Rahul.Arora2

TRIZ is a Russian acronym for the Theory of Inventive Problem Solving. There are certain universal principles of creativity that form the basis for innovation. TRIZ identifies & codifies these principles & uses them to make the creative process more predictable. In simpler words, whatever problem than an individual or a team is solving, somebody, somewhere has already solved it. Thus inventive problem solving involves finding that solution & adapting it to the problem in hand.   There are two central concepts that form an integral part of TRIZ i.e. generalizing problems & solutions, and eliminating contradictions. The first concept can be explained as shown below:-     Here, a specific problem is taken & is generalized to one of the TRIZ general problems. From the TRIZ general problems, you identify the general TRIZ solution that is required, & then one considers how to apply the same to the specific problem. TRIZ is basically a collection of 40 principles & 76 standard solutions which can be leveraged to solve any kind of problem.   The other concept which talks about eliminating contradictions & it explains the fact that there are contradictions at the root of most of the problems, thus it is important to eliminate these contradictions in order to effectively solve the problem.   TRIZ has two main categories of contradictions i.e. Technical Contradictions & Physical Contradictions. Let us understand both of these categories:-   Technical Contradictions :   These are the classical engineering trade-offs, where you can’t reach the desired state because something else in the system prevents it i.e. when something gets better, something else automatically gets worse. Some of the examples can be:-   The product gets stronger, but the weight increases. Service is customized to each customer, but the service delivery system gets complicated. Training is comprehensive, but keeps employees away from their BAU.   The key technical contradictions are summarized in the TRIZ Contradiction Matrix which is a matrix that is organized in the form of 39 improving parameters & 39 worsening parameters with each cell entry giving the most used inventive principles that may be used to eliminate the contradiction. The contradiction matrix is leveraged using a four step process:-   Use the 39 parameters to identify the critical features in the problem. Identify the contradictions between the parameters where one causes problems with other. identify the principles that can be used to resolve the contradictions. Use the numbers from the matrix to look up the resolution principles & use these principles to find solutions to the problem.   Below is an excerpt of the contradiction matrix:-       Physical Contradictions :   These are the situations in which an object or system suffers contradictory, opposite requirements. Some of the examples are:-   Software should be complex i.e. have many features, but simple i.e. easy to learn. Coffee should be hot to be enjoyed, but also cool so as to avoid burning the tongue of the drinker. An umbrella should be large to keep the rain off, but small so as to be easily moved in the crowd.   Physical contradictions are solved with the TRIZ Separation Principles, these separation principles are as explained below:-   Separation in Time : Changing the property, response or behavior vs time. Here the concept is to separate the opposite requirements in time. Here one can try to schedule the system operation in such a way that requirements, functions that contradict each other take effect at different times. One classical example can be traffic lights that are used to sequence the flow of traffic at different points of time. Separation in Space : Changing the property, response or behavior based on special location. Here the objective is to separate requirements in space. Here try to partition the system into sub-systems & then assign each contradictory function or condition to a different sub-system. One common example is bifocal lenses for eyes where you have sections for far vision & near vision at separate locations within the same lens. Separation between Part & Whole : Changing the property so as to make it different in the sub-system/system/super-system. Here the concept is to separate the opposite requirements within a whole object or its parts. Here we try to partition the system & assign one of the contradictory functions to a sub-system or several sub-systems. One common example can be a bicycle chain which has rigid links but is flexible at the system level. Separation between Conditions : Changing the property, response or behavior on condition. Here the concept of separating opposing requirements of a condition can resolve contradictions in which a helpful process takes place when special conditions exist. Consider changing the system or the environment so that only the helpful process can take place. One common example can be Ice Skates where ice which is initially solid but when ice skating, the ice below the skates melts for a fraction of a second, therefor enabling the skaters to slide.   When deciding which separation principle to use, 40 inventive principles can be used as guidelines to implement solution. Below is an excerpt of some of these inventive principles:-     An interesting thing to note that can never be a situation when you have a physical or a technical contradictions only. Both are two different but interrelated views of the same problem & thus can’t exist separately. Below visual will help us to understand the above statement.       Let us now see this conversion through an example:-   Let us first define the negative effect of a problem for eg: Long travel time, the cause for this problem is that the car stops at a traffic light, the positive effect of this problem is that it will avoid collision with other cars. Thus the technical contradiction in this will be "Long travel time vs Avoiding collision with other cars”. Thus in this case we have specified both technical contradiction & one side of the physical contradiction (i.e. the cause).

