Chi-Square

1
Chi-Square

• CJ 526 Statistical Analysis in Criminal Justice

2
Parametric vs Nonparametric

• Parametric DV Interval/Ratio
• Nonparametric DV nominal/ordinal

3
Chi-Square Test for Goodness of Fit

• One sample, DV is at Nominal/Ordinal Level of
  Measurement
• This test , the chi-square good of fit,
  determines whether the sample distribution fits
  some theoretical distribution

4
Null Hypothesis

• Population is evenly distributed the uniform
  distribution
• Or
• Some other distribution, such as the normal
  distribution
• The sample distribution is not different from the
  theoretical distribution (such as the uniform
  distribution or the normal distribution)

5
Observed and expected frequency

• Observed number of individuals from the sample
  who are classified in a particular category
• Expected frequency the frequency for a
  particular category that is predicted from the
  null hypothesis

6
Chi-Square Statistic

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

7
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

8
Interpretation

• If the null hypothesis is retained, the sample
  distribution is like that of the theoretical
  distribution
• If H0 is rejected, distribution is different from
  what is expected

9
Report Writing Results Section

• Null hypothesis retained The results of the
  chi-square goodness of fit test were not
  statistically significant
• Null hypothesis rejected
• The results of the Chi-Square Test for Goodness
  of Fit involving ltDVgt were statistically
  significant, ?2 (df) ltvaluegt, p lt .05.

10
Report Writing Discussion Section

• It appears as if the ltsamplegt is (or is not)
  distributed as expected.
• Depends on the result

11
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 (uniform distribution)

12
Example
O E O-E (O-E)2 /E
60 40 20 400 10
40 40 0 0 0
20 40 20 400 10
13
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

14
Example

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

15
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)

16
Distribution of Intelligence in Dr. Zeldas School
Below 60 119
60-85 150
86-100 687
101-115 32
116-130 12
131 0

17

1. Number of Samples 1
2. DV IQ categories
3. Target Population delinquents enrolled in the
   state training school

18

• Inferential Test Chi-Square Test for Goodness
  of Fit
• 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

19

• 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
• If the p-value of the obtained test statistic is
  less than .05, reject the null hypothesis

20
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
21

• X2 (5) 1169, p lt .001
• Reject H0

22
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

23
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 (5) 1169, p lt .
  001.

24
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.

25
Chi-Square Test for Independence

• Used to assess the relationship between two or
  more variables

26
Null Hypothesis

• No relationship between the two variables
  (independent of one another)
• Or
• Alternative the two variables are related to
  one another

27
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 (review bivariate
  table)

28
Example

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

29
Background

• Number of samples 1
• IV Gender
• DV Type of sentence received
• Nominal
• Target Population convicted burglars

30

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

31
Create a bivariate table
probation jail total
male 14 80 94
female 46 20 66
60 100 160
32
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

33
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
34

• X2 (1) 48.3, p lt .001
• Reject H0

35
Probation Jail Total
Male 14 (35) 80 (59) 94
Female 46 (25) 20 (41) 66
60 100 160
36
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

37
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 (1) 48.3, p lt .001.

38
Discussion Section

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

39
Assumptions

• Independence of Observations