Analysis of Two Way Table

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  • Topic: Arithmetic mean, Missing values, Expectation-maximization algorithm
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Two- Way Table is a set of data in which the observations are written yij i= 1,2 ,…,I j= 1,2,…,J and displayed in a rectangular array as shown below
TABLE1. Format and notation for a Two-Way Table
iJ
1J
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...
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