Group 12_Unit 12_Design and Analysis of Single Factor Experiments PDF

UNIT 12 DESIGN AND ANALYSIS OF SINGLE FACTOR EXPERIMENTS PRESENTED BY: START NOW GROUP 12 TABLE OF CONTENT - TOPICS 01 Designing Engineering Experiments 02 Completely Randomized Single-Factor Experiment 2.1 Analysis of Varia...

UNIT 12 DESIGN AND ANALYSIS OF SINGLE FACTOR EXPERIMENTS PRESENTED BY: START NOW GROUP 12 TABLE OF CONTENT - TOPICS 01 Designing Engineering Experiments 02 Completely Randomized Single-Factor Experiment 2.1 Analysis of Variance (ANOVA) 2.2 Muplitple Comparisons following ANOVA 2.3 Residual Analysis and Model Checking 2.4 Determining Sample Size TABLE OF CONTENT 03 The Random Effects Model 3.1 Fixed versus random factors 3.2 ANOVA and Variance Components 04 Randomized Complete Block Design 4.1 Design and Statistical Analysis 4.2 Multiple Comparisons TABLE OF CONTENT 04 Randomized Complete Block Design 4.3 Residual Analysis and Model Checking  DESIGNING ENGINEERING EXPERIMENTS REPORTED BY: PLETE, MICAH A.  In engineering, statistically based experimental design approaches are useful for finding new phenomena, creating and bringing to market new technologies, and enhancing already- existing goods and procedures. These tests help engineers determine which process variables have the most effects on performance, which might result in important breakthroughs. These experimental design methods are typically applied in: Evaluation and comparison of basic design configurations Evaluation of different materials Selection of design parameters so that the product will work well under a wide variety of field conditions (or so that the design will be robust) Determination of key product design parameters that affect product performance  Every experiment involves a sequence of activities: 1. Conjecture—the original hypothesis that motivates the experiment. 2. Experiment—the test performed to investigate the conjecture. 3. Analysis—the statistical analysis of the data from the experiment. 4. Conclusion—what has been learned about the original conjecture from the experiment.  COMPLETELY RANDOMIZED SINGLE- FACTOR EXPERIMENT Completely randomized single-factor experiment: An experimental design where one variable (factor) is manipulated across several levels (treatments), and the order of applying treatments to experimental units is randomized. Treatment The different levels or conditions of the factor being studied Replicates The multiple observations made at each treatment level. Example: Tensile Strength A manufacturer of paper used for making grocery bags is interested in improving the tensile strength of the product. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp and that the range of hardwood concentrations of practical interest is between 5 and 20%. A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15%, and 20%. They decide to make up six test specimens at each concentration level, using a pilot plant. All 24 specimens are tested on a laboratory tensile tester, in random order The role of randomization in this experiment is extremely important. By randomizing the order of the 24 runs, the effect of any nuisance variable that may influence the observed tensile strength is approximately balanced out. For example, suppose that there is a warm-up effect on the tensile testing machine; that is, the longer the machine is on, the greater the observed tensile strength. If all 24 runs are made in order of increasing hardwood concentration, any observed differences in tensile strength could also be due to the warm-up effect. Graphical interpretation of the data is always useful. Box plots show the variability of the observations within a treatment (factor level) and the variability between treatments. We now discuss how the data from a single-factor randomized experiment can be analyzed statistically.  Created with Canva ANALYSIS OF VARIANCE Completely Randomized Design Reported By: Posadas, H.X. THE GOAL WE WANT TO DO Understand what is 01 Analysis of Variance; and Perform Hypothesis 02 Testing with ANOVA. Reported By: Posadas, H.X. ANOVA ANALYSIS OF VARIANCE A statistical method for analyzing differences among group means in a sample. ANOVA ANALYSIS OF VARIANCE It splits the total variability in the data into two parts: systematic factors, treatment and experimental conditions, and random factors, measurement errors, or individual differences.  