 Welcome to session 30 on course on Quality Control and Improvement with Minitabh and Professor Indrajit Mukherjee from Shailesh J. Mehta School of Management at Bombay. So, earlier in the last session what we have seen is that we have taken two categorical factors and we have seen the influence on the CTQs like adhesive strength and we have taken and we have seen that the interaction is not so signal. So, we have experimented and we have analyzed the data like this. So, and the same thing we will do for another experimentation to see that when there is a continuous variable what else we can do. So, here in in this case battery life experimentation this is taken from Douglas Montgomery's books on Design and Analysis of Experiments. And in this case the engineer is interested into maximize the battery life and two factors are considered over here one is the material type that is being selected over here and other one is temperature various levels of temperatures are selected over here. So, temperature is a continuous variable. So, this variable what we are mentioning over temperature is a continuous variable, but material type is basically a categorical variable. So, this is continuous variable and this is a categorical variable ok. So, in this case and battery life what we are expecting is continuous variable. So, this is also a continuous variable like that ok. So, we want we want to see that one and experiment was conducted at three levels in each ah for material type A, a material type which is a factor A and this is the temperature which is factor B and temperature is at three levels, discrete levels over here and material types are at three levels which is and this is a symmetric design because levels are ah levels are ah same for material type and also temperature like that and also the number of ah replicates that is collected over here. So, this is a balanced design we we can also see that number of replicates is same and this is a symmetric design this is a symmetric design because the levels are also same over here. So, ah basically ah number of combinations is ah two factors we have and each is at three levels. So, total number of experimental trials is 9 over here and in each trial I have four ah replicates over here. So, 9 to 4 ah that is 36 number of observations we are having and everything is done at keeping ah ah the keeping in mind that it is randomized experimentation. So, every data is collected over here. So, whenever you have collected the data ah what is required is that we need to analyze the data over here ok. So, what we will do is that we will just see how to analyze this data. So, when one is categorical one is ah basically continuous variable in the predictor side and the other one is CTQ is continuous variable over here. So, all assumptions normally assumptions and everything is true whenever I am doing design of experiments. So, we have to verify that one whenever we get the residual and we have to cross check all the assumptions that we have done in regression also ok. So, in this case ah what we will do is that this data is in MINITAB and I have already I have already located this data over here C6, C7, C8 this is plate material temperature and battery life over here. So, first we can see that whether the variance is same throughout for every plate material and temperature combinations whether the variance ah whether the variance is same or not. So, that check or ah we can do that. So, we go to ANOVA analysis over here and test for equal variance then I say that my response factor over here is ah basically we can delete this one and we we can write response factor is basically ah response is ah temperature battery life over here and the factor is plate material and temperature operations over here and options we will go for ah other than normality. So, we can also do normality test which is ah which will give me ah other results Bartlett's test. So, in this case if I go with this and I will see the Levin's test results like that and this is the Levin's test and multiple comparison test that we have done. So, ah we can take any of this over here. So, I will go by Levin's test let us say and 608 is quite ah significantly more than 0.05. So, in this case we can say that there is no as such deviation, but at 15 although we are seeing, but overall results is showing it is not so significant ah the variance is not changing ah significantly ok throughout the experimentation. So, we are satisfied with this. So, immediately what we will do is that we will go for ANOVA analysis or balanced ANOVA. So, I will go to ANOVA balanced ANOVA over here. So, then when I go to balanced ANOVA then I will write which are the which is the response. So, I will give response as bit material type then I will give plate material and temperature over here and also plate material and interactions I want to see the interactions of multiplication of this with ah temperature over here ah temperature of operation. So, in options I do not have to do anything and in graphs way we can see normal plot and the residual plots per se speed and storage what we can do is that residual we can save and then we can click ok over here. And then we can see the ANOVA analysis that is given over here. So, in this case I will copy this one and take it to excel and try to see enlarge this one and see what is the result outcome and it is saying that plate material type when you are changing this one this is having a significant impact on the battery life. So, ah there is at least two levels where when I change the material from 1 to 2 or 2 to 3 like that. So, it is impacting basically the mean expected value of the battery life and temperature is also when I change that one it is also significantly influencing like that. And also we can