Chapter 10 Design of Experiments and Analysis of Variance Elements - - PowerPoint PPT Presentation

Chapter 10

Design of Experiments and Analysis of Variance

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

Elements of a Designed Experiment

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
• bserves treatments and responses, but

does not determine treatments

• Many experiments are a mix of designed

and observational

Elements of a Designed Experiment

Dependent Variable Independent Variable Sample Population of Interest

Single-Factor Experiment

Elements of a Designed Experiment

Two-factor Experiment

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:

1 2

: ...

k

H      :

a

H A t lea st tw o o f th e trea tm en t m ea n s d iffer

The Completely Randomized Design

• 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

The Completely Randomized Design

Sum of Squares for Treatments (SST)

Measure of the total variation between treatment means, with k treatments Calculated by Where

 

2 1 k i i i

S S T n x x



 



th i th i

n n u m b e r o f o b s e r v a tio n s in i tr e a tm e n t g r o u p x m e a n o f m e a s u r e m e n ts in i tr e a tm e n t g r o u p x

• v e r a ll m e a n o f a ll m e a s u r e m e n ts

  

The Completely Randomized Design

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 2

2 2 2 1 1 2 2 1 1 1

...

k

n n n j j k j k j j j

S S E x x x x x x

  

      

  

     

2 2 2 1 1 2 2

1 1 ... 1

k k

S S E n s n s n s       

The Completely Randomized Design

Mean Square for Treatments (MST)

Measure of the variability among treatment means

Mean Square for Error (MSE)

Measure of sampling variability within treatments

1 S S T M S T k   S S E M S E n k  

The Completely Randomized Design

F-Statistic

Ratio 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, ) M S T F w ith d f k n k M S E   

The Completely Randomized Design

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

The Completely Randomized Design

A Format for an ANOVA summary table

ANOVA Summary Table for a Completely Randomized Design

Source

d f S S

M S

F

T reatm ents 1 k  SST

1 S S T M S T k   M S T M S E

E rror n k  SSE

S S E M S E n k  

T otal 1 n 

 

S S T o ta l

The Completely Randomized Design

ANOVA summary table: an example from Excel

The Completely Randomized Design

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

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
• f means that can be compared, with c calculated

as

 

1 2 k k c  

Multiple Comparisons of Means

• 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

• f 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

Multiple Comparisons of Means

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

The Randomized Block Design

Two-step procedure for the Randomized Block Design:

• 1. Form b blocks (matched sets of

experimental units) of p units, where p is the number of treatments.

• 2. Randomly assign one unit from each

block to each treatment. (Total responses, n=bp)

The Randomized Block Design

     

2 1 2 1 2 1

( ) ( )

i B i

k T i b i n i i

S S T b x x S S B p x x S S T o ta l x x S S E S S T o ta l S S T S S B

  

        

  

Partitioning Sum

• f Squares

The Randomized Block Design

1 1 S S T M S T k S S E M S E n b k      

Calculating Mean Squares Hypothesis Testing Setting Hypotheses

1 2

: ... :

k a

H H A t le a s t tw o tr e a tm e n t m e a n s d iffe r      

M S T F M S E 

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

The Randomized Block Design

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

The Randomized Block Design

Conducting an ANOVA for a Randomized Block Design

1. Ensure design consists of blocks, random assignment

• f treatments to units in block

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

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

Factorial Experiments

Partitioning Total Sum of Squares

• Usually done with

statistical package

Factorial Experiments

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 B 6. Compare the treatment means

Factorial Experiments

SPSS ANOVA Output for a factorial experiment