TREATMENTS: A broad term traced back to the early applications of experimental design methodology in agricultural sciences. These are the different levels of a single factor. KEY TERMS TO REMEMBER  FACTOR: The independent variable of the experimental design which is the cause of the variability within the results. The factor may or may not have a significant effect on the response. KEY TERMS TO REMEMBER  RESPONSE: A random variable that is a result or observation from the application of the treatment or factor to the replicates. KEY TERMS TO REMEMBER  ENVIRONMENTAL UNIT The environment where the experimental design is conducted may have extraneous variables that may affect the experiment's results. KEY TERMS TO REMEMBER  BETWEEN-GROUP VARIANCE The variance or spread of scores between the different treatments.  TREATMENT SUMS OF SQUARES It represents the between-group variability. It is the sum of the squared differences between the group means and the grand mean.  WITHIN-GROUP VARIANCE The variance or spread of results within each treatment (level of specified factor).  ERROR SUMS OF SQUARES It represents the within-group variability. It’s the sum of the squared differences between each observation and its group mean.  TOTAL SUMS OF SQUARES The sum of the Treatment Sums of Squares and the Error Sums of Squares. It represents the total variability in the data. DEGREES OF FREEDOM The number of values that have the freedom to vary when computing a statistic. The degrees of freedom of a group would be the number of observations in one group (n) minus one.  MEAN SQUARE It represents the average squared deviation. It is calculated by dividing the sum of squares by the corresponding degrees of freedom.  F-STATISTIC The test statistic for Analysis of Variance (ANOVA). It is the ratio of the between-group variance to the within- group variance.  F-STATISTIC If the between-group variance is significantly larger than the within- group variance, the F- statistic will be large and likely significant.  P-VALUE In this context, it is the probability of obtaining a test statistic as extreme as the one that was actually observed, assuming a true null hypothesis.  P-VALUE If the p-value is less than the significance level, then the null hypothesis is rejected in favor of the alternative hypothesis. POST-HOC ANALYSIS A secondary analysis performed after the first statistical analysis test. Generally, these are conducted after performing an ANOVA, specifically when the null hypothesis is rejected.  POST-HOC ANALYSIS This analysis is conducted to determine which of the specific groups’ means (levels) are significantly different from each other.  UNIT XII - Group 12 COMPLETELY RANDOMIZED DESIGN Presented By: Posadas, H.X.  UNIT XII COMPLETELY RANDOMIZED DESIGN The experimental design in where observations are taken at random, and the environment is as uniform or controlled as possible. Analysis of Variance of CRD  WAYS TO CHOOSE TREATMENTS FIXED-EFFECT MODEL VS RANDOM-EFFECT MODEL  RANDOM-EFFECTS MODEL Choosing treatments as random samples from a larger population of treatments. The variability of the treatment effects is estimated.  RANDOM-EFFECTS MODEL The conclusions, based on the sample treatments, of this experimental design could be extended to all other treatments in the population, regardless of whether they were explicitly considered in the experiment.  FIXED-EFFECTS MODEL It involves choosing specific treatments and testing the hypothesis on the treatment means. The conclusions of this process cannot be extended to other similar treatments in the population that were not considered. EXPECTED VALUES OF SUMS OF SQUARES: SINGLE FACTOR EXPERIMENT Reported By: Dondiog, Hope P.  CONCEPT OVERVIEW We can gain considerable insight into how the analysis of variance works by examining the expected values of SSₜᵣₑₐₜₘₑₙₜₛ and SSₑ. This will lead us to an appropriate statistic for testing the hypothesis of no differences among treatment me Reported By: Dondiog, Hope P.  THE EXPECTED VALUE OF THE TREATMENT SUM OF SQUARES IS Reported By: Dondiog, Hope P.  THE EXPECTED VALUE OF THE ERROR SUM OF SQUARES IS Reported By: Dondiog, Hope P.  DEGREES OF FREEDOM Reported By: Dondiog, Hope P.  ERROR MEAN SQUARE Reported By: Dondiog, Hope P.  ANOVA F-TEST Reported By: Dondiog, Hope P.  