Chapter 10 Design of Experiments and Analysis of Variance

Chapter 10

Design of Experiments and Analysis of Variance

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Elements of a Designed Experiment

•Response variable

Also called the dependent variable

•Factors (quantitative and qualitative)

Also called the independent variables

•Factor Levels

•Treatments

•Experimental Unit

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Designed vs. Observational Experiment•In a Designed Experiment, the analyst determines the treatments, methods of assigning units to treatments.•In an Observational Experiment, the analyst observes treatments and responses, but does not determine treatments•Many experiments are a mix of designed and observational

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Dependent Variable

Independent Variable

Sample

Population of Interest

Single-Factor Experiment

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Two-factor Experiment

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The Completely Randomized Design

Achieved when the samples of experimental units for each treatment are random and independent of each other

Design is used to compare the treatment means:

0 1 2: ... kH :aH At least two of the treatment means differ

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• The hypotheses are tested by comparing the differences between the treatment means to the amount of sampling variability present

• Test statistic is calculated using measures of variability within treatment groups and measures of variability between treatment groups

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Sum of Squares for Treatments (SST)Measure of the total variation between treatment means, with k treatments

Calculated by

Where

2

1

k

iii

SST n x x

th

i

thi

n number of observations in i treatment group

x mean of measurements in i treatment group

x overall mean of all measurements

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Sum of Squares for Error (SSE)Measure of the variability around treatment means attributable to sampling error

Calculated by

After substitution, SSE can be rewritten as

1 22 2 2

1 1 2 21 1 1

...knn n

j j kj kj j j

SSE x x x x x x

2 2 21 1 2 21 1 ... 1k kSSE n s n s n s

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Mean Square for Treatments (MST)Measure of the variability among treatment means

Mean Square for Error (MSE)Measure of sampling variability within treatments

1

SSTMST

k

SSEMSE

n k

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F-StatisticRatio of MST to MSE

Values of F close to 1 suggest that population means do not differ

Values further away from 1 suggest variation among means exceeds that within means, supports Ha

, ( 1, )MST

F with df k n kMSE

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Conditions Required for a Valid ANOVA F-Test: Completely Randomized Design

1. Independent, randomly selected samples.

2. All sampled populations have distributions that approximate normal distribution

3. The k population variances are equal

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A Format for an ANOVA summary table

ANOVA Summary Table for a Completely Randomized Design

Source df SS MS F

Treatments 1k SST 1

SSTMST

k

MST

MSE

Error n k SSE SSE

MSEn k

Total 1n SS Total

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ANOVA summary table: an example from Minitab

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Conducting an ANOVA for a Completely Randomized Design

1. Assure randomness of design, and independence, randomness of samples

2. Check normality, equal variance assumptions

3. Create ANOVA summary table

4. Conduct multiple comparisons for pairs of means as necessary/desired

5. If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error

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Multiple Comparisons of Means

•A significant F-test in an ANOVA tells you that the treatment means as a group are statistically different.•Does not tell you which pairs of means differ statistically from each other•With k treatment means, there are c different pairs of means that can be compared, with c calculated as 1

2

k kc

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•Three widely used techniques for making multiple comparisons of a set of treatment means•In each technique, confidence intervals are constructed around differences between means to facilitate comparison of pairs of means•Selection of technique is based on experimental design and comparisons of interest•Most statistical analysis packages provide the analyst with a choice of the procedures used by the three techniques for calculating confidence intervals for differences between treatment means

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Guidelines for Selecting a Multiple Comparisons Method in ANOVA

Method Treatment Sample Sizes Types of Comparisons Tukey Equal Pairwise Bonferroni Equal or Unequal Pairwise Scheffe Equal or Unequal General Contrasts

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The Randomized Block Design

Two-step procedure for the Randomized Block Design:

1. Form b blocks (matched sets of experimental units) of k units, where k is the number of treatments.

2. Randomly assign one unit from each block to each treatment. (Total responses, n=bk)

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2

1

2

1

2

1

( )

( )

i

Bi

k

T

i

b

i

n

ii

SST b x x

SSB k x x

SS Total x x

SSE SS Total SST SSB

Partitioning Sum of Squares

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1

1

SSTMST

kSSE

MSEn b k

Calculating Mean Squares

Hypothesis Testing

Setting Hypotheses

0 1 2: ...

:k

a

H

H At least two treatment means differ

MSTF

MSE

Rejection region: F > F, F based on (k-1), (n-b-k+1) degrees of freedom

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Conditions Required for a Valid ANOVA F-Test: Randomized Block Design

1. The b blocks are randomly selected, all k treatments are randomly applied to each block

2. Distributions of all bk combinations are approximately normal

3. The bk distributions have equal variances

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Conducting an ANOVA for a Randomized Block Design

1. Ensure design consists of blocks, random assignment of treatments to units in block

2. Check normality, equal variance assumptions 3. Create ANOVA summary table4. Conduct multiple comparisons for pairs of means as

necessary/desired

5. If H0 not rejected, consider possible explanations, keeping in mind the possibility of a Type II error

6. If desired, conduct test of H0 that block means are equal

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Factorial Experiments

Complete Factorial Experiment

•Every factor-level combination is utilized

Schematic Layout of Two-Factor Factorial Experiment

Factor B at b levels Level 1 2 3 … b

1 Trt.1 Trt.2 Trt.3 … Trt.b 2 Trt.b+1 Trt.b+2 Trt.b+3 … Trt.2b 3 Trt.2b+1 Trt.2b+2 Trt.2b+3 … Trt.3b

…

…

…

… …

…

Factor A at a Levels

a Trt.(a-1)b+1 Trt.(a-1)b+2 Trt.(a-1)b+3 … Trt.ab

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Partitioning Total Sum of Squares

•Usually done with statistical package

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Conducting an ANOVA for a Factorial Design

1. Partition Total Sum of Squares into Treatment and Error components

2. Test H0 that treatment means are equal. If H0 is rejected proceed to step 3

3. Partition Treatment Sum of Squares into Main Effect and Interaction Sum of Squares

4. Test H0 that factors A and B do not interact. If H0 is rejected, go to step 6. If H0 is not rejected, go to step 5.

5. Test for main effects of Factor A and Factor B6. Compare the treatment means

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SPSS ANOVA Output for a factorial experiment

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Chapter 10 Design of Experiments and Analysis of Variance