see that the plate material type and temperature of operation interaction between these two is also significant because this is also less than 0.05. So, interaction is prominent ah individually they are prominent. So, that means, all need to be considered when when we are trying to determine what is the optimal combinations like that. Whenever interaction is prominent in that case we cannot ignore this interaction interactions between the variables while we are determining the optimal combinations of plate material and temperature like that ok. So, ah what we have to do is that whenever we have seen this one and normal probability plot seems to be satisfactory. So, there is no problem as such we can also check that one because C14 is the residual. So, we can just check that one whether everything is fine. So, in this case normality test we are doing and residual 2 that because earlier there was some other residual that was saved. So, in this case what we are seeing is that p value of this is more than 0.05. So, it is satisfactory basically. So, we do not have any problem in normality assumptions like that. So, we can delete these two over here and then what we can do is that we have done this one and so, ah we can see two plots over here one is known as main effect plot one is known as interaction plot like that ok. So, ah what we have to do is that to get the best possible settings that that we that we have to see ah. So, in this case I have to go to stat ANOVA analysis and there there will be a main effect plot and one interaction plot over here because interaction is prominent I am going by interaction plot. So, what I will do is that I will I mention which is the response and I want to see where where the response is maximized and then I will give the two variables one is plate material and one is temperature over here and I will click display full interaction plot over here similarly what we have done last time also. So, in this case I clicked ok and whenever the interaction is present you see that the graphical ah representation is somewhat different as what we have seen not similar to the what we have seen when we are talking about as it adhesive strength maximization like that with two categorical variables what we have seen and what we will see if there is interaction you you can expect that there is this lines will cross each other. So, what you are seeing is that lines are crossing each other. So, in this case ah what is expected is that whenever whenever such such kind of scenario of interaction is prominent ah the lines will cross each other basically and ah they will not be parallel they will not be parallel like that. So, over here what is expected is that you can see that I will see one one half of this. So, let us see upper upper right hand side corner I will see over here. So, in this case what you see is that ah to maximize the battery life this is the highest point that we are observing this this ah ah brown point that you are seeing over here in 15. So, ah and this line what it is shows is that plate material type 2. So, plate material type 2 and 15 degree gives you the best ah combination over here this gives you the best combination nearest to this is ah we have material type 3 and 15 degree that is also. So, what we are seeing is that we are seeing maximum temperature or maximum battery life condition is appearing when we are taking a combination at lower temperature the 15 degree and material type 2 or 3 like that 2 or 3 any of this material ok. Visually what we are seeing over here. So, in this case there is no problem in seeing this one although ah this plate material type 2 is the is the ah preferred one because this is, but we have to make a multiple comparison test and figure out whether this is different from this one. So, whether this point what combination of 15 and 2 is different from 15 and 3 like that. So, we have seen multiple comparison test that is possible and we can see that one and, but we have to consider over here something else ah what was given in the problem what was given in the problem it was mentioned that ah which material will be robust to temperature change while we are selecting keep into mind that which material I should select which is robust to temperature changes like that. So, if you see this ah diagrammatically over here what you observe is that for material type 3 what you see there is a flat region from 15 to 70 over here that means this ah battery life does not change much ah if you are following plate material type 3 over here, but if you are taking a plate material 2 the slope of this line is drastically falling what we can see as compared to material type 3. So, the slope is higher than material type 3. So, material type 3 is more robust within the temperature zone of 15 to 70 like that if you are considering that temperature range. So, it is insensitive to the change in temperature if I change from 15 to 70 any range within this assuming the continuity of the of the or ah we can say that within this range we we can expect. So, ah we are just extrapolating our ah interpretation over here. So, 15 to 70 ah because we have done on discrete points. So, but we we can we can just see the interpolation over here to be to be more or less flat what we are seeing over here. So, in this case what we are trying to say is that this this is more flat. So, if you have to take reference on robustness now which material I should use I I should go for material 3 over here in the range of 15 to 70 and beyond 70 also see material type ah 3 is giving you higher value of ah battery life as compared to any other material material 2 or ah material 3, but anyhow material 1 is of we can we can ignore this one. So, ah for a 2 ah what you see from 70 to 125 also it is lower than 70 