An extension on hypothesis testing, this lesson reviews the multiple regression and GLM as part of measuring statistical relationships.

1. Section & Lesson #:
Pre-Requisite Lessons:
Complex Tools + Clear Teaching = Powerful Results
Hypothesis Testing: Relationships
(Compare 2+ Factors)
Six Sigma-Analyze – Lesson 28
An extension on hypothesis testing, this lesson reviews the multiple
regression and GLM as part of measuring statistical relationships.
Six Sigma-Analyze #27 – Hypothesis Testing: Relationships
(Compare 1 to 1)
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted by any means
(electronic, mechanical, photographic, photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.

2. Why do we need hypothesis testing?
o Remember, our project goal is to resolve a problem by first building a transfer function.
• We don’t want to just alleviate symptoms, we want to resolve the root cause.
 Remember Hannah? We don’t want to alleviate the arthritis pain in her leg, but heal the strep throat.
• If we don’t know what the root cause is, then we need to build a transfer function.
 By building a transfer function, we can know what changes (improvements) should fix the root cause.
o Remember, the Transfer Function is defined as Y = f(X).
• This is described as “output response Y
is a function of one or more input X’s”.
• It’s part of the IPO flow model where we
described the IPO flow model as one or
more inputs feeding into a process that
transforms it to create a new output.
o How does a transfer function fit with hypothesis testing?
• Hypothesis testing tells us which X’s (inputs) are independently influencing the Y (output).
 When we reject a null hypothesis, we’re building evidence proving which X’s are “guilty” of driving the Y.
 We’ll compile all the evidence in the Improve phase of DMAIC and begin to fix those root causes.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
Y = f(X)
Input (X) > Process > Output (Y)

3. Review Hypothesis Testing: 4 Step Process
o Remember, the 4 high-level steps for hypothesis testing begin/end with being practical:
o As the heart of hypothesis testing, steps 2 & 3 can be drilled to the following 6 steps:
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
3
Practical Statistical
Problem Problem
Solution Solution
 

Practical Problem
State the problem as a
practical Yes/No question.
 Statistical Problem
Convert the problem to an
analytical question identifying
the statistical tool/method.

Practical Solution
Interpret the analytical
answer in a practical way.
Statistical Solution
Interpret the results of the
hypothesis test with an
analytical answer.

1. Define the objective.
2. State the Null Hypothesis (H0) and Alternative Hypothesis (Ha).
3. Define the confidence (1-α) and power (beta or 1-β).
4. Collect the sample data.
5. Calculate the P-value.
6. Interpret the results: accept or reject the null hypothesis (H0).

4. Is the data type for both values discrete?
What are you measuring?
Is the data normal?
SpreadCentral Tendency
Compare 1:Standard
1 Proportion Test
Compare 1:1
2 Proportion Test
Compare 2+ Factors
Chi2 Test
Compare 1:Standard
1 Sample T Test
Compare 1:1
2 Sample T Test
or Paired T Test
Compare 2+ Factors
One-way ANOVA Test
Compare 1:Standard
1 Sample Wilcoxon
or 1 Sample Sign
Compare 1:1
Mann-Whitney Test
Compare 2+ Factors
Mood’s Median Test
or Kruskal-Wallis Test
Compare 1:Standard
1 Variance Test
Compare 1:1
2 Variance Test
Compare 2+ Factors
Test for Equal Variances
Yes No
Yes No
Proportions
Compare 1:1
Pearson Correlation
or Fitted Line Plot
Compare 2+ Factors
Multiple Regression or
General Linear Model
Is the data type for both values continuous?
Relationships
No Yes
Review Finding the Right Statistical Test
o What statistical test do I use for my hypothesis testing?
• The type of statistical test depends on the data to be tested, as described in the chart below:
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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Lesson
Current

5. Multiple Regression: Introduction
o When should I use it?
• To build a regression equation by comparing more than two
continuous values to see how correlated they are.
o How do I find it in Minitab?
• Stat > Regression > Regression…
o What are the inputs for the test?
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
5
Now
Display residuals by
selecting “Four in
one” residual plots.
Select the continuous values to be compared.

6. Multiple Regression: Several Variables Example
o Example: Several sample values
• Background:
 Use all the arbitrary continuous values in Minitab Sample Data file.
• Practical Problem:
 Is there a relationship between any of the continuous values? If so, how strong?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: ρ1 = 0, ρ2 = 0, ρ3 = 0, etc. and Ha: ρ1 > 0, ρ2 > 0, ρ3 > 0, etc.
 Define the confidence (1-α) and power (1-β):
– For confidence, we’ll accept the default of 95% (which
means α = 5%) and power of 90% (which means β = 10%).
 Type the statistical problem into Minitab:
– In Minitab, go to Stat > Regression > Regression…
– In the “Variables” box, select MetricA, MetricB, MetricC,
MetricD, and MetricE from the list of columns.
– Click the “Graphs” button and select “Four in one”.
• Statistical Solution:
 Refer to the session window results.
– Each continuous value is tested against MetricA.
– Since P-value is < 0.05 (α), then we reject H0.
– r2(adj) of 94.84% suggests a strong correlation.
– Of the variables tested, only MetricE has P-value < 0.05 (α).
• Practical Solution:
 About 95% of a relationship can be explained between
the tested factors, but most of it is driven by MetricE.
– This can be proven by re-running this test with just MetricE
and the P-value and r2 are likely to remain unchanged.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
6
Now

7. General Linear Model (GLM): Introduction
o When should I use it?
• To evaluate the correlation between multiple factors - at least two
continuous variables and any number of discrete factors.
o How do I find it in Minitab?
• Stat > ANOVA > General Linear Model…
o What are the inputs for the test?
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
7
Now
Enter at least one other continuous value
Enter the Y output (a continuous value)
Enter all other metrics to test

8. General Linear Model: Interpreting Results
o How do I interpret the GLM results?
• Below is an example of the out from a GLM test.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
8
Now
A list of all
unique values for
each factor.
Just like an ANOVA,
each factor is
separately evaluated.
A list of coefficients
for the covariate
A list of all unusual
observations within
the model.
Low P-value indicates
significance and high F-
value more significance.
R2(adj) describes how
much of the model can
be explained by the
tested factors.

9. General Linear Model: Several Variables Example
o Example: Several sample values
• Background:
 Use most of the arbitrary continuous & discrete values in Minitab Sample Data file.
• Practical Problem:
 Is there a relationship between the continuous & discrete values? If so, how strong?
• Statistical Problem:
 State the null (H0) and alternative (Ha) hypotheses:
– H0: ρ1 = 0, ρ2 = 0, ρ3 = 0, etc. and Ha: ρ1 > 0, ρ2 > 0, ρ3 > 0, etc.
 Define the confidence (1-α) and power (1-β):
– For confidence, we’ll accept the default of 95% (which
means α = 5%) and power of 90% (which means β = 10%).
 Type the statistical problem into Minitab:
– In Minitab, go to Stat > ANOVA > General Linear Model…
– In the “Responses” box, select MetricA.
– In the “Model” box, select CategoryA, CategoryB, Clerk,
and Month from the list of columns.
– Click the “Covariates” button and select MetricC.
• Statistical Solution:
 Refer to the session window results.
– Since only MetricC and CategoryA have P-value < 0.05 (α),
then we reject H0 for them; it’s confirmed by high F-values.
– r2(adj) of 88.99% suggests a strong correlation.
• Practical Solution:
 About 89% of a relationship can be explained by
the MetricC an CategoryA factors.
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
9
Now

10. Practical Application
o Refer to the critical metric (output Y) and at least 5 factors (input X’s) you identified in
a previous lesson for applying to this hypothesis testing.
• For any all factors that are continuous values, try applying the multiple regression. Or if you have
a least two continuous values, then try applying all factors to the general linear model (GLM).
 Other factors in your organization can be used for this exercise.
• Before running either of these tests, do you expect there to be a relationship? If so, how strong?
• After running either of these tests, does a statistical relationship exist? If so, how strong is it?
• If the answers to the above 2 questions are different, then how does that affect how you’d
typically measure and communicate the relationship of those factors in the organization?
 For example, does the relationship between the factors affect financial decisions (e.g., how people are
compensated), or process changes (e.g., how the process may be modified), or other critical actions?
 If so, then how should the results from this test be used to influence your organization?
– Should they change how the factors are compared (e.g., across different times, locations, groups, etc.)?
– Should they change how each factor is measured?
– Should they change how they react when they compare these metrics this way?
Copyright © 2011-2019 by Matthew J. Hansen. All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted by any means (electronic, mechanical, photographic,
photocopying, recording or otherwise) without prior permission in writing by the author and/or publisher.
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