ASQ CSSBB Certified Six Sigma Black Belt – Improve

1. Simple Linear Regression

In this video we will learn about improve Phase improve Phase Overview and Objectives By the end of this phase you will be able to explain simple linear regression. Describe Multiple Linear Regression sample linear Regression sample Linear Regression Session overview and Objectives By the end of this session you will be able to to define correlation analysis. Explain Regression Equation Correlation correlation the popular linear correlation coefficient row measures the strength of the linear relationship between the paired x and yvalues in a population. Rho is a population parameter. The sample linear correlation coefficient r measures the strength of the linear relationship between the paired X and y values in a sample. R is a sample statistic.

Few important points. A positive value for r implies that the line slopes upward to the right. A negative value for r implies that the line slopes downward to the right. Note that r equals zero implies no linear correlation, not simply no correlation. A pronounced curvilinear pattern may exist when r equals one or r equals negative one. All points fall on a straight line. When r equals zero, they are scattered and give no evidence of a linear relationship. Any other value of r suggests the degree to which the points tend to be linearly related. Simple linear regression correlation coefficient of Determination r squared the coefficient of determination is r bar. The square of the linear correlation coefficient is r squared. It can be shown that r squared equals r squared.

Correlation versus Causation there could be a logical correlation such as car weight and gas mileage. The student should be aware that a number of other factors carburetor type, car design, air conditioning, passenger weights, speed, et cetera could also be important. The most important cause may be a different or collinear variable. For example, car and passenger weight may be collinear. There can also be such a thing as a nonsensical correlation. For example, it rains after my car is washed. Regression Equations regression equation regression analysis is used to construct relationships between a dependent or response variable y and one or more independent or predictor variables X’s.

The goal is to determine the values of parameters for a function that cause that function to best fit a set of data. Observations regression Approach the dependent variable y that needs to be predicted is identified. The multiple regression analysis that focuses on the independent variables it wants to use as predictors. The XS is carried out through the analysis. The relationship between the y and the XS is identified as a mathematical formula. A model Regression Graph in linear regression, the function is a straight line. Equation regression analysis fits a straight line to data points so they are distributed evenly along the line. A curve linear relationship is one that is described by a curve, not a straight line. Summary simple Linear Regression in this session you learned about correlation analysis. Regression Equation.

2. Multiple Regression Analysis

In this video we will discuss about designed experiments. Designed experiments session Overview and Objectives by the end of this session, you’ll be able to define experimental objectives, identify experimental methods, explain different experimental design considerations, experiment objectives, designed experiments introduction classical experiments focus on one fat, one factor at a time at two or three levels and attempt to hold everything else constant, which is impossible to do in a complicated process. When Doe is properly constructed, it can focus on a wide range of key input factors or variables and will determine the optimum levels of each of the factors. It should be recognized that the Pareto principle applies to the world of experimentation. That is, 20% of the potential input factors generally make 80% of the impact on the result. The classical approach to experimentation, changing just one factor at a time, has shortcomings. Too many experiments are necessary to study the effects of all the input factors. The optimum combination of all the variables may never be revealed. The interaction the behavior of one factor may be dependent on the level of another factor. Between factors cannot be determined unless carefully planned and the results studied. Statistically conclusions may be wrong or misleading even if the answers are not actually wrong.

Nonstatistical experiments are often inconclusive. Many of the observed effects tend to be mysterious or unexplainable. Time and effort may be wasted through studying the wrong variables or obtaining too much or too little data. Designed Experiments Introduction design of experiments overcomes these problems by careful planning. In short, Doe is a methodology of varying a number of input factors simultaneously in a carefully planned manner such that their individual and combined efforts on the output can be identified. Advantages of Doe include many factors can be evaluated simultaneously, making the Doe process economical and less interruptive to normal operations. Sometimes, factors having an important influence on the output cannot be controlled noise factors but other input factors can be controlled to make the output insensitive to noise factors in depth. Statistical knowledge is not always necessary to get a big benefit from standard planned experimentation. One can look at a process with relatively few experiments. The important factors can be distinguished from the less important ones. Concentrated effort can then be directed at the important ones. Since the designs are balanced, there is confidence in the conclusions drawn.