to 125 battery life what is given by the material material 3 over here. So, ah without much hesitation what we can do is that ah we can select material type 3 ah if I want a robust ah robust material which is which is insensitive to temperature changes like that we will go for material 3 like that. So, this is interaction plot what we can see over here and ah we can make a comparison test also. So, if you want to make a comparison test go to analysis and the general linear model comparison test over here. So, you have to just mention that battery life is 1 2 case test and I want to see plate material type 3 and temperature and I want to make a comparison because that is ah prominent what we have seen. So, here ah what you what you what you can see is that ah grouping information is given. So, 2 and 15 and 3 and 15 are not different as such, but based on the robustness assumptions what we are doing is that we are adopting ah 3 with 3 as the material type because ah although there is significant no significant difference because the later code remains same. So, but we will select go for 3 because that is more robust with the change in temperature like that. So, this comparison test is also possible to see which levels to select which combination of the levels to select like that that is possible and also we can fit a ah regression model that that we told that ah we can do general linear models like that fit general linear models over here. So, I have battery life which I want to predict and plate material type and temperature is done and ah we want to select the models that interactions also is selected over here what you can observe and when I click ok and I click ok over here. So, in options ah we have not given anything so that is not required and in graphical representation this normal ah standardized residual we we can do that and we can see the residual plots also and then ah like what we do in regression. So, if I click ok over here now what you observe is that the same results what we have get analysis of variance remains same. So, and ah regression r square adjusted is around 69 r square predicted is 58 ah although not drastically changing like that, but predictive behavior is not so much what we expect ah. So, in this case, but we have generated a regression model which is significant over here and we have we have developed that based on the ah ANOVA analysis results that we are getting and then what we can do is that use this prediction we we can predict anything like that. So, I have selected a material type 3 let us say and at 15 degree what is expected. So, I will go to ANOVA analysis and I go to general linear model and predict. If I go to predict it will say what material type you want to predict. So, I will say 3 and temperature range let us say I want to predict what will happen in 15 ah give me some what will be the battery life. So, in this case I will click ok and it will give me some possible values over here. So, this is the predicted fit values. So, the regression model that is ah general linear model that we have fitted over here it is predicting about 144 is the values and it will give you a prediction interval and confidence interval like that ok. So, 144 ah is expected when when this combination is run at a separate and we can read on this one. So, expected value is around this one, but there is a prediction interval that is given over here. So, there will be ah for a given value of x ah x conditions over here which is ah 3 and 15 combinations. So, for that some prediction interval is also provided over here. Now, one of the variable is continuous variable over here one is discrete or categorical variable over here and the CTQ is continuous variable that we can see. So, there is another option what we can do is that we can also see graphically ah surface plot of this. So, we have seen interaction. So, how does it look when there is interaction how the graph looks in a 3D dimensions if you want to see then we can see this by 3D surface plot and when you go to that there is wire frame surface like that I go by surface let us say surface plots like that and in this case you can you can identify that which is the variable. So, battery life is the variable which will be y axis and which will be x axis I let us say y axis is temperature over here and x axis is plate material you can change ok. Then surface options over here we can we can see the methods of this mesh over here y mesh numbering over here mesh numbering. So, in this case and this this is not so no much required to change the settings over here if you click ok what will happen is that you will get a plot like this you will get a plot like this and which shows that there is some curvature in the in the graph what you see over here. So, this can be rotated also you can see by rotation of this. So, you can see rotation on z axis over here and also you can rotate on x axis you can rotate on x axis and so you can see the surface over here. So, if you see the surface like this so you can see that it is not plain surface what is expected because interaction is present. So, in this case you can expect some amount of curvature that is present in the model. So, that is prominent over here that you can see and you can change the direction over here you can also change the graphical customize the surface like that. So, in this case if you want to change the color like this you can change the color like that that is possible over here. So, you can change the color then you can see which is which is the lower surface which is the upper surface like that and the curvature that is present in the surface like that. So, one is categorical variable one is continuous variable and better life is continuous. So, we can see the plots of this this is possible