Chapter : Factorial Designs. Two factor factorial designs ( levels factors ) This situation is similar to the randomized block design from the previous chapter. However, in addition to the effects within the groups (A) and within the blocks (B), we now also consider so-called interaction effects between the two parameters A & B. Schematically speaking we have: with n = # data points; c = # of groups (factor A); r = # of groups (factor B); and n` = # of repeat experiments. We now calculate the various sum of squares: sum of squares total; SST = r c i= j= k= i= j= k= ( Xi jk X ) with X = r c r c X i jk sum of squares factor A; SSA = c r i= j= k= ( Xi X ) with Xi = c c X i j k
i= k= ( X j X ) with X j = r sum of squares factor B; SSB = r sum of squares interaction AB; c j= SSAB = r r c ( Xi j X i X j + X ) i= j= X i j k Random error sum of squares error; SSE = r c ( X jk X ) with X = i ij ij i= j= k= k= X i jk The mean sum of squares is derived by dividing the values by the corresponding degree of freedom. The present case is called the "Two-Factor ANOVA with Replications" You may realize that the present calculations are very similar to the randomized block situation in the previous chapter. However, please realize that it is necessary for this type of analysis to have repeat measurements. Otherwise MSE will become infinite and no F-ratios can be calculated.
The textbook by Montgomery had another way of simplifying the actual calculation of the sum of squares. You may want to decide for yourself if one formula is easier than the other.
Use Laundry excel file as an example. (manipulate the data to show weak and strong AB-interactions)
The AB-interaction effects are sometimes difficult to see in a table format. Thus, graphical representations might be more appropriate. Plot the averages for the various repeat groups. Parallel lines indicate a lack of interaction, whereas a strongly different slope or even crossing lines are indications of strong interactions. Similar to the previous chapter we can run the Tukey-Kramer or the Fischer- LSD test to evaluate individual differences between multiple levels within one factor (more than levels).!! remember to use the correct degree of freedoms!! Depending on how the levels of one or both factors have been selected, we have to use different processes for our F-ratio analysis! (see following table) (Excel assumes fixed-levels)
Residual Analysis: We test the experimental data against the assumed model (fit-equation). In case the model describes the situation correctly and there are no 'outlier', the residuals should follow a straight line in a normal probability plot. We will use the data from "Laundry" as an example. st, we have to define our model (fit-equation) The ANOVA analysis indicated that the AB-interaction effect is relative minor. Thus, we will first consider a simple linear effect of A, detergent brand, and B, temperature. Warm Hot predicted values residuals Warm Hot X 4 8 X A + X B X X -.3-0. X 6 9 Average: X 0.7 0.9 X 5 7 6.7 5.3 8. X -0.3 -. X 9 X -3.3 0.9 X 7 0 X.7.9 Y 7 Y -.5-0.3 Y 9 0 Y 0.5 -.3 Y 0 8 9.9 8.5.3 Y.5-3.3 Y 7 Y -.5 0.7 Y 3 Y 3.5.7 Average 6.9 9.7 8.3 Residuals Rank # P-% Z-value -.3 5 0.5-0.755 0.7 0.575 0.89-0.3 8 0.375-0.39-3.3 0.05 -.960.7 7 0.85 0.935 -.5 3 0.5 -.50 0.5 0.55 0.063.5 6 0.775 0.755 -.5 3 0.5 -.50 3.5 0 0.975.960-0. 0 0.475-0.063 0.9 4 0.675 0.454 -. 7 0.35-0.454 0.9 4 0.675 0.454.9 9 0.95.440-0.3 8 0.375-0.39 -.3 5 0.5-0.755-3.3 0.05 -.960 0.7 0.575 0.89.7 7 0.85 0.935 The normal probability plot shows a reasonable linear relation. Thus, our model assumption seems to fit and no obvious outliers are present.