The factors can usually be set at the optimum levels for verification. If important factors are overlooked in an experiment, the results will indicate that they were overlooked. Precise statistical analysis can be run using standard computer programs. Frequently, results can be improved without additional costs other than the costs associated with the trials. In many cases, tremendous cost savings can be achieved. Designed Experiments Design Principles Experimental Objectives Choosing an experimental design depends on the objectives of the experiment and the number of factors to be investigated. Some experimental design objectives are discussed below. Comparative Objective If several factors are under investigation but the primary goal of the experiment is to make a conclusion about whether a factor, in spite of the existence of other factors, is significant.

Then the experimenter has a comparative problem and needs a comparative design solution screening Objective The primary purpose of this experiment is to select or screen out the few important main effects from the many lesser important ones. These screening designs are also termed main effects or fractional factorial designs. Response surface Method Objective this experiment is designed to let an experimenter estimate interaction and quadratic effects and therefore give an idea of the local shape of the response surface under investigation. For this reason they are termed response surface method. RSM Designs RSM designs are used to find improved or optimal process settings, troubleshoot process problems and weak points make a product or process more robust against external influences.

Optimizing Responses when factors are proportions of a Mixture Objective If an experimenter has factors that are proportions of a mixture and wants to know the best proportions of the factors to maximize or minimize a response then a mixture design is required. Optimal Fitting of a Regression Model Objective if an experimenter wants to model a response as a mathematical function either known or empirical, of a few continuous factors to obtain good model parameter estimates then a regression design is necessary. Response surface mixture and regression designs are not featured as separate entities in this primer. It should be noted that most good computer programs will provide these models. The best design sources are often full factorial in some cases with replication and screening designs. Experimental Methods Designed Experiments experimental methods first order refers to the power to which a factor appears in a model.

Fractional an adjective that means fewer experiments than the full design calls for. Full factorial describes experimental designs which contain all combinations of all levels of all factors. No possible treatment combinations are omitted. Input factor an independent variable which may affect a dependent response variable and is included at different levels in the experiment inner array in taguchi style. Fractional factorial experiments these are the factors that can be controlled in a process. Interaction an interaction occurs when the effect of one input factor on s the output depends on the level of another input factor. Level a given factor or a specific setting of an input factor.

Four levels of a heat treatment may be 100 degrees Fahrenheit, 120 degrees Fahrenheit, 140 Degrees Fahrenheit and 160 degrees Fahrenheit. Main Effect An estimate of the effect of a factor independent of any other factors. Mixture Experiments Experiments in which the variables are expressed as proportions of the whole and sum to 10 multicolinearity. This occurs when two or more input factors are expected to independently affect the value of an output factor but are found to be highly correlated. For example, an experiment is being conducted to determine the market value of a house. The input factors are square feet of living space and number of bedrooms. In this case, the two input factors are highly correlated. Larger residences have more bedrooms.

Nested experiments an experimental design in which all trials are not fully randomized. There is generally a logical reason for taking this action. For example, in an experiment, technicians might be nested within labs. As long as each technician stays with the same lab, the texts are nested. It is not often that techs travel to different labs just to make the design balanced. Optimization involves finding the treatment combinations that give the most desired response. Optimization can be maximization as, for example, in the case of product yield or minimization in the case of impurities. Orthogonal a design is orthogonal if the main and interaction effects in a given design can be estimated without confounding the other main effects or interactions.

A full factorial is said to be balanced or orthogonal because there are an equal number of data points under each level of each factor. Outer array in a taguchi style fractional. Factorial experiment these are the factors that cannot be controlled in a process. Paired comparison the basis of a technique for treating data so as to ignore sample to sample variability and focus more clearly on variability caused by a specific factor effect. Only the differences in response for each sample are tested because sample to sample differences are irrelevant. Designed Experiments experimental Methods response surface methodology the graph of a system response plotted against one or more system factors. Response surface methodology employs experimental design to discover the shape of the response surface and then uses geometric concepts to take advantage of the relationships discovered. Response variable the variable that shows the observed results of an experimental treatment, also known as the output or dependent variable. Screening experiment a technique to discover the most probable important factors in an experimental system. Most screening experiments employ two level designs. A word of caution about the results of screening experiments if a factor is not highly significant, it does not necessarily mean that it is insignificant. Robust design a term associated with the application of Toguchi experimentation in which a response variable is considered robust or immune to input variables that may be difficult or impossible to control.