over here and then what we have is that we can also have a. So, this is the surface plot that we are seeing over here. So, what I have done is that I have just changed the surface over here. So, customize this one and I have made the surface like this and let us say this is the surface over here. Now, if you take a top view of this surface if you take a top view of this surface then what we get is a contour plot that what you what is known as contour plot, contour plot which is also very important when we are talking about optimization when we talk about optimization we also use that contour plot to see and locate where the. So, it is like you are taking a top view of a mountain like that. So, where the mountains will have different surfaces like that and on the top from top what you what you see is that and the altitude if you see mountain altitude that is the y axis what you see z axis that you are seeing over here. So, that will be in sudden planes it will be same altitude will be same like that. So, contour plot is an important plot which can also be a minute have also gives you that options. So, if you go to graph and you go to contour plot over here and you say battery life and temperature and plate material types then generally the variables should be continuous treated as continuous variable and we do the contour plot, but for your sake of simplicity we are doing this we will take another example where all the variables are continuous and that case it will be more relevant, but just showing you the options that we have. So, here there is options of contour plot like this. So, where the battery life will be maximized like that. So, battery life over here on the right hand side what you see is that greater than 180 what is expected in the green zone that is the dark green zone that you are seeing over here, plate material 2 and 3 approximately that one and temperature range over here. So, this is around 15 that we we can expect that it will be maximized like that. So, contour plot is possible, but we as one of the one of the variable is discrete over here. So, let us not do this one over here, but I just showed you that there is an option of contour plot which is also used for when defining the reason which is where the optimality lies basically, where to see basically. So, this is contour plot options is there and these are the things that you can do when one is categorical, one is continuous variable. So, these are the possibilities what we have. Then we have another example over here at the at the end what we are having is that we are having another example over here where temperature and pressure is given. So, this is also another example. So, this example and this is a two factor experimentation. So, where impurity is impurity is our concern we want to minimize the impurities over here and we have different combinations of temperature and pressure that was run. So, temperature is at 1 to 3 levels over here, temperature is at 3 levels over here and this is having a level 4 levels over here. So, asymmetric design what you 5 5 levels over here. So, this is 5 levels over here. So, asymmetric design one is at 5 level, one is at 3 level, 5 into 3 there are 15 experimental trials. So, overall there are replicates like that. So, that total number of experiments. So, for every possibility so, total number of trials is 15 over here and there is no replicates over here in this design there is no replication. So, there is no replication that is given over here. So, we want to analyze this data, we want to analyze this data and see what is happening if I am taking this data set what is happening. So, what how to analyze this one. So, if I discard this one only thing you have to remember there is no replicate over here that means we are not repeating the trials over here. So, in this case this is the data set that we are having temperature pressure and impurities over here. So, what we will do is that let us go directly to analysis of variance and let us try to do and see that because there is one replicate. So, we can also go for balanced experimentation over here only thing is that what will be the outcome that that is of concern for us. So, in this case impurity and the model variable is temperature and pressure over here and let us say I want to estimate the interactions also over here temperature and pressure and this options we will give over here and graphical options we will place like this and then we place ok over here. What will happen is that you will see that if I copy this one as picture and if I paste this over here you will find the estimation is not possible over here as there is no replicates. So, temperature is at two levels over here ok degree of freedom is 2 pressure degree of freedom is 4 and pressure degree of freedom because there are five levels over here. So, total degree of freedom is 6 that is consumed over here by these two factors over here and a total 15 experimental trials was done because 5 into 3 15 trials are done and so, 14 degree of freedom is the total degree of freedom. So, if you subtract 6 from 14 8 degree of freedom and if you if you place temperature and pressure with 8 degree of freedom then there is no error degree of freedom. If there is no error degree of freedom I cannot calculate mean square over here. If I cannot calculate mean square I cannot calculate the F values and P values over here. So, it is not possible to see temperature and pressure information over here. So, in this case we cannot do this we cannot do this. So, we have to confine to now what whether the interaction is present or not. So, how do we calculate interaction? It is already given in Montgomery's books how to calculate interaction in case of single replicates what is to be done. So, I am not going to