Effect size indicates the practical significance of a research outcome. it tells you how meaningful the relationship between two variables or the difference between groups is. A large effect size means that a research finding has practical significance.
  While statistical significance shows that an effect exists in a study, practical significance shows the effect is large enough to be meaningful in the real world. Statistical significance is denoted by p-value, whereas the practical significance is represented by effect size.   Statistical significance alone can be misleading as it is influenced by sample size i.e. increasing the sample size will always make it more likely to find a statistical significant effect, no matter how small the effect truly is in the real world. In contrast to this, effect size is independent of the sample size which makes it relevant to showcase in order to represent the practical significance of a finding.   Let us understand the difference in statistical & practical significance through an example:-   In a study, we are comparing two weight loss methods with 13000 subjects each in two groups. One group let’s say uses method I of weight loss & the other group uses method II of weight loss. Now basis the results, the mean weight loss in Kg for one group is 10.6 kg with standard deviation of 6.7 kg, which is marginally higher compared to the mean weight loss in Kg for the other group which is 10.5 kg with a standard deviation of 6.8 kg. Statistically these results are significant at p=0.01, however a difference of only 0.1 kg between the groups is negligible & doesn’t really tell you that which of the weight loss method is more effective. Here adding a measure of practical significance can showcase the differences in the two methods.   There are various measures of effect size. Let us see some of the common ones:-   Cohen’s d :   Cohen’s d is designed for comparing two groups, it basically takes the difference between two means & expresses them in standard deviation units. It shows how many standard deviation lie between two means. Cohen’s d is calculated with the below formula:-   d = (x̅1 - x̅2) / s where x1-bar is the mean of one group, x2-bar is the mean of the other group & s is the standard deviation. In general, greater the value of cohen’s d, the larger the effect size.   Considering the above weight loss example, let us calculate cohen’s d for both the groups:-   d = (10.6 - 10.5) / 6.8 = 0.015, now with this value of cohen’s d, there’s limited to no practical significance that one group findings are more effective than the other group’s findings.   Pearson’s r :   It is also known as the correlation coefficient & it measures the extent of a linear relationship between two variables. The main premise is to compute how much of the variability of one variable is determined by the variability of the other variable. A value of pearson’s r closer to -1 or +1 indicates a larger effect size.   Below is the representation of the magnitude of the effect size in terms of both Cohen’d d as well as Pearson’s r methods:-   Effect Size : Small, Cohen’s d : 0.2, Pearson’s r : +/- 0.1 to 0.3 Effect Size : Medium, Cohen’s d : 0.5, Pearson’s r : +/- 0.3 to 0.5 Effect Size : Large, Cohen’s d : >=0.8, Pearson’s r : >= 0.5 or <= -0.5    It is always helpful to calculate effect size before commencing any study & post data collection completion. The reason behind this statement is that within an expected effect size, one can figure out the minimum sample size required in order to have enough statistical power to detect an effect of that magnitude. If we don’t ensure enough power in a study, we may not be able to detect a statistically significant result even though it has practical significance, thus it is helpful to perform a power analysis, so that one can use a set effect size & significance level to determine the required sample size.Once data is collected, one can calculate & report the actual effect size. 

A Waterfall Chart is basically a variation of a bar graph that shows how an initial value changes due to other factors over a period of time. It is also known as a Waterfall Graph or a Bridge Chart particularly in finance parlance.   Its purpose is to show a before & after picture of your data, it depicts each step in the journey & shows which factors increases or decreases the progression. Its was made popular by McKinsey & Company, thus many people consider it to be a financial charting tool, however it has its applications in other areas as well.   Let us see below a common example of a waterfall chart being leveraged in financial sector in order to study the effect of various revenue streams on the overall profit of an organization.     Waterfall chart has its application in business excellence as well, especially while driving improvement projects. Let us see some examples in order to further explore this aspect:-   In this example, the waterfall chart is leveraged to showcase the roadmap of achieving the targeted goal broken down in terms of various solution levers. Here we will start off with the initial baseline of the project metric & then showcase each solution lever along with its projected impact on the metric & finally arriving at the targeted value of the project metric.
 
  This example showcases the application of waterfall chart at a program level in order to showcase the 5 year projection journey of the program metric by showcasing the projected reduction of the program metric over a period of 5 years.   Let us have a look at one more example where we are showcasing the impact of various factor X’s on the output Y of a regression model. Here we are starting with mapping the intercept coefficient value & then also showing the coefficient values of all the relevant X’s & finally arriving at the projected Y.

