Chi-Square by pengxuebo

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									Chi-Square

   CJ 526 Statistical Analysis in
         Criminal Justice
Parametric vs Nonparametric
   Parametric DV: Interval/Ratio
   Nonparametric DV: nominal/ordinal
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
Null Hypothesis
1.   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)
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
Chi-Square Statistic
   Sum of
   (Observed - Expected)2
       divided by
   Expected
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
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
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 <DV> were
    statistically significant, 2 (df) =
    <value>, p < .05.
Report Writing: Discussion
Section
   It appears as if the <sample> is (or is
    not) distributed as expected.
   Depends on the result
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)
Example
O    E    O-E   (O-E)^2 /E

60   40   20    400     10

40   40   0     0       0

20   40   20    400     10
Chi square
   Chi square = 20
   D.f. = 2
   See p. 726
   Chi square = 20, p < .01
   The distribution is significantly different
    from the expected distribution
Example
   Dr. Zelda, a correctional psychologist, is
    interested in determining whether the
    intelligence of delinquents enrolled in a
    state training school is normally
    distributed
   Distribution of Intelligence in the
   General Population

                                 Percentage of
       IQ Range        Z-score      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)
Distribution of Intelligence in
Dr. Zelda’s School
Below 60               119
60-85                  150
86-100                 687
101-115                 32
116-130                 12
131+                    0
1.   Number of Samples: 1
2.   DV: IQ categories
3.   Target Population: delinquents
     enrolled in the state training school
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
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
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
X2 (5) = 1169, p < .001
Reject H0
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
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 < . 001.
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.
Chi-Square Test for
Independence
   Used to assess the relationship between
    two or more variables
Null Hypothesis
   No relationship between the two
    variables (independent of one another)
     Or
     Alternative: the two variables are related
      to one another
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)
Example
   Dr. Cyrus, a forensic psychologist, is
    interested in determining whether
    gender has an effect on the type of
    sentence that convicted burglars
    receive
Background
1. Number of samples: 1
IV: Gender
DV: Type of sentence received
     1.   Nominal
Target Population: convicted burglars
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
Create a bivariate table
         probation   jail   total

male     14          80     94

female   46          20     66

         60          100    160
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
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
X2 (1) = 48.3, p < .001
Reject H0
         Probation   Jail      Total

Male     14 (35)     80 (59)   94

Female   46 (25)     20 (41)   66

         60          100       160
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
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 < .001.
Discussion Section
   It appears as if gender has an effect on
    the type of sentence received.
Assumptions
   Independence of Observations

								
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