Engr 610 applied statistics fall 2007 week 8
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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 610 Applied Statistics Fall 2007 - Week 8

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Engr 610 applied statistics fall 2007 week 8

ENGR 610Applied StatisticsFall 2007 - Week 8

Marshall University

CITE

Jack Smith


Overview for today

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 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

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

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

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

Z Test on Proportion

  • Using normal approximation to binomial distribution


P value

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

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

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

H0: µ1 = µ2


T test for the difference between two means unequal variances

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 Satterthwaiteapproximation to the degrees of freedom (df)

  • Use Excel Data Analysis tool!


F test for the difference between two variances

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

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

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

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

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

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

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

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 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

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

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

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 variation1

Partitioning of Total Variation

  • Total variation

  • Among-group variation

  • Among-block variation

(Grand mean)

(Group mean)

(Block mean)


Partitioning cont d

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 variances1

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 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 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

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 means1

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

Homework

  • Work through Appendix 10.1

  • Work and hand in Problems

    • 10.27

    • 10.28 (except part c)

  • Read Chapter 11

    • Design of Experiments: Factorial Designs


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