Reliability Engineering And Design Of Experiments

A seminar report on:Nested And Nested-Factorial Experiments

Presented By:Kaisar Katchi(Roll No. 05)

Under The Guidance Of:Professor S. B. Rane

Sardar Patel College of Engineering, Andheri (West)

ContentsTopicPage

The Nested Experiment2

The Nested-Factorial Experiment6

Repeated-Measures Design10

Conclusion12

Multiple Choice Questions13

References17

The Nested Experiment

Factor: any item that may affect a measured variable. Nested Experiment: levels of one factor are nested within (or are subsamples of) another factor. Also called hierarchical experiments. Nested-factorial Experiment: an experiment where both factors and nested factors exist.

In a nested experiment, factors are contained, or dependant, on other factors. That is, they are unique enough to cause significant changes in the experimental reading on their own, but they are also encapsulated inside other factors, which are under consideration.Consider an experiment to study the effect of the readings of two machines designed to observe the strain values on the supports used for fixing cathode ray tubes.Further, each machine has two heads that individually measure the strain reading.

The following rules are decided for the experiment: Four readings are observed per machine, that is, two per head (k). The heads can be chosen from any number of heads. Hence, those heads are a random sample. But, heads are not crossed (no interchanging; same head is not used for both machines). Machines can be considered as fixed. That is, not random.With the experimental parameters set, the experiment is conducted, and the following observations are noted:

MachineHead A1 2 B3 4

6 13 10 2

2 3 9 1

Head Total 8 16 19 3

Machine Total 24 22

Experimental ModelYijk = + Mi + Hj(i) + k(ij)

Where: j = 1, 2 (number of heads per machine) i = 1, 2 (number of machines) k = 1, 2 (number of readings per head)

An Expected Mean Square Table is constructed from the data:

Source 2Fi 2Rj 2RK EMS

(Machine) Mi 0 2 2 2 + 4H2 + 16M

(Head) Hj(i) 1 1 2 2 + 4H2

(Error) k(ij) 1 1 1 2

Data Analysis:

The sum of squares for the factors is calculated: SS (total) = 62 + 22 + 132 + 32 + 102 + 92 + 22 + 12 - (462/8) = 139.5 SS (machines) = ((242 + 222)/4) (462/8) = 0.5 SS(Head for Machine A) = ((82 + 162)/2) (242/4) = 16 SS(Head for Machine B) = ((192 + 32)/2) (222/4) = 64 SS(Total for Heads) = 16 + 64 = 80 Error sum of squares = 139.5 0.5 80 = 59. Degrees of Freedom of heads in machine A = 2 -1 = 1. For both machines, degree of freedom = 2 x 1 = 2.

ANOVAThe analysis of variance yields:Source df SS MS EMS F

(Machine) Mi 1 139.5 139.5 2 + 4H2 + 16M 4.72

(Head) Hj(i) 2 80 40 2 + 4H2 1.35

(Error) k(ij) 2 59 29.5 2

Totals 5 278.5

Experiment Conclusions:From SS (SSHeads) calculation, we see significant difference between the heads of Machines A and B. These results suggest a more careful adjustment between heads within the machines.

The Nested-Factorial Experiment

In a factorial experiment, multiple factors affect our variable; such as the effect of temperature and altitude on current flow in an integrated circuit.When both factors and nested factors appear in the same experiment, it is known as a nested-factorial experiment.Consider an experiment to test a new method of loading bullets in a gun, to improve the loading speed.Three groups of testers are selected, categorized by their build: slight, average and heavy build. Each group is divided into three teams.Each team uses two methods: old and new. Each method is used twice.

Experimental ModelYijkm = + Mi + Gj + MGij + Tk(j) + MTik(j) + m(ijk)

Where: Mi = methods, i = 1, 2. Gj = groups, j = 1, 2, 3. Tk(j) = teams within groups, k = 1, 2, 3 for all j. MTik(j) = interaction between methods and teams within groups. m(ijk) = random error, m = 1, 2 for all i, j, k.

