NG BB 37 Multiple Regression

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National Guard
Black Belt Training
Module 37

Multiple Regression

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CPI Roadmap – Analyze
8-STEP PROCESS
6. See
1.Validate 2. Identify 3. Set 4. Determine 5. Develop 7. Confirm 8. Standardize
Counter-
the Performance Improvement Root Counter- Results Successful
Measures
Problem Gaps Targets Cause Measures & Process Processes
Through

Define Measure Analyze Improve Control

ACTIVITIES TOOLS
• Value Stream Analysis
• Identify Potential Root Causes • Process Constraint ID
• Reduce List of Potential Root • Takt Time Analysis
Causes • Cause and Effect Analysis
• Brainstorming
• Confirm Root Cause to Output
• 5 Whys
Relationship
• Affinity Diagram
• Estimate Impact of Root Causes • Pareto
on Key Outputs • Cause and Effect Matrix
• FMEA
• Prioritize Root Causes
• Hypothesis Tests
• Complete Analyze Tollgate • ANOVA
• Chi Square
• Simple and Multiple
Regression

Note: Activities and tools vary by project. Lists provided here are not necessarily all-inclusive. UNCLASSIFIED / FOUO

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Learning Objectives
 Understand how to identify correlation with multiple
variables
 Learn how to create a mathematical model for the
effect of multiple inputs on an output variable
 Understand and identify multicollinearity
 Understand how to use best subsets to identify the
best model
 Examine unusual observations to learn more about
the data

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Multiple Regression
 In Simple Linear Regression, we
had:
Y = f(X)
 Y = B0 + B1X

 In Multiple Linear Regression,
we have:
 Y = B0 + B1X1 + B2X2 + X3
B3X3 X1
X5

 We’d like to identify which, if any,
of the predictor variables are X2
X4 Y
useful in predicting Y


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When Should I Use Multiple Regression?
Independent Variable (X)
Continuous Attribute
Dependent Variable (Y)

Continuous

Regression ANOVA
Attribute

Logistic Chi-Square (2)
Regression Test

The tool depends on the data type. Regression is typically used with a continuous
input and a continuous response but may also be used with count or categorical
inputs and outputs.

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Basic Steps for Regression Modeling
STEPS OBJECTIVES KEY QUESTIONS

1
Process Flowchart To identify KPIVs and Which KPIVs will significantly
SIPOC KPOVs improve which KPOVs?

2 Does it look like there is
Scatter Plot, To visualize the data
Histogram C&E relationship?

3 How strong is the C&E
To qualify the C&E relationship
Correlation, Test
(Strength, % Variability, P-value) relationship?
Hypothesis

4 To quantify the C&E relationship What is the prediction
Regression Analysis (Method of Least Squares) equation?

5 Is there anything suspicious
To validate the model selected
Residual Analysis with the model selected?

KPIV = Key Process Input Variables KPOV = Key Process Output Variables

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Example: Production Plant
 A chemical engineer is investigating the amount of
silver required in the high volume production of contact
switches for a new Army radio. Although only a small amount of
silver is deposited on the switches, a larger amount is wasted
through a multiple step process. She has collected data and
would like to develop a prediction model. A-06 Production
Plant
 Step 1: The variables identified as KPIVs are given below:
 X1 = Average temperature of rinse bath (degrees C)
 X2 = Speed of reel that feeds the switches through the line (inches/min)
 X3 = Thickness of silver deposit (angstroms)
 X4 = Water consumed (gallons per day) What questions
 Y = Amount of silver consumed (pounds/day)
would you ask
about this data?
Source: Applied Regression Analysis, Draper and Smith


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Visualize the Data!

Step 2:
Visualize the Data

Data file: A-06 Production
Plant.mtw
Select Graph>Matrix Plot


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Step 2: Visualize the Data!
Looking for relationships between variables...

