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Design Of Experiment is a statistical tool first developed by Sir R A Fisher, Ronald Aylmer in 1920, while estimating the impact of different input variable, designated as Factors, on single output variable designated as Levels. Factorial design can be used to interpolate the main effects (effect of dependent variable on independent variable) and interaction effects (effect of interaction between dependent variables on the independent variable). As such, design of experiment is classified as: Full Factorial Design of Experiment A full factorial design involves all possible factor combinations in a design of experiment, and, most importantly, varies the factors (input variables) simultaneously rather than one factor at a time. A confounding variable is a variable, that has effect on supposed cause and the supposed effect. A confounding variable is corelated to independent variable, and shares causal relationship with dependent variable. Higher day Temperature may be considered as a confounding variable having association with ice cream eating tendency of people and sun cream usage by people. Confounding effect refers to eliminating the block effect of treatment factor and considering treatment effects being contributed by estimation of main effects and interaction effects from linear combination of the experimental observations. So, in a 23-factorial design of experiment, the estimation for three-way interaction (ABC interaction) is eliminated, by confounding with the block. The confounding effect is generally observed, when full factorial designs are run in blocks and, the block size is smaller than the number of different treatment combinations. Blocking in Design of Experiments refers to arranging data samples/units in groups or blocks that are similar to one another. Replication in Design of Experiments refers to repeating experimental situation by replicating the experimental unit. Replication allows estimate of variance for each experimental unit. This permits experimenter to reduces variability in experimental results, increasing experimentation design significance and the confidence level of experimental output. Trail design of experiment for “five factor into two level” Full Factorial Design of Experiment Trail Factor A Factor B Factor C Factor D Factor E Notation 1 - - - - + e 2 + - - - - a 3 - + - - - b 4 - - + - - c 5 - - - + - d 6 + + + - + abce 7 - + + + + bcde 8 + + - + + abde 9 + - + + + acde 10 + + + + - abcd 11 + + - - - ab 12 + - - + - ad 13 + - + - - ac 14 + - - - + ae 15 - + + - - bc 16 - + - + - bd 17 - + - - + be 18 - - + + - cd 19 - - + - + ce 20 - - - + + de 21 + + + - - abc 22 + - + + - acd 23 + - + - + ace 24 + + - - + abe 25 + + - + - abd 26 - + + + - bcd 27 - + + - + bce 28 - + - + + bde 29 + + + + + abcde 30 + - - + + ade 31 - - + + + cde 32 - - - - - - Trail Factor A Factor B Factor C Factor D Factor E Notation 1 -1 -1 -1 -1 1 e 2 1 -1 -1 -1 -1 a 3 -1 1 -1 -1 -1 b 4 -1 -1 1 -1 -1 c 5 -1 -1 -1 1 -1 d 6 1 1 1 -1 1 abce 7 -1 1 1 1 1 bcde 8 1 1 -1 1 1 abde 9 1 -1 1 1 1 acde 10 1 1 1 1 -1 abcd 11 1 1 -1 -1 -1 ab 12 1 -1 -1 1 -1 ad 13 1 -1 1 -1 -1 ac 14 1 -1 -1 -1 1 ae 15 -1 1 1 -1 -1 bc 16 -1 1 -1 1 -1 bd 17 -1 1 -1 -1 1 be 18 -1 -1 1 1 -1 cd 19 -1 -1 1 -1 1 ce 20 -1 -1 -1 1 1 de 21 1 1 1 -1 -1 abc 22 1 -1 1 1 -1 acd 23 1 -1 1 -1 1 ace 24 1 1 -1 -1 1 abe 25 1 1 -1 1 -1 abd 26 -1 1 1 1 -1 bcd 27 -1 1 1 -1 1 bce 28 -1 1 -1 1 1 bde 29 1 1 1 1 1 abcde 30 1 -1 -1 1 1 ade 31 -1 -1 1 1 1 cde 32 -1 -1 -1 -1 -1 - Anova: Single Factor SUMMARY Groups Count Sum Average Variance Factor A 32 0 0 1.032258 Factor B 32 0 0 1.032258 Factor C 32 0 0 1.032258 Factor D 32 0 0 1.032258 Factor E 32 0 0 1.032258 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 0 4 0 0 1 2.430002 Within Groups 160 155 1.032258 Total 160 159 We replace trail data observation with “-“ sign replaced by “-1” and “+” sign by “+1” for better representation of presence of factor in design of experiment. Further to reach four blocks of data arrangement from 32 data observation points, confounding is shared for two higher effects ADE and BCE. After confounding, the data is categorized into four blocks, as per following scheme: ADE BCE Block -1 -1 1 1 -1 2 -1 1 3 1 1 4 The net impact of above changes is represented as follows: