Edme7 One-Factor Tutorial

Pat

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¾¾Simple Comparative ExperimentsOne-Way ANOVAOne-way analysis of variance (ANOVA)F-test and t-test1. Mark Anderson and Pat Whitcomb (2000), DOE Simplified, Productivity Inc., Chapter 2.One-Factor Tutorial 1Welcome to Stat-Ease’s introduction to analysis of variance (ANOVA). The objective of this PowerPoint presentation and the associated software tutorial is two-fold. One, to introduce you to the Design-Expert software before you attend a computer-intensive workshop, and two, to review the basic concepts of ANOVA. These concepts are presented in their simplest form, a one-factor comparison. This form of design of experiment (DOE) can compare two levels of a single factor, e.g. two or more vendors. Or perhaps to compare the current process versus a proposed new process. Analysis of variance is based on F-testing and is typically followed up with pairwise t-testing.Turn the page to get started!If you have any questions about these materials, please e-mail [e-mail] or call 612-378-9449 and ask for statistical support.1General One-Factor Tutorial(Tutorial Part 1 – The Basics)Instructions:1. Turn on your computer.®2. Start Design-Expert version 7.3. Work through Part 1 of the “General One-Factor Tutorial.”4. Don't worry yet about the actual plots or statistics. Focus on learning how to use the software.5. When you complete the tutorial return to this presentation for some explanations.One-Factor Tutorial 2First use the computer ...

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• Publié par: Inyio
• Nombre de lectures: 26
• Langue: English

Extrait

Simple Comparative Experiments One-Way ANOVA

¾One-way analysis of variance (ANOVA) ¾F-test and t-test

1. Mark Anderson and Pat Whitcomb (2000),DOE Simplified, Productivity Inc.,Chapter 2.

One-Factor Tutorial

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Welcome to Stat-Eases introduction to analysis of variance (ANOVA). The objective of this PowerPoint presentation and the associated software tutorial is two-fold. One, to introduce you to the Design-Expert software before you attend a computer-intensive workshop, and two, to review the basic concepts of ANOVA. These concepts are presented in their simplest form, a one-factor comparison. This form of design of experiment (DOE) can compare two levels of a single factor, e.g. two or more vendors. Or perhaps to compare the current process versus a proposed new process. Analysis of variance is based on F-testing and is typically followed up with pairwise t-testing. Turn the page to get started! If you have any questions about these materials, please e-mail [e-mail] or call 612-378-9449 and ask for statistical support.

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General One-Factor Tutorial (Tutorial Part 1  The Basics)

Instructions: 1.Turn on your computer. 2.Start Design-Expert ® version 7. 3.Work throughPart 1 of the General One-Factor Tutorial.  4.Don't worry yet about the actual plots or statistics. Focus on learning how to use the software. 5.When you complete the tutorial return to this presentation for some explanations.

One-Factor Tutorial

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First use the computer to complete the General One-Factor Tutorial (Part 1  The Basics) using Design-Expert software. This tutorial will give you complete instructions to create the design and to analyze the data. Some explanation of the statistics is provided in the tutorial, but more complete explanation is given in this PowerPoint presentation. Start the Design-Expert tutorial now. After youve completed the tutorial, return here and continue with this slide show.

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One-Factor Tutorial ANOVA

Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 2212.11 2 1106.06 12.57 0.0006 A 2212.11 2 1106.06 12.57 0.0006 Pure Error 1319.50 15 87.97 Cor Total 3531.61 17

One-Factor Tutorial

In this tutorial Pure Error SS = Residual SS

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Model Sum of Squares (SS): SS Model = Sum of squared deviations due to treatments. = 6(153.67-162.72)2+6(178.33-162.72)2+6(156.17-162.72)2= 2212.11 Model DF (Degrees of Freedom): The deviation of the treatment means from the overall average must sum to zero. The degrees of freedom for the deviations is, therefore, one less than the number of means. DF Model = 3 treatments - 1 = 2. Model Mean Square (MS): MS Model = SS Model /DF Model = 2212.11/2 = 1106.06 Pure Error Sum of Squares: SS Pure Error = Sum of squared deviations of response data points from their treatment mean. = (160-153.67)2+ (150-153.67)2+...+ (156-156.17)2) = 1319.50 Pure Error DF are (nt  1) degrees of freedom for the deviations within each treatment.: There DF Pure Error =∑(nt - 1) = (5 + 5 + 5) = 15 Pure Error Mean Square: MS Pure Errorl = SS Pure Error /DF Pure Error = 1319.50/15 = 87.97 F Value for comparing treatment variance with error variance.: Test F = MS Model /MS Pure Errorl = 1106.06/87.97 = 12.57 Prob > F probability equals the tail The of observed F value if the null hypothesis is true.: Probability area of the F-distribution (with 2 and 15 DF) beyond the observed F-value. Small probability values call for rejection of the null hypothesis. Cor Total: Totals corrected for the mean. SS = 3531.61 and DF = 17

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One-Way ANOVA

Corrected total sum of = Between + Within    squared deviations treatment sum of treatment sum of from the grand mean squares squares SS Corrected  SStotal = Model + SS Residuals ∑ k n ∑ t ( y ti −y ) 2 = ∑ k n t ( y t −y ) 2 + k ∑ n ∑ t ( y ti −y t ) 2 t=1 i= 1 t=1 t =1 i=1 SS Model = = where: SS≡"Sum of Squares"  F  SS Residuals df fd l lsua M i o de d s e R SM M R S Mod    lsuasedi le qu MS≡"Mean S are"

One-Factor Tutorial

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Lets breakdown the ANOVA a bit further. The starting point is the sum of squares: It becomes convenient to introduce a mathematical partitioning of sums of squares that are analogous to variances. We find that the total sum of squared deviations from the grand mean can be broken into two parts: that caused by the treatment (factor levels) and the unexplained remainder which we call residuals or error. Note on slide: k = # treatments n = # replicates

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Corrected Total SS

,6 SS corrected total = ∑ 3 6, ∑ 6 ( y ti −y ) 2 =3531.61 t=1 i=1

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This is the total sum of squares in the data. It is calculated by taking the squared differences between each bowling score and the overall average score, then summing those squared differences. Next, this total sum of squares is broken down into the treatment (or Model) SS and the Residual (or Error) SS. These are shown on the following pages.