Second order reverse to the power to which one or more factors appear in a model. Sequential experiments experiments are done one after another, not at the same time. This is often required by the type of experimental design being used. Sequential experimentation is the opposite of parallel experimentation. Simplex a geometric figure that has a number of vertexes corners equal to one or more than the number of dimensions in the factor space. Simplex design a spatial design used to determine the most desirable variable. Combination proportions in a mixture. Test coverage the percentage of all possible combinations of input factors in an experimental test treatments in an experiment the various factor levels that describe how an experiment is to be carried out. A PH level of three and a temperature level of 37 degrees. Celsius describes an experimental treatment. Experiment design Considerations designed experiments experiment Design considerations Situations where experimental design can be effectively used include choosing between alternatives selecting the key factors affecting a response response surface modeling to hit a target, reduce variability, maximize or minimize a response make a process robust despite uncontrollable noise factors.

Seek multiple goals. Experiment design considerations doe steps getting good results from a Doe involves a number of steps. Set objectives select process variables select an experimental design. Execute the design. Check that the data are consistent with the experimental assumptions. Analyze and interpret the results. Use or present the results may lead to further runs or does. Experiment. Design considerations. Important. Practical considerations in planning and running experiments are check the performance of gauges or measurement devices first. Keep the experiment as simple as possible. Check that all planned runs are feasible. Watch out for process drifts and shifts during the run. Avoid unplanned changes. For example, switching operators at halftime. Allow some time and backup material for unexpected events obtained. Buy in from all parties involved. Maintain effective ownership of each step in the experimental plan.

Preserve all the raw data. Do not keep only summary averages. Record everything that happens. Reset equipment to its original state after the experiment. Experiment design considerations select and scale the process variables. Process variables include both inputs and outputs, for example, factors and responses. The selection of these variables is best done as a team effort. The team should include all important factors based on engineering and operator judgment. Be bold but not foolish in choosing the low and high factor levels. Avoid factor settings for impractical or impossible combinations. Include all relevant responses. Avoid using responses that combine two or more process measurements. When choosing the range of settings for input factors, it is wise to avoid extreme values. In some cases, extreme values will give runs that are not feasible. In other cases, extreme ranges might move the response surface into some erratic region.

The most popular experimental designs are called two level designs. Two level designs are simple and economical and give the most information required to go to a multilevel response surface experiment if one is needed. However, two level designs are something of a misnomer. It is often desirable to include some centerpoints for quantitative facts during the experiment. centerpoints are located in the middle of the design box. Summary Designed Experiments In this session, you learned about experimental objectives. Experimental methods. Different experimental design considerations.

3. Designed Experiments

In this video, we will learn about multiple regression analysis. Multiple Regression Analysis session Overview and Objectives By the end of this session, you’ll be able to define nonlinear regression, explain multiple linear regression, calculate confidence and predictions, describe residual analysis, discuss data transformation, and boxcocks nonlinear regression. Cluster Analysis Cluster analysis is used to determine groupings or classifications for a set of data. A variety of rules or algorithms have been developed to assist in group formations. The natural groupings should have observations classified so that similar types are placed together. A file on attributes of high achieving students could be grouped or classified by IQ, parental support, school system study habits, and available resources. Cluster analysis is used as a data reduction method in an attempt to make sense of large amounts of data from surveys, questionnaires polls, test questions, scores, etc. Nonlinear regression, canonical Correlation Analysis and Manova canonical analysis tests the hypothesis that effects can have multiple causes, and causes can have multiple effects.

This technique was developed by Hoteling in 135, but was not widely used for over 50 years. The emergence of personal computers and statistical software has led to its fairly recent adoption. Canonical correlation analysis is a form of multiple regression to find the correlation between two sets of linear combinations. Each set may contain several related variables. The relating of one set of independent variables to one set of dependent variables will form linear combinations. The largest correlation values for sets are used in the analysis. The pairings of linear combinations are called canonical variates, and the correlations are called canonical correlations, also called characteristic roots. There may be more than one pair of linear combinations that could be applicable for an investigation. The maximum number of linear combinations would be limited by the number of variables in the smaller set. Most researchers involve only two sets. An analysis of variance is used for many independent x variables to solve one dependent y variable. This method tests whether the mean differences among groups on a single dependent y variable is significant.