ANOM or Analysis of Means is a systematic procedure for analyzing the difference among groups or sub-groups in a visual form. it allows the data to be graphically visualized.    It is a graphical variation of ANOVA or Analysis of Variance. The graph shows the decision limits, overall mean & mean for each group. If a point in the chart falls outside of the decision limit for any given group, it will thus showcase a significant difference between the group mean & the population mean. Below is an example of this graphical representation:-       In the above visual, the centre line represents the overall mean & the dots represent the means of different groups, also the line connecting these dots with the overall mean center line represents the difference of the group mean with the overall mean. UDL & LDL represents the upper & lower limit values of the decision limits. Also one can see that there is a large difference between the mean defect rate of Eastpointe & Saginaw sites when compared with the overall defect rate for the entire company.   While conceptually both ANOM & ANOVA serve a common objective, there are still marked differences between the two approaches. Let us try to understand these differences basis certain criteria:-   Framing the Hypothesis :-   In case of ANOM below are the hypothesis that can be framed :   Ho : Means of all groups are equal. Ha : Mean of at-least one group is not equal to the population mean.   Below are the hypothesis in case of ANOVA :   Ho : Means of all groups are equal. Ha : Mean of at-least one group is not equal to other group means.   Distribution Assumptions :-   While ANOVA only takes data which belongs to a normal distribution, ANOM can take into consideration data belonging to both Normal & Binomial Distribution.   Calculation Approach :-   ANOM calculates the overall mean of all the data from all the samples & then measures the variation of each group mean from the overall mean. Here the identity of the sources of variation is retained.   ANOVA takes into account two calculations while assessing variations i.e. Variation between groups is summarized into Mean Squares Between or MSB, variation within each group is summarized into Mean Squares Within or MSW. Here the individual identities of the groups are somehow lost.   Flexibility of Result Interpretation :-   ANOVA tell us whether or not there is a statistically significant difference amongst the group means, however it cannot tell whether which of the group(s) is different from the others. Here we generally leverage tests like Fisher LSD or Tukey Post Hoc test in order to identify the statistically significant difference creating group in terms of absolute difference.   ANOM on the other hand in addition to telling whether there is a significant statistical difference between the group mean & the overall mean, also tells which group mean is having a significant difference when compared to the overall mean & can be visually represented as well.    Now let us take an example of a bank where we want to see the impact of performing wire transfer to four countries eg : India, Brazil, USA & France on the wire transfer cycle time. Here we want to analyze the whether there is a significant variation in cycle times when compared to the overall cycle time. By applying ANOM in this case, we will first able to find out the variation in mean cycle times for each country with respect to the overall cycle time & also will be able to find out that the transactions to which country is generating the most variation when compared to the overall cycle time. This would not have been possible while using ANOVA as we would not have been able to figure out the country(s) are contributing the most to the variation in wire transfer cycle time.   Thus to conclude in this example, that ANOM not only reveals the statistically significant difference amongst the wire transfer cycle times for different countries, but also identifies those countries which are contributing to these differences which would not have been possible through ANOVA.

Lindley’s Paradox, developed by Sir Harold Jeffrey, showcased the conflict between the frequentist & bayesian approaches to hypothesis testing. It refers to the fact that with the increase in sample size (keeping a constant p-value eg p < 0.05), there seems to be a conflict between p-values & baye’s factors i.e. the p-value suggests that the null hypothesis (Ho) should be rejected, however the baye’s factor indicates towards the null hypothesis (Ho) out-predicting the alternative hypothesis (Ha) & this would ultimately result in Ho being rejected as per the frequentist approach & accepted basis the bayesian approach simultaneously.   Let us try to understand this concept through an example:-   Suppose a bank which processes loan applications receives applications for home loan. Also generally the bank receives all kinds of loan applications in two batches on a regular basis i.e. one batch containing 25% home loan applications & the second batch containing 50% home loan applications. Now the bank wants to figure out which of these two batches the received applications belong to.   Thus in order to do that, let’s say the bank takes a random sample of 48 applications & observed that 36 of these random samples are home loan applications which amounts to 75%. Thus going by the above result we can conclude that the applications belong to the second batch i.e. which contains 50% home loan applications.   Now let us apply hypothesis testing & go with the first hypothesis i.e. Testing whether the applications belong to the first batch which contains 25% home loan applications. Let us calculate the populations parameters i.e µ & σ.  µ = np = 48*0.25 = 12 σ = sqrt(np(1-p)) = sqrt(48*0.25*(1-0.25)) = sqrt(48*0.25*0.75)) = 3   Now at 99% confidence level (or 0.01 significance level), the range is 12 +/- 3*3 i.e. from 3 to 21. Here findings of 36 samples taken above is nowhere close to this range thus making us reject the null hypothesis i.e. the applications belong to the batch containing 25% home loan applications.   Now let us also test the hypothesis whether the applications received belong to the second batch containing 50% home loan applications. Let us again calculate the populations parameters i.e µ & σ.  µ = np = 48*0.50 = 24 σ = sqrt(np(1-p)) = sqrt(48*0.50*(1-0.50)) = sqrt(48*0.50*0.50)) = 3.5   Now at 99% confidence level (or 0.01 significance level), the range from 13.5 to 34.5 which does not include the sample result of 36, which again will lead us to reject the null hypothesis that the applications received belong to the second batch i.e one containing 50% home loan applications.   Now basis the results, the possibility of the received applications belonging to both the batches got rejected which is the underlying premise of lindley’s paradox.   Let us now also see the different ways through which we can mitigate the same:-   One approach is to lower down the alpha level as a function of the sample size, thus one should get the best result with any value of alpha that makes the ratio of critical value to the standard error increase with increase in sample size.  
Another approach is to set the baye’s factor (which is basically the ratio of the probability of data under both null & alternate hypothesis i.e. p(data|Ha) / p(data|Ho)) to 1 which implies equal evidence for both null & alternate hypothesis. Next is to adjust the alpha level in a way that the baye’s factor at the critical test statistic value is not greater than 1.

