TABLE1. Format and notation for a Two-Way Table

iJ

1…J

1

.

.

.

I y11 . . . y1J

. . .

. . .

. . .

yI1 . . . yIJ

This data structure involves three variables:

the row factor, which has I levels

the column factor, which has J levels

the response y, which we have I x J observations. The cell is the intersection of a row and column.

Simple Additive Model

yij = u + ai + bj + eij

where u - overall typical value; “common value”

ai - row effect

bj - column effect

eij – departure of yij from the purely additive model; random fluctuation

MOTIVATION:

We are primarily concerned with the techniques of analysis that are resistant, so that isolated violent disturbances in a small number of cells will not much affect the common value, row effects, column effects, and, as a consequence, will be reflected in the residuals.

Example:

TABLE2. Infant Mortality Rates in US, 1964-1966

RegionEducation of Father (in yrs)

≤ 89-111213-15≥ 16

Northeast25.325.318.218.316.3

North Central32.129.018.824.319.0

South38.831.019.315.716.8

West25.421.120.324.017.5

Source:UREDA

TABLE3. Students GPA’s based on the type of their major and their class status MajorClass Status

FreshmanSophomoreJuniorSenior

Science2.83.13.22.7

Humanities3.33.53.63.1

Other3.03.22.93.0

Source: http://www.ltcconline.net

TABLE4. Life (in hours) of batteries by material type and temperature Material

TypeTemperature (˚C)

Low( -10˚C)Medium (10 ˚C)High (45˚C)

11303420

215013625

313817496

Source: Montgomery (2001)

MEDIAN POLISH

Was described by Tukey in 1970

Simplest method of analyzing two-way tables

An iterative process of subtracting medians until all rows and columns have zero median

Begin with rows and then work with columns or the other way around (arbitrary choice)

Notes:

Δ represents a change

n is assumed to be 1 initially

We denote the fit and residuals at the end of n iterations by yij=m(n) + ai(n) + bj(n) + eij(n)

Initial conditions before the first iteration: m(0)=0, ai(0)=0 where i=1 to I, bj(0)=0 where j=1 to J

Method:

1st Sweep

Compute for row medians (new row medians).

From each row observation, deduct row median.

Call the row medians as previous row medians. Get the median of previous row medians. This is Δma(1).

Compute for the column medians in the present table. Call these new column medians. m(1)=Δma(1)

Compute for[a(1)]=[0]+[new row medians]-[Δma(1)]

[b(1)]=[0]+[new column medians]-[0]

2nd Sweep

From each column of the present table, deduct column medians.

Compute for row medians in the resulting table in (1). Call these new row medians.

Compute Δmb(2)=med(bj(1)). Note Δmb(1)=0.

From each row, deduct row median.

Compute for column medians in the updated table. Call these new column medians.

Add the new row medians to [a(1)] to get the 'previous row median'.

Compute Δma(2)=med of previous row median.

m(2)=m(1)+Δma(2)+Δmb(2)

Compute for[a(2)]=[a(1)]+[new row medians]-[Δma(2)]

[b(2)]=[b(1)]+[new column medians]-[Δmb(2)]

3rd Sweep

Follow 2nd sweep each time updating Δma(.), Δmb(.), m(.), [a(.)], [b(.)]

Example:

1. Consider the two-way table below with I=J=3

1st Sweep

1st polish on rows

LegsWingsBodyNew med(row)Previous

Waterfowl6311(6)0

Gallinaceous324(3)0

Raptorial900(0)0

Previous000(0)0

(Source: The Analysis of Contingency Tables in Archaeology; L-legs W-wings B-body)

After 1st polish on rows, 1st...