New Statistical Tests

1
New Statistical Tests

• Cross Tabulation and Chi Square and
• Analysis of Variance and F Test

2
To date.

• We have examined statistical tests for
  differences of means, proportions, regression
  coefficients and correlation coefficients.
• These statistics are all measured at the interval
  level.

3
New Tests

• Now we wish to examine statistical tests for
  questions involving nominal and ordinal
  variables. To do so we introduce the Chi Square
  Test.
• We also wish to extend the logic of the t test to
  more than two groups. To do so we introduce the F
  Test.

4
Chi Square Test

• Remember the cross tabulated tables and graphs
  from earlier in the semester.

5
Concentration of Occupational Groups by Ward
6
Cross Tabulation

• Are the occupational patterns different in the
  four wards?
• Or.are the patterns a result of chance? (null
  hypothesis)
• How would we decide?

7
Some Terms and Assumptions

• Cell frequency number in the body of the table
• Marginal total total of the row or the column
• Row percent the proportion of cases in the cell
  for the particular row.
• Column percent the proportion of cases in the
  cell for the particular column
• Expected frequency the number of cases expected
  based upon the marginal proportions
• Deviation the difference between the expected
  frequency and the actual frequency

8
Illustration Frequencies and Marginals
9
Row and Column Percents
10
Expected and Actual Frequencies
Frequencies OCC (rows) by WARD (columns)
14 18 20 22
Total ----------------------------
------------- profcler 9 55
30 57 151 prop
27 33 54 45 159
skilled 90 16 149 114
369 skillpart 13 12
40 26 91 unskilled 175
12 71 46 304
-----------------------------------------
Total 314 128 344
288 1074
Expected values OCC (rows) by WARD (columns)
14 18 20
22 -----------------------------------
------ profcler 44.15 18.00 48.36
40.49 prop 46.49 18.95
50.93 42.64 skilled 107.88
43.98 118.19 98.95 skillpart
26.61 10.85 29.15 24.40
unskilled 88.88 36.23 97.37
81.52 -------------------------
----------------
11
Expected Frequencies

• Row Total Column Total/ Grand Total
• Or
• Row Proportion Column Total
• Or
• Column Proportion Row Total

12
Chi Square

• Chi Square (Sum of (Expected Observed)2) /
  Expected Frequency
• Chi Square Table http//www.uwm.edu/renlex/chis
  quare.html

13
Degrees of Freedom for Chi Square

• Degrees of Freedom (r-1) (c-1)
• So, 2 by 2 table has 1 degree of freedom
• 3 by 2 table has (3-1)(2-1) 2 degrees of freedom

14
Strength of Relationships

• Phi Square root of (Chi Square/N)
• Cramers V Square root of (Chi
  Square/nmin(r-1, c-1))
• Contingency Coefficient Square root of (Chi
  Square/(Chi Squaren))

15
Analysis of Variance

• Extends the logic of the T test to more than 2
  groups
• To be continued.