Multi-Factored Experiments

Transcription

1 Design and Analysis of Multi-Factored Experiments Advanced Designs -Hard to Change Factors- Split-Plot Design and Analysis L. M. Lye DOE Course 1 Hard-to-Change Factors Assume that a factor can be varied, with great difficulty, in an experimental setup (such as a pilot plant), although it cannot be freely varied during normal operating conditions. Assume further that each of two factors has two levels and the design is to have a factorial structure, and it is imperative that the number of changes of the hard-tochange factor be minimized. We can minimize the number of level changes of one factor simply by keeping the level constant in pairs of consecutive runs. That is, either the high level is used on consecutive runs and then the low level on the next two runs, or the reverse. L. M. Lye DOE Course 2 1

2 This means we that we have restricted randomization, as there are 6 possible run orders of that one factor without any restrictions, but with the restriction, there are only 2 possible run orders (+ +, --) or (- -, + +) Restricted randomization increases the likelihood that extraneous factors (i.e. factors not included in the design) could affect the conclusions that are drawn from the analysis. Furthermore, this will also cause bias in the statistics that are used to assess significance. i.e. normal ANOVA based on a completely randomized design may give the wrong conclusions. L. M. Lye DOE Course 3 Although hard-to-change factors have not been discussed extensively in textbooks, it is safe to assume that such factors occur very frequently in practice. Sometimes there may be no hard-to-change factors at all in the experiments, but the experimenters or technician who wants to save time may not have followed the randomized design as prescribed by the experimental design. Hence it is very important for the analyst performing the statistical analysis to know exactly how the experiments were performed. Were the runs randomized as prescribed or were the runs made convenient to save time. How the experiments were carried out can have serious consequences on the results. Significant effects may turn out to be insignificant or vice versa if is not properly analyzed. The software will not know unless you tell it. L. M. Lye DOE Course 4 2

3 Split-Plot Design with Hard-to-Change Factors For example, all of you know a 2 3 full factorial design. Most would choose to run the 8 treatment combinations in a completely randomized order as given say by Design-Expert. Unfortunately, limitations involving time, cost, material, and experimental equipment can make it inefficient and, at times, impossible to run a completely randomized design. In particular, it may be difficult to change the level for one of the factors. E.g oven temperature may take many hours to stabilize. In this case, the hard-to-change factor is typically fixed at one level and then run the combinations of the other factors the split-plot design (or Split-Unit design) L. M. Lye DOE Course 5 Recognizing a Split-Plot Design Split-plot experiments began in the agricultural industry because one factor in the experiment usually a fertilizer or irrigation method can only be applied to large sections of land called whole plots. The factor associated with this is therefore called a whole plot factor. Within the whole plot, another factor, such as seed variety, is applied to smaller sections of the land, which is obtained by splitting the larger section of land into subplots. This factor is therefore referred to as the subplot factor. These same experimental situations are also common in industrial settings. L. M. Lye DOE Course 6 3

4 3 Main Characteristics of Split-Plot Designs The levels of all the factors are not randomly determined and reset for each run. Did you hold a factor at a particular level and the run all the combinations of the other factors? The size of the experimental unit is not the same for all experimental factors. Did you apply one factor to a larger unit or group of units involving combinations of the other factors? There is a restriction on the random assignment of the treatment combinations to the experimental units. Is there something that prohibits assigning the treatments to the units completely randomly? L. M. Lye DOE Course 7 Effect of restricted randomization on statistical analysis Consider a very simple example of 2 factors each at 2 levels. If the 4 combinations are run in random order, we have a 2 2 design. Now assume that one of the factors is hard-to-change. One of the levels of this factor is selected randomly and then used in combination with each of the 2 levels of the other factor, which are also randomly selected. Then this process is repeated for the second level of the first factor. So assuming A is the hard-to-change factor the run combinations could be as follows: A 2 B 2, A 2 B 1, A 1 B 1, A 1 B 2. L. M. Lye DOE Course 8 4

5 Notice that this sequence of runs could have of course risen from the completely randomized experiment, but the data would still have to be analyzed differently because of the restricted randomization in the second case. That is, there are only 8 possible sequences of treatment combinations with the restriction, whereas there are 24 possible sequences without the restriction. Another key point: With complete randomization, each run is completely reset, whereas, with restricted randomization, the hard-to-change factor was not reset. If the data were analyzed as a 2 2 completely randomized design using Design-Expert, we will get the wrong answer! Why is this so? L. M. Lye DOE Course 9 Split-Plot Designs have 2 error terms Recall that in a 2 k design, each effect is estimated with the same precision. i.e. they have the same standard error. This does not happen with a split-plot p design as subplot factors are generally estimated with greater precision (smaller errors) than are whole plot factors. This is because there is greater homogeneity among subplots than are the whole plots, especially if the whole plots are large. E.g. Smaller pieces cut from a sheet of plywood are more homogeneous than between 2 different sheets of plywood. i.e. pieces within a sheet has less variability than between 2 sheets of plywood. L. M. Lye DOE Course 10 5