UNIT XII TENSILE STRENGTH AN EXAMPLE OF ANALYSIS OF VARIANCE WITH A COMPLETELY RANDOMIZED DESIGN Presented By: Posadas, H.X. QUESTION: THE PAPER TENSILE STRENGTH EXPERIMENT A paper manufacturer that makes grocery bags is interested in improving the product's tensile strength. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp and that the range of hardwood concentrations of practical interest is between 5 and 20%. ANOVA QUESTION: THE PAPER TENSILE STRENGTH EXPERIMENT A paper manufacturer that makes grocery bags is interested in improving the product's tensile strength. Product engineering thinks that tensile strength is a function of the hardwood concentration in the pulp and that the range of hardwood concentrations of practical interest is between 5 and 20%. ANOVA QUESTION: THE PAPER TENSILE STRENGTH EXPERIMENT A team of engineers responsible for the study decides to investigate four levels of hardwood concentration: 5%, 10%, 15%, and 20%. Using a pilot plant, they decide to make up six test specimens at each concentration level. All 24 specimens are tested on a laboratory tensile tester in random order. ANOVA QUESTION: THE PAPER TENSILE STRENGTH EXPERIMENT Consider the paper tensile strength experiment, which is a CRD. We can use the analysis of variance to test the hypothesis that different hardwood concentrations do not affect the mean tensile strength of the paper. ANOVA  GIVEN: Treatment Observation/Replicates Hardwood Total Averages 1 2 3 4 5 6 Concentration 5% 7 8 15 11 9 10 60 10.00 10% 12 17 13 18 19 15 94 15.67 15% 14 18 19 17 16 18 102 17.00 20% 19 25 22 23 18 20 127 21.17 Grand Total, Grand Mean 383 15.96 ANOVA HYPOTHESIS for at least one i (hardwood concentration) ANOVA CREATED WITH CANVA EQUATIONS SIGNIFICANCE LEVEL ANOVA  FOR SUM OF FORMULAS SQUARES ANOVA CREATED WITH CANVA EQUATIONS  FOR DEGREES FORMULAS OF FREDDOM ANOVA CREATED WITH CANVA EQUATIONS  FOR MEAN FORMULAS SQUARES ANOVA CREATED WITH CANVA EQUATIONS  FORMULA for F-value ANOVA CREATED WITH CANVA EQUATIONS  ANOVA TABLE Sources of Degrees of Sum of Squares Mean Square F-value p-value Variation Freedom Treatments SS(Treatment) df1 MS(Treatment) F-value p-value Error SS(E) df2 MS(E) Total SS(T) df(T) ANOVA  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS REJECTION CRITERIA Reject null hypothesis if: ANOVA CREATED WITH CANVA EQUATIONS  SOLUTION: ANOVA CREATED WITH CANVA EQUATIONS ANSWER Reject null hypothesis because: ANOVA CREATED WITH CANVA EQUATIONS  SUMMARY: ANOVA TABLE Sources of Sum of Degrees of Mean F-value p-value Variation Squares Freedom Square Hardwood 382.79 3 127.6 19.60 3.59 E-6 Concentration Error 130.17 20 6.51 Total 512.96 23 ANOVA GROUP 12 - UNIT XII USING MINITAB: STATISTICAL SOFTWARE Let’s get to know how to use Minitab for the Analysis of Variance (ANOVA). START NOW Presented By: Posadas, Harvelle Xyrell CREATED WITH THE MINITAB STATISTICAL SOFTWARE TAKEN FROM MINITAB STATISTICAL SOFTWARE TAKEN FROM MINITAB STATISTICAL SOFTWARE TAKEN FROM MINITAB STATISTICAL SOFTWARE  CONFIDENCE INTERVAL ON A TREATMENT MEAN CONFIDENCE INTERVAL ON A TREATMENT MEAN CONFIDENCE INTERVAL ON A DIFFERENCE IN TREATMENT MEANS Reported By: Alo, Richard L.  CONFIDENCE INTERVAL ON A TREATMENT MEAN CONFIDENCE INTERVAL- the range within which the true treatment effect is likely to lie. Reported By: Alo, Richard L. A 100(1- Α) PERCENT CONFIDENCE INTERVAL ON THE MEAN OF THE ITH TREATMENT I IS DATA TABLE EXAMPLE:  CONFIDENCE INTERVAL ON A DIFFERENCE IN TREATMENT MEANS A 100(1-Α) PERCENT CONFIDENCE INTERVAL ON THE DIFFERENCE IN TWO TREATMENTS MEANS IS Reported By: Alo, Richard L.  EXAMPLE: A 95% CI ON THE DIFFERENCE IN MEANS IS COMPUTED FROM EQUATION 13-12 AS FOLLOWS:  EXAMPLE: A 95% CI ON THE DIFFERENCE IN MEANS IS COMPUTED FROM EQUATION 13-12 AS FOLLOWS:  EXAMPLE: A 95% CI ON THE DIFFERENCE IN MEANS IS COMPUTED FROM EQUATION 13-12 AS FOLLOWS:  EXAMPLE: A 95% CI ON THE DIFFERENCE IN MEANS IS COMPUTED FROM EQUATION 13-12 AS FOLLOWS:  COMPUTING FORMULAS FOR ANOVA: SINGLE FACTOR WITH UNEQUAL SAMPLE SIZES Reported By: Alo, Richard L.  EXAMPLE: IT IS SUSPECTED THAT HIGHER-PRICED AUTOMOBILES ARE ASSEMBLED WITH GREATER CARE THAN LOWER-PRICED AUTOMOBILES. TO INVESTIGATE WHETHER THERE IS AN BASIS FOR THIS FEELING, A LARGE LUXURY MODEL A, A MEDIUM-SIZE SEDAN B, AND A SUBCOMPACT HATCHBACK C WERE COMPARED FOR DEFECTS WHEN THEY ARRIVED AT THE DEALER'S SHOWROOM. ALL CARS WERE MANUFACTURED BY THE SAME COMPANY. THE NUMBER OF DEFECTS OF THE THREE MODELS IS RECORDED IN THE TABLE BELOW. TEST THE HYPOTHESIS AT THE 0.05 LEVEL OF SIGNIFICANCE THAT THE AVERAGE NUMBER OF DEFECTS IS THE SAME FOR THE THREE MODELS. EXAMPLE: EXAMPLE: MULTIPLE COMPARISONS FOLLOWING THE ANOVA PRESENTED BY: SACNAHON , O. J.  