in this video we will discuss what a full factorial design is how to create one and how to analyze the results a full factorial design is a systematic method for examining the effects and the possible interactions of several factors on a response variable the aim is to analyze the relationship between several input variables and an output variable the input variables of the system are called factors and the output variable is called response variable of course there can also be several response variables therefore factors are the variables believ to affect the response each factor now has at least two levels these levels are the specific values a factor can take and we want to find out whether a change in the levels of the factors has an influence on the response let's look at an example when you paddle a bike the wheels rotate around a central Rod known as the axle to ensure that the wheel can spin there are bearings now imagine you're paddling your bike on a smooth road if the bearings are in good condition the wheels will spin easily the so-called frictional torque in this case is low if the bearing is not in a good condition you have to push harder to keep the same speed because there's more frictional torque with the help of a full factorial design we would now like to find out what influences the frictional torque of a bearing in this case of course the frictional torque is our Target variable possible factors could be lubrication or temperature for lubrication there could be the factor levels oiled and greased and for temperature there could be the levels low medium and high now we want to know what influence the different levels have on the response variable a full factorial design can help us here especially if we have not just two but three four or five variables a full factorial design can lead to Great savings depending on the field of study the system under investigation can be a process a machine or a product but of course also humans for example if you want to investigate the influence of medication but now we still have four open questions how to estimate the number of experiments needed how does a full factorial design work how can a full factorial design help to reduce the number of experiments how to analyze the results let's start with the first question experiments cost time and money the number of runs should therefore be kept as small as possible but beware if the number of runs is too small there is a high risk that relevant differences will not be recognized let's take a look at our example we would like to investigate which factors have an influence on the frictional torque of a bearing let's start with one factor lubrication we would like to know whether it has an influence on the frictional torque if a bearing is oiled or greased to find out we take a random sample of 10 bearings we oil one half of the bearings and grease the other half now we can measure the frictional torque of the five oil bearings and the five re bearings but stop why 10 bearings in most cases each run costs a lot of money perhaps we can manage with fewer runs so how many experiments do we need in order to find out if the lubricant has an effect on the frictional torque let's just start with 10 bearings we can now calculate the mean value of the frictional torque of the oiled and greased bearings then we can calculate the difference between the two mean values in this sample we can see that there is a difference between oiled and greased however we can already see that the frictional torque in the oiled and greased bearings is highly scattered if we take another random sample of 10 bearings it is possible that this time the difference will be greater but it is also possible that the difference will be in exactly the opposite direction in other words the frictional torque of the bearings scatters the wi the spread the more difficult it is to identify a specific difference or effect fortunately the variability of the mean value can be reduced by increasing the sample size the larger the sample size the more precise is the estimation of the mean therefore the smaller the effect and The Wider the spread of the response variable the larger the sample needs to be but how much larger how can you estimate the number of runs needed you can use this formula as an approximation here n is the number of runs Sigma is the standard deviation and Delta is the effect to be determined for example if we have a standard deviation of 3 Newton mm and a difference of 5 Newton mm is relevant for us then we need 22 runs if we only have a standard deviation of 2 Newton mm we only need 10 runs and with a standard deviation of one newton mm we get 2.4 so we need two runs with greased bearings and two runs with oil bearings but how can a full factorial design help you reduce the number of experiments let's assume that the calculation of the number of runs results in 16 experiments so eight runs with oil bearings and eight runs with grease bearings but what if we have a second Factor let's say in addition to lubrication we have temperature with low and high then we need another eight runs to take this factor into account so we need 16 runs to check if the lubricant has an effect and 16 runs to check if the temperature has an effect this gives us a total of 24 runs now the question arises is it possible to achieve this with fewer runs and that brings us to the full factorial design the question is why should we limit ourselves to testing one factor at a time instead we could devise a design that incorporates the fourth potential combination which is grease and high temperature of course we still need 16 runs per Factor we get this by making four runs with each of the four combinations then we have eight runs with oil and eight with grease and on the other side eight with low temperature and eight with