ENGR 610 Applied Statistics Fall 2007 - Week 8

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# ENGR 610 Applied Statistics Fall 2007 - Week 8 - PowerPoint PPT Presentation

ENGR 610 Applied Statistics Fall 2007 - 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.

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### ENGR 610Applied StatisticsFall 2007 - 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 (Zc) based on chosen level of significance, 
• Typically  = 0.05 (95% confidence), where Zc = 1.96 (area = 0.95/2 = 0.475)
•  = 0.01 (99%) and 0.001 (99.9%) are also common, where Zc = 2.57 and 3.29
• Null hypothesis rejected if sample Z > Zc or < -Zc, where
Z Test ( known) - One-tailed
• Critical value (Zc) based on chosen level of significance, 
• Typically  = 0.05 (95% confidence), but where Zc = 1.645 (area = 0.95 - 0.50 = 0.45)
• Null hypothesis rejected if sample Z > Zc, where
t Test ( unknown) - Two-tailed
• Critical value (tc) based on chosen level of significance, , and degrees of freedom, n-1
• Typically  = 0.05 (95% confidence), where, for exampletc = 2.045 (upper area = 0.05/2 = 0.025), for n-1 = 29
• Null hypothesis rejected if sample t > tc or < -tc, 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 tc
• 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

• Random samples from independent groups with normal distributions, but with equal and unknown1 and 2
• Using the pooled sample variance

H0: µ1 = µ2

• Random samples from independent groups with normal distributions, with unequal and unknown1 and 2
• Using the Satterthwaiteapproximation 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
• FL(,n1-1,n2-1)  F  FU(,n1-1,n2-1), where FL(,n1-1,n2-1) = 1/FU(,n2-1,n1-1), and where FU 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 (Di)
• Z Test for known population D
• t Test for unknown D (with df = n-1)

H0: D = 0

2 Test for the Difference among Two or More Proportions
• Uses contingency table to compute
• (fe)i = nip or ni(1-p) are the expected frequencies, where p = X/n, and (fo)i are the observed frequencies
• For more than 1 factor, (fe)ij = nipj, where pj = Xj/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
• H0: all treatment group means are equal
• H1: 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

nj = observations for group j

Xij = 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 H0 if F > FU(,c-1,n-c) [or p<]
• FU from Table A.7
• One-Way ANOVA Summary
Tukey-Kramer Comparison of Means
• Critical Studentized range (Q) test
• qU(,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)

Xij = 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 H0 if F > FU(,c-1,(r-1)(c-1))
• FU from Table A.7
• Two-Way ANOVA Summary
F Test for Block Effects
• F = MSBL / MSE
• Reject H0 if F > FU(,r-1,(r-1)(c-1))
• FU 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.
• nj (without blocking)  RE*r (with blocking)
Tukey-Kramer Comparison of Means
• Critical Studentized range (Q) test
• qU(,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
• 10.27
• 10.28 (except part c)