ENGR 610 Applied Statistics Fall Week 8 Marshall University CITE Jack Smith

Overview for Today Review Hypothesis Testing, Ch 9 Go over homework problem: 9.69, 9.71, 9.74 Design of Experiment, Ch 10 One-Factor Experiments Randomized Block Experiments Homework assignment

Critical Regions Critical value of test statistic (Z, t, F,  2,…) Based on desired level of significance (  ) Acceptance (of null hypothesis) region Rejection (alternative hypothesis) region Two-tailed or one-tailed

Z Test (  known) - Two-tailed Critical value (Z c ) based on chosen level of significance,  Typically  = 0.05 (95% confidence), where Z c = 1.96 (area = 0.95/2 = 0.475)  = 0.01 (99%) and (99.9%) are also common, where Z c = 2.57 and 3.29 Null hypothesis rejected if sample Z > Z c or < -Z c, where

Z Test (  known) - One-tailed Critical value (Z c ) based on chosen level of significance,  Typically  = 0.05 (95% confidence), but where Z c = (area = = 0.45) Null hypothesis rejected if sample Z > Z c, where

t Test (  unknown) - Two-tailed Critical value (t c ) based on chosen level of significance, , and degrees of freedom, n-1 Typically  = 0.05 (95% confidence), where, for example t c = (upper area = 0.05/2 = 0.025), for n-1 = 29 Null hypothesis rejected if sample t > t c or < -t c, where t

Z Test on Proportion Using normal approximation to binomial distribution

p-value Use probabilities corresponding to values of test statistic (Z, t,…) Compare probability (p) directly to  instead of, say, t to t c If the p-value  , accept null hypothesis If the p-value < , reject null hypothesis Does not assume any particular distribution (Z-normal, t, F,  2,…)

Z Test for the Difference between Two Means Random samples from independent groups with normal distributions and known  1 and  2 Any linear combination (e.g. the difference) of normal distributions (  k,  k ) is also normal CLT: Populations the same

t Test for the Difference between Two Means (Equal Variances) Random samples from independent groups with normal distributions, but with equal and unknown  1 and  2 Using the pooled sample variance H 0 : µ 1 = µ 2

t Test for the Difference between Two Means (Unequal Variances) Random samples from independent groups with normal distributions, with unequal and unknown  1 and  2 Using the Satterthwaite approximation to the degrees of freedom (df) Use Excel Data Analysis tool!

F test for the Difference between Two Variances Based on F Distribution - a ratio of  2 distributions, assuming normal distributions F L ( ,n 1 -1,n 2 -1)  F  F U ( ,n 1 -1,n 2 -1), where F L ( ,n 1 -1,n 2 -1) = 1/F U ( ,n 2 -1,n 1 -1), and where F U is given in Table A.7 (using nearest df)

Mean Test for Paired Data or Repeated Measures Based on a one-sample test of the corresponding differences (D i ) Z Test for known population  D t Test for unknown  D (with df = n-1) H 0 :  D = 0

 2 Test for the Difference among Two or More Proportions Uses contingency table to compute (f e ) i = n i p or n i (1-p) are the expected frequencies, where p = X/n, and (f o ) i are the observed frequencies For more than 1 factor, (f e ) ij = n i p j, where p j = X j /n Uses the upper-tail critical  2 value, with the df = number of groups – 1 For more than 1 factor, df = (factors -1)*(groups-1) Sum over all cells

Other Tests  2 Test for the Difference between Variances Follows directly from the  2 confidence interval for the variance (standard deviation) in Ch 8. Very sensitive to non-Normal distributions, so not a robust test. Wilcoxon Rank Sum Test between Two Medians

Design of Experiments R.A. Fisher (Rothamsted Ag Exp Station) Study effects of multiple factors simultaneously Randomization Homogeneous blocking One-Way ANOVA (Analysis of Variance) One factor with different levels of “treatment” Partitioning of variation - within and among treatment groups Generalization of two-sample t Test Two-Way ANOVA One factor against randomized blocks (paired treatments) Generalization of two-sample paired t Test