that details what I will do is that I want to see whether the interaction is prominent or not graphically whether I can see that one. So, if I go to stat and ANOVA analysis over here. So, interaction plots is possible over here. So, or we can directly go to general linear model and we can we can go to factor plots also and we can see the plots like that. So, temperature over here this is this will be impurities that we want to check and then temperature and pressure are the variables and in options titles we are not doing anything. So, graphically we are just depending in main effect plot and interaction plot and when you do this this is the surface. So, you must have seen the surface is also like this. So, zigzag pattern what we have seen and there are two locations what you can see where impurities will be minimized for pressure one is at 30 and one is at 40 like that and temperature is 125 seems to be the condition what what we can expect and and when you when you when you when you do this. So, in this case interaction plot is not given, but we can we can go over here and go to interaction plots over here and using this what we will do is that we will use inquiries and we will give temperature and pressure over here display full interaction plot. If you do that you will find some displays like this in in the interaction plot when we are going by that. So, this is the zigzag pattern what we have seen. So, in this case what you see is that patterns are not intersecting each other most of the time that the patterns are more or less parallel like that. So, we can expect that there there is no interactions as such, but in the Montgomery's books also this was proved that there is no interaction between the variables temperature and pressure over here. So, this things are confirmed by a separate test separate test that is and Minitab does not give you options for that, Intab does not give you options for that, but what I am trying to say is that two variables are there I can graphically see whether interaction is present or not and we can also calculate that for using the mathematical model that is given by given in Montgomery's books that we can use to calculate the interaction effects ok and interaction is not significant over here. So, in this case what we can do is that we can we can we can graphically also see that what is the what is the 3D surface plot that we are generating over here. So, in this case maybe wire frames also you can use. So, instead of this impurity we will take and temperature and pressure we will take over here and I click ok I will get some surface what zigzag patterns that we have seen. So, in this case this graphs can be changed and I can customize the surface patterns over here. So, maybe some colors we can use over here and wire colors also we can use like this and we can click ok like that. So, this is the surface that we are generating over here and we can rotate the surface like that we can rotate the surface and we can see what is happening when I rotate this one. So, this is possible we can change the color. So, this is too dark surface color we can change it to yellow let us say and we can change this one. So, this is the this is the surface that is been generated over here. So, there is too lower peak that we can see 1 at 30 and 1 at 40 approximately around 40. So, this is the surface plot and when we when we when we can also draw the contour plot over here. So, when you draw the contour plot it will be more easier to see. So, in this case what I will do impurity is over here and then we will change this to temperature. So, this is temperature over here and this will be pressure over here and I click ok in the contour plot what you will observe is that there are two pressure points over here. So, in this case what you see around 30 and around 40 we are getting a impurity which is less than 2 like that. So, this is the darkest one and this is the darkest position over here. So, around 40 and 30 the optimality somewhere we can we can see the combination which is giving you the optimality is over here. So, in this case the combination can be 30 125 or 40 125 like that because what we have seen is that based on the because there is no interaction. So, we can have a main effect plot and we can see. So, in this case ANOVA analysis main effect plot we want to we want to see which which is the combination we should use. So, temperature and pressure like this and I click ok main effect plot. So, in this case 30 or 40 any of these options and temperature is coming out to be this if you enlarge this one the best combination is coming out where where I want to minimize the impurities over here what I can do is that take impurity because interaction is not prominent. So, temperature is around 125 and pressure can be either 30 or 40 like that these are the two combinations that we can think of. Minitab gives you another option over here because the variables are continuous over here the two variables are continuous and also the CTQ is continuous over here. Minitab also Minitab also gives you an option for optimization of this what should be the combination of temperature and pressure that will that will give you lowest impurities like that. So, in this case what you can do is that this ANOVA analysis GLM there is a option of there is a option of response optimizer over here and there is also a predictor that you can also use over here. So, predicting any of this so if I am using battery not battery life over here impurities what we have done. So, I can take a combination of let us say temperature over 125 and pressure is approximately let us say 30 and what is the predicted value what we can see over here is around feed value is 0.93, 0.93 3. So, near to we can assume let us say near to 1 like