Analysis of Variance: Formulation and Outputs

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I. One-Way Classification or One-Factor Experiments

 

 

Scenario

 

One-factor experiments are made of image28u independent groups (or treatments), each of which is represented by image51u observations.

 

Statistics for Equal Number of Observations

 

 

Formula

df

Mean Square

F

Treatment Mean

image30u

N/A

N/A

N/A

Grand Mean

image31u

N/A

N/A

N/A

Variation Between Treatments

image32u

image56u

image44u

image35u

Variation Within Treatments

image36u

image37u

image38u

image35u

Total Variation

image39u

image40u

N/A

N/A

 

Statistics for Unequal Number of Observations

 

 

Formula

df

Mean Square

F

Treatment Mean

image41u

N/A

N/A

N/A

Grand Mean

image42u

N/A

N/A

N/A

Variation Between Treatments

image43u

image56u

image34u

image35u

Variation Within Treatments

image46u

image47u

image48u

image35u

Total Variation

image49u

image50u

N/A

N/A

 

Output

 

ANOVA -- One-Factor Experiments

<Empty Line>

 

Count

Sum

Mean

Variance

Treatment 1

 

 

 

 

…

 

 

 

 

Treatment a

 

 

 

 

<Empty Line>

Variation

SS

df

MS

F

P

F-Critical

Null Hypothesis

Between Treatments

v

v

v

v

v

v

v

Within Treatments

v

v

v

 

 

 

Total

v

v

 

 

 

 

 

 

 

 

II. Two-Way Classification or Two-Factor Experiments

 

 

Scenario for Experiments without Replication

 

Two-factor experiments are made of image28u groups (or treatments) each of which is said to consist of image51u blocks of measurements.

 

Statistics for Experiments without Replication

 

 

Formula

df

Mean Square

F

Treatment Mean

image52u

N/A

N/A

N/A

Block Mean

image53u

N/A

N/A

N/A

Grand Mean

image54u

N/A

N/A

N/A

Variation Between Treatments

image55u

image56u

image57u

image1u

Variation Between Blocks

image2u

image18u

image4u

image5u

Residual or Random Variation

image6u

image20u

image8u

N/A

Total Variation

image9u

image40u

N/A

N/A

 

Output for Experiments Without Replication

 

ANOVA -- Two-Factor Experiment without Replication

<Empty Line>

 

Count

Sum

Mean

Variance

Treatment 1

 

 

 

 

…

 

 

 

 

Treatment a

 

 

 

 

Block 1

 

 

 

 

…

 

 

 

 

Block b

 

 

 

 

<Empty Line>

Variation

SS

df

MS

F

P

F-Critical

Null Hypothesis

Between Treatments

v

v

v

v

v

v

v

Between Blocks

v

v

v

v

v

v

v

Residual

v

v

v

 

 

 

 

Total

v

v

 

 

 

 

 

 

 

Scenario for Experiments with Replication

 

Same as above but each block entry consists of image11u repetitions, or replications.

 

Statistics for Experiments with Replication

 

 

Formula

df

Mean Square

F

Cell Mean

image12u

N/A

N/A

N/A

Treatment Mean

image13u

N/A

N/A

N/A

Block Mean

image14u

N/A

N/A

N/A

Grand Mean

image15u

N/A

N/A

N/A

Variation Between Treatments

image16u

image56u

image57u

image1u

Variation Between Blocks

image17u

image18u

image4u

image5u

Interaction Effects

image19u

image20u

image21u

image22u

Residual or Random Variation

image23u

image24u

image25u

N/A

Total Variation

image26u

image27u

N/A

N/A

 

Output for Experiments with Replication

 

ANOVA -- Two-Factor Experiment with Replication

<Empty Line>

a

Count

Sum

Mean

Variance

Treatment 1

 

 

 

 

…

 

 

 

 

Treatment N

 

 

 

 

Block 1

 

 

 

 

…

 

 

 

 

Block N

 

 

 

 

<Empty Line>

Variation

SS

df

MS

F

P

F-Critical

Null Hypothesis

Between Treatments

v

v

v

v

v

v

v

Between Blocks

v

v

v

v

v

v

v

Interaction

v

v

v

v

v

v

v

Residual

v

v

v

 

 

 

 

Total

v

v