The following data represents the number of bullets loaded per minute (more is better):GROUPIIIIII

Team 1 2 3 4 5 6 7 8 9

Method I 20.2 26.2 23.8 22 22.6 22.9 23.1 22.9 21.8

24.1 26.9 24.9 23.5 24.6 25 22.9 23.7 23.5

Method II 14.2 18 12.5 14.1 14 13.7 14.1 12.2 12.7

16.2 19.1 15.4 16.1 18.1 16 16.1 13.8 15.1

EMS Table:Source 2Fi 3Fj 3Rk 2Rm EMS

Mi 0 3 3 2 2 + 2MT 2 + 18M

Gj 2 0 3 2 2 + 4T 2 + 12G

MGij 0 0 3 2 2 + 2MT 2 + 6MG

Tk(j) 2 1 1 2 2 + 4T 2

MTik(j) 0 1 1 2 2 + 2MT 2

m(ijk) 1 1 1 1 2

The Sample Squares are computed:Group I II III

SS (Cell) 256.18 199.96 246.96

SS (Method) 214.19 196.83 242.10

SS (Team) 35.74 1.62 1.90

SS (MxT Interaction) 6.25 1.51 10.72

ANOVA:An Analysis of Variance leads to:Source df SS MS EMS

Mi 1 651.95 651.95 2 + 2MT 2 + 18M

Gj 2 16.05 8.02 2 + 4T 2 + 12G

MGij 2 1.19 0.60 2 + 2MT 2 + 6MG

Tk(j) 6 39.26 6.54 2 + 4T 2

MTik(j) 6 10.72 1.79 2 + 2MT 2

m(ijk) 18 41.59 2.31 2

Totals 35 760.76

To calculate the effect of Methods and Teams, the group readings are retabulated:Methods1Teams2 3 Method Totals 4 Teams5 6 M Totals 7 Teams8 9 M Totals

I 20.2 26.2 23.8 22 22.6 22.9 23.1 22.9 21.8

24.1 26.9 24.9 23.5 24.6 25 22.9 23.7 23.5

44.3 53.1 48.7 146.1 45.5 47.2 47.9 140.6 46 46.6 45.3 137.9

II 14.2 18 12.5 14.1 14 13.7 14.1 12.2 12.7

16.2 19.1 15.4 16.1 18.1 16 16.1 13.8 15.1

30.4 37.1 27.9 95.4 30.2 32.1 29.7 92 30.2 26 27.8 84

Team Totals 74.7 90.2 76.6 241.5 75.7 79.3 77.6 232.6 76.2 72.6 73.1 221.9

Experiment Conclusion:The mean number of bullets per minute for methods 1 (new) and 2 (old) are 23.58 and 15.08 respectively, showing significant improvement.The significant difference between teams within the groups is concentrated in group 1 (SS 35.74). Mean for teams in group I are: 18.68, 22.55, 19.15, suggesting that team 2 from group 1 is exceptionally faster than the other two teams.

Repeated-Measures Design

Consider an experiment to determine the weight lifting capacity of strength trainers before and after a new training method. Here, the same subjects repeat the experiment; hence, the same experimental unit is repeated. There are two repeated measures on each subject. Such an experiment is called a repeated-measures experiment.Seven trainers undergo the experiment, and the following results are obtained:

Subjects 1 2 3 4 5 6 7

Pretest 100 110 90 110 125 130 105

Post-test 115 125 105 130 140 140 125

Experiment Model:

Yij = + Si + j(i) Degress of Freedom for Si: 6 and j(i): 7.Model for the within-subject experiment, that is, before and after test results:

Yij = + Si + Tj + STij Degrees of Freedom for Si: 6, Tj: 1 and Stij: 6.

ANOVAAn analysis of variance for square of samples leads to:Source of Variation df SS MS F

Between Subjects (Si) 6 2084.71 347.45

Within Subjects 7 901.00 128.71

Tests (Tj) 1 864.29 864.29 145

Residual 6 35.71 5.96

Totals 13 2985.71

If treated as a two-factor factorial experiment (that is, treating the members as two factors: one before the new training method, one after), with one observation per treatment, gives us the following EMS table:Source df 7Ri 2Fj1Rk EMS

Si 6 1 2 1 2 + 2S 2

Tj 1 7 0 1 2 + 2ST 2 + 7T

STij 6 1 0 1 2 + 2ST 2

k(ij) 0 1 1 1 2

Hence, repeated measures experiments can be treated as nested factorial experiments.Conclusion

Examples of factors in the nest may be: Farms within townships. Classes within schools. Heads within machines. Samples within batches etc.As more factors are added to experiment, only the mathematical model needs to be expanded.