that. So, impurity is less than 2 what we have seen in control plot also. So, that is the prediction and prediction seems to be ok and GLM models also we can we can develop that there is a option of generalized linear model I can feed this one. So, only thing is that I have to use this impurity over here and I have to see what temperature and pressure over here and the model will not take interaction. So, that is not required. So, in this case we will have a generalized linear model and in this case we can make a prediction out of this. So, you know what does this mini-tap does it automatically for you when I am when I am mentioning predict. So, it is developing the equations and based on that it is predicting basically. So, what I am interested in response optimization that means this is a response surface that is developed with the CTQ and temperature and pressure over here. So, I want to identify which is I do not want to see that graphically and all these things I want I do not want to go to that I want to get the optimal condition. So, mini-tap uses an optimization techniques to do that. So, response optimizer. So, what I want to do is that impurity I want to optimize. So, I want to minimize let us say impurity is over here. So, in setup what you have to do is that this mini-tap automatically takes, but you can change the upper limit over here. So, if you want upper limits can be changed and target maybe we can make it impurity I want to be 0 and this may be upper bound may be 20 I can I can just and this depends on you. So, there is no as such hard and fast rule over here. So, in this case options what you have to do is that temperature and pressure whether you want to apply constraints over here because optimization we are doing and it will it will search around the surface that will be generated and based on that. So, regression that response surface will be generated. So, I can say constraint to the region. So, within the region of experimentation let us constraint to the region over here. So, in this case I will click ok and graphically if you want to see optimization plot will be given and if you want to store something we can store and then we click ok what will happen is that it will give you a best combination like that. So, the best combination that is given by this mini-tap software is solution 1. One of the solution is temperature combination of 125 and pressure at 40 that is giving me one one indicator over here what what I will just highlight over here which we will discuss afterwards and this is the indicator that mini-tap uses which is known as desirability which is known as desirability and this is the last quantity. It is close to one indicates that that is the best solution we are we are near to the best solution basically ok. We are trying to minimize and if the score is near to 1. So, this is one of the measures that is used which is known as desirability and one of the measure here it is composite desirability, but we have only one CTQs that is why we will get the same measures of desirability and composite desirability over here that we will discuss afterwards. But my intention was to show that mini-tap can give you some solution if the data set is continuous variable we are having where CTQ is continuous and temperature and pressure is continuous. Mini-tap can search in the surface on the surface using an optimization algorithm and it can give you the best possible combination that is temperature at 125. So, this you can see temperature at 125 and pressure at 40 and this you can see that this is red, red highlights that you are seeing over here that is the best combination. Mini-tap has figured out like that and it will give you a y values of approximately 0.9333 ok. So, that is near to 1 we can assume like that impurity is less like that. So, mini-tap gives you and this is the best option mini-tap using the search algorithm that is using, it will give you some best condition like that. So, what we have done over here in two way analysis of variance we have taken factors which are categorical, we have taken factor where categorical variables and continuous variables are together, we have taken an example where the response CTQ is having. So, the experiment was done and asymmetric design was used and in this case there is no replicate basically n equals to 1 means there is no replicate and how to analyze that one, how to figure out whether interaction is there or not and how to see and how to optimize the data sets like that and find out the best combination of temperature and pressure like that. So, those things are discussed over here more complex relationship and understanding of these theories can be seen using you can you can see Montgomery's book like that. So, we will stop over here and we will move on to a new topic. So, now we are entering into design of experiments, but before that what is required is that we need to know one important topic which is known as measurement system analysis. So, we will just discuss measurement or system analysis and why it is required, why it is so important because before I go into design of experiment this is one of the area we should also understand because that is very helpful because if the instrument is not correct then the measurements that we are taking has no values basically. So, until and unless we measure accurately and that has to be ensured. So, after experimentation this CTQ values that you are getting if the instrument is not correct these values will be different and the results will be different. So, the purpose will not be served I will not get the optimal condition and pseudo optimal condition we will get. So, that is required. So, what is that measurement system analysis we will try to see in our next session. So, thank you for listening.