This dialog box comes up
first

Select Matrix of Plots – Simple
Since we have only one (Y)
variable and no groups

Click on OK to go the next
Dialog box


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Double click on all of the
variables you want to include in
the Matrix, to place them in the
Graph variables box

Select Matrix Options to move
on to the next dialog box


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Select Lower left to place all
the graph labels to the
lower left of the boxes

Click on OK here and on the
previous dialog box to get
the matrix


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Correlation Table
There appear to be some relationships
between certain variables and the response.

Matrix Plot of Temp, Speed, Thickness, Water, Amt of Ag

Temp

12

10 Speed Is this
8
14.0
good or
13.5
bad? Response
Thickness
13.0
Variable
170
(Y)
160
Water
150

21

20 Amt of Ag
19
55 60 65 8 10 12 13.0 13.5 14.0 150 160 170


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Quantify the Relationships Between Variables
Step 3: Quantify the relationship

Select Stat>Basic
Statistics> Correlation


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Correlation Matrix
Evaluating coefficients of correlation among predictors...

Double click on all of the
variables you want to
include, to place them in
the Variables box

Check to display p-values
(default setting)

Click on OK to get the
Correlation Matrix in your
Session Window


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Correlation Matrix

The TOP number in
each pair is the
Pearson
Coefficient of
Correlation,
(r-Value)
While the BOTTOM
number is the
p-Value

Predictor variable pairwise correlations larger than .5-.7 are signs of
trouble ... Multicollinearity. We will explain more shortly.

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Finding the Regression Equation...
Step 4: Develop a prediction model

Select: Stat>
Regression>
Regression


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Finding the Regression Equation... (Cont.)

Double click on C5 Amt of AG
and place it in the Response:
variable box, then double
click on all the variables you
want to place in the Predictors:
box.

Select Options to go to next
dialog box.


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Finding the Regression Equation... (Cont.)

In this dialog box, the only
thing you have to do is check
Variance inflation factors

Click on OK here and on
previous dialog box to get the
regression analysis in your
Session Window


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Regression Equation
Minitab displays the following regression equation:
Amt of Ag = 5.7 - 0.0156 Temp + 0.239 Speed + 0.44 Thickness
+ 0.0449 Water

Predictor Coef SE Coef T P VIF
Constant 5.72 10.83 0.53 0.607
Temp -0.01558 0.02616 -0.60 0.563 1.276
Speed 0.2393 0.2644 0.90 0.383 10.997
Thickness 0.443 1.033 0.43 0.675 11.671
Water 0.04495 0.01481 3.04 0.010 1.731

S = 0.412748 R-Sq = 80.9% R-Sq(adj) = 74.5% The P-values indicate
whether a particular
predictor is significant
This new model R-Sq (adj) adjusts for degrees in presence of other
explains 80.9% of of freedom due to variables predictors in the
response variability that have no real value. It model
should be used when
comparing models

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Interpreting P-values
 The P columns give the significance level
for each term in the model
 Typically, if a P value is less than or equal
to 0.05, the variable is considered significant
(i.e., null hypothesis is rejected)
 If a P value is greater than 0.10, the term is removed
from the model. A practitioner might leave the term in
the model, if the P value is within the gray region
between these two probability levels


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Regression Equation
Regression output in Minitab’s Session Window

Amt of Ag = 5.7 - 0.0156 Temp + 0.239 Speed + 0.44 Thickness
+ 0.0449 Water

Predictor Coef SE Coef T P VIF
Constant 5.72 10.83 0.53 0.607 Variance Inflation Factor
Temp -0.01558 0.02616 -0.60 0.563 1.276
Speed 0.2393 0.2644 0.90 0.383 10.997
Thickness 0.443 1.033 0.43 0.675 11.671
Water 0.04495 0.01481 3.04 0.010 1.731

S = 0.412748 R-Sq = 80.9% R-Sq(adj) = 74.5%

High VIF values are signs of trouble (VIF > 10)