There are two common statistical approaches that are being followed when it comes to statistical testing i.e. The Frequentist Approach, which is based on the observation of data at a given moment or instance & The Bayesian Approach, which is basically a forecasting approach & it involves analyzing prior information.   The frequentist approach is also described as experimental or inductive as it relies on observations while the bayesian approach is theoretical or deductive as it enables to combine the information provided by data with a priori knowledge from previous studies or expert opinions.   Let us take a very simple example to understand both the concepts:-   Let us toss a coin 10 times, now when it comes to frequentist approach, the probability of getting either a head or a tail is 0.5, now let’s say we get heads on 7 out of 10 tosses, then the probability of getting the heads will be 7/10 i.e. 0.7.   Now let’s say we have a prior information through previous experiments of expert experience that heads will come 6 out of 10 times thus we have a prior probability of 0.6, now we will compare the outcome of the experiments with this prior probability.      Thus we can say that the objective of the frequentist approach is to explore the data collected in order to identify a significant effect that could only be explained through by the hypothesis of the experiment & for the bayesian approach the focus is on comparing two hypothesis by comparing the data collected at the time of the experiment with the prior information available therefore assessing the chances that one was true comparison to other.   As an organization performing experiments & relying on statistical analysis for analyzing the results of these experiments, it is thus important to understand the difference between the above two approaches on the basis of different parameters which are as shown below:-   In terms of analyzing the test data :-   Frequentist approach requires the experiment to be completed first by collecting sufficient samples before analyzing the data, this limits the test to be an offline experiment.   Bayesian approach analysis can be performed during the experiment while collecting the data. Also it is an online experiment as the analysis results get updated when new batch of data gets ingested.   Sample Size :-   Frequentist approach requires calculating the sample size prior to conducting the test, also the number of samples among test groups needs to be balanced.   Bayesian approach does not require a pre-defined sample size & also there is no need to have same number of samples amongst the test groups thus allowing an imbalanced sample size.   Test results explanation :-   For the frequentist approach, conclusions can be made like “We reject/ fail to reject the hypothesis that group A is better then group B. This conclusion is based on the observation of the historical data collected during the test. This approach uses p-value in order to quantify the confidence of the business conclusions.   For the bayesian approach, we introduce the element of probability while making an interpretation of results such as “ There is a 98% probability that group A is better than group B”. Thus this probabilistic result quantifies the confidence of the business conclusions.   Leveraging Test Results :-   Frequentist approach gives summary statistics of the samples collected during the experiment period, thus cannot be used for making any conclusions about the future unseen data.   Bayesian approach leverages the parameters of the distribution from the data & gives the posterior predictive distribution for unobserved, future values on the observed data.   Duration of the Test :-   In the frequentist approach, the duration of the experiment can be estimated basis the designed sample size as it is easy to estimate how long an experiment will be conducted.   In the bayesian approach, the duration of the experiment cannot be estimated as more samples coming every day helps to get more confidence conclusions, but cannot estimate how long a specific experiment would take.   Granularity of input data :-   In the frequentist approach, the level of granularity of the input data is at the very base level for eg: data collected basis each user / ID & also it depends on the duration for which the test is conducted.   In the bayesian approach, the level of granularity of the data depends on the frequency of the updating the test results, for eg : in case you are testing the Click through rate & the results are updated every 24 hours, one needs to calculate the number of total seen events & number of click events every day in order to arrive at the daily click through rate.   Performing Multiple Comparison :-   Frequentist approach leverages bonferonni adjustment in case when multiple variants are required to be tested at the same time.   Bayesian approach uses hierarchical bayesian methods for cases involving multiple variants.   Testing Approach :-   The frequentist approach recommends different tests based on the distribution(s) that a variable of variable(s) follows.   The bayesian approach leverages conjugate families for variables following different distributions for eg : Click through rate would leverage the beta distribution conjugate wherein prior parameters need to be set for the beta distribution, collected data is updated basis the baye’s rules in order to get the posterior of the parameters, then samples are taken from the posterior distribution & inferences are made on the test results accordingly. 

Regression analysis is one of the most common forecasting method, one of the most critical assumption while leveraging regression analysis is that the error terms are independent or random i.e they are not correlated. However in most business scenarios, these error terms tend to be correlated. This correlation of error terms of a regression forecasting model is termed as Autocorrelation or Serial Correlation.
 

