1. Analysis of Discrete Variables
Gender (x1 = male, x2 = female)
Educ. level (x1 = low, x2 = middle, x3 = high)
5-point scales (x1 = 1, x2 = 2, ..., x5 = 5)
Diagnosis (x1 = Neurosis, x2 = Schizophrenia, ...)

2. Distribution of Discrete Variables: The General Casex x x
.... x 1 2 3 k p p p
.... p 1 2 3 k

3. An Example of Discrete Distribution
xi: 1 2 3
pi:
0.20
0.35
0.40
0.05

4. Analysis of 1 discrete variable in 1 population

5. Distribution Fitting
Hypothetical example: Which do you prefer best: Coke (x1), Pepsi (x2), or Fanta (x3)?
Null hypothesis: H0: P(x1) = P(x2) = P(x3) = 1/3
Obtained frequencies (ni):
Out of 150 Ss n1 = 80, n2 = 50, n3 = 20
Expected frequencies (ni): If H0 were true, one would expect for each ni.

6. Distribution Fitting with 2-test
The greater the difference between obtained (ni) and expected (ni) frequencies, the greater the likelihood that H0 is false.
A possible measure of the difference:
c2 = (n1 - n1)2/n1 + (n2 - n2)2/n (ng - ng)2/ng
If H0 is true, the distribution of c2 is approximately chi-square, with df = g - 1.

7. Calculations ni: 80 50 20 S=150 ni: 50 50 50 S=150
= 36 > = c20.01 (df = 2)
Thus we reject H0 and say:
‘The three proportions differ significantly.’

8. 2-test c2 < c20.05 c2 ³ c20.05 Keep H0
Condition: ni ³ 5
H0: P(x1) = p1, P(x2) = p2 , ... , P(xg) = pg
X-sample
0,6
c2 (df=1)
0,4
0,2
(df =g - 1)
0.95
0.05 c2 1 2 3
0.05
c2 < c20.05
c2 ³ c20.05
Keep H0
HA: For at least one i: P(xi) ¹ pi

9. Comparing 2 Populations by means of 1 Discrete Variable
Example: Is there a difference between males and females with respect to education level (EL)?
H0: The distribution of EL is the same among males and females
P(xi|Males) = P(xi|Females), (i = 1, 2, 3)
x1 = Low, x2 = Middle, x3 = High

10. Two-Way Frequency Table
Low Middle High Total
Male 16 32 32
n1=80
Female 18 45 27
n2=90
Total N=170

11. Two-Way Frequency Table: Row Percentages
Low Middle High Total
Male
20%
40%
40%
100%
Female
20%
50%
30%
100%
Total % % %

12. 2-test for Comparing Groups
If H0 is true then
Follows c2-distribution with df=(g-1)·(h-1).
Decision
c2 < c20.05: Keep H0 (p > .05 n. s.).
c2 ³ c20.05 : Reject H0 (p < .05 significant).

13. Comparison of Males and Females
Number of rows: g = 2
Number of columns: h = 3
Degrees of freedom: df = (2-1)×(3-1) = 2
Critical values:
c20.1 = 4.605; c20.05 = 5.991; c20.01 = 9.210
Computed chi-square value: c2 = 2.155
Decision: Keep H0 (p > .10 n. s.).

14. General Case Condition: nij ³ 5 df = (g-1)×(h-1) nij= (ni×mj)/N
Samples
X=x
X=x
X=x3
...
Total 1 2
Sample 1 n n n n 11 12 13 1
Sample 2 n n n n 21 22 23 2
nij= (ni×mj)/N
...
Total m m m N 1 2 3
df = (g-1)×(h-1)
Condition: nij ³ 5

15. Comparing the Distribution of 2 Variables in 1 Population
Example: Lecture about the disadvantages of smoking. Outcome: 8 of 36 students give up smoking, 3 start smoking. Any effect?
H0: The proportion of smokers does not change.
Indicatior of change: x1= positive change, x2 = negative change
H0: P(x1) = P(x2)

16. Computation: McNemar’s test
FrequencyTable:
Smoking
Time 2: No
Time 2: Yes
Time 1: No a
b = 8
Time 1: Yes
c = 3 d
Computation: McNemar’s test
Condition: (b+c)/2 ³ 5, that is b+c ³ 10

17. More General Cases X is arbitrary, two related samples: Bowker’s test
X is dichotomous, h related samples: Cochran’s Q test

18. Relationship of 2 Discrete Variables
Girls at 15
Makes friends easily
Test of independence = Comparisons

19. Table of Row Percentages
Girls at 15
Makes friends easily

20. Table of Column Percentages
Girls at 15
Makes friends easily

21. Strength of Relationship
Cramér’s contingency coefficient:
Ordinally scaled variables: Kendall’s G
Dichotomous variables: G= Yule’s Q