 So, it was said that main factor A was aliased with BCD, if we look at this particular table we can see that the blue entries for A and BCD or the entries in blue for A and BCD are exactly identical, minus 1 here, minus 1 here, 1 here, 1 here, 1 1, minus 1, minus 1, 1 1, minus 1, minus 1, minus 1, minus 1, 1 1, so that is matching perfectly and if you also want to look at the red, okay before we go to the red entries, let us look at the blue entries and all these blue entries are corresponding to the blue entries of the design generator ABCD. Now looking at the red entries, the red entries correspond to the negative 1 or minus 1 entries in the design generator ABCD, so when you look here, you can see the red entries are related in such a way that A is equal to minus BCD, so with that relation 1 for A and minus 1 for BCD and for the second setting B, if A is at a lower level minus 1, BCD is at plus 1, so that corresponds to the second fraction. In the second fraction, we are all taking the negative values for ABCD, the second fraction corresponds to the negative entries in the ABCD column and that is the reason why you have the relation A is equal to minus BCD for the second fraction, the first fraction A is equal to plus BCD, in the second fraction A is equal to minus BCD. We also know that B was interacting or aliased rather with ACD interaction, looking at the blue entries, we can see that all the blue entries are matching, whereas the red entries are related by B is equal to minus ACD minus 1 plus 1, 1 minus 1 minus 1 plus 1. So the red entries correspond to the second fraction, the aliased effects are related by a negative value. In the principal fraction, we are having the main effect aliased with the 3 factor interactions directly. In the second fraction, the main effects are aliased with the 3 factor interactions with a negative relationship. Similarly for other 2 factor interactions, 2 factor interactions are also aliased with each other. That table I do not have, but you can easily show that for example AB column would be exactly matching with CD column for the principal fraction, whereas AB column would be related by related to minus CD in the second fraction. I leave these things as self-exercises. So let us again refer to the table of contrast and identify the linear combination of contrast for A and BCD in the half corresponding to the first fraction. So please look at the table, linear combination of contrast for A and BCD. We just now saw that A and BCD are identical. So you cannot separate the 2 contributions and you have to live with both of them. So if you look at the contrast for either A or BCD, the effect obtained from the contrast is going to represent both A and BCD. So you will have minus 1, minus 1 plus AB plus AC minus BC plus AD minus BD minus CD and AB plus ABCD. I request you to write it down and then compare with what I am going to show in the next slide. So its effect of factory A is decided by minus 1 plus AB plus AC minus BC plus AD minus BD minus CD plus ABCD. Please note that there are only 8 entries here 3, 5 and 3, so that makes it 8. Not only effect of factory A is decided by the contrast as shown here, the same contrast defines the effect of factor BCD also. So these 8 entries are representing the combined effects of A and BCD. Similarly we can check for the other effects and the linear contrast was only shown. In order to calculate the effect, you have 4 positive entries and then 4 negative entries. You are finding the difference between the 4 positive entries and the 4 negative entries and when you average, hence you have to divide it by 4. If you look at the second fraction, please look at the red entries 1 minus 1 minus 1. So ABC, ABCD and so on, please write it again on the paper. The calculation for the contrast of A corresponding to the second fraction, so you will get A minus B minus C, A minus B minus C plus ABC minus D plus ABD plus ACD minus BCD. You may verify this and this A is not only uniquely determined but A will also be matching with minus BCD. So the effect of factor minus BCD is also decided by the same contrast. So for AD, the 2 factor interaction, you go back to the principal fraction given in blue colour and AD, you have to go all the way back to the original table. So it would be 1 minus AB minus AC, 1 minus AB minus AC plus BC plus AD minus BD minus ACD plus ABCD and so on. So that is the effect of AD but that is not only the unique contrast for AD, the same contrast applies for BC as well. You can go back and check it out that the contrast in the first fraction are the same for both BC and AD. So similarly you can find the aliases for other 2 factor interactions. So now the important question is how do you find the effect of A and effect of BCD or how do you resolve the combined effects of A and BCD or separate the BC with AD. So what I am trying to say here is we know that main factors are aliased with 3 factor interactions and 2 factor interactions are aliased with other 2 factor interactions. We want to separate out this combined effect so that we uniquely get the effects of A, B, C