For multiple independent x variables and multiple dependent y factors, that is, two or more YS and one or more x’s, the multiple analysis of variance is used. Minova tests whether mean differences among groups of a combination of YS are significant or not. The concept of various treatment levels and associated factors is still valid. The data should be normality, distributed, have homogeneity of the covariance matrices, and have independence of observations. Multiple Linear Regression Multiple linear regression Multivariate analysis is concerned with two or more dependent variables y one, y two being simultaneously considered for multiple independent variables x one, x two, et cetera. Recent advances in computer software and hardware have made it possible to solve more problems using multivariate analysis. Some of the software programs available to solve multivariate problems include SPSS, S Plus, SAS, and Minitab. Multivariate analysis has found wide usage in the social sciences, psychology, or educational fields. Applications for multivariate analysis can also be found in the engineering, technology and scientific disciplines.

We will learn the highlights of the following multivariate concepts or techniques principal component analysis, factor analysis, discriminate function analysis, cluster analysis, canonical correlation analysis, multivariate analysis of variance, multivariate analysis, principal components analysis, principal components analysis, PCA principal components analysis or PCA and factor analysis. Or FA are two related techniques used to find patterns of correlation among many possible variables or subsets of data and to reduce them to a smaller, manageable number of components or factors. The researcher attempts to find the primary components or factors that account for most of the sources of variance. PCA refers to subsets as components and FA uses the term factors. Minimum of 100 observations should be used for PCA. The ratio is usually set at approximately five observations per variable. If there are 25 variables, then the ratio of five to one requires five observations. Variables times 25 variables equals 125 observations. Perhaps two principal components will explain 95% of the variance. The other 3 may only contribute 5%. Factor Analysis factor analysis is a data reduction technique to identify factors that explain variation. It is very similar to the principal components analysis technique. That is, factor analysis attempts to simplify complex sets of data, reducing many factors to a smaller set. However, there is some subjective judgment involved in describing the factors. In this method of analysis. The output variables are linearly related to the input factors.

The variables under investigation should be measurable, have a range of measurements, and be symmetrically distributed. There should be four or more input factors for each dependent variable. Factor analysis undergoes two stages factor extraction factor Rotation confidence and prediction. Confidence and prediction. Continuous data. Large samples use the normal distribution to calculate the confidence for the mean. Z plus slash minus z alpha by two, multiply by standard deviation divided by the square root of N where X equals sample average. Mu equals the population standard deviation. N equals sample size. Row row by two equals the normal distribution value for a desired confidence level. Continuous data. Small samples use the normal distribution to calculate the confidence for the mean. X plus slash minus T alpha by two, multiply by S divided by square root of N, where X equals sample average, s equals the population standard deviation, n equals sample size, t row by two equals the t distribution value for a desired confidence level and N minus one degrees of freedom.

Point and interval estimates are discussed in the explanation of minitab usage later. Residual Analysis Residual Analysis residuals are estimates of experimental error obtained by subtracting the observed response from the predicted response. The predicted response is calculated from the chosen model. After all the unknown model parameters have been estimated from the experimental data. Residuals can be thought of as elements of variation unexplained by the fitted model. Since this is a form of error, the same general assumptions apply to the group of residuals that one typically uses for errors. In general, one expects them to be normally and independently distributed with a mean of zero and some constant variance.