. Factorial designs with 3 or more factors The most basic set-up of these factorial designs uses each factor at only two levels (low and high). And for a complete factorial test set-up all possible combinations between these factors at the two levels have to be analyzed. It is also necessary to perform at least one repeat test for each experiment in order to evaluate all possible interactions. A test without repeat measurements is possible when certain interactions are assumed to be negligible. General outline for a 3 -factorial experiment:
Detergent-example: Each factor has "high" and "low" values, e.g. Brand X is 'low" and Brand Y is 'high" and Warm is 'low' whereas Hot is 'high'. The corner notations now indicate which factors are at a high or low level, e.g. corner 'bc' represents the high values Hot and Liquid for the nd (b) and 3 rd (c) factor, whereas Brand X st (a) factor is at the low level.
So, how do we calculate the mean square values?? We could use the same approach as shown earlier. For example: sum of sum of sum of sum of squares total; squares factor A; SST = squares interaction AB; squares SSABC = Random error sum of ABC; a b c i= j= k= l= SSA = b c a i= SSAB = c i= j= k= l= ( Xi jkl X ) with X = a b c a b c j= k= l= ( Xi X ) with Xi = b c b c a b ( X i j Xi X j + X ) i= j= a b c ( Xi jk Xi X j X k + Xi j X i k X j k + 5 X ) i= j= k= squares error; SSE = a b c ( X jkl X ) with X = i ijk ijk i= j= k= l= Apparently these equations become quite cumbersome and your calculations are prone to error. However, there is a slightly easier way to calculate these sum of squares. This is described as "Contrasts". The following table might come in handy. l= X X i jkl X i jkl i j k l
In order to calculate the Contrast- or Effect-A we calculate the summation over all experimental data according to the '+' and '-' signs shown in the previous table. Thus, data from the experiment labeled as () are counted negative, whereas data from the experiment a are used with a positive value, etc.. This summation results in the so-called 'Contrast' for A. Dividing the contrast by # of factors - results in the so-called 'Effect'. The sum of squares is now: SS = # ( Contrast ) of factors For our Laundry example the following examples may demonstrate this process. Effect A = bc 3-3- = 3- [8.4 The Sum of Squares is now : [ Contrast] = [ a + ab + ac + abc () b c ] + 8.8 + 0.7 +.+.7 +.9 + 3.4 + 4.6 4.8 5.0 7.8 8. 9.9 0.3.6 3.] = SSA = ( 8.8) 3 =.09 8.8 8 = =.35
The 3-parameter interaction ABC is calculated as follows: Effect ABC = = [ a + b + c + abc () ab ac bc] = 3- = 3- [8.4 The Sum of Squares is now : + 8.8 + 7.8 + 8. + 9.9 + 0.3 + 3.4 + 4.6 4.8 5.0 0.7..7.9.6 3.] = SSABC = The Sum of Square Total is given as: SS total a b = c i= j= k= l= X ijkl ( 0.4) 3 abc = 0.0 ( X ) =. 78 0.4 8 and the Sum of Squares Error is the difference between all individual sum of squares and the SS total. The following table summarizes the results: = 0.05 Interaction effect df SS MS F P Brand.090.090 40.5 0.000 Temperature 7.040 7.040 7.68 0.000 Type 67.40 67.40 46.9 0.000 Brand x Temperature 0.490 0.490 3. 0.6 Brand x Type 3.60 3.60.9 0.00 Temperature x Type 0.040 0.040 0.5 0.68 Brand x Temperature x Type 0.00 0.00 0.06 0.807 Error 8.60 0.58 Total 5.780.780 It is obvious that all single effects and the Brand x Type interaction effect are significant.
