Analysis of Two Way Table
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 i J 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
Region Education of Father (in yrs) ≤ 8 9-11 12 13-15 ≥ 16
Northeast 25.3 25.3 18.2 18.3 16.3
North Central 32.1 29.0 18.8 24.3 19.0
South 38.8 31.0 19.3 15.7 16.8
West 25.4 21.1 20.3 24.0 17.5 Source:UREDA
TABLE3. Students GPA’s based on the type of their major and their class status
Major Class Status Freshman Sophomore Junior Senior
Science 2.8 3.1 3.2 2.7
Humanities 3.3 3.5 3.6 3.1
Other 3.0 3.2 2.9 3.0 Source: http://www.ltcconline.net
TABLE4. Life (in hours) of batteries by material type and temperature
Material
Type Temperature (˚C) Low( -10˚C) Medium (10 ˚C) High (45˚C)
1 130 34 20
2 150 136 25
3 138 174 96 Source:
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
Region Education of Father (in yrs) ≤ 8 9-11 12 13-15 ≥ 16
Northeast 25.3 25.3 18.2 18.3 16.3
North Central 32.1 29.0 18.8 24.3 19.0
South 38.8 31.0 19.3 15.7 16.8
West 25.4 21.1 20.3 24.0 17.5 Source:UREDA
TABLE3. Students GPA’s based on the type of their major and their class status
Major Class Status Freshman Sophomore Junior Senior
Science 2.8 3.1 3.2 2.7
Humanities 3.3 3.5 3.6 3.1
Other 3.0 3.2 2.9 3.0 Source: http://www.ltcconline.net
TABLE4. Life (in hours) of batteries by material type and temperature
Material
Type Temperature (˚C) Low( -10˚C) Medium (10 ˚C) High (45˚C)
1 130 34 20
2 150 136 25
3 138 174 96 Source: