DMAIC:%20Improve

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DMAIC Improve

• Robert Setaputra

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Objective

• Ready to develop, test, and implement solutions
  to improve the process by reducing variation in
  the critical output variables caused by the vital
  few of input variables.

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Small note

• In many cases, it is difficult to completely
  separate the activities in Measure, Analyze, and
  Improve.

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Design of Experiment (DOE)

• DOE is a collection of statistical methods for
  studying the relationships between independent
  variables, and their interactions (also called
  factors, input variables, or process variables)
  on a dependent variable (or CTQ).

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Design of Experiment (DOE)
23.5 24.6
Factors
Replications
Levels
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Design of Experiment (DOE)

• Full factorial
• All possible combinations
• No prior knowledge about the subject
• 2k k factors each with 2 levels
• 22 2 factors each with 2 levels
• Fractional factorial
• Excluding some combinations
• Preferred when it is costly to do experiments
• 2k-1 k-1 factors each with 2 levels

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Design of Experiment (DOE)

• ANOVA One Factor
• ANOVA Two Factor
• Remember Gage RR with ANOVA?

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Correlation Coefficient

• The sample correlation coefficient (r) measures
  the degree of linearity in the relationship
  between X and Y
• -1 lt r lt 1

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Correlation Analysis
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Notes on Correlation Coefficient

• Correlation is a measure of linear association
  and not necessarily causation
• Just because two variables are highly correlated,
  it does not mean that one variable is the cause
  of the other, and vice versa.

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Notes on Correlation Coefficients
How about this one? Do you think there is no
correlations between X and Y? Remember that rxy
only measures linear correlation.
Obviously, the above shows no correlations
between X and Y
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Example

• A golfer is interested in investigating the
  relationship, if any, between driving distance
  and 18-hole score

Average Driving Distance (yds.)
Average 18-Hole Score
69 71 70 70 71 69
277.6 259.5 269.1 267.0 255.6 272.9
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Example (contd)
x
y
69 71 70 70 71 69
-1.0 1.0 0 0 1.0 -1.0
277.6 259.5 269.1 267.0 255.6 272.9
10.65 -7.45 2.15 0.05 -11.35 5.95
-10.65 -7.45 0 0 -11.35 -5.95
Average
267.0
70.0
-35.40
Total
Std. Dev.
8.2192
.8944
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Example

• Correlation Coefficient

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Regression Analysis

• Simple Regression Analysis
• One predictor and one response.
• Multiple Regression Analysis
• Two or more predictors and one response.

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Simple Linear Regression

• Analyzes the relationship between two variables
• It specifies one dependent (response) variable
  and one independent (predictor) variable

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Simple Linear Regression
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Regression Model and Parameters

• Unknown parameters are
• b0 Intercept
• b1 Slope
• The assumed model for a linear relationship is
• yi b0 b1xi ei for all observations
  (i 1, 2, , n)

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Estimations

• The fitted model used to predict the expected
  value of Y for a given value of X is
• yi b0 b1xi
• The fitted coefficients are
• b0 the estimated intercept
• b1 the estimated slope

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Formulas

• yi b0 b1xi
• where

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Example

• Reed Auto periodically has a special week-long
  sale. As part of the advertising campaign Reed
  runs one or more television commercials during
  the weekend preceding the sale. Data from a
  sample of 5 previous sales are shown below.

Number of TV Ads
Number of Cars Sold
1 3 2 1 3
14 24 18 17 27
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Example

• Slope
• Intercept
• Estimated regression equation

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Assessing the Fit

• Relationship Among SST, SSR, SSE

SST SSR SSE
where SST total sum of squares SSR
sum of squares due to regression SSE
sum of squares due to error
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R2 or Coefficient of Determination

• R2 is a measure of relative fit based on a
  comparison of SSR and SST.
• 0 lt R2 lt 1
• R2 1 means that the regression fits perfectly
  (x can 100 explain the variations in y).

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R2 or Coefficient of Determination
R2 SSR/SST
where SSR sum of squares due to
regression SST total sum of squares
Note that in a simple regression, R2 (r)2
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Example

• In Reed Auto Example, the coefficient of
  determination, R2 is

R2 SSR/SST 100/114 .8772
The regression relationship is very strong 88
of the variability in the number of cars sold can
be explained by the linear relationship between
the number of TV ads and the number of cars sold.
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Hypothesis Testing

• We need to determine whether x is statistically
  significant to y
• To test for the significance, we must conduct a
  hypothesis test to determine whether the value of
  b1 is different than zero or not.

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Regression Using Excel (Reed Auto previous TV
ads example)
gtgt Tools gtgt Data Analysis gtgt Regression
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Interpreting the result

• The regression equation is
• y 10 5x
• The above means that when x 2, the model
  predicts y (that is ) to be 20.
• R2 0.8772 means that X could explain 87.72
  variations in Y.

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Interpreting the result

• Is the slope (b1) statistically significant?
• p-value for b1 is 0.01898. Using a 0.05, we
  reject Ho (since a gt p-value). Therefore we
  conclude that the slope is not equal to zero. It
  means that X is statistically influencing Y.

• The above question can be rewrite as
• Is the slope (b1) statistically different than
  zero?
• We know that the slope is 5. But our interest is
  to check whether this value, 5, is statistically
  different than zero or not.

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Reading ANOVA table

• Note that in this case K 1

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Multiple Regression

• Multiple regression is simply an extension of
  bivariate regression.
• Multiple regression includes more than one
  independent variable.
• Same concepts as in Bivariate Analysis.

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Multiple Regression

• Y is the response variable and is assumed to be
  related to the k predictors (X1, X2, Xk)
• Regression Model
• Estimated Regression Equation

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Example (Y is Price)
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Example (contd)

• Is SqFt significantly affecting Price?

p-value for b1 is 1.42561E-14 or 1.426 x 10-14
or 0.0000. Using a 0.05, we reject Ho (since a
gt p-value). Therefore we conclude that the slope
is not equal to zero. It means that SqFt is
statistically influencing Price.
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Example (contd)

• Is LotSize significantly affecting Price?

p-value for b1 is 0.00011462. Using a 0.05, we
reject Ho (since a gt p-value). Therefore we
conclude that the slope is not equal to zero. It
means that LotSize is statistically influencing
Price.
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Reading ANOVA table