and D and also AB, BC, CD and so on. So with this in mind obviously the solution is to do the first fraction and analyze it and then do the next fraction, analyze it and in order to segregate the different effects you combine the 2 fractions. Now sort of summarizing we created the first fraction by using the plus ones in the ABCD column. Similarly the second and last fraction here in the 2 power 4 minus 1 design was created by using minus ones in the ABCD column. In this case the design generator is i is equal to minus AB, CD for the second fraction the design generator was i is equal to minus AB, CD in the first fraction the design generator was represented by i is equal to plus AB, CD. I only represents the matrix of pluses. So you have factor A is aliased with minus BCD, factor B is aliased with minus ACD and factor C with minus ABC. How is it possible? We just put A here minus A squared, A squared will be all plus ones. So we can take it out, 1 into any number is that number so we can just take out A squared and we will have A is equal to minus BCD. Similarly B will be equal to minus ACD and C will be equal to minus ABD. So there is a typo I will just correct it immediately right. So C is with minus ABD. So now the full factorial design may be recreated by combining the 2 fractions so that the aliasing terms get cancelled out. And let us denote the linear combination of effect A in the first fraction as LA and in the second fraction as L prime A. So this is the linear combination. We know that in the first case A was aliased with BCD and in the second fraction A was aliased with minus BCD and when you combine the 2 fractions the BCD in the first fraction will cancel out with the minus BCD in the second fraction so you will be able to recover A alone. So you have LA is equal to A plus BCD and that is given by this entire contrast divided by 4, L prime A is given by A minus BCD and that is the contrast corresponding to that and when we combine LA with L prime A you will get 2A plus BCD minus BCD plus BCD and minus BCD will cancel and you will be having 2A. So 2A is represented by 1 by 4 of the sum of these 2 entities. So A will be 1 by 8 of a linear combination of both of these. So LA is given by this and LA prime is given by this we add the 2. So we get 2A we get 2A and that was equal to 1 by 4 of this total sum bracket is missing let me just put the brackets here so that there is no problem here you go. So we have these entities and the effect of A is given by this and that would be 1 by 8 into this entire thing. If we had done the 2 power 4 design without this fractional approach this is the contrast we would have had and since there are 8 pluses and 8 minuses we are taking the average and then for that we are dividing it by 8. Similarly, the other effects may be DLIS by combining the 2 fractions. Now we come to an important concept called as resolution and the resolution of the fractional factorial design is important when we talk of fractional factorial designs of higher resolution they refer to clearer designs or those which hide less and give more information. So what is the meaning of resolution? Let us denote the resolution of a design by S by the symbol S. Let the factor type or order of the factor B F is equal to 1 for a main factor 2 for a binary factor and design of is of resolution S if factor F is not aliased with another factor less than S minus F. It is quite simple resolution is denoted as S and the order of the factor is denoted by F we calculate S minus F and say that a design is of resolution S if factor F is not aliased with another factor less than S minus F. So we will plug in some values for S and F and you will soon see what this means. If the order of the factor is 1 that means it is a main factor and the resolution used is 3 we use Roman numerals to represent the resolution S is equal to 3 then 3 minus 1 is equal to 2 it also means that the main factors are not aliased with each other. So if F is equal to 1 in resolution 3 we have S is equal to 3 the resolution 3 is represented by Roman numerals so 3 minus 1 is equal to 2 and that means if you have 3 minus 1 is equal to 2 less than 2 will be 1 so the main factor is not aliased with another main factor okay so be a bit careful about this we are only saying that the main factor cannot be aliased with the factor of order less than S minus F which means that the main factor is not aliased with another main factor this also means that the main factor is aliased with a factor of order 2. So when you have a design generator I is equal to ABC factor A will be aliased with BC factor B will be aliased with AC and factor C will be aliased with AB this can be easily obtained from the design generator. So you can see that the main factors are not aliased with each other but the main factor main one main factor is aliased with another factor of order 2 okay so this is for resolution 3 we can obviously have resolutions of higher orders like 4 and 5. If you look at F is equal to 2 then in resolution 3 S minus F is equal to 1 and so binary interaction is not aliased with the factor less