Chi-Square

1
Chi-Square

• CJ 526 Statistical Analysis in Criminal Justice

2
Parametric vs Nonparametric

• Parametric
• DV Interval/Ratio

3
Nonparametric

• Nonparametric
• DV Nominal/ordinal

4
Chi-Square Test for Goodness of Fit

• One sample, DV is at Nominal/Ordinal Level of
  Measurement
• Determines whether the sample distribution fits
  some theoretical distribution

5
Null Hypothesis

• Population is evenly distributed
• Or
• Some other distribution, such as the normal
  distribution

6
Observed Frequency

• Number of individuals from the sample who are
  classified in a particular category

7
Expected Frequency

• The frequency value for a particular category
  that is predicted from the null hypothesis and
  the sample size

8
Chi-Square Statistic

• Sum of
• (Observed - Expected)2
• divided by
• Expected

9
Degrees of Freedom

• df C - 1
• where C is the number of categories
• The degrees of freedom are the number of
  categories that are free to vary

10
Interpretation

• If H0 is rejected, distribution is different from
  what is expected

11
Report Writing Results Section

• The results of the Chi-Square Test for Goodness
  of Fit involving ltIVgt were (not) statistically
  significant, ?2 (df) ltvaluegt, p lt .05.

12
Report Writing Discussion Section

• It appears as if the ltsamplegt is ltnotgt
  distributed as expected.

13
Example

• Concerned about health, neither concerned or not
  concerned, not concerned about health
• Could assume that a sample would be equally split
  among these three categories i.e., 120 subjects,
  40 would say concerned, 40 neither, 40 not
  concerned

14
Example
O E O-E (O-E)2 /E
60 40 20 400 10
40 40 0 0 0
20 40 20 400 10
15
Chi square

• Chi square 20
• D.f. 2
• See p. 726
• Chi square 20, p lt .01
• The distribution is significantly different from
  the expected distribution

16
Example

• Dr. Zelda, a correctional psychologist, is
  interested in determining whether the
  intelligence of delinquents enrolled in a state
  training school is normally distributed

17
Distribution of Intelligence in the General
Population
IQ Range Z-score Percentage of General Population
Below 60 -3 .0228 (23)
60-85 -2 .1359 (136)
86-100 -1 .3413 (341)
101-115 1 .3413 (341)
116-130 2 .1359 (136)
131 3 .0228 (23)

18
Distribution of Intelligence in Dr. Zeldas School
Below 60 119
60-85 150
86-100 687
101-115 32
116-130 12
131 0

19

1. Number of Samples 1
2. Nature of Samples N/A
3. N/A
4. IV School enrolled in
5. DV IQ categories
6. Target Population all delinquents enrolled in
   the state training school

20

1. Inferential Test Chi-Square Test for Goodness
   of Fit
2. H0 The distribution of frequencies of the IQ
   categories for the sample will not be different
   from the population distribution of frequencies
   of the IQ categories

21

1. H1 The distribution of frequencies of the IQ
   categories for the sample will be different from
   the population distribution of frequencies of the
   IQ categories
2. If the p-value of the obtained test statistic is
   less than .05, reject the null hypothesis

22
Calculations
O E O-E (O-E)2 /E
119 23 96 9216 401
150 136 14 196 1
687 341 346 119716 351
32 341 309 95481 280
12 136 124 15376 113
0 23 23 529 23
23

1. X2 (5) 1169, p lt .001
2. Reject H0

24
SPSS Chi-Square Goodness of Fit Test

• Weight Cases
• Data, Weight Cases
• Check Weight Cases by
• Move weighted variable over to Frequency Variable
• Analysis
• Analyze, Nonparametric Statistics, Chi-Square
• Move DV to Test Variable List
• Enter Expected Values

25
Results Section

• The results of the Chi-Square Test for Goodness
  of Fit involving the distribution of IQ
  categories for the state training school were
  statistically significant, X2 (6) 682.646, p lt
  . 001.

26
Discussion Section

• It appears as if the distribution of frequencies
  of the IQ categories for students enrolled in the
  state training school is different from the
  population distribution of frequencies of the IQ
  categories.

27
Chi-Square Test for Independence

• Used to assess the relationship between two or
  more variables

28
Null Hypothesis

• No relationship between the two variables
• Or
• Alternative the two variables are related to
  one another

29
Degrees of Freedom

• df (R - 1)(C - 1),
• Where R is the number of rows and C is the number
  of columns in a bivariate table

30
Example

• Dr. Cyrus, a forensic psychologist, is interested
  in determining whether gender has an effect on
  the type of sentence that convicted burglars
  receive

31
Dr. Cyrus Results
Probation Jail Prison
Females 37 42 14
Males 23 16 58
32

1. Number of Samples 2
2. Nature of Samples Independent
3. N/A
4. IV Gender
5. DV Type of sentence received
6. Nominal
7. Target Population all convicted burglars

33

1. Inferential Test Chi-Square Test for
   Independence
2. H0 There is no relationship between gender and
   type of sentence received
3. H1 There is a relationship between gender and
   type of sentence received

34
Create a bivariate table
probation jail total
male 14 80 94
female 46 20 66
60 100 160
35
Calculate expected values

• For each cell, row total times column total,
  divided by the total number of subject
• i.e., for the first cell, (94 x 60)/160 35
• (66x60)/160 25, (94x100)/160 59, (66x100)/160
  41

36
O E (O-E) (O-E)2 /E
14 35 21 441 12.6
80 59 21 441 7.5
46 25 21 441 17.6
20 41 21 441 10.6
37

1. If the p-value of the obtained test statistic is
   less than .05, reject the null hypothesis
2. X2 (2) 48.3, p lt .001
3. Reject H0

38
Probation Jail Total
Male 14 (35) 80 (59) 94
Female 46 (25) 20 (41) 66
60 100 160
39
SPSS Chi-Square Test of Independence

• Analyze
• Descriptive Statistics
• Crosstabs
• Move DV into Columns
• Move IV into Rows
• Statistics
• Chi-Square
• Cells
• Percentage
• Rows
• Columns

40
Results Section

• The results of the Chi-Square Test for
  Independence involving gender as the independent
  variable and type of sentence received as the
  dependent variable were statistically
  significant, X2 (2) 41.745, p lt .001.

41
Discussion Section

• It appears as if gender has an effect on the type
  of sentence received.

42
Assumptions

• Independence of Observations