When the null hypothesis H0=T1=T2=...=Ta=0 is rejected in the ANOVA, we know that some of the treatment or factor level means are different. However, ANOVA doesn’t identify which means are different.  To determine this, we need to perform multiple comparisons post-hoc tests. There are actually various methods used for multiple comparisons, but we will use the simple one, which is the Fisher’s least significant difference (LSD) and a graphical method. The Fisher LSD method compares all pairs of means with the null hypothesis H0:μi=μj (for all i and j ) using the t-statistic. Assuming a two-sided alternative hypothesis, the pair of means i and j would be declared significantly different if | Ȳi - Ȳj | > LSD, where LSD, the least significant difference, is Where; tα/2 :The t-critical value from the t- distribution table with α = 0.05 a(n-1) and N-a: are the degrees of freedom within groups from the ANOVA table. MSE: The mean squares within groups from the ANOVA table. n: The sample sizes of each group If the sample sizes are different in each treatment, the LSD is defined as; Example We will apply the Fisher LSD method to the hardwood concentration experiment. There are α=4 means, n=6, MSE=6.51, and t0.025,20=2.086. The treatment means are; Ȳ1=10.00 psi Ȳ2=15.67 psi Ȳ3=17.00 psi Ȳ4=21.17 psi Solution LSD=t0.025,20√[2(6.51) /6] LSD=2.086√(2.17) LSD=3.073 Therefore, any pair of treatment averages that differs by more than 3.073 implies that the corresponding pair of treatment means are different. The comparisons among the observed treatment averages are as follows: Results of Fisher’s LSD method From this analysis, we see that there are significant differences between all pairs of means except 2 and 3. This implies that 10% and 15% hardwood concentration produce approximately the same tensile strength and that all other concentration levels tested produce different tensile strengths. GRAPHICAL COMPARISON OF MEANS FOLLOWING ANOVA Reported By: Dondiog, Hope P.  CONCEPT OVERVIEW to visualize whether the observed means could plausibly come from a common normal distribution If all treatment means were indeed equal, they would behave like observations from a single normal distribution Reported By: Dondiog, Hope P.  CALCULATE THE MSE From the ANOVA output, you get the Mean Square Error (MSE), which estimates the variance within the groups. Suppose: MSE=Within-group variance=σ² Reported By: Dondiog, Hope P.  ESTIMATE THE STANDARD DEVIATION OF MEANS: From the ANOVA output, you get the Mean Square Error (MSE), which estimates the variance within the groups. Suppose: Standard deviation = √MSE/n ​ Reported By: Dondiog, Hope P.  PLOT THE TREATMENT MEANS: Plot the observed treatment means ȳ₁, ȳ₂ …ȳₐ on the x-axis. Reported By: Dondiog, Hope P.  PLOT THE TREATMENT MEANS: Overlay a Normal Distribution Curve: The normal distribution curve should have a mean equal to the grand mean of all treatment means​ȳ. The standard deviation of this distribution is √MSE/n ​. Plot this normal distribution curve on the same graph as the treatment means. Reported By: Dondiog, Hope P.  VISUALIZE THE COMPARISON: If the treatment means fit well under the normal distribution curve, it suggests that the means could come from the same population. If not, it indicates that some treatment means are significantly different from the others. Reported By: Dondiog, Hope P.  EXAMPLE Let’s illustrate with a hypothetical example where: Treatment Means: ȳ₁=5, ȳ₂=10, ȳ₃=15, ȳ₄=20 MSE (Mean Square Error): 26.51 Number of Observations per Treatment (n): 6 Reported By: Dondiog, Hope P.  ESTIMATE THE STANDARD DEVIATION OF MEANS: Standard deviation = √MSE/n ​ = √26.51/6 ≈ 2.10 Reported By: Dondiog, Hope P.  PLOT THE TREATMENT MEANS: Plot Treatment Means: Plot 5,10,15, 20 on the x-axis. Reported By: Dondiog, Hope P.  PLOT THE TREATMENT MEANS: Overlay a Normal Distribution Curve: Mean of the normal distribution = Grand Mean of Treatment Means. For simplicity, assume the grand mean is 12.5. The standard deviation is 2.10. Reported By: Dondiog, Hope P.  