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Problems with Several Predictor Variables
 Sometimes the Xs are correlated (dependent). This condition is
known as Multicollinearity
 Multicollinearity can cause problems (sometimes severe)
 Estimates of the coefficients are affected (unstable, inflated
variances)
 Difficulty isolating the effects of each X
 Coefficients depend on which Xs are included in the model
 High multicollinearity inflates the standard error estimates,
which increases the P values
 If case of extreme multicollinearity, Minitab will throw out one
term and give you notice


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Graphical Representation of Multicollinearity

Variation
Explained by
X1 Total
Variation
in Y
Variation
Explained by
X2

• Overlap represents correlation
• X1 and X2 are both correlated with Y
• X1 and X2 are highly correlated
• If X1 is in the model, we don’t need X2, and
vice versa


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Assessing the Degree of Multicollinearity
 We use a metric called Variance Inflation Factor (VIF):
1
VIF  2
Select
1  Ri Stat>Regression>Regression>Options>
Display variance inflation factors
Where:
 Ri2 is the R2 value you get when you regress Xi against the other X’s
 A large Ri2 suggests that a variable is redundant
 Rule of Thumb:
 Ri2 > 0.9 is a cause for concern (high degree of collinearity) (VIF > 10)
 0.8 < Ri2 < 0.9 (moderate degree of collinearity) (VIF > 5)

 For the Production Plant data, Minitab gives us:

VIF
Temp 1.276
Speed 10.997 Two VIF’s are a bit large, but in this case with a R-sq.
Thickness 11.671 of 80.9%, some multicollinearity can be tolerated
Water 1.731

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Some Cautions About the Coefficients
 Remember the prediction equation obtained earlier:
Amt of Ag  5.7  0.0156 Temp.  0.239 Speed  0.44 Thickness  0.0449 Water

 Relative importance of predictors cannot be
determined from the size of their coefficients:
 The coefficients are scale dependent
 The coefficients are influenced by correlation among
the predictor variables
 If a high degree of multicollinearity exists, even the
signs of the coefficients may be misleading


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Residual Analysis
Step 5: Validate the selected model

Select Stat>
Regression>
Regression

Is there anything
suspicious with
this model?


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Residual Analysis (Cont.)

Double click on C5 Amt of AG
and place it in the Response
variable box, then double
click on all the variables you
want to place in the Predictors
box

Select Graphs to go to next
dialog box


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Select Four in one to get all four
Residual plots on one graph, or
you can pick and choose the plots
You want

Click on OK here and on previous
Dialog box to get Residual plots


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Not too bad overall…
Residual Plots for Amt of Ag
Normal Probability Plot Versus Fits
99
N 17
AD 0.249 0.50
90 P-Value 0.705
0.25

Residual
Percent

50 0.00
If you want to see
-0.25
10 the value for any
-0.50
1
observation, just
-1.0 -0.5 0.0 0.5 1.0 19.5 20.0 20.5 21.0 hold your cursor
21.5
Residual Fitted Value
over that point
Histogram Versus Order
4
0.50
3 0.25
Frequency

Residual

2 0.00

-0.25
1
-0.50
0
-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 2 4 6 8 10 12 14 16
Residual Observation Order


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How to Address Multicollinearity
 Eliminate one or more input variables
 We’ll look at a technique called Best Subsets
Regression
 Collect additional data
 Use process knowledge to determine the principal
relationship
 Use DOE to further assess the multicollinearity
 If neither are significant then eliminate both from the
analysis


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Best Subsets Regression
 Rather than relying on the p-values alone, the
computer looks at all possible combinations of
variables and prints the resulting model
characteristics
 Statistics like adjusted R-Sq and MSError will improve
as important model terms are added, then worsen as
“junk” terms are added to the model


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Best Subsets Regression Considerations
 Objective: We want to select a model with predictive
accuracy and minimum multicollinearity
 Seek compromise between:
 Overfitting (including model terms with only
marginal, or no, contribution)
 Underfitting (ignoring or deleting relatively
important model terms)
 What are some problems with overfitting?
overfit underfit
 What are some problems with underfitting?