6 In fact for agricultural experiments, it can be shown that the variance of any whole plot effect estimate must exceed the variance of any subplot effect estimate for any 2 k design. The statistical model for a split-plot design can be written as: y ij ( τβ ) ε SP = μ + τ + ε + β + + i WP j ij L. M. Lye DOE Course 11 It has been shown mathematically that the whole plot factors and their interactions have a variance of: 4 N k 2 2 ( 2 σ +σ ) 1 Whereas subplot factors and their interactions have a variance of: ( 2 ) 4 σ 0 N Here the subscript 1 and 0 denote the whole plot and subplot, respectively. 0 L. M. Lye DOE Course 12 6

7 Example Assume that factor A is a hard-to-change factor and factor B is not hard to change, with the experiment such that material (e.g. a board) is divided id d into two pieces and the two levels of factor A applied to the two pieces, one level to each piece. Then the pieces are further subdivided and each of the two levels of factor B and applied to the subdivided pieces. Three pieces of the original length (e.g. three full boards) are used. The data will be analyzed assuming a fully randomized design like a regular 2 2 design, and then correctly using a split-plot design. L. M. Lye DOE Course 13 Data and Analyses A B Observations If the data are improperly analyzed as 2 2 design with three replications, the results are as follows: Two-way ANOVA: Y versus A, B Source DF SS MS F P A B AB Error Total L. M. Lye DOE Course 14 7

8 The proper analysis of the data as having come from a splitplot design is not easily achieved. Most DOE software cannot handle split-plot design directly. Design-Expert can handle some types of split-plot p design but it must be done manually. Minitab 16 can do it directly now. However, statistical packages can be tricked into performing the correct analysis by assuming a nested model and forcing a nested model analysis. Minitab s General Linear Model routine can do the analysis if the data are set up properly. Best papers to read are: Kowalski and Potcner (2003), Kevin Potcner and Kowalski (2004), Bisgaard, Fuller, and Barrios (1995). You can download these from the course website. L. M. Lye DOE Course 15 Proper statistical analysis: split-plot analysis General Linear Model: Y versus A, B, WP Factor Type Levels Values A fixed 2-1, 111 WP(A) random 6 1, 2, 3, 4, 5, 6 need to set up this column B fixed 2-1, 1 Analysis of Variance for Y, using Sequential SS for Tests Source DF Seq SS Seq MS F P A WP(A) * * WP error term B A*B times higher than CRD Error subplot error term Total L. M. Lye DOE Course 16 8

9 A somewhat different picture emerges when the data are analyzed correctly. The p-value for AB is more than 3 times than of for CRD. The difference in the conclusions drawn with the wrong analysis and the conclusions made with the proper analysis can be much greater than the difference in this example. As illustrated by Potcner and Kowalski (2004), a significant main effect in the complete randomization analysis can become a non-significant whole-plot main effect when the split-plot p analysis is performed. And, a non-significant main effect in the complete randomization analysis can become a significant subplot main effect when the split plot analysis is performed. L. M. Lye DOE Course 17 Several hard-to-change factors Sometimes there may be instances where there are several hard-to-change factors and one or more easy to change factors. For example, 4 of the factors (A, B, C, and D) may be hard-to-change whereas E may be easy to change. Or we may have 3 hard-to-change factors and say 6 easy to change factors, etc. In addition, there is only one replication. This means that there are no pure error terms. For these situations, the split plot design can be analyzed using two separate half-normal probability plots. One for the whole plot effects and one for the subplot effects. This can be done on Design-Expert manually. L. M. Lye DOE Course 18 9

10 Dividing into Whole Plots and Subplots Let s consider 2 examples: Example 1: 5 factors (A, B, C, D, E). A, B, C, D are hard-tochange factors, and E is easy to change. Whole plot group: A, B, C, D and interactions involving only these factors. Subplot group: E, and all interactions involving E only. E.g. AE, BE, CDE, etc. Example 2: 9 factors (A, B, C, D, E, F, G, H, J). A, B, C are hard-to-change, and D, E, F, G, H, J are easy to change. Whole hl plot group: A, B, C, and all llinteractions i involving li only these 3 factors Subplot group: D, E, F, G, H, J and all interactions involving these factors. E.g. AD, DE, etc, but not AB, BC, or ABC. L. M. Lye DOE Course 19 Effects and Half-normal plots The effects of each factor and its interaction are determined in exactly the same way as in regular factorial design. Once the whole plot group and subplot group have been decided, a half-normal plot of effects are used to determined the significant effects for each group. Hence, two half-normal plots are constructed. The significant effects from both groups are then combined to give the final model and prediction equation. If the half normal plot is done for all the effects without separating them into the two groups, it is likely that the subplot effects will be buried in the whole plot error terms. Hence significant subplot effects maybe missed if the split plot nature of the experiment is not taken into account by the analysis. Example Plasma.dx7 L. M. Lye DOE Course 20 10

11 Summary When it is not convenient or not economical to do a completely randomize experiment due to one or more hard-to-change factors, we have a restricted randomization case. A common and often used approach is a split plot experiment which has a whole plot group of effects and a subplot group of effects leading to two error terms in the ANOVA or two half-normal plots for the unreplicated case. If not analyzed properly, significant subplot effects may be masked by the larger whole plot errors thus giving the wrong conclusions and wrong model. It is thus crucial that you know exactly how the experiment was carried out in practice. L. M. Lye DOE Course 21 11