ANALYSIS If All Means Align Well: The normal distribution curve should pass through or be close to all treatment means, indicating that they could come from the same distribution. If Means Deviate Significantly: The treatment means would not align well with the normal curve, indicating significant differences among the treatment levels. Reported By: Dondiog, Hope P.  DETERMINING SAMPLE SIZE REPORTED BY: TALATALA, VINCENT SAMPLE SIZE DETERMINATION The process of choosing the right number of observations from a larger group to use in a sample. An important aspect of designing an experiment is to know how many observations are needed to make conclusions of sufficient accuracy and with sufficient confidence.  OPERATING CHARACTERISTIC CURVES (OC CURVES) THESE ARE USED AS GUIDES IN DETERMINING THE SAMPLE SIZE. THIS IS A PLOT OF THE PROBABILITY OF A TYPE II ERROR (β) FOR VARIOUS SAMPLE SIZES AGAINST VALUES OF THE PARAMETER UNDER TEST. THESE CURVES PLOT β AGAINST Φ. TO DETERMINE THE SAMPLE SIZE (n) USING AN OC CURVE WE REQUIRE THE PARAMETERS:  TO FIND THESE PARAMETERS WE USE THE GENERAL FORMULA: since the value for error variance (σ^2) is usually unknown, the usual process of using these curves is by defining the ratios we wish to detect from We then plot our assumed and known values in the table: We can find β by plotting our Φ and v2 in our OC curve to its corresponding α and with this data we can conclude whether the sample size we assumed is enough. EXAMPLE: EXAMPLE: EXAMPLE: EXAMPLE: EXAMPLE: EXAMPLE: EXAMPLE: Conclusion: At least n = 6 replicates must be run in order to obtain a test with the required power (0.90).  THE RANDOM-EFFECTS MODEL REPORTED BY: VIÑEGAS  FIXED VERSUS RANDOM FACTORS Fixed factor - A statistical model in which the model parameters are fixed or non-random quantities. Random factor - A statistical model where the model parameters are random variable.  IF A FIXED EFFECTS MODEL IS USED THAT WOULD MEAN THE SAME PEOPLE ARE USED IN EACH TRIAL OF THE STUDY. THAT BEING SAID, IF A RANDOM EFFECTS MODEL IS USED IT IS MORE GENERALIZABLE BECAUSE DIFFERENT PARTICIPANTS ARE USED EACH TIME.  ANOVA AND VARIANCE COMPONENTS Reported By: Badar, Cielo Marie C.  ANOVA AND VARIANCE COMPONENTS THE RANDOM EFFECTS ANOVA MODEL IS SIMILAR IN APPEARANCE TO THE FIXED EFFECTS ANOVA MODEL. HOWEVER, THE TREATMENT MEANS ARE CONSTANT IN THE FIXED-EFFECT ANOVA MODEL WHEREAS IN THE RANDOM-EFFECTS ANOVA MODEL THE TREATMENT MEAN ARE RANDOM VARIABLES.  The Linear Statistical Model If the variance of the treatment effects by independence the variance of response is  TESTING HYPOTHESES For the random-effects model, testing the hypothesis that the individual treatment effects are zero is meaningless. It is more appropriate to test hypotheses about. Specifically,  In random effects, the ANOVA decomposition of total variability is still valid; that is, However, the expected values of the mean squares for treatments and error are somewhat different than in the fixed-effects case. In the random-effects model for a single-factor, completely randomized experiment, the expected mean square for treatments is: The expected mean square for error is: Analysis of Variance Method procedure to estimate the variance components ( and ) in the model it consists of equating the expected mean square to their observed values in the ANOVA table and solving for the variance components Estimators of the Variance Components: EXAMPLE In Design and Analysis of Experiments, 7th edition (John Wiley, 2009), D. C. Montgomery describes a single-factor experiment involving the random-effects model in which a textile manufacturing company weaves a fabric on a large number of looms. The company is interested in loom-to- loom variability in tensile strength. To investigate this variability, a manufacturing engineer selects four looms at random and makes four strength determinations on fabric samples chosen  STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA TSS = 4(97.5-95.45)^2 + 4(91.5-95.45)^2 + 4(95.8-95.45)^2 + 4(97.0-95.45)^2 = 89.19 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA Loom 1 = (98-97.5)^2 + (97-97.5)^2 + (99-97.5)^2 + (96-97.5)^2 = 5 Loom 1 = (98-97.5)^2 + (97-97.5)^2 + Loom 3 = (95-97)^2 + (96-97)^2 + (99-97.5)^2 + (96-97.5)^2 = 5 (99-97)^2 + (98-97)^2 = 10 Loom 2 = (91-91.5)^2 + (90-91.5)^2 + (93-91.5)^2 + (92-91.5)^2 = 5 Loom 3 =( 96-95.8)^2 + (95-95.8)^2 + (97-95.8)^2 + ( 95-95.8)^2 = 2.75 Loom 1 = (98-97.5)^2 + (97-97.5)^2 + Loom 3 = (95-97)^2 + (96-97)^2 + (99-97.5)^2 + (96-97.5)^2 = 5 (99-97)^2 + (98-97)^2 = 10 Loom 2 = (91-91.5)^2 + (90-91.5)^2 + (93-91.5)^2 + (92-91.5)^2 = 5 SSE = 5+5+2.75+ 10 Loom 3 =( 96-95.8)^2 + (95-95.8)^2 + (97-95.8)^2 + ( 95-95.8)^2 = 2.75 SSE= 22.75 Loom 1 = (98-97.5)^2 + (97-97.5)^2 + Loom 3 = (95-97)^2 + (96-97)^2 + (99-97.5)^2 + (96-97.5)^2 = 5 (99-97)^2 + (98-97)^2 = 10 Loom 2 = (91-91.5)^2 + (90-91.5)^2 + (93-91.5)^2 + (92-91.5)^2 = 5 SSE = 5+5+2.75+ 10 Loom 3 =( 96-95.8)^2 + (95-95.8)^2 + (97-95.8)^2 + ( 95-95.8)^2 = 2.75 SSE= 22.75 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA + ANALYSIS OF VARIANCE FOR THE STRENGTH DATA + SStotal = SST + SSE SStotal = 89.19 + 22.75 = 111.94 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA SStotal = SST + SSE SStotal = 89.19 + 22.75 = 111.94 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA df (SST)= k-1 df (SST)= 4-1 k= number of levels df (SST)= 3 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA df (SSE)= n-k k= number of levels n= number of observations ANALYSIS OF VARIANCE FOR THE STRENGTH DATA df (SSE)= n-k df (SSE)= 16-4 k= number of levels n= number of observations df (SSE)= 12 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA df (SSE)= n-k df (SSE)= 16-4 k= number of levels n= number of observations df (SSE)= 12 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA + df (SStotal)= df (SST) + df (SSE) df (SStotal)= 3 + 12 df (SStotal)= 15 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MST = SST / K- 1 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MST = SST / K- 1 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MST = SST / K- 1 MST = 89.19 / 3 MST = 29.73 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MSE = SSE / n-k ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MSE = SSE / n-k ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MSE = SSE / n-k MSE =22.75 / 12 MSE = 1.90 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA MSE = SSE / n-k MSE =22.75 / 12 MSE = 1.90 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA ANALYSIS OF VARIANCE FOR THE STRENGTH DATA 29.73 = ANALYSIS OF VARIANCE FOR THE STRENGTH DATA 29.73 = 1.90 ANALYSIS OF VARIANCE FOR THE STRENGTH DATA 29.73 = = 15.68 1.90 FROM THE ANALYSIS OF VARIANCE, WE CONCLUDE THAT THE LOOMS IN THE PLANT DIFFER SIGNIFICANTLY IN THEIR ABILITY TO PRODUCE FABRIC OF UNIFORM STRENGTH. THE VARIANCE COMPONENTS ARE ESTIMATED BY FROM THE ANALYSIS OF VARIANCE, WE CONCLUDE THAT THE LOOMS IN THE PLANT DIFFER SIGNIFICANTLY IN THEIR ABILITY TO PRODUCE FABRIC OF UNIFORM STRENGTH. THE VARIANCE COMPONENTS ARE ESTIMATED BY THEREFORE, THE VARIANCE OF STRENGTH IN THE MANUFACTURING PROCESS IS ESTIMATED BY CONCLUSIONS: MOST OF THE VARIABILITY IN STRENGTH IN THE OUTPUT PRODUCT IS ATTRIBUTABLE TO DIFFERENCES BETWEEN LOOMS.  RANDOMIZED COMPLETE BLOCK DESIGN JESTRELL DAGANGON  Blocking and ANOVA In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study. as much as possible, we would like to avoid such factors, but, on many occasions, that’s just not possible.  Blocking and ANOVA In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study. as much as possible, we would like to avoid such factors, but, on many occasions, that’s just not possible. Examples: Multiple equipment setups  Blocking and ANOVA In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study. as much as possible, we would like to avoid such factors, but, on many occasions, that’s just not possible. Examples: Multiple equipment setups Different personnel  Blocking and ANOVA In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study. as much as possible, we would like to avoid such factors, but, on many occasions, that’s just not possible. Examples: Multiple equipment setups Different personnel Different raw materials  Blocking and ANOVA In many practical situations, there are “nuisance” factors that influence the results of an experimental program but are not what we are attempting to study. as much as possible, we would like to avoid such factors, but, on many occasions, that’s just not possible. Examples: Multiple equipment setups Different personnel Different raw materials Temporal conditions change  Blocking and ANOVA Example: You are to test bus tires from 4 manufacturers. you can place only 4 tires on a bus, and you have a sample of 5 tires from each manufactures.  