Nested and nested-factorial experiments may hence be designed in the following method:Experiment Design Analysis

Two or more Factors:

A. Factorial (crossed) Completely randomizedYijkm = + Ai + Bj + ABij + k(ij), For more factorsGeneral case ANOVA with interactions

B. Nested (hierarchical) Completely randomizedYijkm = + Ai + Bj(i) + k(ij) Nested ANOVA

C. Nested factorial Completely randomizedYijkm = + Ai + Bj(i) + Ck + ACik + BCjk(i) + m(ijk) Nested-factorial ANOVA

Multiple Choice Questions1. Nested experiments are a form of:a) Screening experimentb) Full-factorial experimentc) Finishing experimentd) Hypothesis test

2. Nested experiments are:a) Limited to a single factorb) Limited to two factorsc) Limited to factors with one level of nestingd) Not limited in terms of factors and levels of nest.

3. Nested experiments are also known as:a) Hierarchial experimentsb) Taguchi experimentsc) Plackett-Burman experimentsd) Reliability experiments

4. Which of the following are examples of nests?a) Farms within townships of a districtb) Ships within docks of a bayc) Samples within batchesd) All of the above

5. Nested-factorial experiments consider:a) Nested factorsb) Independent factorsc) Nested and independent factorsd) None of the above

6. Which of the following will require nested-factorial experiment?a) Crops within farms within district for multiple seasonsb) Local and visiting ships within docks in a bayc) a and bd) Number of students in ME Machine Design course in SPCE

7. Nested-factorial experiment must contain:a) Factors and nested factorsb) Only independent factorsc) Non-interacting factorsd) Only interacting factors

8. Which of the following is true?a) Nested-factorial experiments neglect errorb) Nested-factorial experiments are partially hierarchicalc) Nested-factorial experiments require complex modelsd) Nested-factorial experiments are recursive

9. A nested experiment will not have anya) Independent factors outside a nestb) Interactions between nests and their factorsc) Interactions in between nested factorsd) All of the above

10. Which of the following is a nested experiment?a) Gear tooth failure analysis per tooth for five different gearsb) Effect of temperature and pressure on molding process.c) Stress failure test for a random sample of pistonsd) Crash test for a batch of 100 trucks, on 3 samples

11. A nested-factorial experiment may havea) Independent factors outside a nestb) Interactions between nests and their factorsc) Interactions in between nested factorsd) All of the above

12. Which of the following is a nested-factorial experiment?a) Gear tooth failure analysis per tooth for five different gearsb) Effect of temperature and pressure on molding process.c) Study of efficiency of a new anticorrosive layer coating on land, sea, and air vehicles, using four different types of construction metals versus old anticorrosive layer coating.d) Crash test for a batch of 100 trucks by four different manufacturers.

13. In the repeated measures example, which is the repeated measure?a) Physical strength of the subjectb) Strength subjectc) The testd) The experiment

14. What must be done to the mathematical model of a nested experiment as more factors are added?a) It must be simply expandedb) It must be significantly changedc) It must be discarded and remodeled d) It remains the same

15. Which of the following most accurately describes a nest?a) A container for factorsb) A factor which contains other factorsc) A repeated valued) None of the above

16. Which of the following is true for nested factors?a) They may have interactions with the nestb) They may be independent of other factors in the nestc) They may significantly affect the nestd) All of the above17. In an experiment to determine principal cause of defects in a system, if x1, x2, x3 are nested factors inside nest x4, and a Pareto chart shows highest magnitude for the x1x4 (interaction) factor, then what is recommended?a) Factor x1 needs to be changed/improved to reduce defectsb) Nest x4 needs to be changed/improved to reduce defectsc) Factor x1 needs to be placed in another nestd) Stratification of the interaction is necessary to determine the exact cause

18. If Ai represents the nest in the model, and B is a nested factor with j levels, how is B represented in the model?a) jBi b) jBi c) Bj(i)d) AjBi

19. There must be no interaction between factors from different nests. a) Trueb) Falsec) Depends on the factors and nest under considerationd) Factors must be carefully chosen to minimize cross-nest interaction, to simplify model.

20. If the ME Machine Design and Thermal Engineering courses both represent nests, which of the following is a better consideration for a factor within the nest that affects the final result of the class?a) The professorb) The studentc) The amount of preparation leave (PL)d) All of the above

ReferencesFundamental Concepts in the Design of Experiments Charles R. Hicks (Oxford University Press, Fourth Edition).1