture 2.2 24

Scatter plot and diagnostic plotfor artificial data

-1 0 1

Fitted values

-2

0

2

4

6

8

10

Deletedresiduals

-2 -1 0 1 2

X

-2

-1

0

1

2

Y


Lecture 2.2 25

Normal plot of residuals

-2 -1 0 1 2

Normal scores

-2

0

2

4

6

8

10

Deletedresiduals


Lecture 2.2 26

Statistical AnalysisSection 8.4

Iterating the analysis

• Revising the fit

– revised prediction formula

– revised diagnostics

• A further iteration


Lecture 2.2 27

Revised fit, case 9 deletedThe regression equation isJobtime = 41.7 – 0.0835 Units + 10.0 Ops + 0.110

T_Ops – 38.2 Rushed

19 cases used, 1 cases contain missing values


S = 13.7952


Lecture 2.2 28

Revised fitExercise

Predict job times for small (U=100, O=5),

medium (U=300, O=10) and

large (U=500, O=15) jobs,

normal and rushed.


Lecture 2.2 29

Revised predictions

Table 8.3 Original and revised predicted job times for small, medium and large jobs

Small Medium Large Original 155 447 969 Normal Revised 138 447 975

Original 130 422 944 Rushed Revised 100 409 937


Lecture 2.2 30

Recall scatter plot for artificial data

-2 -1 0 1 2

X

-2

-1

0

1

2

Y


Lecture 2.2 31

Revised diagnostics, case 9 deleted

-2 -1 0 1 2

Normal scores

-4

-2

0

2

4

6

Deletedresiduals

200 300 400 500 600 700 800

Fitted values

-4

-2

0

2

4

6

Deletedresiduals


Lecture 2.2 32

Revised fit, cases 9, 11, 16 deleted

The regression equation isJobtime = 44.2 – 0.0693 Units + 9.83 Ops + 0.108 T_Ops

– 38.0 Rushed


Predictor Coef SE Coef T PConstant 44.216 9.080 4.87 0.000Units –0.06931 0.02853 –2.43 0.032Ops 9.8286 0.8873 11.08 0.000T_Ops 0.107795 0.004114 26.20 0.000Rushed –37.960 3.857 –9.84 0.000

S = 7.41272


Lecture 2.2 33

Revised diagnostics, cases 9, 11, 16 deleted

-2 -1 0 1 2

Normal scores

-3

-2

-1

0

1

2

3

Deletedresiduals

200 300 400 500 600

Fitted values

-3

-2

-1

0

1

2

3

Deletedresiduals


Lecture 2.2 34

Coefficient estimates from three fits

Coefficient Units Ops T_Ops Rushed

Original fit 77 –0.15 7.2 0.11 –25 Revised fit 42 –0.08 10 0.11 –38 Final fit 44 –0.07 9.8 0.11 –38

Final s.e. 9 0.03 0.9 0.004 4


Lecture 2.2 35

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.


Lecture 2.2 36







4. t-tests



Lecture 2.2 37

The model fitting and testing procedure

• Step 1: Initial data analysis:

• Step 2: Least squares fit and interpretation:

• Step 3: Diagnostic analysis of residuals:

• Step 4: Iterate fit and check:


Lecture 2.2 38

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 2.2 39

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 2.2 40

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 2.2 41

Step 4: Iterate fit and check

• determine cases for deletion

– repeat steps 2 and 3 until checks are passed.


Lecture 2.2 42

Homework 2.2.2You 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 2.2 43







4. t-tests



Lecture 2.2 44

t-tests

First fit


– 24.9 Rushed


S = 37.4612


Lecture 2.2 45

Revised fit, case 9 deletedThe regression equation isJobtime = 41.7 – 0.0835 Units + 10.0 Ops + 0.110

T_Ops – 38.2 Rushed



S = 13.7952


Lecture 2.2 46

Revised fit, cases 9, 11, 16 deleted


– 38.0 Rushed



S = 7.41272


Lecture 2.2 47

Homework 2.2.3

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

Compare, contrast and explain.


Lecture 2.2 48







4. t-tests



Lecture 2.2 49

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 2.2 50


Regression Sum of Squares measuresexplained variation

Residual Sum of Squares measuresunexplained (chance) variation

Total Variation = Explained + Unexplained

Check it!

%Total

ExplainedR2


Lecture 2.2 51


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 2.2 52

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 2.2 53

Reading

SA §§ 8.2 - 8.6, § 1.6

Extra Notes: Degrees of Freedom

R2 and Adjusted R2

(Further Interpretation of the Correlation Coefficient)

diploma in statistics introduction to regression lecture 2.21 introduction to regression lecture 2.2...

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