Blocking and ANOVA Example: You are to test bus tires from 4 manufacturers. you can place only 4 tires on a bus, and you have a sample of 5 tires from each manufactures. How many total buses will you need in order to perform the experiment?  Blocking and ANOVA Example: You are to test bus tires from 4 manufacturers. you can place only 4 tires on a bus, and you have a sample of 5 tires from each manufactures. How many total buses will you need in order to perform the experiment? You will need to allocate one tires from each manufacturer to each of 5 buses  Blocking Design “Classic” example: tire wear evaluation Tires from manufactures: 1, 2, 3, 4 : main factor Placed on city buses that run different routes Bus designation: A, B, C, D, E = Blocking Factor  Blocking Design “Classic” example: tire wear evaluation Tires from manufactures: 1, 2, 3, 4 : main factor Placed on city buses that run different routes Bus designation: A, B, C, D, E = Blocking Factor  Blocking and ANOVA This raises the issue that tire wear may vary from bus to bus for various reasons (distance traveled, character of bus route, driver "cowboy factor",...). But we wish to study tire wear, so we must separate the bus factor from the tire factor. We know and have control over the bus allocations, so this is called a "blocking" factor.  Blocking and ANOVA This raises the issue that tire wear may vary from bus to bus for various reasons (distance traveled, character of bus route, driver "cowboy factor",...). But we wish to study tire wear, so we must separate the bus factor from the tire factor. We know and have control over the bus allocations, so this is called a "blocking" factor. Blocking essentially removes some of the error that otherwise would be assigned to the treatments and assigns it to a known 'nuisance factor'.  Blocking and ANOVA This raises the issue that tire wear may vary from bus to bus for various reasons (distance traveled, character of bus route, driver "cowboy factor",...). But we wish to study tire wear, so we must separate the bus factor from the tire factor. We know and have control over the bus allocations, so this is called a "blocking" factor. Blocking essentially removes some of the error that otherwise would be assigned to the treatments and assigns it to a known 'nuisance factor'. IMPORTANT: A blocking factor can be thought of as just another treatment!  Blocking Design “Classic” example: tire wear evaluation Tires from manufactures: 1, 2, 3, 4 : main factor Placed on city buses that run different routes Bus designation: A, B, C, D, E = Blocking Factor  Blocking Design “Classic” example: tire wear evaluation Tires from manufactures: 1, 2, 3, 4 : main factor Placed on city buses that run different routes Bus designation: A, B, C, D, E = Blocking Factor  Blocking Design Since all tire types (called "treatment levels") are assigned to each bus (each "block") [each block is big enough to hold all the treatment levels and the assignment of levels can be randomized within the blocks - tire location on each bus can be randomly assigned], this is called a randomized complete block design [RCBD]. It is the simplest one to analyze. RANDOMIZED COMPLETE BLOCK DESIGN RANDOMIZED COMPLETE BLOCK DESIGN Random order of experiment, but within blocks RANDOMIZED COMPLETE BLOCK DESIGN Random order of experiment, but within blocks NOTE: No replicate observations in this design Linear Statistical Model  ANOVA SUMS OF SQUARES IDENTITY: RANDOMIZED BLOCK EXPERIMENT REPORTED BY: PLETE, MICAH A.  The sums of squares identity for the randomized complete block design is or symbolically Furthermore, the degrees of freedom corresponding to these sums of squares are For the randomized block design, the relevant mean square are EXPECTED MEAN SQUARES: RANDOMIZED BLOCK EXPERIMENT REPORTED BY: PLETE, MICAH A. Expected Mean Squares: Randomized Block Experiment  To test the null hypothesis that the treatment effects are all zero We would reject the null hypothesis at the -level of significance if the computed value of the test statistic COMPUTING FORMULAS FOR ANOVA: RANDOMIZED BLOCK EXPERIMENT REPORTED BY: PLETE, MICAH A.  