than 1 so the binary interaction is not aliased with the factor less than 1 means it is aliased with the factor 1 you cannot have a factor less than 1 okay. The rule says that for resolution 3 the binary interaction cannot be aliased with the factor less than 1 so it cannot be aliased with the factor 0 it does not exist so the binary interaction is aliased with one of the main factors. I think this rule is a bit difficult to remember on the long run but it is quite useful and anyway you do not need to remember this rule because you can always look at the design generator and see how the main effects are aliased and the binary interaction effects are aliased. So that is not really required all you need to do have is the design generator the proper appropriate design generator for the resolution and these are available in standard textbooks so you do not have to even remember all of them. So there can be resolutions of 3, 4 and 5 and resolutions 3 and 4 are applied in factor screening experiments factor screening means you are considering a large number of factors and by doing fractional factorial design you are screening out some factors or eliminating some factors. So when you have resolution 3 and you are constructing a half fraction that means it is represented by 2 power 3-1 design and the complete notation for half fraction of a 2 power 3 design of resolution 3 is given by 2, 3 Roman numerals as subscript and 3-1 as superscript. This tells we are conducting a half fraction of a 2 level factorial design of resolution 3 and the number of factors in our consideration is 3. Resolution 3 the summary is no main effects are aliased with one another but they would be aliased with 2 factor interactions that is what the rule also said s is equal to 3 and f is equal to 1 and so it cannot be aliased with factor less than 3-1. So a main effect cannot be aliased with another main effect but they would be each main effect would be aliased with another 2 factor interaction so quite clear some 2 factor interactions could be aliased with other 2 factor illustrations sorry let me just correct the typo here it should be 2 factor interactions. Some 2 factor interactions could be aliased with each other with could be aliased with other 2 factor interactions and this happens in a 2 level factorial design of resolution 3 involving a quarter fraction of a full 2 power 5 design or let me put it again we are looking at a quarter fraction of 2 level factorial design of resolution 3 involving 5 factors that means essentially we are doing only 8 experiments in such cases some 2 factor interactions could be aliased with other 2 factor interactions. When we construct half of a 2 power 4 design it is of resolution 4 with the defining relation i is equal to a, b, c, d this is what we saw in the beginning of the demonstration of the fractional factorial design. So the design generator i is equal to a, b, c, d and that is defined as one half fraction of resolution 4 for a 2 level factorial design involving 4 factors. In this case we know by now no main effects are aliased with each other but or even with 2 factor interactions this is very good now the resolution has increased so the main effects are not in aliased with other main effects or main factors and they are also not aliased with any other 2 factor interactions but they would be aliased with 3 factor interactions. On the other hand 2 factor interactions are aliased with other 2 factor interactions okay this again we saw here s is equal to 4 the resolution is 4 and the number of the f is equal to 2 for example the binary interaction and so s-f would be 4-2 which is 2. So no 2 factor interaction could be aliased with the single factor so hence a 2 factor hence a 2 factor interaction would be aliased with other 2 factor interaction when we have a resolution 5 design we construct 2 power 5-1 it is of resolution 5 with the defining relation i is equal to a, b, c, d, e remember there are 5 factors starting from a up to e. Here we denote this fractional design as 2 of resolution 5 2 level factorial design of resolution 5 involving 5 factors and the fraction is a half fraction. In resolution 5 design you can easily understand that the no main effects are aliased with one another or even with 2 factor interactions but they would be aliased with 4 factor interactions that is a very big benefit 2 factor interactions are aliased with 3 factor interactions. We can even construct a smaller fraction and the number of factors increase it is tempting and makes more economical sense not to just consider a half fraction but even consider a 1 by 4 fraction and if the number of factors are really large a 1 by 8 fraction may also be suitable and that we may represent as 1 by 4 2 power n 1 by 8 2 power n 4 can be written as 2 power 2 8 can be written as 2 power 3 and so this will become 2 power n-2 this will become 2 power n-3 as shown here. In a 2 power 6 design involving 64 runs there are 6 main factors 6 c 2 2 factor interactions 6 c 2 would be 6 into 5 divided by 2 which is 30 by 2 which is 15 and then you also have 6 c 3 3 factor interactions 