high temperature we now have a total of 16 runs before that we had 24 runs we now need fewer experiments and get even more information why more information we now also know whether there is an interaction between temperature and lubrication for example oil bearings may show A variation in frictional torque at different temperatures which is not observed with greased bearings this information would have been lost previously now when we have three factors instead of two the savings are even higher if we test one of the three factors at a time we need 32 runs if we now run two experiments for each combination in a full factorial design we still only need 16 runs however for each factor we still have eight tests per Factor level for example for the lubrication Factor we have eight runs with oil and AD runs with grease of course we can also create full factorial designs with more than two levels for example the temperature Factor could have three Factor levels low medium and high however even with a full factorial designed with two levels in each factor the number of runs required increases very quickly as the number of factors increases in a full factorial design the number of experiments or runs is n equal to 2 to the power of K where n is the number of runs and K is the number of factors here is a small overview if we have three factors for example we have to make at least eight runs with seven factors it is already at least 128 runs and with 10 factors it is already at least 1,24 runs therefore full factorial designs are generally only used up to a maximum of six or seven factors please note that this table applies to a full factorial design where each factor only has two levels otherwise there will be even more runs and now I'll show you how you can create a full factorial design online with data Tab and how you can then analyze and interpret the results to do this simply go to data.net click on plus and then on doe here you can choose which type of design you want to create we want to create a full factorial design in this case we assume that each factor has only two levels in the second case there is no restriction and you can Define the number of levels for each factor we will simply use the design with two levels next we can find the number of factors we take three factors and now we can Define the names and the values of the factor levels let's say temperature with low and high lubrication with oil and grease and material with steel and caramic now we can see the created test plan each row is now an experiment the first experiment we do for example with temperature low with lubrication oil and material deal if we want we can still choose to order randomly so that there is as little influence as possible regarding the order depending on how many runs we need we can determine the number of applications we'll just take two each combination is now performed twice so we don't have eight runs but 16 now we can export the table to Exel and carry out the individual experiments we then write down the friction torque obtained for each experiment note this values are just fictional once we have performed all experiments we can copy the data back to data Tab and analyze them to do this we click on doe analysis and then select all factors and the target value now we can click on the two Factor interactions and the three Factor interaction we now get the regression coefficients with coded values the paror diagram the regression coefficients with uncoded values and the results of the analysis of variance what is the difference between coded and uncoded regression coefficients in our example we have categorical Factor levels if we copy these into Data tab data tab simply numbers the different categories with values for the temperature factor for example the category low has the value one and high the value two of course we can also change this we can also give low the value minus one and high the value one and this is exactly what is done automatically with the coded coefficients here the categories are automatically set to minus one and one how are the results interpreted the first step is to analyze which factors and interactions have a significant influence on the frictional torque a factor interaction has a statistically significant influence if the P value in this table is smaller than 0.05 in this case the temperature Factor has a P value of less than 0.05 as well as the lubrication and the material all main effects therefore have a significant influence on the frictional torque we can also see that the interaction between temperature and lubrication also has a significant influence however the other interactions have a P value of 0.5 which is greater than 0.05 so these interactions have no significant influence on the frictional torque exactly the same can be seen in this par diagram in the diagram the amount of the T value is plotted for temperature for example the value of T is 2616 which we can also see in the table above the red line indicates the critical T value I.E Above This value we obtain a P value of less than 0.05 and there for a significant value the results show that the last three interactions are not significant in order to obtain the final results the nonsignificant factors and interactions are now removed from the model so we remove these three interactions from the model we've now created our model and determined the significant factors and interactions in addition data tab displays the results in diagrams and you can view the results of the analysis of variance finally the optimal input parameters can be calculated ated in order to minimize or maximize the target value of course this makes particular sense if we have matric factors which in the best case have been measured with three Factor levels each I hope you enjoyed the video and see you next time