  From the above visual we can clearly deduce there is an underlying pattern being formed by the error terms when they are correlated thus indicating autocorrelation.   There are two common scenarios pertaining to autocorrelation i.e. Positive Autocorrelation & Negative Autocorrelation.  Positive autocorrelation exists when, the positive errors are associated with the positive errors of comparable magnitude & negative errors are associated with negative errors of comparable magnitude. Negative autocorrelation exists when, the positive errors are associated with the negative errors of comparable magnitude & negative errors are associated with positive errors of comparable magnitude.      There are several possible problems that can arise due to autocorrelation:-   The estimates of the regression coefficients will become inefficient as they will no longer have the minimum variance property. The variance of the error terms will be underestimated by the mean square error value. The true standard deviation of the estimated regression coefficient will also be underestimated. The confidence intervals & the tests using the t & F distribution will no longer be strictly applicable.   One of the most common way to test whether autocorrelation is present in a regression model is by leveraging the Durbin Watson Test, which is calculated basis the below equation:-   where n is the number of observations.   Durbin Watson test involves finding the difference between the successive values of error i.e. (et - et -1) & it formulates the below hypothesis:-   H0 : ρ = 0 (There is no autocorrelation) Ha : ρ != 0 ((There is autocorrelation)   The Durbin Watson statistic ranges from 0 to 4 & consists of two values dU & dL. If DW > dU, we fail to reject H0 hence no autocorrelation exists & if DW < dL, we reject H0 & there is autocorrelation.   Several approaches are leveraged in order to overcome the autocorrelation problem. Some of these are:-   By adding independent variables, as one of the most common reason autocorrelation exists in a regression forecasting model is that one or more important independent or predictor variable have not been included in the analysis. For eg : In a model which predicts the sales of new homes might contain autocorrelation & exclusion of the variable “mortgage interest rate” might be a factor contributing to autocorrelation, thus adding this variable to the model might reduce the autocorrelation significantly. Transforming the variables will also help in significantly reducing the autocorrelation. One such method of transforming variables is the first differences approach, which involves subtracting each value of the independent variable X from each succeeding time period value of that same variable X. This difference thus becomes the transformed X variable, the same process is used to obtain the transformed Y. The regression analysis is then conducted on these transformed X & Y variables in order to compute a revised model free of autocorrelation. Another way is to use the percentage changes from period to period & regressing these new variables. Another important approach is to leverage autoregression models which leverages the relationship of values Yt to previous period values i.e. Yt-1, Yt-2 etc. Here the independent variables are time lagged versions of the dependent variable & is represented as Y-hat = b0 + b1Yt-1 + b2Yt-2 +….        

The basic premise of conducting Attribute Agreement Analysis is to assess whether there is consistency amongst the appraisers in terms of assessing an attribute which is non-measurable in nature (i.e. Nominal, Ordinal, Binary etc) in terms of three aspects:-   Agreement of appraisers within themselves i.e. Repeatability Agreement of appraisers between themselves i.e. Reproducibility Agreement of appraisers with the standard i.e. Accuracy   There are two popular measures of appraiser consistency / reliability i.e. Fleiss Kappa & Krippendorff’s Alpha values. However both are equally consistent measures when it comes to assessing the reliability of your measurement system, there are slight differences in these two measures. These differences are as mentioned below:-   Fleiss Kappa is based on the concept of the ratio calculated between observed agreement(Pa) & agreement expected by chance(Pe) whereas Krippendorff's Alpha is based on the concept of ratio calculated between observed disagreement(Pa) & disagreement expected by chance(Pe). Mathematically both are calculated by the below formula:-     Ranges for both the measures is from -1 to 1 with 1 indicating perfect agreement, 0 indicating no agreement & -1 denoting inverse agreement in case of Fleiss Kappa with an acceptable threshold value of 0.75 resembling significant agreement. However in the case of Krippendorff's Alpha an alpha value of 1 denoting perfect disagreement, 0 being no disagreement with an acceptable threshold value of 0.80 for significant disagreement.  
Fleiss Kappa is most suitable in case of nominal data while Krippendorff's Alpha has high flexibility as it can work with nominal, ordinal as well as metric data.  
In case of missing data Krippendorff's Alpha is the preferred option rather than Fleiss Kappa which cannot handle missing values & these missing values must be excluded from the data. Krippendorff's Alpha is said to be much more robust even if we have 50% of the values missing in our data & provides unbiased results.     Based on the above facts it would be preferable to use Krippendorff's Alpha as the preferred statistic for measuring inter-appraiser reliability in situations where we have data other than nominal data, have multiple appraisers choosen randomly & the attribute agreement data is having missing values.  