6 into 5 30 30 into 4 120 120 by 6 is 20 23 factor interactions 15 4 factor interactions 64 is equal to 62 6 5 factor interactions and 1 6 factor interaction let us see whether the total adds up to 63 6 plus 15 21 plus 20 41 41 plus 15 56 32 62 rather plus 163 the remaining one corresponds to beta 0 in the model. So we can see that most of the number of effects are consumed by higher order terms even a 2 power 6-1 fraction may be costly in terms of the cost to benefit ratio. So we may even consider 2 power 6-2 fractional factorial design that means a quarter fraction of a 2 power 6 fractional factorial design. So how do we do that the general first step is to write down the 2 power 4 design in the usual manner involving factors a b c and d create a design table in the standard order and so this is the standard order and we are imagining as if there are only 4 factors we are having a 2 power 6 case which is involving 64 experiments we are looking at a quarter fraction. So since we are considering a quarter fraction that means that is 1 by 4 so consider the first 4 factors for convenience a b c and d write down the design matrix the standard order this is the standard order and you then define the remaining 2 unaccounted factors namely e and f e is equal to a b c and f is equal to b c d e is equal to a b c and f is equal to b c d you can also say f is equal to a b c and e is equal to b c d there is no problem for illustration I am taking e is equal to a b c and f is equal to b c d and so you can generate smaller fractions 2 power 6 minus 2 fractional factorial and the additional factors were created from e is equal to a b c and f is equal to b c d. So e is equal to a b c means the setting in the design matrix for e would be minus 1 then I do a b c it becomes plus 1 so I can fill up these 2 columns and what is that actually mean so how do I get the first fraction how do I actually do the experiments before we analyze let us see the practical issue here how do we get the first fraction. There are 4 fractions here because we are considering a 1 by 4 of a 2 power 6 design so you should have 4 fractions and how do you identify the 4 fractions so you have a b c and b c d so you can write all those elements corresponding to plus 1 in this column the complete set okay you write the full factorial design involving 2 power 4 sorry 2 power 6 case and all the plus 1 in a b c and all the plus 1 in b c d will constitute the first fraction the second may be all the pluses in a b c all the minuses in b c d will help to contribute to the second fraction and the third fraction would be minuses in a b c and pluses in b c d that combination and the final fourth fraction would be negative in e and negative in f or negative in a b c and negative in b c d so with this combination you should be able to get the or identify the 4 fraction settings so the complete defining relation for a 2 power 6-2 design is i is equal to a b c e so if I do e squared I get i e into e would be all pluses so i is equal to a b c e f squared will be b c d f so i is equal to b c d f i is equal to a b c e so that is the complete defining relation for a 2 power 6-2 design i is equal to a b c e b c d f and if I multiply the 2 I get a b squared c squared let me see what are all the repeating terms a is not repeated if I multiply these 2 so a survives b and c do not survive because b squared and c squared will result so we have a d e f so you have a d e f here. So the complete defining relation for the design is given by i is equal to a b c e is equal to b c d f equals a d e f each of these terms is referred to as the word so once you have the complete defining relation to detect the alias of an effect simply multiply the effect throughout with all the words that are present in the complete defining relationship. So for this particular case I multiply a here I will get a is equal to b c e equals a b c d f and equal to d e f that means a is aliased with b c e a is also aliased with a b c d f and a is aliased with d e f and that is what is given here similarly the alias of effect a b you can find easily a b is equal to c e equals a c d f and b d e f similarly for effect a b c is aliased with a b c is equal to e equals a d f is equal to b c d e f and so on how did we get here you have a b c so if I put a b c here a will cancel out and then you have b c d e f in other words a b c is aliased with the 5 factor interaction also the length the number of letters of the shortest word in the complete defining relationship is the resolution of the 2 power k minus p design k is number of factors and p is the order of the fraction and so the shortest word in the defining relationship gives you the resolution and so this is the shortest word that would be 4 so we were constructing a resolution of 4 in this particular case please do not say the resolution is 3 the resolution is 4 because you have to have the generator i and then the different generators so you have i is equal to a b c e equals b c d f is equal to a d e f the shortest word here is of length 4. So now let us take 1 by 4 fraction of 2 power 7 2 level factorial