Fractional Factorial Design (DoE) Simply explained

what is a fractional factorial design a fractional factorial design is a type of experimental design used to analyze the effects of several factors or variables on a response variable what is the difference to a full factorial design in a full factorial design every possible combination of factor levels is examined conversely a fractional factorial design strategically excludes certain combinations this redu reduces the number of experiments while still capturing essential information about the system's behavior let's take a look at an example let's assume we want to know what has an influence on the frictional torque of a

bearing the frictional torque is then a response variable possible factors could be lubrication temperature and the bearing material the lubrication could have the factor levels or characteristics oil and grease the temperature could have the level low and high and the bearing material steel and keramic in a full factorial design we would now test all possible combinations of factor levels we would therefore test the effect of lubrication with oil temperature low and material Steel on the frictional torque the effect of lubrication oil temperature low and material caramic on the frictional torque the effect of oil high

and steel on the frictional torque and so on and so forth this makes it possible not only to find out whether the individual factors have an effect on a frictional torque but also to find out whether there are so-called interaction effects between the factors however the number of runs required increases very quickly as the number of factors increases therefore let us now take a look at the fractional factorial design the fractional factorial design is used for screening experimental designs I.E if you have more than approximately four to six factors of course reducing the number of

runs means reducing information in fractional factorial designs the resolution is reduced what is the resolution the resolution is a measure of how well a doe can distinguish between different effects more precisely the resolution indicates how much the main effects and interaction effects are confounded in a design but what are main effects and interaction effects and what does confounded mean in design of experiments the term effect refers to the impact that a certain factor or a combination of factors has on the response variable of an experiment essentially they measure how much the response variable changes when

you change the factors a main effect is now the influence of a single Factor on the response variable for example what influence the lubrication of a bearing has on the frictional torque interaction effects occur when the effect of one factor on the response variable depends on the level of another factor for example the effect of the lubricant on the frictional torque could depend on the temperature but what does that mean let's say we have an average frictional torque value of 102 Newton mm for the bearings with oil and an average value of 108 Newt mm

for the bearings with grease then we have a main effec of lubrication of 6 Newton mm but now we can break this down into high and low temperatures at high temperature we could get 98 for oil and 102 for grease so the difference between oil and grease is only 4 Newton mm at low temperature we could get 104 and2 so a difference of eight so the lubrication factor is influenced by the temperature and we have an interaction between lubrication and temperature the interaction leads to a difference of 2 Newton mm to the original result we

therefore have an interaction effect of 2 new mm full factorial designs take all interactions into account in our bearing friction example in addition to the lubricant and temperature factors we also looked at the interaction between lubricant and temperature however as the number of factors increases numberous interactions rapidly emerge for example if we have the five factors a b c d and e we get the interaction between two factors between three factors between four factors and between all five factors now of course the question is do we really need all the interactions or can we reduce

the resolution and this is exactly what the fractional factorial design does in a fractional factorial design interaction can be confounded with other interactions or with main effects or factors what does confounded mean effects of different factors or the effect of interaction of factors cannot be separated from each other the extent to which the number of runs can be reduced at the expense of resolution is shown in this table the resolution is usually indicated by Roman numers EG 3 4 5 and so on and so forth here on a diagonal we see the full factorial designs

we'll go through what resolutions three four and five mean in a moment for example if we have six factors we need at least 64 runs for a full factorial design if we choose a fractional factorial design with a resolution of six we need 32 runs with a resolution of four we need 16 runs and with a resolution of three we need only eight runs but what does that mean and how does does it work the full factorial design is always used as the starting point let's take a look at the example with eight runs let's

say we have the factors a b and c with a full factorial design we can now test whether factor a b or c have an effect we can test whether the interactions with two factors have an effect and whether the interaction with all three factors has an effect if we now want to test not just three factors with a runs but an additional fourth Factor say Factor D then we must drop the information from one of the interactions for example the interaction of a b and c if we want to test a fifth factor with

a trials let's say Factor e then we will have to drop another interaction for example the interaction between B and C but I didn't express myself correctly we are not actually dropping the information we are mixing the new factor with the interaction we've confounded the factor with the interaction what does that mean it means that we are unable to determine if an observed effect is attributable to factor D or to the interaction of a b and c similarily we can't distinguish if an effect results from Factor e or from the interaction of A and B

of course it is much less critical to mix one factor with an interaction of three factors then with an interaction of two factors and now we have a good transition to the resolution what do the resolutions 3 four and five mean at resolution three main effects can be confounded with interactions of two factors for example Factor D with the interaction of the factors A and B experiments with resolution three should therefore be considered as critical they can only be used if the interaction of two factors is significantly small smaller than the effects of the main

factors otherwise the interaction of two factors can significantly distort the result of one factor experiments at resolution four are much less critical here only the main effects are confounded with the interactions of the three factors and the more factors are involved in an interaction the smaller the effect is likely to be furthermore in resolution four interactions with two factors are confounded with interactions with other two factors experiments at resolution five are not considered critical main effects are only confounded with interactions of four factors in the same way two Factor interactions are only confounded with interactions

of three factors but how do you confound a factor and an interaction let's take a look at this example here we have the full factorial design of the three factors a a b and c these eight runs are carried out in total we still only consider factors with two levels minus one stands for one level and one for the other for our frictional torque example the test plan would then look like this for example for the factor temperature minus one is the low temperature and one is the high temperature if we now run the experiments

we obtain a value for the response variable for each run if the factor a is one or minus one this has a certain effect on the target value exactly the same if the factor B is one or minus one the interaction effect now tells us whether there is an additional effect if Factor A and B are simultaneously one or minus one or if both go in exactly opposite direction so on one side we have the pairings with the same sign and on the other side the pairings with an unequal sign now we can check whether

there is a difference in a response variable between the values in the green group and the values in the red group if yes then there is an interaction between A and B however if we know in advance that there is only a very small interaction if any then we can use these combinations to test a fourth Factor D to do this we simply multiply A and B now we always have a one if the factors a and B have the same sign and minus one if they have a different sign now of course a problem

may arise when analyzing the results if there is a difference between the green and the red values in the response variable we cannot say whether this effect comes from the interaction between a and b or whether the effect comes from the factor D if we are already sure that there can be no interaction between A and B this is of course not a problem then we can be sure that the difference is due to factor D in the same way we can now take the interaction of a and c and also measure Factor e and

the interaction of A and B and C and measure factor F therefore in this case we measure six factors with only eight runs but we can no longer distinguish the factor D from the interaction A and B we can no longer distinguish the factor E from the interaction A and C and we can no longer distinguish factor F from the interaction a b and c and now I'll show you how you can create a fractional factorial design online with data tab to do this simply go to data.net click on plus and then on doe here

you can select which design you would like to create eg a full factorial design or a fractional factorial design so let's click on fractional factorial design next you can determine the number of factors and the number of runs let's just take four factors and eight runs so we have a resolution of four here can now enter the names of the factors and Define the factor levels let's say temperature with low and high lubrication with oil and grease material with steel and keramic and speed with low and high you will then be shown the test plan

and can export it now you can carry out the test afterwards you can copy all the data back to data Tab and evaluate the results I hope you enjoyed the video see you soon