Pat Hammett, University of Michigan 1 Two-Variable Analysis: Simple Linear Regression/ Correlation Topics I. Scatter Plot (X-Y Graph) II. Simple Linear Regression III. Correlation, R IV. Assessing Model Accuracy, R 2 V. Regression Abuses / Misinterpreting Correlation
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Two-Variable Analysis: Simple Linear Regression/ Correlationmeonline.engin.umich.edu/courses/blackbelt/6s... · II. Simple Linear Regression • Simple Linear Regression examines
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Pat Hammett, University of Michigan 1
Two-Variable Analysis:Simple Linear Regression/
Correlation
TopicsI. Scatter Plot (X-Y Graph)
II. Simple Linear Regression
III. Correlation, R
IV. Assessing Model Accuracy, R 2
V. Regression Abuses / Misinterpreting Correlation
Pat Hammett, University of Michigan 2
I. Scatter Plot
• Used to visualize relationship between two variables.
• Common results:Ø Linear relationshipsØ non-linear relationshipsØ No Relationships (robustness)
Scatter Plot• Shows the relationship between X (predictor)
and Y (response) given a range of X.
XIndependent Variable
Predictor Variable
YDependent VariableResponse Variable
Pat Hammett, University of Michigan 3
Example 1: Coating Thickness(From: “SPC of a Phosphate Coating Line”, Wire, J. J. Intl, May 1997, pp. 78-81.)
• Suppose you measure the efficiency of a phosphate coating operation for steel versus coating tank temperature.Ø What is the response, what is the predictor?
• Open the excel file, tanktemp.xls, which has this data file.Ø Compute the range of Y (efficiency) if you reduce
the tank temperature from 170-188 to 180-182.
Ø Is the range of Y (efficiency) smaller, larger, or the same as over the full range of X?
Ø Construct a scatter plot of this new data set? Do you still think a relationship exists?
Lecture Exercise 1:Effect on Y by reducing Variation in X
• Coating Example:
• Note: if a strong relationship exists (positive or negative) between X and Y, then reducing variation in X should result in a variation reduction in Y.
Temp (Range X)
Efficiency Ratio
(Range Y)
170-188 2.24
180-182 1.84
Pat Hammett, University of Michigan 15
Efficiency Vs. Reduced Range in Temperature
• Over the smaller range of the input (temperature), this relationship weakens.
Temperature Vs. Coating Efficiency
00.5
11.5
22.5
3
179 180 181 182 183
Temperature
Pho
spha
te C
oatin
g E
ffici
ency
Rat
io
Lessons from Coating Example
• Relationships between Y and X variables may change depending on the range of X.
• Scatter plots provide good visualization of relationships between variables, but we need a metric to assess Strength of Relationship.Ø For Two variables – we use simple linear
regression to develop a model in order to assess the strength of relationship using correlation.
Pat Hammett, University of Michigan 16
II. Simple Linear Regression
• Simple Linear Regression examines the relationship between two variables: Ø one response (y), andØ one predictor (x).
• If two variables are related, a regression equation may be used to predict a response value given a predictor value with better than random chance.
Simple Regression Equation
• Y = βo + β1X1
Ø Y = dependent variable (response)Ø X1 = independent variable (predictor)Ø β0 = intercept; the value of Y when X = 0.Ø β1 = slope; the predicted change in output Y per
unit change of input X.
• Alternatively,• Y = mX + b (m is slope, and b is y-intercept)
Pat Hammett, University of Michigan 17
Computing Slope and Intercept• We typically use software to compute the slope
and y-intercept. In Excel, we may use:Ø =slope(y-array,x-array); =intercept(y-array,x-
Lecture Exercise 3:Model Prediction and Correlation
• Suppose you are in charge of a Design for Six Sigma project to determine the appropriate pressure settings for bicycle tires?Ø Currently you produce 37 mm tires.
• One of your response variables is the coefficient of rolling friction (Cr).
• Note: lower the Cr, the better the ride.
Lecture Exercise 3:Bicycle Tire Analysis Data
• Experiment: Ø Response:
§ coefficient of rolling friction (Cr).
Ø Predictor:§ tire pressure,
Ø Target: Cr < 0.006
• Perform the following:Scatter plot (pressure Vs. Cr), fitted regression line,Correlation (R), and Assess model accuracy with R2