design 2 power 7 would be 128 experiments and you are going to have a quarter fraction that means each fraction would have 32 experiments so when this design is of resolution 4 we represent it as 2 level factorial design of resolution 4 involving 7 factors and performed through quarter fractions 4 fractions are involved so to look at the construction of the design we will have 32 experiments so we will first run it as a usual 2 power 5 design how many factors we have we have 7 what are those factors a b c d e that would be 5 f and g come corresponding to the 6th and 7th factor for convenience let us start with a b c d e and run it as a proper 2 power 5 design and what are the design generators we define the generators as f is equal to a b c d and g is equal to b c d e the two remaining factors are f and g so f we alias with a b c d and g we alias with b c d e we can have that is yeah a b c d and a b d e we can have f as a b c d and g we can even have it as a b d e so you are not really constrained to specific cases so other possibilities also sometimes there now we have 4 fractions and each fraction may be identified according to the 4 combination arising from plus or minus a b c d e and plus or minus a b d e this is what we saw earlier in the earlier example so the first fraction would be the entries corresponding to plus a b c d and plus a b d e and then it will be plus a b c d entries and plus sorry minus a b d e entries then you can have minus a b c d and plus a b d e minus a b c d and minus a b d e will complete the last fraction so you write down the standard design here and then you put f is equal to a b c d and g is equal to b c d e so if you are next taking a case involving a 1 by 8 fraction of the full set we have 1 by 2 power 3 into 2 power 7 number of factors is 7 and we are looking at 2 power 3 that means 1 8th of a fraction so when you choose a resolution as 4 we represent it as 2 level factorial design of resolution 4 with the 7 factors and we are considering a 1 by 8 fraction so we will run the experiments as usual in the 2 power 4 full factorial mode and define the different design generators as e is equal to a b c f is equal to b c d and g is equal to a c d this is very interesting and ingenious also so we are first considering only the first 4 factors a b c d and then the remaining factors are set at a b c b c d and a c d we are not putting e is equal to a b we are trying to get the aliasing with the highest order interaction which is possible so we have e is equal to a b c f is equal to b c d and g is equal to a c d so we have the defining the different design generators are a b c e b c d f and a c d g this is the defining relation rather sorry so these are the design generators and the defining relation is given by i is equal to a b c e b c d f and a c d g the length of the shortest word here is 4 so we are we have constructed a 2 power fractional factorial design and 8 fractions we can get by looking at these combinations e is equal to plus a b c f is equal to plus b c d g is equal to plus a c d will constitute the first fraction e is equal to plus a b c f is equal to plus b c d g is equal to minus a c d will constitute the second fraction and so you can have 2 into 2 into 2 8 possibilities to complete your 8 fractions a complete table of choice of words is given in standard text like Montgomery and Runger 2011 or Montgomery 2009 so you do not have to remember anything you have to just see the number of runs you can economically carry out in your workplace and then identify the appropriate resolution and then set up the design matrix find the contrast and calculate the effects identify which factor is identified is aliased rather with the other factors so once you have done this you can sequentially conduct the different fractions and get more and more information from your set of experiments sometimes you may even stop after finishing the first 2 fractions saying that I have now pretty good idea about the process it does not it is a law of diminishing returns so after the first 2 fractions I may not really need the third fraction even if you save on one fraction that means you do only 3 out of 4 fractions that means you have done the experiments efficiently without overdoing them sometimes even overdoing experiments is not good the best way to understand this would be through an example and I will be covering examples for factorial designs and fractional factorial designs in the next lecture please go through what I have said they are pretty straight forward also refer to the standard textbooks I have referred to look identify the different tables and see how you may use them the important thing is to identify the number of fractions the resolution of the design the design generators the defining relationship and what the different fractions are sometimes when you are having 2 power 7-3 1 by 8 fraction you have to set up the 8 fractions correctly there are software which also help you do this one of them is mini tab so what we will do is do a few problems in both factorial designs and fractional factorial designs thank you.