The computing formulas for the sums of squares in the analysis of variance for a randomized complete block design are ANOVA TABLE Source Of Sum of Degrees of Mean Variation Squares Freedom Square Treatments Blocks Error Total Example 13.5 : Fabric Strength An experiment was performed to determine the effect of four different chemicals on the strength of a fabric. These chemicals are used as part of the permanent press finishing process. Five fabric samples were selected, and a RCBD was run by testing each chemical type once in random order on each fabric sample. The data are shown in Table 13-12. We will test for differences in means using an ANOVA with WHEN IS BLOCKING NECESSARY? EXPERIMENTAL VARIABLES (also known as independent variables or factors) are the variables that the experimenter intentionally changes or manipulates to observe their effect on the dependent variable. CONFOUNDING VARIABLE (also known as confounders) are extraneous variables that are not of primary interest but can affect the dependent variable. These variables can create a false impression of a relationship between the experimental variables and the dependent variable by influencing both. If not controlled for, confounding variables can lead to incorrect conclusions about the relationship between the experimental and dependent variables.. ,  BLOCKING IS NECESSARY It is used to control for the variability caused by nuisance factors, which are variables that are not of primary interest but can affect the outcome of an experiment. Removing the noise to make sure that when the test happens that your end results will be accurate “BLOCK WHAT YOU CAN , RANDOMIZE WHAT YOU CANNOT” REDUCING VARIABILITY: When there are sources of variability that can be controlled or measured but are not of primary interest, blocking can help reduce the impact of these variables on the results. SUBJECT-RELATED FACTORS: When experiments involve human or animal subjects, blocking can control for subject-related variability such as age, gender, or health status. REDUCING VARIABILITY: When there are sources of variability that can be controlled or measured but are not of primary interest, blocking can help reduce the impact of these variables on the results. SUBJECT-RELATED FACTORS: When experiments involve human or animal subjects, blocking can control for subject-related variability such as age, gender, or health status.  MULTIPLE COMPARISON WE WILL ILLUSTRATE FISHER’S LSD METHOD _ CHEMICAL TYPE TREAMENT AVEAGES Yi 1 1.14 2 1.76 3 1.38 4 3.56 FIGURE 13-10 PRESENTS THE RESULTS GRAPHICALLY. THE UNDERLINED PAIRS OF MEANS ARE NOT DIFFERENT.THE LSD PROCEDURE INDICATES THAT CHEMICAL TYPE 4 RESULTS IN SIGNIFICANTLY DIFFERENT STRENGTHS THAN THE OTHER THREE TYPES DO. CHEMICAL TYPES 2 AND 3 DO NOT DIFFER, AND TYPES 1 AND 3 DO NOT DIFFER. THERE MAY BE A SMALL DIFFERENCE IN STRENGTH BETWEEN TYPES 1 AND 2.  RESIDUAL ANALYSIS AND MODEL CHECKING Presented By: Degamon, Jonn Jasper The analysis of variance assumes that the observations are normally and independently distributed with the same variance for each treatment or factor level. These assumptions should be checked by examining the residuals. A Residual is the difference between observation yij and its estimated (or fitted) value from the statistical model being studied, denoted as ŷij. In any designed experiment, it is always important to examine the residuals and to check for violation of basic assumptions that could invalidate the results. As usual, the residuals for the RCBD are just the difference between the observed and estimated (or fitted) values from the statistical model, say, and the fitted values are The fitted value represents the estimate of the mean response when the ith treatment is run in the jth block. The residuals from the chemical type experiment are shown in Table 13-15. Figures 13-11, 13-12, 13-13, and 13- 14 present the important residual plots for the experiment. These residual plots are usually constructed by computer software packages.  Residuals from the Randomized Complete Block Design There is some indication that fabric sample (block) 3 has greater variability in strength when treated with the four chemicals than the other samples. Chemical type 4, which provides the greatest strength, also has somewhat more variability in strength. Follow up experiments may be necessary to confirm these findings, if they are potentially important. THANK YOU! Call +63975-362-6084 Musuan, University Town YOU MAY ACCESS THE PRESENTATION IN CANVA USING THE FOLLOWING LINK https://tinyurl.com/tps268c5