Trail Factor A Factor B Factor C Factor D Factor E Notation ADE BCE Block 1 -1 -1 -1 -1 1 e 1 1 4 2 1 -1 -1 -1 -1 a 1 -1 2 3 -1 1 -1 -1 -1 b -1 1 3 4 -1 -1 1 -1 -1 c -1 1 3 5 -1 -1 -1 1 -1 D 1 -1 2 6 1 1 1 -1 1 abce -1 1 3 7 -1 1 1 1 1 bcde -1 1 3 8 1 1 -1 1 1 abde 1 -1 2 9 1 -1 1 1 1 acde 1 -1 2 10 1 1 1 1 -1 abcd -1 -1 2 11 1 1 -1 -1 -1 ab 1 1 4 12 1 -1 -1 1 -1 ad -1 -1 1 13 1 -1 1 -1 -1 ac 1 1 4 14 1 -1 -1 -1 1 ae -1 1 3 15 -1 1 1 -1 -1 bc -1 -1 1 16 -1 1 -1 1 -1 bd 1 1 4 17 -1 1 -1 -1 1 be 1 -1 2 18 -1 -1 1 1 -1 cd 1 1 4 19 -1 -1 1 -1 1 ce 1 -1 2 20 -1 -1 -1 1 1 de -1 1 3 21 1 1 1 -1 -1 abc 1 -1 2 22 1 -1 1 1 -1 acd -1 1 3 23 1 -1 1 -1 1 ace -1 -1 1 24 1 1 -1 -1 1 abe -1 -1 1 25 1 1 -1 1 -1 abd -1 1 3 26 -1 1 1 1 -1 bcd 1 -1 2 27 -1 1 1 -1 1 bce 1 1 4 28 -1 1 -1 1 1 bde -1 -1 1 29 1 1 1 1 1 abcde 1 1 4 30 1 -1 -1 1 1 ade 1 1 4 31 -1 -1 1 1 1 cde -1 -1 1 32 -1 -1 -1 -1 -1 -1 -1 -1 1 The above design is said to have Level V resolution, since generator term ABCDE is said to have five alphabets A typical example of full factorial design of experiment is 2k design, where 2 represents the levels or output independent variable and k represents factor or input dependent variable effect on 2 defined output conditions. Total Run Count utilizing 25 design = 5+10+10+5+1 =31+one minus term = 32 runs Fractional Factorial Design of Experiment To overcome larger size of the design arising from Full Factorial run, fractional factorial design of experiment, is implemented in an experiment by researchers. Fractional Factorial design provide an alternative approach to screening of design of experiment, with a lower number of run count tested, for arriving at correct design of experiment by researcher. Fractional Factorial Design of Experiment refers to situation, when experimenter implements selected subset or "fraction" of the total runs, estimated for the full factorial design. The Factional Factorial design is implemented to generate confounding between the main effects and 2-way interactions. As result of confounding generated from limited runs of experiment, effects of other higher order interaction, cannot be studied independently and assumed to be negligible. This facilitates early estimation of main effects and interaction effects for the researcher. The Fractional Factorial design of experiment are generalized by term lk − p Where, là is the number of levels in each treatment factor. kà is the number of treatment factors. pà is the number of interactions that are confounded. The fraction of trials required is generalized using term 1/(lp). The effective HALF Fractional Factorial design of experiment for five factorial two level doe would have 25-1 = 16 runs. The effective Quarter Fractional Factorial design of experiment for five factorial two level doe would have 25-2 = 8 runs. HALF Fractional Factorial design of experiment for five factorial two level DOE Treatment_Combination I A B C D E ABCDE A + 1 -1 -1 -1 -1 1 B + -1 1 -1 -1 -1 1 C + -1 -1 1 -1 -1 1 D + -1 -1 -1 1 -1 1 E + -1 -1 -1 -1 1 1 abcde + 1 1 1 1 1 1 AB + 1 1 -1 -1 -1 -1 AC + 1 -1 1 -1 -1 -1 AD + 1 -1 -1 1 -1 -1 AE + 1 -1 -1 -1 1 -1 BC + -1 1 1 -1 -1 -1 BD + -1 1 -1 1 -1 -1 BE + -1 1 -1 -1 1 -1 CD + -1 -1 1 1 -1 -1 CE + -1 -1 1 -1 1 -1 DE + -1 -1 -1 1 1 -1 Design Generators: E = ABCD, E = ABCD gives us the basis for the resolution of the design as V degree resolution. Alias Structure I + ABCDE, A + BCDE, B + ACDE, C + ABDE, D + ABCE, E + ABCD AB + CDE, AC + BDE, AD + BCE, AE + BCD, BC + ADE, BD + ACE, BE + ACD, CD + ABE, CE + ABD, DE + ABC A 16-run 25 Half Fractional factorial design can conveniently rewritten as: Trail Run Factor A Factor B Factor C Factor D Factor E=ABCD Treatment Combination 1 -1 -1 -1 -1 1 e 2 1 -1 -1 -1 -1 a 3 -1 1 -1 -1 -1 b 4 1 1 -1 -1 1 abe 5 -1 -1 1 -1 -1 c 6 1 -1 1 -1 1 ace 7 -1 1 1 -1 1 bce 8 1 1 1 -1 -1 abc 9 -1 -1 -1 1 -1 d 10 1 -1 -1 1 1 ade 11 -1 1 -1 1 1 bde 12 1 1 -1 1 -1 abd 13 -1 -1 1 1 1 cde 14 1 -1 1 1 -1 acd 15 -1 1 1 1 -1 bcd 16 1 1 1 1 1 abcde Anova: Single Factor SUMMARY Groups Count Sum Average Variance Factor A 16 0 0 1.066667 Factor B 16 0 0 1.066667 Factor C 16 0 0 1.066667 Factor D 16 0 0 1.066667 Factor E=ABCD 16 0 0 1.066667 ANOVA Source of Variation SS df MS F P-value F crit Between Groups 0 4 0 0 1 2.493696 Within Groups 80 75 1.066667 Total 80 79