ANOVA Cheat Sheet

ANOVA(Analysis of Variance) Cheat Sheet Null Hypothesis, Ho : 1  2  3  ...  k

k

AlternateHypothesis, Ha : At least one k is different ANOVA looks at three sources of variability 1) Total variability among observations 2) Variability between group means (factor) 3) Random variation within each group (error) TOTAL = Between + Within    Between Within

ANOVA Table Source

MSwithin 



 ( y

ij

j 1 i 1

MS(Factor) MS(Error)

Df

MS

F

p

SS(Factor)

k-1

SS(Factor) (k-1)

MS(Factor) MS(Error)

Within or Error

SS(Error)

k(n-1)

SS(Error) k(n-1)

Area under F curve from calculated F to 

Total

SS(Total)

MSbetween 

n = number of observations at each level(sample size per treatment) k = number of levels N = Total number of observations In ANOVA, the degrees of freedom(Df) are as follows: Dftotal = N-1 = # of observations - 1 Dffactor = k-1 = # of levels - 1 Dferror = Dftotal - Dfeverything else

Two-Way ANOVA

Studies the effect of two factors and their interaction at various levels on a response variable

j 1

j

Degrees of Freedom within

 y )2

(k  1)

SS between

Degrees of Freedom between

Note: Figure below assumes alpha level of .05 (5%) for illustration. Your selected alpha should “fit” your problem

Fcalculated > Fcritical means there is less than a 5% chance that the larger between treatment variation occurred by chance alone, thus reject the null hypothesis, Ho. Else, if Fcalculated < Fcritical, you cannot reject the null hypothesis based on the data.

Minitab ANOVA Options (Stat/DOE/Factorial/Analyze Factorial Design very similar to Balanced ANOVA) Studies the effect of one factor at various levels on a response variable

 n( y

SS within

ANOVA Assumptions 1) Equal variance at all treatments 2) Process distribution is normal 3) Runs are independent (replicates)

N-1

One-Way ANOVA

 y j )2

k (n  1) k

SS

Between or Factor

n

Fcritical @ 5%

Balanced ANOVA

General Linear Model Fully Nested ANOVA

Studies the impact of 2 or more factors and there interactions at various levels on a response variable. The levels of factors are structured such that there are an equal number of levels and observations within each level for each factor. Studies the impact of 2 or more factors and interactions at various levels one a response variable. Number of levels and observations may vary. Factors may be a mixture nested and crossed relationship. User must specify factors, interactions and nested/crossed relationships of interest. Studies the impact of 2 or more factors. Factors are structured in a hierarchical structure such that one factor is nested (or unique to) the factor above it. No interactions are obtained.

Curve changes as a function of the numerator and denominator DOF

Use for COV

5% of the total area is from Fcritical to 

Fcalculated

The calculated p value represents the area under the curve from Fcalculated to 

Represents the amount of risk you’re willing to take of being wrong when you say that you’ve found this factor to have a statistically significant effect

11-5-02