# design and analysis experiment

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### Question Description

this class is design and analysis experiment

see the files attached

only the highlighted questions

and i attached slides for chapter4

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Design of Engineering Experiments – The Blocking Principle • Text Reference, Chapter 4 • Blocking and nuisance factors • The randomized complete block design or the RCBD • Extension of the ANOVA to the RCBD • Other blocking scenarios…Latin square designs Chapter 4 1 The Blocking Principle • Blocking is a technique for dealing with nuisance factors • A nuisance factor is a factor that probably has some effect on the response, but it’s of no interest to the experimenter…however, the variability it transmits to the response needs to be minimized • Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc.), different experimental units • Many industrial experiments involve blocking (or should) • Failure to block is a common flaw in designing an experiment (consequences?) Chapter 4 2 The Hardness Testing Example • Text reference, pg 139, 140 • We wish to determine whether 4 different tips produce different (mean) hardness reading on a Rockwell hardness tester • Gauge & measurement systems capability studies are frequent areas for applying DOE • Assignment of the tips to an experimental unit; that is, a test coupon • Structure of a completely randomized experiment • The test coupons are a source of nuisance variability • Alternatively, the experimenter may want to test the tips across coupons of various hardness levels • The need for blocking Chapter 4 3 The Hardness Testing Example • Suppose that we use b = 4 blocks: • Notice the two-way structure of the experiment • Once again, we are interested in testing the equality of treatment means, but now we have to remove the variability associated with the nuisance factor (the blocks) Chapter 4 4 Extension of the ANOVA to the RCBD • Suppose that there are a treatments (factor levels) and b blocks • A statistical model (effects model) for the RCBD is  i  1, 2,..., a yij     i   j   ij   j  1, 2,..., b • The relevant (fixed effects) hypotheses are H 0 : 1  2  Chapter 4  a where i  (1/ b) j 1 (    i   j )    i b 5 Extension of the ANOVA to the RCBD ANOVA partitioning of total variability: a b  ( y i 1 j 1 ij a b  y.. )   [( yi.  y.. )  ( y. j  y.. ) 2 i 1 j 1 ( yij  yi.  y. j  y.. )]2 a b i 1 j 1  b ( yi.  y.. ) 2  a  ( y. j  y.. ) 2 a b   ( yij  yi.  y. j  y.. ) 2 i 1 j 1 SST  SSTreatments  SS Blocks  SS E Chapter 4 6 Extension of the ANOVA to the RCBD The degrees of freedom for the sums of squares in SST  SSTreatments  SSBlocks  SSE are as follows: ab  1  a  1  b  1  (a  1)(b  1) Therefore, ratios of sums of squares to their degrees of freedom result in mean squares and the ratio of the mean square for treatments to the error mean square is an F statistic that can be used to test the hypothesis of equal treatment means Chapter 4 7 ANOVA Display for the RCBD Manual computing (ugh!)…see Equations (4-9) – (4-12), page 124 Design-Expert analyzes the RCBD Chapter 4 8 Manual computing: Chapter 4 9 Chapter 4 10 Vascular Graft Example (pg. 126) • To conduct this experiment as a RCBD, assign all 4 pressures to each of the 6 batches of resin • Each batch of resin is called a “block”; that is, it’s a more homogenous experimental unit on which to test the extrusion pressures Chapter 4 11 Chapter 4 12 Vascular Graft Example Design-Expert Output Chapter 4 13 Residual Analysis for the Vascular Graft Example Chapter 4 14 Residual Analysis for the Vascular Graft Example Chapter 4 15 Residual Analysis for the Vascular Graft Example • Basic residual plots indicate that normality, constant variance assumptions are satisfied • No obvious problems with randomization • No patterns in the residuals vs. block • Can also plot residuals versus the pressure (residuals by factor) • These plots provide more information about the constant variance assumption, possible outliers Chapter 4 16 Multiple Comparisons for the Vascular Graft Example – Which Pressure is Different? Also see Figure 4.2 Chapter 4 17 Other Aspects of the RCBD See Text, Section 4.1.3, pg. 132 • The RCBD utilizes an additive model – no interaction between treatments and blocks • Treatments and/or blocks as random effects • Missing values • What are the consequences of not blocking if we should have? • Sample sizing in the RCBD? The OC curve approach can be used to determine the number of blocks to run..see page 133 Chapter 4 18 Random Blocks and/or Treatments σ2β Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 19 Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 20 Chapter 4 Design & Analysis of Experiments 8E 2012 Montgomery 21 The Latin Square Design • Text reference, Section 4.2, pg. 158 • These designs are used to simultaneously control (or eliminate) two sources of nuisance variability • A significant assumption is that the three factors (treatments, nuisance factors) do not interact • If this assumption is violated, the Latin square design will not produce valid results • Latin squares are not used as much as the RCBD in industrial experimentation Chapter 4 22 The Rocket Propellant Problem – A Latin Square Design • • • • This is a 5  5 Latin square design Page 159 shows some other Latin squares Table 4-12 (page 162) contains properties of Latin squares Statistical analysis? Chapter 4 23 Statistical Analysis of the Latin Square Design • The statistical (effects) model is  i  1, 2,..., p  yijk    i   j   k   ijk  j  1, 2,..., p k  1, 2,..., p  • The statistical analysis (ANOVA) is much like the analysis for the RCBD. • See the ANOVA table, page 160 (Table 4.10) • The analysis for the rocket propellant example follows Chapter 4 24 Chapter 4 25 Chapter 4 26 Graeco-Latin Squares = =

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