The Concept of q-value
  Whenever we are running hypothesis test on a sample of data values that are drawn from the same population, there are chances that we will be getting a test statistic that is very extreme compared to the hypothesized value which indicates that the sample values belongs to a different population while in reality it belongs to the same population from where it is drawn. This extreme value is termed as False Positive(FP) & the probability of getting this false positive or Type-I error is expressed as p-value for that single test, the threshold of this p-value is generally kept at 0.05 (i.e. significance level) & it is desirable to have the p-value to be less than 0.05 or 5%.   Let us now apply this to a multiple testing scenario where we will perform multiple hypothesis tests taking different samples from the same population, now for all these tests we set a threshold p-value of 0.05, it will mean that 5% of all the tests conducted will result in false positives(FP) (i.e. let’s say if we have conducted 1000 tests thus generating 1000 p values there are chances that 50 of those tests will result in p-value<0.05) although all the different samples taken for different tests belong to the same population. If we try to plot the p-values generated from all the tests in a histogram this will result in a shape similar to uniform distribution where each bin represents a p-value range & the frequency represents the number of tests in which p-values fall within this range.   In case of tests conducted by taking samples from two different populations & try to plot the histogram for different p-values generated from these tests we will get the histogram shape that will be right skewed as most of the p-values would be falling within the p<0.05 & thus will be true positives. So let’s say that we conducted 1000 tests & found that 80% of those result in true positives i.e. 800 of them are True Positives(TP) (having p-value < 0.05) thus significant & 200 of them are not significant. Now out of these 800 true positives or significant tests there are false positives as well & this is not possible to identify with the p-value alone.   In order to overcome this limitation we will be leveraging the concept of q-value which leverages the concept of False Discovery Rate (FDR) (defined as FDR = FP / (FP+TP)). q values are basically the p-values that are adjusted by leveraging an optimized FDR approach. Thus let’s say if we have a q-value of 0.05 which means that 5% of the true positives obtained after performing multiple tests are actually false positives. So taking the reference of the above scenario if 800 are true positives (or significant) initially identified then 5% of them i.e. 40 will turn out to be false positives after adjusting the p-values basis the FDR approach. The range of q -values lies between 0 & 1.   Generally the cut off for significance for FDR is < 0.05 which means less than 5% of the significant results will be false positives.   There is one popular method i.e. The Benjamin-Hochberg Method which adjusts the q-value in a way that limits the number of false positives that are reported as significant or True positives by making the p-values larger for eg before the FDR correction, the p-value may be 0.04(significant) & post the FDR correction it may become 0.06(not significant).   Let us now see an example in order to understand the above method of adjusting the p-value:-    Let’s say we have conducted 10 tests taking 10 pairs of sample each time from the same distribution & below are the p-values obtained for each test :   0.91 0.11 0.71 0.31 0.51 0.41 0.61 0.21 0.81 0.01   First step is to order the p-values from smallest to largest & rank these values as shown below:   p-values : 0.01 0.11 0.21 0.31 0.41 0.51 0.61 0.71 0.81 0.91 rank :          1      2     3      4.     5     6      7      8     9     10   Here we have one false positive which is the first one i.e. p=0.01 which is < 0.05. let us now see whether this false positive is significant or not.   Next step is to calculate the adjusted values starting from the last value ranked 10th Here the largest FDR adjusted p-value is same as the largest p-value.   For the next largest adjust p-value we will choose the smaller of the two options i.e. the previous adjusted value & current p-value x (total number of p-values / p-value rank whose adjusted p-value is to be calculated) Thus for calculating the adjusted p-value for the 9th ranked p-value it will be smaller of previous adjusted p-value which is 0.91 or current p-value at 9th rank i.e. 0.81 x (total no. of p-values i.e. 10 / p-value rank whose adjusted p-value is being calculated i.e. 9) which comes out to be 0.90, thus between 0.91 & 0.90 we will choose the smaller value i.e. 0.90.   Similarly we will be repeating this for the other ranked p-values as well & the final output will be as shown below:-    adjusted p-values : 0.10 0.55 0.70 0.77 0.82 0.85 0.87 0.89 0.90 0.91   Now as you can see the false positive value of 0.01 is now converted into an equivalent adjusted p-value or in other words q-value of 0.10 which is no longer significant as it is now > 0.05.   Thus we can see how we can leverage q-value in order to perform adjustments on p-value in order to separate the true positives from the false positives.

Thematic Analysis is a method of analyzing qualitative data. It is generally applied to a set of text such as verbatim or transcripts where it is closely examined in order to identify common themes.    It is a good approach to leverage whenever you are trying to find something meaningful(eg: people’s views, opinions etc) from a set of qualitative data.    It is basically a data analysis process which involves deep diving through a dataset, create coding, identifying patterns, , deriving themes & then finally create a narrative.   Following are the steps to perform Thematic Analysis:- First step is to familiarise yourself with the data i.e. performing an initial exploratory analysis of the data in order to identify meanings & patterns in the data. After familiarising with the data, create the initial codes that represent the meanings & patterns identified in the data. Decide the code, go through the data again & identify the excerpts, then apply the appropriate codes to them. Also add new codes as deemed fit. Bring together all the excerpts associated with a code & collate them under that appropriate code, repeat the same for other codes as well. Group the collated codes containing the excerpts into suitable themes . Evaluate & revise the themes, also ensure that each theme has data to support it & each theme must be unique. Once themes are finalised the final step is to create the narrative in order to share your findings to the audience.   Based on the above understanding one of the common applications of thematic analysis in lean six sigma is VOC analysis. The objective here is to identify major recurring themes that reflect the concerns or problems raised by the customer & eventually this will help to identify the key needs of the customer basis which we can focus our improvement efforts accordingly.   Let us take the example of a bank having rolled out a survey to its customers in order to get their valuable feedback on the overall performance of the services thy are delivering. Now once we have received the responses from the bank’s customers the bank decides to use Thematic Analysis in order to analyze those responses & post the analysis it was found that there were two major themes identified from the data points one towards the accuracy & other towards the timeliness of the Bank’s wire transfer process. Thus by leveraging this analysis they identified two major improvement areas i.e. To improve the accuracy of the wire transfer process & to reduce the overall time to completion of the wire transfer process.

