1. Introduction to Inferential Statistics

2. Taking out the “loosey-goosey”
So far we’ve assessed relationships between variables two ways:
Categorical variables: tables and proportions (percentages)
Continuous variables: scattergrams and simple correlation (r)
Higher rank 
more stress
Higher income  fancier cars
r = -.6
r2 = .36
Higher income  less crime

3. Inferential statistics
Higher rank  less cynicism?
Alas, results are often ambiguous
Maybe the difference in cynicism between officers and supervisors is slight
Maybe the student lot has better cars than we expected
Maybe the r between age and weight is a small -.17 (r2 of .03)
Can we still confirm or reject the hypotheses?
Inferential statistics (see next slide) are an extension of procedures that we’ve already used
Provide more precise assessments
Tell us the chance - the probability that there is no relationship between variables
This allows us to properly “infer” (project) our results to populations
Cynicism
Rank
Low
High
Officers
75%
25%
100%
Supervisors
81%
19%
Higher income  nicer cars?
DV - Car value
IV - Income
LOW
MED
HIGH %
LOW (student lot)
30%
20%
50%
100%
HIGH (F/S lot)
40%
60%
Age  weight?
r = -.17
r2 = .03

4. Inferential statistics (a.k.a. “test statistics”)
Independent and dependent variables are continuous
Regression (r2 and R2)
b statistic - interpreted as unit change in the DV for each unit change in the IV
Independent variables are nominal or continuous; dependent variable is nominal
Logistic regression, generates “b” and exp(b) (a.k.a. odds ratio)
Independent and dependent variables are categorical
Chi-Square (X2)
Categorical dependent and continuous independent variables
Difference between the means test (t statistic)
Procedure
Level of Measurement
Statistic
Interpretation
Regression
All variables
continuous
r2, R2 b
Proportion of change in the dependent variable accounted for by change in the independent variable.
Unit change in the dependent variable caused by a one-unit change in the independent variable
Logistic
regression
DV nominal & dichotomous,
IV’s nominal or continuous
exp(B)
(odds ratio)
Don’t try - it’s on a logarithmic scale
Odds that DV will change if IV changes one unit, or, if IV is dichotomous, if it changes its state.
Chi-Square
All variables categorical
(nominal or ordinal) X2
Reflects difference between Observed and Expected frequencies.
Difference between means
IV dichotomous, DV continuous t
Reflects magnitude of difference.

5. variance - the “real” relationship
General procedure
Types of hypotheses
Working hypothesis – what a regular hypothesis is called
Null hypothesis – its opposite: the presumption that any apparent relationship between variables is caused by chance.
Draw one or more samples and code the independent and dependent variables
Use a test statistic to assess the working hypothesis
The computer calculates a coefficient for the test statistic (e.g., r2 = .20)
These coefficients are the sum of two components
“Systematic” variance: The actual, “systematic” relationship between variables
“Error” variance: An apparent relationship, caused by sampling error. It shrinks as sample size increases.
Systematic
variance - the “real” relationship
The big question
Once we remove the error component, is enough “real” relationship left to reject the null hypothesis?
Error variance

6. Test statistics and the null hypothesis
To reject the null hypothesis, the test statistic coefficient (e.g., r2 = .20) must be sufficiently large, after subtracting sampling error, to reject the null hypothesis
How much “room” is required? Enough to yield a probability of less than five in one- hundred (< .05) that a relationship between variables was produced by chance.
If so, the relationship between variables is deemed “statistically significant” and the null hypothesis of no relationship is rejected as FALSE.
For significant relationships, one to three asterisks usually appear next to the test statistic coefficient (e.g., .25*, .36**, .41***). More asterisks = greater confidence that a relationship is “systematic” – meaning not caused by chance.
* Probability less than 5 in 100 that a coefficient was produced by chance (p< .05)
** Probability less than 1 in 100 that a coefficient was produced by chance (p< .01)
*** Probability less than 1 in 1,000 that a coefficient was produced by chance (p< )
If the coefficient is too small, no asterisk (*) is awarded. The relationship is non- significant. The working hypothesis is rejected and the null hypothesis is TRUE.
Instead of asterisks, sometimes the actual probability that a coefficient was produced by chance is given, often in a column labeled “p”.
Significant relationships are denoted by p’s less than .05
Good
Better
Best

7. How do we know if the null hypothesis is true?
Null hypothesis: No relationship between variables. Any apparent effect was produced by chance.
You can use the table for your test statistic to check if the coefficient is sufficiently large to reject the null
To reject the null, the test statistic (e.g., R2, t, b, X2) must be so large that the probability the null is true is less than five in one-hundred (< .05)
In your articles, look for asterisks in the tables that depict relationships between variables. If there is no asterisk in the column for the test statistic (here it’s a b) the null for that relationship is true.
Remember - usually one asterisk (*) means the probability the null is true is less than 5 in 100 (p <.05). Two asterisks (**) is better (p <.01, probability the null is true is less than one in 100). Three (***) is great (p <.001, probability the null is true is less than one in 1,000.)
Null hypothesis is true Reject null hypothesis
Chi-Square table
Dependent variable: satisfaction with police

8. A caution on hypothesis testing…
Probabilities (that the null hypothesis is true) are the most common way to evaluate relationships.
The smaller the probability, the more likely that the null hypothesis (meaning, no relationship) is false, meaning that the greater the likelihood that the working hypothesis is true
But this process has been criticized for suggesting misleading results. (Click here for a summary of the arguments.)
We normally use p values to accept or reject null hypotheses. Its real meaning is subtle:
Formally, a p <.05 means that, if an association between variables was tested an infinite number of times, a coefficient as large as the one actually obtained (say, an r2 of .30) would come up less than five times in a hundred if the null hypothesis of no relationship was actually true.
For our purposes, as long as we keep in mind the inherent sloppiness of social science, and the difficulties of accurately quantifying social science phenomena, it’s sufficient to use p-values to accept or reject null hypotheses.
We should always be skeptical of findings of “significance,” particularly when very large samples are involved.
When sample size is large - say, a thousand - even weak relationships can show up as statistically significant. (More on this later.)

9. Examples of tables from articles, panels 1-6

10. 1
Hypothesis: Alcohol consumption  Victimization Method: Logistic regression Statistics: b and Odds Ratio (Exp b)
Richard B. Felson and Keri B. Burchfield, “Alcohol and the Risk of Physical and Sexual Assault Victimization,” Criminology (42:4, 2004)

11. 2
Hypothesis: Race and class  Satisfaction with police Method: Logistic regression Statistics: b and Exp b (odds ratio)
Yuning Wu, Ivan Y. Sun and Ruth A. Triplett, “Race, Class or Neighborhood Context: Which Matters More in Measuring Satisfaction With Police?,” Justice Quarterly (26:1, 2009)

12. 3
Hypothesis: Low self control  More contact with police Method: Logistic regression Statistics: b and Exp b (odds ratio)
Kevin M. Beaver, Matt DeLisi, Daniel P. Mears and Eric Stewart, “Low Self-Control and Contact with the Criminal Justice System in a Nationally Representative Sample of Males,” Justice Quarterly (26:4, 2009)

13. 4
Hypothesis: Gender and race of victim  Imposition of death sentence Method: Logistic regression Statistics: b (“coefficient”) and odds-ratio (Exp b)
Marian R. Williams, Stephen Demuth and Jefferson E. