Criminology 320 Lecture 7 - Crosstab and Chi Square 2024 PDF

CRIMINOLOGY 320 LECTURE 7: CROSS-TAB TABLES AND CHI-SQUARE TESTS UPCOMING SCHEDULE Nov. 18: Crosstabs & Chi- Square Nov. 25: t-Test & ANOVA Monday, Dec. 2: Correlations Thursday, Dec. 5: Regression INDEPENDENT AND DEPENDENT VARIABLE Dependent Variable (D.V.) The variable we want to e...

CRIMINOLOGY 320 LECTURE 7: CROSS-TAB TABLES AND CHI-SQUARE TESTS UPCOMING SCHEDULE Nov. 18: Crosstabs & Chi- Square Nov. 25: t-Test & ANOVA Monday, Dec. 2: Correlations Thursday, Dec. 5: Regression INDEPENDENT AND DEPENDENT VARIABLE Dependent Variable (D.V.) The variable we want to explain or predict We do not change the dependent variable, but instead just measure how it is affected by the I.V. The D.V. is the effect Independent Variable (I.V.) The variable that we believe causes change in the dependent variable We adjust or change the independent variable to see how it affects the D.V. The I.V. is the cause D.V. AND I.V. EXAMPLE Research Question: Does the amount of crime committed by a teenager vary by gender? D.V. (effect): Crime Rates  We want to explain this I.V. (cause): Gender  We think this might cause change Stated another way… To answer our research question, we might first look at crime rates (D.V.) for teenage males (I.V.), then we would change our I.V. to teenage females, and look at crime rates (D.V.) for teenage females (I.V.) and compare the results! Or yet another way… The Independent Variable can influence the Dependent variable (gender causes changes in crime rate)… … but the Dependent Variable will never influence the Independent Variable (your crime rate will not change your gender!) OVER THE NEXT 4 WEEKS… Chi-Square (Nominal/Ordinal) Two variables (cross-tabulation) t-Tests (Interval/Ratio) Two sample means compared against one another Analysis of Variance (Interval/Ratio) Determines the differences between three or more samples Correlation Determines the degree of relationship between two or more variables Regression (Interval/Ratio) Measures the relationship between one D.V. and multiple I.V.’s LOOKING FORWARD Type of Analysis Descriptive statistics Used to describe the sample Often used to describe a single variable (e.g., mean age) Inferential statistics Used to describe the population Often used to describe the interaction between variables (e.g., correlation between age and crime) Number of Variables Univariate statistics Analysis of one variable Bivariate / multivariate statistics Analysis of two (bivariate) or more (multivariate) variables HOW DO WE SELECT OUR STATISTICAL TEST? What is our research question? What level of measurement is our data? Do we have a normal distribution of scores? PARAMETRIC VS NONPARAMETRIC: IS YOUR DATA NORMAL? Parametric Tests More accurate results Requires interval/ratio level data Requires normal distribution (skew & kurtosis scores within the accepted +/- 1 range) Nonparametric Tests Not as powerful or accurate Does not require normal distribution Can be used with nominal and ordinal data CHI-SQUARE TEST OF SIGNIFICANCE AND CROSS-TABULATIONS BIVARIATE ANALYSIS Association Present? Strength of Association? Direction of Association? Nature of Association? BIVARIATE ANALYSIS: CROSS-TABULATION TABLES Cross-tabs examine the bivariate relationship with simple and easy to read tables Compares the frequencies in an independent variable and a dependent variable Uses the Chi-Square (χ2) test to check for statistical significance Useful for nominal and ordinal level data CROSS-TABULATION TABLES Columns Number of categories or attributes of the Independent Variable (I.V.) Rows Number of categories or attributes of the Dependent Variable (D.V.) Cells Shows the intersection of the I.V. and the D.V. CROSS-TAB EXAMPLE Research Question: Are offenders with schizophrenia more likely to have a prior record that includes a violent offence? I.V. and D.V. I.V. (or cause): Offender with schizophrenia D.V. (or effect): Prior violent offence Possible Outcomes? No schizophrenia – No violent offence record No schizophrenia – Violent offence record Schizophrenia – No violent offence record Schizophrenia – Violent offence record  THE RESULTS… Schizophrenia Total 0 No 1 Yes Prior charge 0 No Count 140 99 239 for a violent % within 42.0% 45.4% 43.4% offence 1 Yes Count 193 119 312 % within 58.0% 54.6% 56.6% Total Count 333 218 551 % within 100% 100% 100% Offenders without schizophrenia were slightly more likely to have a conviction for a violent offence (58.0%) than an offender with schizophrenia (54.6%), however; it should be noted that the difference (3.4%) was very small. ANOTHER EXAMPLE Research Question: Are police more likely to arrest a suspect at the scene if there are children present? I.V. and D.V. I.V. (or cause): Were children present? D.V. (or effect): Was the suspect arrested? Possible Outcomes? No Children – No Arrest No Children – Arrest Children – No Arrest Children - Arrest  AND THE SPSS RESULTS ARE… An arrest occurred substantially more often when children were present (72.1%) than when no children were present (46.4%)  USING CROSS-TABS Cross-Tabs can show if a relationship between variables exist – Look for patterns! Cross-Tabs can also show the strength of that relationship: Generally, anything over 10% is of interest and worthy of a closer look – perhaps through a correlation test Cross-Tabs can show the direction of a relationship if ordinal (must be able to be ranked!) data is used Fewer categories are usually better (this may mean recoding variables to collapse categories)  WHAT DO WE LOOK FOR? Is there a relationship? (10% rule) If so, what is the strength of the relationship? (10%? 20%? 50%?) If the data is ordinal, what is the direction of the relationship?  TESTING FOR SIGNIFICANCE: THE CHI-SQUARE ANALYSIS THE STEPS FOR TESTING A STATISTIC 1. A statement of the null and research hypothesis 2. Set the level of significance, or α (Normally 0.05) 3. Select appropriate test statistic 4. Compare obtained P value and pre-set significance (.05) 5. Decision time! Reject or Accept the Null Hypothesis? 6. Interpret your results  CHI-SQUARE: A NONPARAMETRIC MEASURE OF CORRELATION Nonparametric Parametric tests tests of of significance: significance: Do not assume Assume that the normality in the variable of interest is variable of interest normally distributed Can analyze nominal and ordinal data Assumes the use of But remember that the interval/ratio level data power of the test is limited  WHAT IS CHI-SQUARE? Chi-Square, or a Pearson’s Chi-Square test, is notarized with χ2 A Chi-Square is a simple analysis that tests the null hypothesis (no relationship) Checks if a sample frequency distribution occurs by random chance A Chi-Square test on a single variable is often referred to as a goodness of fit test A Chi-Square test on two variables, usually shown in a cross-tab, is referred to as a test of independence Other Chi-Square tests: Yate’s, Fisher’s, Phi, Cramers V, Cochran-Mantel- Haenszel, McNemar’s, Linear-by-Linear, Time Series Analysis, Likelihood Ratio, etc…  CHI-SQUARE: EXPECTED VERSUS OBSERVED Compares expected values to the actual observed values This relies on the laws of probability of an event occurring Expected Value (fe): If there were no relationship between variables (random chance) Observed Value (fo): What proportion of cases fall in each cell Delta: The difference between the expected and the observed  TWO-SAMPLE CHI-SQUARE TESTS ARE HOCKEY FANS BINGE DRINKERS? How would I answer this research question? With a two-sample chi-square test! TWO-SAMPLE CHI-SQUARE TEST Level of Significance = p