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Evaluating Candidate Models
 Four things to look at when evaluating candidate models:
1. R2 (large R2 is desired, although R2 increases as we add more
predictors to the model, so this should only be used for
comparing models with the same number of terms)
2. Adjusted R2 (large is desired)
3. Mallows Cp statistic (small Cp desired, close to the number of
terms in the model)
4. s (the estimate of the standard deviation around the regression)
 Generally, the best three models are selected and checked for
significance of all factors and residual assumptions


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More on the Mallows C-p Statistic
 In practice, the minimum number of parameters needed in
the model is when the Mallows’ C-p statistic is a minimum

 Rule of Thumb:
 We want C-p  number of input variables


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Minitab data set: Production Plant

Select Stat>
Regression>
Best Subsets


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Best Subsets Regression (Cont.)

Enter Response variable

Enter Predictor variables
(Input Variables)

Click on OK to get analysis
in Session Window


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Best Subsets Regression: Amt of Ag versus Temp, Speed, Thickness, Water

Response is Amt of Ag

T
h
i
c
S k W
T p n a
e e e t
Mallows m e s e
Vars R-Sq R-Sq(adj) Cp S p d s r
1 64.4 62.0 9.4 0.50387 X
1 62.3 59.8 10.7 0.51836 X
2 80.0 77.2 1.5 0.39047 XX
What Model(s)
2 78.8 75.8 2.3 0.40200 X X are the best
3 80.6 76.1 3.2 0.39959 X X X candidates?
3 80.3 75.8 3.4 0.40237 X X X
4 80.9 74.5 5.0 0.41275 X X X X


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T
h
R-Sq: Look for the highest value i
when comparing models with the c
S k W
same number of input variables
T p n a
e e e t
Mallows m e s e
1 64.4 62.0 9.4 0.50387 X
1 62.3 59.8 10.7 0.51836 X
2 80.0 77.2 1.5 0.39047 XX
2 78.8 75.8 2.3 0.40200 X X
3 80.6 76.1 3.2 0.39959 X X X
3 80.3 75.8 3.4 0.40237 X X X
4 80.9 74.5 5.0 0.41275 X X X X


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T
R-Sq (adj): Look for the h
i
highest value when comparing c
models with different number S k W
of input variables T p n a
e e e t
Mallows m e s e
1 64.4 62.0 9.4 0.50387 X
1 62.3 59.8 10.7 0.51836 X
2 80.0 77.2 1.5 0.39047 XX
2 78.8 75.8 2.3 0.40200 X X
3 80.6 76.1 3.2 0.39959 X X X
3 80.3 75.8 3.4 0.40237 X X X
4 80.9 74.5 5.0 0.41275 X X X X


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T
Cp: Look for models where Cp is h
small and close to the number of i
c
input variables in the model S k W
T p n a
e e e t
Mallows m e s e
1 64.4 62.0 9.4 0.50387 X
1 62.3 59.8 10.7 0.51836 X
2 80.0 77.2 1.5 0.39047 XX
2 78.8 75.8 2.3 0.40200 X X
3 80.6 76.1 3.2 0.39959 X X X
3 80.3 75.8 3.4 0.40237 X X X
4 80.9 74.5 5.0 0.41275 X X X X


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T
h
S: We want S, the estimate of i
the standard deviation about c
the regression, to be as small S k W
as possible T p n a
e e e t
Mallows m e s e
1 64.4 62.0 9.4 0.50387 X
1 62.3 59.8 10.7 0.51836 X
2 80.0 77.2 1.5 0.39047 XX
2 78.8 75.8 2.3 0.40200 X X
3 80.6 76.1 3.2 0.39959 X X X
3 80.3 75.8 3.4 0.40237 X X X
4 80.9 74.5 5.0 0.41275 X X X X


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Once the Candidate Models Are Identified
 Evaluate the candidate models under a “microscope”
 Outliers
 High leverage
 Influential observations
 Residuals
 Prediction quality
 Once a model has been selected, find the new
regression equation
 Test its predictive capability for observations NOT
originally used in the modeling


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Regression with Reduced Model
We select the best model with two variables, Speed & Water,
and run Minitab again to obtain the new regression equation:

Select Stat>
Regression>
Regression


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Regression with Reduced Model (Cont.)