The larger the number of factors is, which are considered for the analysis, then more experiments have to be preformed. As a result of that quite often repeat experiments are not performed or a fractional factorial design is being used. First we evaluate the "no-repeat" process. The following is a 5 factorial design:
The factorial analysis gives the following results: (see excel file, "cake-analysis" for further details) Interactions Sum of Squares df MS F P-value A.4878.4878 3.03 0.0 B 0.9453 0.9453.9 0.86 C 4.888 4.888 9.93 0.007 D 0.053 0.053 0.03 0.86 E 0.0003 0.0003 0.00 0.980 AB 0.378 0.378 0.8 0.604 AC 0.058 0.058 0. 0.748 AD.403.403.5 0.33 AE 0.0703 0.0703 0.4 0.7 BC 7.9003 7.9003 6.07 0.00 BD 0.378 0.378 0.8 0.604 BE 0.4753 0.4753 0.97 0.34 CD 0.6903 0.6903.40 0.54 CE 0.053 0.053 0.05 0.84 DE 38.5003 38.5003 78.3 0.000 ABC 0.478-0.478 0.478 ABD 0.9453-0.9453 0.9453 ABE 0.378-0.378 0.378 ACD 0.0003-0.0003 0.0003 ACE.403 -.403.403 ADE.4003.4003 3.9 0.000 BCD 0.0378-0.0378 0.0378 BCE 0.053-0.053 0.053 BDE 0.553-0.553 0.553 CDE.403 -.403.403 ABCD 0.3003-0.3003 0.3003 ABCE 0.68-0.68 0.68 ABDE 0.68-0.68 0.68 ACDE 0.0533-0.053 0.053 BCDE.3033 -.303.303 ABCDE 0.6383-0.638 0.638 Error 7.3747 5 0.496 Total 75.337
In graphical form the results may be presented as follows:.3 Fractional factorial designs The number of necessary experiments increases substantially for increasing number of variables (factors). To reduce this number of experiments, we 'assume' certain types of interaction to be negligible and use subsets of the complete factorial design. We distinguish between different types of fractional designs: Resolution III designs; where main effects are confounded with two-way interactions (e.g. A confounded with BC) Resolution IV designs; where main effects are not confounded with two-way interactions (e.g. CD is confounded with BC) Resolution V designs; where main effects and two-way interactions are not confounded with each other, but higher order interactions are confounded (e.g. ACD confounded with BCEF)
Example of a 4 factorial design: A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD () - - - - + + + + + + - - - - + a + - - - - - - + + + + + + - - b - + - - - + + - - + + + - + - c - - + - + - + - + - + - + + - d - - - + + + - + - - - + + + - ab + + - - + - - - - + - - + + + ac + - + - - + - - + - - + - + + ad + - - + - - + + - - + - - + + bc - + + - - - + + - - - + + - + bd - + - + - + - - + - + - + - + cd - - + + + - - - - + + + - - + abc + + + - + + - + - - + - - - - abd + + - + + - + - + - - + - - - acd + - + + - + + - - + - - + - - bcd - + + + - - - + + + - - - + - abcd + + + + + + + + + + + + + + + For example, we decide to run only the experiments, where Effect ACD has a positive sign in the previous table. We can then see that the effects A & CD as well as C & AD or D & AC are confounded (They have the same sign sequence and we are unable to distinguish /analyze them individually) Vice versa, if we had picked the positive experiments for column AC, then B & ABC, D & ACD, as well as BC & ABCD are confounded. We should also realize that the effect AC can no longer be analyzed at all. If we decide to use only the positive experiments in the ABCD column, we would then have the two-parameter interactions AB & CD, AC & BD, AD & BC confounded.
In other cases you may use the same concept to reduce experimental variations due to time or instrument limitations. For example, if you cat run all experiment on the same day. You know that results from the same date are very consistent, but repeat tests on different dates show larger variations. You then perform groups (or blocks) of tests according to the same principle as used before and you realize that for example, that a possible significance of the ABCD interaction is caused by your set-up. In graphical form the block confounding is shown in the following graph: The open and the closed circle experiments show the two blocks. It is further possible to split the total number of experiments into ¼ blocks. However, one has to be aware that more and more information is being lost in that process. E.g. Block # Block # Block #3 Block #4 () a d ad ab b abd bd bc c bcd cd ac abc acd abcd For additional and more detailed information on factorial designs (e.g. factors at more than levels) see: O.L. Davis, The Design and Analysis of Industrial Experiments, Oliver and Boyd, London (967)