These are the assumptions behind ANOVA and classical regression analysis. This means that an analyst should expect a regression model to err in predicting a response in a random fashion. The model should predict values higher and lower than actual with equal probability. In addition, the level of the error should be independent of when the observation occurred in the study, or the size of the observation being predicted, or even the factor settings involved in making the prediction. The overall pattern of the residuals should be similar to the bell shape pattern observed when plotting a histogram of normally distributed data. Graphical methods are used to examine residuals. Departures from assumptions usually mean that the residuals contain structure that is not accounted for in the model. Identifying that structure and adding a term representing it to the original model leads to a better model. Any graph suitable for displaying the distribution of a set of data is suitable for judging the normality of the distribution of a group of residuals. The three most common types are histograms, normal probability plots and dot plots, data transformation and boxcocks. Data transformation and boxcocks. Boxcocks transformation is a critical tool that is used to transform data from any distribution to normal distribution. It functions on a trial and error basis. For example, if a given set of data does not follow a normal distribution, the data can be transformed using boxcock’s transformation and checked. If data is successfully transformed to normal distribution, in case it is not done that, the data can again be further transformed using boxcock’s transformation, and we can check if the data is now normally distributed. Summary Multiple Regression Analysis In this session, you learned about nonlinear regression, multiple linear regression, confidence and predictions, residual analysis, data transformation, transformation and boxcocks.

4. Full-Factorial Experiments

In this video, we will discuss about full factorial experiments. Full Factorial Experiments section Overview and Objectives By the end of this session, you will be able to define a two k full factorial design, identify the linear and quadratic mathematical models, describe balanced and orthogonal designs, explain fit, diagnose, model, and center points. Two k full Factorial Experiments Full Factorial Experiments Full Factorial Design A full factorial design combines the levels for each factor with all the levels of every other factor. It covers all combinations and provides the best data. However, it consumes most time and resources. Fractional Factorial Design A fractional factorial design does not take into account each and every factor.

If a full factorial design uses too many resources or if a slightly nonorthogonal array is acceptable, a fractional factorial design is used to analyze the data from a Doe. The team must first evaluate the statistical significance by computing the one way ANOVA or for more than one factor, the Nway ANOVA. The practical significance can be evaluated through the study of sum of squares, pie charts, pareto diagrams, main effects plots, and normal probability plots. The factorials are also known as two k. factorials two equals number of levels. K equals number of factors. Full factorial equals two k fractional factorial equals two k minus one.

Linear and Quadratic Mathematical Models linear and quadratic mathematical models level A level is a value or a setting for a factor an assignment of high or low for the different states of the x’s inputs being used in the Doe. Runs runs are calculated through the use of design matrix, also known as array, the table of treatment combinations that will be used to set the x’s for the levels that are defined and collect the YS outputs that will be analyzed. Repetition replication when an experiment needs to run for more than once, the subsequent run might be repetition or replication. For repetition, the factors are not reset. For replication, the factors are reset. Replicates give a better estimate of experimental error, but cost more.

Randomization when the runs are ordered randomly and there are concerns about the possibility of unknown external factors affecting variation, it is called randomization. Balanced in Orthogonal Designs balanced in Orthogonal Designs orthogonality the term orthogonality is a mathematical property of a matrix indicating that the Doe is a very good design. Generally, orthogonality describes independence among factors. An orthogonal design matrix is balanced both vertically and horizontally. For each factor, there are equal numbers of high and low values, vertical and horizontal balance aliasing fractional factorial designs reduce the number of runs by screening factors. A disadvantage of that can be that some of the effects of those factors might be confounded or mixed together so that it cannot be estimated separately. This is called aliasing of effects.

Main effect versus Interaction Effects traditional one factor at a time experimentation tests only single factors known as main effects. Doe tests for interaction between or among factors are known as interaction effects fit diagnose and model center points. Full factorial experiments fit diagnose, model and center points. Yates order is a standard according to which design matrices are generally organized, with k being the number of factors. The KTH column consists of two k one minus signs the low level of the factors, followed by two k one plus signs the high level of the factor.

So for a full factorial design with three factors, the design matrix is as shown in the diagram. The diagram talks about the Yates order, which is used to fit diagnose models and center points of the given experiment. A has low as well as high variables minus and plus, respectively, b has minus, minus, plus, plus, etc, and C has minus, minus, minus, minus and plus plus plus and plus, respectively. Summary Designed Experiment In this session, you’ve learned about two k full factorial design. The linear and quadratic mathematical models balanced in orthogonal designs fit diagnose, model and center points.