Enter Amt of Ag as the
Response

Enter only Speed and Water
as Predictors

Click on OK to get analysis
in Session Window


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Regression with Reduced Model (Cont.)
Session window of Minitab yields the following regression
equation for the reduced model:
Amt of Ag = 9.92 + 0.357 Speed + 0.0425 Water
Predictor Coef SE Coef T P
Constant 9.919 1.694 5.86 0.000
Speed 0.35689 0.08544 4.18 0.001
Water 0.04253 0.01206 3.53 0.003
S = 0.3905 R-Sq = 80.0% R-Sq(adj) = 77.2%

…to compare with the previous model:
Amt of Ag = 5.7 - 0.0156 Temp. + 0.239 Speed
+ 0.44 Thickness + 0.0449 Water
Constant 5.72 10.83 0.53 0.607
H20 Temp -0.01558 0.02616 -0.60 0.563
Speed 0.2393 0.2644 0.90 0.383
Thick. 0.443 1.033 0.43 0.675
Water 0.04495 0.01481 3.04 0.010
S = 0.4127 R-Sq = 80.9% R-Sq(adj) = 74.5%


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Unusual Observations

Session window of Minitab also gives us the following output:

Unusual Observations
Obs Speed Amt of A Fit SE Fit Residual St Resid
3 11.5 21.0000 20.3784 0.2477 0.6216 2.06R

R denotes an observation with a large standardized residual

An unusual observation means a large standard residual

Let’s see what would happen if we
eliminated such an observation
from our collected data!


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Impact of the Unusual Observation
Without the Unusual Observation, the Session window of Minitab
yields the following regression equation:
Constant 8.610 1.567 5.49 0.000
Speed 0.23698 0.08960 2.64 0.020
Water 0.05775 0.01226 4.71 0.000
R-Sq goes up a little
S = 0.3383 R-Sq = 85.0% R-Sq(adj) = 82.7%
because we’ve gotten rid
of “noise” in the model
…to compare with the regression equation of our
previous reduced model
Constant 9.919 1.694 5.86 0.000
Speed 0.35689 0.08544 4.18 0.001
Water 0.04253 0.01206 3.53 0.003
S = 0.3905 R-Sq = 80.0% R-Sq(adj) = 77.2%


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Takeaways
 Regression analysis can be used with historical data as well
data from designed experiments to build prediction models
 Care must be exercised when using historical data
 Correlation does not imply a cause and effect relationship
 There may be serious problems with multicollinearity and
high leverage observations
 There are several diagnostic tools available to evaluate
regression models:
 Fit: R2, adjusted R2, Cp, S
 Unusual observations: residual plots, leverage, CooksD
 Multicollinearity: VIFs (Variance Inflation Factors)


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Considerations in Regression
 Set goals before doing the analysis (what do you want to learn,
how well do you need to predict, etc.).
 Gather enough observations to adequately measure error and
check the model assumptions.
 Make sure that the sample of data is representative of the
population.
 Excessive measurement error of the inputs (Xs) creates
uncertainty in the estimated coefficients, predictions, etc.
 Be sure to collect data on all potentially important explanatory
variables.


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Regression Checklist
 Scatterplots (Y vs. X)
 Histograms and/or Boxplots of Ys and Xs
 Coefficients
 Significance (p < .05 - .10)
 R2 and adjusted R2
 S
 Residuals (no obvious pattern)
 Unusual Y values (standardized residuals > 2)
 Unusual X values (leverage > 2p/n)

 Overfitting vs. underfitting (C-p  number of input variables in model)
 Multicollinearity (VIF > 5-10)


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What other comments or questions
do you have?

