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Assumptions for Chi square

ASSUMPTIONS

A. Independent variables

*Example of non-independence: Height after experiment by height before experiment

Q: Can Chi square be used on this data?

A: No, because the variables are not independent (height before and after are

dependent).

*Note: This will generally not be a problem in this course.

  1. Every fe >5 recall

fe = (row total)(column total)
N

*The smallest sample size you can possibly have and meet this assumption is 20.

CHURCH AFFILIATION

ALIENATION SCORES
V

D

Q

R

T
100 00 11 2
80 01 22 5
70 42 00 6
61 36 10 11
30 23 10 6
21 11 00 3
2 1013 53 33

Q: Can Chi square be used on this data?

A: = (fo - fe)2
------------fe

Where: fo = number in a cell

fe = (row total)(col total)
-------------N

But computing every fe and then summing them up is quite time consuming. Instead compute the smallest fe and, if it is greater than five, then all others will satisfy the assumption.

Using table above: Smallest fe = 2(2) = 4 = .12
---------------------------------33----33

Steps for meeting chi square assumptions:

1. Compute smallest fe =(smallest row total)(smallest column total)
------------------------------------------ N

If smallest fe is less than 5, it is necessary to collapse the table, assuming it is larger than a

2 x 2 table.

2. Locate the smallest subtotal in the entire table (i.e. the smallest row total or the smallest column total). If this smallest subtotal is for a subcategory of a nominal variable, then combine it with the next smallest subtotal for a subcategory of a nominal variable.

3. If the smallest subtotal is for a subcategory of an ordinal or interval variable, then combine it with the adjacent subcategory of the variable with the smallest subtotal.

4. After making a combination of a row total with another row total or a column with another column, check again to see if the smallest fe is greater than 5. If not, proceed to collapse the table again by the procedure described above, assuming it is still larger than a 2 x 2 table.

5. Continue collapsing the table only until the smallest fe is greater than 5. Stop collapsing at this point and compute Chi square. Be sure to compute the degrees of freedom on the collapsed table rather than the original table.

6. If the table is collapsed until it is a 2 x 2 table, and the smallest fe is still smaller than 5, then it can be collapsed no further. It is not legitimate to use Chi square on the data.

Notice: This fe is < 5 so look for the smallest subtotal in the table. If the data is nominal, combine the smallest subtotal with the next smallest in that category. If the data is ordinal, combine the subtotal with the row or column on either side. This is called "collapsing the table for Chi Square". It is a very frequent task in SPSS with large range variables .

CHURCH AFFILIATION

ALIENATION SCORES
V&T

D

Q

R
101 00 12
82 01 25
70 42 06
61 36 111
30 23 16
21 11 03
5 1013 533

Here column V and column T were combined. Now figure fe again:

Smallest fe= [2(5)]/33 = 10/33 = .33, but this is still <5 so recombine the smallest sub-total.

CHURCH AFFILIATION

ALIENATION SCORES
V&T

D

Q

R
10 & 83 01 37
70 42 06
61 36 111
30 23 16
21 11 03
5 1013 533

Here row 10 and row 8 were combined. Note that this is ordinal data so it could have only been combined with either the row above or below this. (Since 10 was the top of the category, combine the row with number 8, which was directly below 10.) The smallest fe = [(3)(5)]/33 = 15/33 = too small; combine again!

CHURCH AFFILIATION

ALIENATION SCORES
V&T

D

Q

R
10 & 83 01 37
70 42 06
61 36 111
3 & 21 34 19
5 1013 533

Here rows 3 & 2 were combined.

fe = [5(5)]/33 = 25/33 = too small! Do again . . .

CHURCH AFFILIATION

ALIENATION SCORES
VT&R

D

Q
10 & 86 01 7
70 42 6
62 36 11
3 & 22 34 9
10 1013 33

Combine columns V, T and R. fe = [6(10)]/33 < 2

This fe is still less than 5, so combine again.

CHURCH AFFILIATION

ALIENATION SCORES
VT&R

D

Q
10,8, & 7 64 313
62 36 11
3 & 22 34 9
10 1013 33

Combine rows 10, 8 and 7.

fe < 3

CHURCH AFFILIATION

ALIENATION SCORES
VT&R

D

Q
10,8, & 76 43 13
6,3, & 2 46 1020
10 1013 33

Combine rows 6, 3, and 2.

fe < 4

CHURCH AFFILIATION

ALIENATION SCORES
VTR&D

Q
10,8, & 710 313
6,3, & 210 1020
20 1333

Combine columns V, R, T and D.

fe = [13(13)]/33 > 5 so done. Notice that this is a 2 x 2 table.

Some things to remember when collapsing tables:

Much information is lost when this is done. Make the fewest number of combinations so that fe > 5.

Do not immediately assume that the table will be 2 x 2.

Short cut: Don't rewrite table after each combination, just do it often enough to keep track of the data.

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