My two cents on this:-
 
Let us understand the concept & limitations of the two conventional experimental designs & how latin square design takes care of those limitations through example from optical lens industry:-
 
Completely Randomised Design (CRD) or One Way ANOVA:
 
In CRD each experimental unit is randomly assigned to one of the treatment levels. For eg: Let us take an example from optical industry where we want to study the Impact of different varnish types (coating formulations) on the final yield of our lens coating process. Here the experimental unit is the lens on which coating will be done. Here each sample will be randomly allocated to a treatment group hence in this case let’s say we have 60 samples & three types of varnishes (let, say X,Y,Z) thus the entire samples will be divided into three groups of 20 each & one group will be subjected to Varnish X, other to Y & the third to Z. This can be shown as:-
 
Varnish X
Varnish Y
Varnish Z
Group B
Group A
Group C
 
We will be taking into account the variability within each unit in the overall sample (SS within) & the variability between groups subjected to the three varnish types X,Y,Z  (SS between)
 
Randomised Block Design (RBD) or Two Way ANOVA:
 
Now in the above example let’s say we observed that the suppliers (let’s say Supplier A,B,C) from which the varnishes (X,Y,Z) are imported also influences the final yield of our coating process. Here the supplier factor will become the blocking variable. In this case the units are first assigned to each block & each unit within the block will be subjected to all the treatments but cannot be assigned to other blocks & other treatments. Thus let’s say we have 180 samples , first we will divide these samples into three groups of 90 I.e. one for supplier A, one for Supplier B & one for supplier C & these three groups will be further subdivided into groups of 30 & one subgroup will be subjected to Varnish X, second with Y & third with Z & likewise for supplier B & C group.
 
 
Block
Varnish X
Varnish Y
Varnish Z
Group 1
Supplier A
Subgroup 1
Subgroup 3
Subgroup 2
Group 2
Supplier B
Subgroup 2
Subgroup 3
Subgroup 1
Group 3
Supplier C
Subgroup 3
Subgroup 1
Subgroup 2
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between)
 
Latin Square Design:
 
Latin square Design takes care of above limitation with the fact that each experimental unit will get all the treatment but that treatment combination will be a square & each treatment combination occurs only once in a row & a column which is the underlying principle of Latin Square Design. Let us see below:-
 
Now considering the above example lets say we have a sample of 60 lenses & these will be divided into groups of 20 basis the supplier levels A,B,C as well as Varnish Types X,Y,Z.
 
Here each group will be subjected to a combination of each supplier & each varnish type but only once.
 
An important assumption to consider in Latin square Design is the levels in each of the factors considered should be the same like in this example where we have three levels of Suppliers (A,B,C) & three levels of medicine (X,Y,Z). Thus in this case it will be a 3x3 latin square .
 
 
Varnish X
Varnish Y
Varnish Z
Supplier A
Group B
Group A
Group C
Supplier B
Group C
Group B
Group A
Supplier C
Group A
Group C
Group B
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between) & the variability due to each combination of block I.e. supplier & Treatment i.e. Varnish & Supplier.

My two cents on this:-
 
Let us understand the concept & limitations of the two conventional experimental designs & how latin square design takes care of those limitations through example from optical lens industry:-
 
Completely Randomised Design (CRD) or One Way ANOVA:
 
In CRD each experimental unit is randomly assigned to one of the treatment levels. For eg: Let us take an example from optical industry where we want to study the Impact of different varnish types (coating formulations) on the final yield of our lens coating process. Here the experimental unit is the lens on which coating will be done. Here each sample will be randomly allocated to a treatment group hence in this case let’s say we have 60 samples & three types of varnishes (let, say X,Y,Z) thus the entire samples will be divided into three groups of 20 each & one group will be subjected to Varnish X, other to Y & the third to Z. This can be shown as:-
 
Varnish X
Varnish Y
Varnish Z
Group B
Group A
Group C
 
We will be taking into account the variability within each unit in the overall sample (SS within) & the variability between groups subjected to the three varnish types X,Y,Z  (SS between)
 
Randomised Block Design (RBD) or Two Way ANOVA:
 
Now in the above example let’s say we observed that the suppliers (let’s say Supplier A,B,C) from which the varnishes (X,Y,Z) are imported also influences the final yield of our coating process. Here the supplier factor will become the blocking variable. In this case the units are first assigned to each block & each unit within the block will be subjected to all the treatments but cannot be assigned to other blocks & other treatments. Thus let’s say we have 180 samples , first we will divide these samples into three groups of 90 I.e. one for supplier A, one for Supplier B & one for supplier C & these three groups will be further subdivided into groups of 30 & one subgroup will be subjected to Varnish X, second with Y & third with Z & likewise for supplier B & C group.
 