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Treatment SS

SS mod el = ∑ 3 n t ( y t −y ) 2 =2212.11 t=1

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Pat Mark Shari

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This is a graph of the between treatment SS. It is the squared difference between the treatment means and the overall average, weighted by the number of samples in that group, and summed together.

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Residual SS

SS residuals = 3 ∑ 6,6 ∑ ,6 ( y ti −y t ) 2 =1319.50 t=1 i=1

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This is the within treatment SS. Think of this as the differences due to normal process variation. It is calculated by taking the squared difference between the individual observations and the treatment mean, and the summed across all treatments. Notice that this calculation is independent of the overall average  if any of the means shift up or down, it wont effect the calculation.

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ANOVA p-values

The F statistic that is derived from the mean squares is converted into its corresponding p-value. In this case, the p-value tells the probability that the differences between the means of the bowlers scores can be accounted for by random process variation.

As the p-value decreases, it becomes less likely the effect is due to chance, and more likely that there was a real cause. In this case, with a p-value of 0.0006, there is only a 0.06% chance that random noise explains the differences between the average bowlers scores. Therefore, it is highly likely that the difference detected is a true difference in Pat, Mark and Sharis bowling skills.

One-Factor Tutorial

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The p-value is a statistic that is consistent among all types of hypothesis testing so it is helpful to try to understand its meaning. This was reviewed in the tutorial and will be covered much more extensively in the Stat-Ease workshop. For now, lets just lay out some general guidelines:  If p<0.05, then the model (or term) is statistically significant.  If 0.05<p<0.10, then the model might be significant, you will have to decide based on subject matter knowledge.  If p>0.10, then the model is not significant.

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Std. Dev 9.38 Mean 162.72 C.V. 5.76 PRESS 1900.08

One-Factor Tutorial

One-Factor Tutorial ANOVA (summary statistics)

R-Squared 0.6264 Adj R-Squared 0.5766 Pred R-Squared 0.4620 Adeq Precision 6.442

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The second part of the ANOVA provides additional summary statistics. The value of each of these will be discussed during the workshop. For now, simply learn their names and basic definitions. Std Dev root of the Pure (experimental) Error. (Sometimes referred to as Root MSE): Square = SqRt(87.97) = 9.38 Mean mean of the response = [160 + 150 + 140 + ... +156] /18 = 162.72: Overall C.V.: Coefficientthe standard deviation as a percentage of the mean. of variation, = (Std Dev/Mean)(100) = (9.38/162.72)(100) = 5.76% Predicted Residual Sum of Squares (PRESS) measure of how this particular model fits each: A point in the design. The coefficients are calculated without the first point. This model is then used to estimate the first point and calculate the residual for point one. This is done for each data point and the squared residuals are summed. R-Squared measure of the amount of variation about the A multiple correlation coefficient.: The mean explained by the model. = -1 [SS Pure Error /(SS Model + SS Pure Error )] = 1 - [1319.50/(2212.11 + 1319.50)] = 0.6264 Adj R-Squared Measure of the amount of variation about the mean: Adjusted R-Squared. explained by the model adjusted for the number of parameters in the model. = 1 - (SS Pure Error /DF Pure Error )/[(SS Model + SS Pure Error )/(DF Model + DF Pure Error )] = 1 - (1319.50/15)/[(2212.11 + 1319.50)/(2 + 15)] = 0.5766   Pred R-Squared measure of the predictive capability of the model. R-Squared. A: Predicted = 1 - (PRESS/SS Total ) = 1 - (1900.08/3531.61) = 0.4620 Adeq Precision the range of predicted : Comparesvalues at design points to the average prediction error. Ratios greater than four indicate adequate model discrimination.

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One-Factor Tutorial Treatment Means

Treatment Means (Adjusted, If Necessary) Estimated Standard Mean Error 1-Pat 153.67 3.83 2-Mark 178.33 3.83 3-Shari 156.17 3.83 Estimated Mean average response for each treatment.: The Standard Error:The standard deviation of the average is equal to the standard deviation of the individuals divided by the square root of the number of individuals in the average. SE = 9.38/√(6) = 3.83

One-Factor Tutorial

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This is a table of the treatment means and the standard error associated with that mean. As long as all the treatment sample sizes are the same, the standard errors will be the same. If the sample sizes differed, the SEs would vary accordingly. Note that the standard errors associated with the means are based on the pooling of within treatment variances.

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ANOVA Conclusions

Conclusion from ANOVA: There are differences between the bowlers that cannot be explained by random variation.

Next Step: Run pairwise t-tests to determine which bowlers are different from the others.

One-Factor Tutorial

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The ANOVA does not tell us which bowler is best, it simply indicates that at least one of the bowlers is statistically different from the others. This difference could be either positive or negative. In a one-factor design, the ANOVA is followed by pairwise t-tests to gain more understanding of the specific results.

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