【正文】 nalysis Of Variance) Compare two or more group averages. Test for equal variances (Ftest, Bartlett’s test, Levene’s test) Compare two or more group variances. Chisquare test Compare two or more group proportions. 24 Regression: Quantifies the Relationship Between X and Y Regression analysis generates a line that quantifies the relationship between X and Y. 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 X (input) Y (output) Appropriate Data for X or Y In Regression Data Type Minitab Format DiscreteOrdinal ranks 1, 2, ..., 5 Numerical DiscreteCount or Percents of defects, % defective Numerical Continuous Amounts Cycle time Numerical The line, or regression equation, is represented by mathematical equation of a straight line as: Y= bo+ b1X bo = intercept (where the line crosses X= 0) b1 = slope (rise over run, or change in Y per unit increase in X) 25 Confidence and Prediction Intervals 200 250 300165175185195205215225T e m p (F )Seal (g/cm2)Y = 1 0 1 . 6 1 1 + 0 . 3 5 4 2 3 7 XR S q = 8 3 . 3 %R e g r e s s i o n9 5 % C I9 5 % P IR e g r e ssi o n P l o tConfidence Interval ? . = An interval likely to contain the “best fit” line. ? Gives a range of the predicted values for the fitted Y if the regression is repeated again (average). ? Based on a given Xvalue ? For a given confidence Prediction Interval ? . = An interval likely to contain the actual Y values for a given X. ? Gives a range of likely actual values for Y (individual). ? Based on a given Xvalue. ? For a given confidence. 26 Three Factors: Cube Layout ? A cube helps us visualize the experimental space covered by 3 factors ? Each corner represents one of the eight sets of experimental conditions Lower left front corner = Recycled paper, no paper clip, ” (This corner represents all the Low levels: –, –, –) Upper right back corner = Copier paper, paper clip, ” wing (This corner represents all the High levels: +, +, +) Yes Copier Paper clip Paper type Wing Length No Recycled 27 Introduction to MSD 23 Experiment ?We will do an experiment that will take you through each of the steps needed to plan, carry out, and analyze a designed experiment Experiment Design Experiment Analysis 1. Identify responses 2. Identify factors 3. Select design 4. Choose factor levels 5. Randomize runs 6. Conduct experiment and collect data 7. Analyze data 8. Draw conclusions 9. Verify results 28 Focused question: What are reasons why MSDs would fail (be nondurable)? Shaded areas indicate potential causes the team thought were most likely to contribute to the problem Environment Procedures Materials People Nondurable MSDs Causes Effect Raw Material Vendor Abel Vendor Noesting Sizes Vendor Quality Small Large Bad Good Purchasing Agent Management Total Cost Warehouse Requester Specs Detail Storage Time Inventory cost High quality Low bid Personality Knowledge Out of date Lacking Packaging Storage Inspection Processing Method Date Type Batch Date Temperature Time Type Quality vendor Consistent Temperature Time Attributes Type Heat Treatment Raw Material Chemicals Inspection Cleaning Purchasing Cheapest No Specs. Analyze MSD Example 29 Main Effects Plot Here is a typical Main Effects plot (this is not the MSD data). ? The Main Effects Plot is an efficient way to see the change in the average response (Y) for each factor ? Use Pvalues from the output to discern which effects are significant (distinguishable from mon cause variation) C B A 90 85 80 75 70 Response (Y) Main Effects Plot (data means) for Response (Y) Low A High A Low B High B Low C High C Positive effect of A Negative effect of B Nonsignificant effect of C Dotted line indicates overall average 30 Interaction Plot 60 80 100 60 80 100 A B C Low A High A Low B High B Interaction Plot (data means) for Response (Y) Low B High B Low C High C Nonsignificant interaction of AB Nonsignificant interaction of AC Significant interaction of BC Average response (Y) Average response (Y) 31 Cube Plots ? Each corner represents a particular experimental condition ? The average response is labeled at each condition ? Compare responses on the faces of the cube for factor effects: ? Left to right = Effect of A ? Bottom to top = Effect of B ? Front to back = Effect of C 81 91 107 93 45 67 83 73 C B A High Low High Low High Low Cube Plot (data means) for Response (Y) 32 Confounding From observing the result, it is impossible to tell if it was caused by A alone, or B alone, or a bination of both. 33 Available Factorial Designs Minitab mands Stat DOE Factorials Create Factorial Design Display Available Designs Rows = number of experimental runs (before replication) Columns = number of factors being investigated 34 16 18 20 22 24 26 42 52 62 7 8 9 Soak Time (min.) Concentration (%) Contour Plot of Plating What is Response Surface Methodology? Response Surface Methodology refers to the design and analysis of experiments that can model curved relationships. For Response Surface Analysis, all X’s must be continuous variables. 40 15 20 Plating (mm) Soak Time (min.) 50 60 25 Concentration (%) x2 x1 y x2 x1 y = Plating (mm) Wire Diagram 35 Central Composite Designs Since we have previously covered the design of