 
Block
Varnish X
Varnish Y
Varnish Z
Group 1
Supplier A
Subgroup 1
Subgroup 3
Subgroup 2
Group 2
Supplier B
Subgroup 2
Subgroup 3
Subgroup 1
Group 3
Supplier C
Subgroup 3
Subgroup 1
Subgroup 2
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between)
 
Latin Square Design:
 
Latin square Design takes care of above limitation with the fact that each experimental unit will get all the treatment but that treatment combination will be a square & each treatment combination occurs only once in a row & a column which is the underlying principle of Latin Square Design. Let us see below:-
 
Now considering the above example lets say we have a sample of 60 lenses & these will be divided into groups of 20 basis the supplier levels A,B,C as well as Varnish Types X,Y,Z.
 
Here each group will be subjected to a combination of each supplier & each varnish type but only once.
 
An important assumption to consider in Latin square Design is the levels in each of the factors considered should be the same like in this example where we have three levels of Suppliers (A,B,C) & three levels of medicine (X,Y,Z). Thus in this case it will be a 3x3 latin square .
 
 
Varnish X
Varnish Y
Varnish Z
Supplier A
Group B
Group A
Group C
Supplier B
Group C
Group B
Group A
Supplier C
Group A
Group C
Group B
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between) & the variability due to each combination of block I.e. supplier & Treatment i.e. Varnish & Supplier.

My two cents on this:-
 
Let us understand the concept & limitations of the two conventional experimental designs & how latin square design takes care of those limitations through example from optical lens industry:-
 
Completely Randomised Design (CRD) or One Way ANOVA:
 
In CRD each experimental unit is randomly assigned to one of the treatment levels. For eg: Let us take an example from optical industry where we want to study the Impact of different varnish types (coating formulations) on the final yield of our lens coating process. Here the experimental unit is the lens on which coating will be done. Here each sample will be randomly allocated to a treatment group hence in this case let’s say we have 60 samples & three types of varnishes (let, say X,Y,Z) thus the entire samples will be divided into three groups of 20 each & one group will be subjected to Varnish X, other to Y & the third to Z. This can be shown as:-
 
Varnish X
Varnish Y
Varnish Z
Group B
Group A
Group C
 
We will be taking into account the variability within each unit in the overall sample (SS within) & the variability between groups subjected to the three varnish types X,Y,Z  (SS between)
 
Randomised Block Design (RBD) or Two Way ANOVA:
 
Now in the above example let’s say we observed that the suppliers (let’s say Supplier A,B,C) from which the varnishes (X,Y,Z) are imported also influences the final yield of our coating process. Here the supplier factor will become the blocking variable. In this case the units are first assigned to each block & each unit within the block will be subjected to all the treatments but cannot be assigned to other blocks & other treatments. Thus let’s say we have 180 samples , first we will divide these samples into three groups of 90 I.e. one for supplier A, one for Supplier B & one for supplier C & these three groups will be further subdivided into groups of 30 & one subgroup will be subjected to Varnish X, second with Y & third with Z & likewise for supplier B & C group.
 
 
Block
Varnish X
Varnish Y
Varnish Z
Group 1
Supplier A
Subgroup 1
Subgroup 3
Subgroup 2
Group 2
Supplier B
Subgroup 2
Subgroup 3
Subgroup 1
Group 3
Supplier C
Subgroup 3
Subgroup 1
Subgroup 2
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between)
 
Latin Square Design:
 
Latin square Design takes care of above limitation with the fact that each experimental unit will get all the treatment but that treatment combination will be a square & each treatment combination occurs only once in a row & a column which is the underlying principle of Latin Square Design. Let us see below:-
 
Now considering the above example lets say we have a sample of 60 lenses & these will be divided into groups of 20 basis the supplier levels A,B,C as well as Varnish Types X,Y,Z.
 
Here each group will be subjected to a combination of each supplier & each varnish type but only once.
 
An important assumption to consider in Latin square Design is the levels in each of the factors considered should be the same like in this example where we have three levels of Suppliers (A,B,C) & three levels of medicine (X,Y,Z). Thus in this case it will be a 3x3 latin square .
 
 
Varnish X
Varnish Y
Varnish Z
Supplier A
Group B
Group A
Group C
Supplier B
Group C
Group B
Group A
Supplier C
Group A
Group C
Group B
 
Here we will be taking into account the variability within each unit in the overall sample (SS within), variability in groups amongst the blocks I.e. supplier A & B (SS blocks) & the the variability between groups basis the three varnish types X,Y,Z (SS between) & the variability due to each combination of block I.e. supplier & Treatment i.e. Varnish & Supplier.