Category 1 Category 2 Category 3 Category k

Obs. 1 x11 x12 x13 … x1k

Obs. 2 x21 x22 x23 … x2k

Obs. 3 x31 x32 x33 … x3k

Obs. 4 x41 x42 x43 … x4k

.

..

Obs. i xi1 xi2 xi3 … xik

No. of obs. n1 n2 n3 … nk

Mean x+1 x+2 x+3 … x+k

Std. Dev. s1 s2 s3 … sk

Overall Mean: x++

ANOVA: Partitioning of the Sum of Squares

• Total sum of squares (TSS)

• Between sum of squares (BSS)

• Within sum of squares (WSS)

TSS = BSS + WSS

ANOVA

• F-statistic

)/(

)1/(

kNWSS

kBSSF

F-distributions

Source: Earickson, RJ, and Harlin, JM. 1994. Geographic Measurement and Quantitative Analysis. USA: Macmillan College Publishing Co., p. 190.

ANOVA Table

Source of Sum of Degrees of Mean SquareVariationSquares Freedom Variance F-Test

Between BSS k - 1 BSS/(k-1) BSS/(k-1)Groups WSS/(N-k)

Within WSS N - k WSS/(N-k)Groups

Total TSS N – 1Variation

Fα, k-1, N-k

Annual Swimming Frequencies

Central Suburbs Rural City Areas 38 58 80 42 66 70 50 80 60 57 62 55 80 73 72 70 39 73 32 73 81 20 58 50

Hypothetical Swimming Frequency Data (Textbook)

Do people in one of the three parts of the city swim more or less often than people in one of the other parts?

Tired of using ANOVA manually?

S-Plus

Variable vs. Grouping Variable

Example: Textbook: #9 pp. 83

Nonparametric Kruskal-Wallis Test

Levene’s Test for Equality of Variances

Tests of Normality: the Shapiro-Wilk Test

Alternative: var.test

Alternative: qqnorm

  Lab #3: Analysis of Variance (ANOVA)