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Quantitative Methods for Measuring and Assessing Uncertainties in Testing Process and Outcomes (Workshop from July 30 to August 10, 2001 at National Bureau of Standards & Metrology and Inspection, Taiwan) Carl Lee Professor of Statistics, Statistical Consultant and University Assessment Coordinator Central Michigan University Mt. Pleasant MI 48859 USA 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 1 Outline of the Workshop Module One: The Measurements and Uncertainties •Basic concepts of measuring uncertainty •Key issues to be considered •Bias and variability Module Two: Graphical and numerical exploration of univariate variables •Planning an inter-laboratory testing •Graphical tools for demonstrating uncertainty •Numerical tools for quantifying uncertainty •Shapes of distributions and relationship to numerical measurements. 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 2 Module Three: Graphical and numerical exploration of bivariate variables •Scatter Plot and relationship with correlation coefficient •Least square fitting of linear regression lines Module Four: Normal distribution and it’s applications to interlaborotary testing •The use of standardized normal table •Empirical Rule – the probability concept of rare event • Techniques for testing Normality assumption •Data transformation to approximate normal 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 3 Module Five: Outlier Detection for one sample case •Box Plot, h-plot and interpretations for one sample case •Numerical approaches for detecting outliers •A statistical model for one-sample case •Use of standardized residuals for outlier detection. •Use of deleted studentized residuals for outlier detection. •Statistical quality control charts for monitoring between-lab means and withinlab variations. 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 4 Module Six: Outlier Detection for Two sample case •Youden Plots and interpretations •Principal Component Analysis •Bivariate normal distribution •Bivariate plots and interpretations •Marginal plot and its interpretations •A general statistical model for two-sample cases 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 5 Module Seven: Reporting and presenting uncertainties •Four conditions in measuring and propagating uncertainty •No information about the variables •Variables are independent •Variables are correlated •Variables follow a certain probability distribution •Combining Uncertainty of the same components measured by different individuals •Standard error of sample mean •Confidence interval of sample mean •Uncertainty due to calibration using least square fitting •Type B uncertainty 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 6 Module Eight:Comparative Study for Inter-laboratory Testing • Comparative Study One: Comparing testing results with a given reference or a given standard • The concept and Procedure for performing the one sample t-test •Comparative Study for Inter-laboratory Testing : two-group cases •Designing experiments for two-sample comparative study •Analysis of paired samples •Analysis of two independent samples 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 7 Module Nine:Techniques for Monitoring Variability and Testing Homogeneity •Comparing variance with a standard or reference •Comparing variances of two groups •Bonferroni’s simultaneous confidence interval for uncertainty •Testing uniformity of variances for more than two groups •Bartlett’s method •Levene’s method •Graphical methods for demonstrating and comparing variances 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 8 Module Ten: Planning Experiments – the general consideration and Comparative Study for more than two groups •General consideration of planning experiments •Local control of errors •Statistical model for One-way ANOVA •Residual Analysis for outliers and diagnosis of assumptions •Data transformation techniques •Post-Hoc Analysis and Sum of square decomposition •Contrasts, simultaneous comparison, pair-wise comparison, comparison with control •Trend analysis, 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 9 Module Eleven: Experiments to study VariancesVariance Components Analysis and Nested Models •Variance Component Estimation, •Intra-correlation •Mixed Models •Nested Models with two variance components •Nested Models with three variance components 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 10 Module Twelve: Designs and Analysis for Factorial Treatments •Statistical Model – Two-factor Case •ANOVA for two-factor Case •Interactions – concepts, computation and interpretation •Post-Hoc Comparison •Techniques of Sum of Squares Decompositions •Statistical Model – Three Factor Case •Analysis for Three-factor Case 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 11 Module Thirteen: Gage R&R Analysis – Quantifying Repeatability and Reproducibility • What is Gage R&R Analysis? • Statistical model for Gage R&R Crossed Design • Estimation of Repeatability and Reproducibility • Using X-bar, R-Chart • Using ANOVA • Statistical Model for Gage R&R Nested Design • Use of Generalized Linear Model to analyze Gage R&R Data • Gage Capability Index 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 12 Module Fourteen : Block Designs and Others •Randomized Complete Block Designs •Statistical Model, Analysis and Interpretation •Others 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 13 Appendix 1. A Simplified Minitab Manual 2. Data sets 3. References 7/30-8/10/2001 Uncertainty Workshop by Carl Lee 14