CRIM 320 Lecture Notes - Null Hypothesis, Joint Probability Distribution, Frequency Distribution

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Crim 320
Week 9
March 5th, Post-Midterm
Association between two categorical variables
Research Questions
- Testing the null hypothesis that two categorical variables (nominal, ordinal) are independent
- Moffitt’s (1993) life-course persistent theory of offending
o Research hypothesis
o H1: early onset offenders are more likely than late-onset offenders to become violent
offenders later on in life
o Null hypothesis
o H0 early onset offenders are not more likely than late-onset offenders to become
violent offenders later on in life
CONTINGENCY TABLE
Definition: The joint frequency distribution of two categorical variables refers to the simultaneous
occurrence of the first variable and another vent form the second variable (Bachman et al, 2004)
2 x 2 Contingency Table
Number of columbs
Number of rows 1
2 rows marginals
Total sample
size
1
A
B
R1
2
C
D
R2
Column marginals C1
C2
n
- Distribution of two categorical variables
- Analysis can only be done on the information which is available for both cases, i.e., if the second
chart has missing information for a number of kids, pay attention to it. Only take valid
percentage into account
- 2x2 able= 4 possible outcomes
o (1) no early onset no fight between 15-18
o (2) early onset no fight between 15-18
o (3) no early onset fight between 15-18
o (4) early onset fight between 15-18
Percentage difference
- A simple way to investigate a relationship between two categorical variables
- For each cell the outcome is divided by the row marginal and multiplied by 100
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- For late onset offenders
o 60.6% did not fight (215/355*100)
o 39.4% did fight (140/355*100)
- For early onset offenders
o 35.3 did not fight (12/34*100)
o 64.7 did fight (22/34*100)
- The percentage difference in prevalence of fighting in late adolescence between the early onset
and late-onset offenders is:
o 25.3% = (64.7% - 39.4%)
o 25% is pretty high but is it statistically significant? Or is it the result of sampling error?
How can we interpret this difference?
- Can we reject the null hypothesis that there is no association between two variables?
o What percentage difference would be expected by chance alone?
o What percentage difference would be large enough to reject the null hypothesis?
- Based on Moffitt’s theories, the variables are expected to be related because early onset
offenders are more likely to be characterized by neuropsychological deficits, which imply low
self control n manifest in committing violent crime when older (fighting between 15-18)
- Not that there is a causation but correlation
CHI-SQUARE TEST OF INDEPENDENCE
- Two-sample chi-square
- Test the null hypothesis that two categorical variables are independent from each other
- Definition: statistical test used for assessing how well the distribution of observed frequencies
of a categorical variable fits the distribution of expected frequencies
Observed frequencies
- Joint distribution of two categorical data in the sample
Expected frequencies
- Joint distribution we would expect if the two categorical data were independent from each
other
Calculating the Expected Frequencies
Formula: Fe = ((CS x RS)/GS) you’d do this for A, B,C, D (GS is the farthest right, and bottom. Same
for all four.)
Fe = expected frequency
CS = Column sum
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