5. Fractional Factorial Experiments

Fractional Factorial Experiments Designs fractional Factorial Experiments Designs Two level Fractional factorial the following seven step procedure is followed select a process, identify the output factors of concern, identify the input factors and levels to be investigated, select a design from a catalog to Gucci, self created, etc. Conduct the experiment under the predetermined conditions, collect the data relative to the identified outputs, analyze the data and draw conclusions. Placid Berman Designs placid Berman designs are used for screening experiments. Placate Berman designs are very economical. The run number is a multiple of four rather than a power of two. Placate Berman geometric designs are two level designs with 4816, 32, 64 and 128 runs and work best as screening designs. Each interaction effect is compounded with exactly one main effect.

All other two level Placket Berman designs 1220, 24, 28, et cetera. Are non geometric designs. In these designs, a two factor interaction will be partially confounded with each of the other main effects in the study. Thus, the non geometric designs are essentially main effect designs when there is reason to believe that any interactions are of little significance. A Three factor Three Level Experiment often a three factor experiment is required after screening a large number of variables. These experiments may be full or fractional factorial generally, the minus and plus levels in two level designs are expressed as zero and one. In most design catalogs.

Three level designs are often represented as zero, one and two confounding effects fractional Factorial Experiments Confounding effects alias an alias occurs when two factor effects are confused or confounded with each other. Balanced Design A fractional factorial design in which an equal number of trials at every level state is conducted for each factor. Block a subdivision of the experiment into relatively homogeneous experimental units. The term is from agriculture where a single field would be divided into blocks for different treatments. Blocking when structuring fractional factorial experimental test trials, blocking is used to account for variables that the experimenter wishes to avoid. A block may be a dummy factor which doesn’t interact with the real factors. Box Bankin when full second order polynomial models are to be used in response surface studies of three or more factors, box banking designs are often very efficient. They are highly fractional.

Three Level Factorial Designs A collinear condition occurs when two variables are totally correlated. One variable must be eliminated from the analysis for valid results. Confounded when the effects of two factors are not separable. In the following example, A, B and C are input factors and columns AB. AC and BC represent interactions multiplication of two factors. Correlation coefficient R a number between negative one and one that indicates the degree of linear relationship between two sets of numbers. Zero indicates no linear relationship. Covariates things which change during an experiment which had not been planned to change, such as temperature or humidity. Randomize the test order to alleviate this problem. Record the value of the covariant for possible use in regression analysis. Curvature refers to non straight line behavior between one or more factors and the response. Curvature is usually expressed in mathematical terms involving the square or cube of the factor. Degrees of freedom. The terms used are DOF, DF, lowercase DF, or V the number of measurements that are independently available for estimating a population parameter.

Design of Experiments Doe the arrangement in which an experimental program is to be conducted and the selection of the levels of one or more factors or factor combinations to be included in the experiment. Factor levels are accessed in a balanced, full or fractional factorial design. The term SDE statistical design of experiment is also widely used. Efficiency. A concept from RA Fischer. He considered one estimator more efficient than another if it had a smaller variance. EVOp stands for evolutionary operation, a term that describes the way sequential experimental designs can be made to adapt to system behavior by learning from present results and predicting future treatments for better response. Often, small response improvements may be made via large sample sizes. The experimental risk, however, is quite low because the trials are conducted in the near vicinity of an already satisfactory process. Experiment A test undertaken to make an improvement in a process or to learn previously unknown information. Experimental error variation in response or outcome of virtually identical test conditions. This is also called residual error.

Experimental Resolution Fractional Factorial Experiments experimental Resolution resolution one an experiment in which tests are conducted adjusting one factor at a time, hoping for the best. This experiment is not statistically sound. Definition totally fabricated by the authors. Resolution two an experiment in which some of the main effects are confounded. This is very undesirable. Resolution three a fractional factorial design in which no main effects are confounded with each other but the main effects and two factor interaction effects are confounded. Resolution four a fractional factorial design in which the main effects and two factor interaction effects are not confounded, but the two factor effects may be confounded with each other. Resolution five a fractional factorial design in which no confounding of main effects and two factor interactions occur. However, two factor interactions may be confounded with three factor and higher interactions. Resolution six, also called resolution V plus. This is at least a full factorial experiment with no confounding.

It can also mean two blocks of 16 runs. Resolution seven can refer to eight blocks of eight runs. Summary fractional factorial experiments in this session, you’ve learned about the different designs, confounding effects, experimental resolution summary fractional factorial experiments in this phase, you’ve learned about simple linear regression. Multiple linear regression.

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