One-Way ANOVA ANOVA = Analysis of Variance However, goal is to discern differences in means One-Way ANOVA = One factor, multiple treatments (levels) Randomly assign treatment groups Partition total variation (sum of squares) SST = SSA + SSW SSA = variation among treatment groups SSW = variation within treatment groups (across all groups) Compare mean squares (variances): MS = SS / df Perform F Test on MSA / MSW H 0 : all treatment group means are equal H 1 : at least one group mean is different

Partitioning of Total Variation Total variation Within-group variation Among-group variation (Grand mean) (Group mean) c = number of treatment groups n = total number of observations n j = observations for group j X ij = i-th observation for group j

Mean Squares (Variances) Total mean square (variance) MST = SST / (n-1) Within-group mean square MSW = SSW / (n-c) Among-group mean square MSA = SSA / (c-1)

F Test F = MSA / MSW Reject H 0 if F > F U ( ,c-1,n-c) [or p<  ] F U from Table A.7 One-Way ANOVA Summary SourceDegrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) Fp-value Among groups c-1SSAMSA = SSA/(c-1)MSA/ MSW Within groups n-cSSWMSW = SSW/(n-c) Totaln-1SST

Tukey-Kramer Comparison of Means Critical Studentized range (Q) test q U ( ,c,n-c) from Table A.9 Perform on each of the c(c-1)/2 pairs of group means Analogous to t test using pooled variance for comparing two sample means with equal variances

One-Way ANOVA Assumptions and Limitations Assumptions for F test Random and independent (unbiased) assignments Normal distribution of experimental error Homogeneity of variance within and across group (essential for pooling assumed in MSW) Limitations of One-Factor Design Inefficient use of experiments Can not isolate interactions among factors

Randomized Block Model Matched or repeated measurements assigned to a block, with random assignment to treatment groups Minimize within-block variation to maximize treatment effect Further partition within-group variation SSW = SSBL + SSE SSBL = Among-block variation SSE = Random variation (experimental error) Total variation: SST = SSA + SSBL + SSE Separate F tests for treatment and block effects Two-way ANOVA, treatment groups vs blocks, but the focus is only on treatment effects

Partitioning of Total Variation Total variation Among-group variation Among-block variation (Grand mean) (Group mean) (Block mean)

Partitioning, cont’d Random error c = number of treatment groups r = number of blocks n = total number of observations (rc) X ij = i-th block observation for group j

Mean Squares (Variances) Total mean square (variance) MST = SST / (rc-1) Among-group mean square MSA = SSA / (c-1) Among-block mean square MSBL = SSBL / (r-1) Mean square error MSE = SSE / (r-1)(c-1)

F Test for Treatment Effects F = MSA / MSE Reject H 0 if F > F U ( ,c-1,(r-1)(c-1)) F U from Table A.7 Two-Way ANOVA Summary SourceDegrees of Freedom (df) Sum of Squares (SS) Mean Square (MS) (Variance) Fp-value Among groups c-1SSAMSA = SSA/(c-1)MSA/ MSE Among blocks r-1SSBLMSBL = SSBL/(r-1)MSBL /MSE Error(r-1)(c-1)SSEMSE = SSE/(r-1)(c-1) Totalrc-1SST

F Test for Block Effects F = MSBL / MSE Reject H 0 if F > F U ( ,r-1,(r-1)(c-1)) F U from Table A.7 Assumes no interaction between treatments and blocks Used only to examine effectiveness of blocking in reducing experimental error Compute relative efficiency (RE) to estimate leveraging effect of blocking on precision

Estimated Relative Efficiency Relative Efficiency Estimates the number of observations in each treatment group needed to obtain the same precision for comparison of treatment group means as with randomized block design. n j (without blocking)  RE*r (with blocking)

Tukey-Kramer Comparison of Means Critical Studentized range (Q) test q U ( ,c,(r-1)(c-1)) from Table A.9 Where group sizes (number of blocks, r) are equal Perform on each of the c(c-1)/2 pairs of group means Analogous to paired t test for the comparison of two- sample means (or one-sample test on differences)

Homework Work through Appendix 10.1 Work and hand in Problems (except part c) Read Chapter 11 Design of Experiments: Factorial Designs