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References
 Neter, Wasserman, and Kutner, Applied Linear Regression Models, Irwin, 1989
 Draper and Smith, Applied Regression Analysis, Wiley, 1981
 Schulman, Robert S., Statistics in Plain English, Chapman and Hall, 1992.
 Gunst and Mason, Regression Analysis and its Application, Marcel Dekker, 1980
 Myers, Raymond H., Classical and Modern Regression with Applications,
Duxbury, 1990
 Dielman, Applied Regression Analysis for Business and Economics, Duxbury,
1991
 Hosmer and Lemeshow, Applied Logistic Regression, Wiley, 1989
 Iglewicz and Hoaglin, How to Detect and Handle Outliers, ASQ Press
 Crocker, Douglas C., How to use Regression Analysis in Quality Control, ASQ
Press


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National Guard
Black Belt Training
APPENDIX
Additional Exercises
 Anthony’s Pizza
 Customer Satisfaction
 A Study of Supervisor
Performance
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Additional Practice Example:
Anthony’s Pizza
 We have received Voice of the Customer feedback
telling us that customers are dissatisfied if we cannot
accurately predict the time of their pizza delivery when
it is beyond the 30 minute target
 We would like to develop a model so that when the
customer calls, we can accurately predict delivery time


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Additional Practice Example:
Six Sigma Pizza
 Our Minitab data can be found in the file Multiple
Regression - Pizza.mpj
 Based on the data that we have collected, we are going to
study the effects of total pizzas ordered, defects, and
incorrect order on delivery time


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Additional Practice Exercise:
Customer Satisfaction
 Bob Black Belt would like to get a better understanding of the
customer satisfaction data
 Use the data provided in the Minitab file A-06 Customer
Satisfaction Data.mtw to create a Regression Model to predict
Overall Satisfaction

Each row of data is a monthly average of how customers rated the
services on a scale of 1-10. For example, in January, the average
of customer ratings for Staff Responsiveness was a 7.9.


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Customer Satisfaction (Cont.)
 Consider Staff Responsiveness, Check-out Speed,
Frequent Guest Program, and Problems Resolved as
possible inputs that could be used to predict Overall
Satisfaction.
 First, study correlation with a Matrix Plot and Correlation
Table
 Next, create the initial Regression Model
 Find the best combination of inputs with Best Subsets
 Finally, run the reduced Regression Model


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A Study of Supervisor Performance
 A recent survey of clerical employees in a large financial organization
included questions related to employee satisfaction with their
supervisors. The company was interested in any relationships between
specific supervisor characteristics and overall satisfaction with
supervisors as perceived by the employees,
 Y = Overall rating of the job being done by the supervisor
 X1 = Handles employee complaints
 X2 = Does not allow special privileges
 X3 = Provides opportunity to learn new things
 X4 = Raises based on performance
 X5 = Too critical of poor performance
 X6 = Rate of advancing to better jobs (employee’s perception
of their own advancement rate)

Source: Regression Analysis by Example, Chatterjee and Price


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A Study of Supervisor Performance
 The survey responses were on a scale of 1-5
 For purposes of analysis, a score of 1 or 2 was considered
“favorable”, while a score of 3, 4, or 5 was considered “unfavorable”
 Data was collected from 30 departments, selected randomly form
the organization. Each department had approximately 35 employees
with one supervisor
 For each department, the data was aggregated and the data
recorded was the percent favorable for each item
 Data file is A-06 Attitude.mtw
 Questions:
 Can we predict the overall supervisor rating using this data?
 What variable(s) have the strongest correlation with the supervisor rating?
 Are there any unusual observations?
 Comments on the data?


NG BB 37 Multiple Regression

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NG BB 37 Multiple Regression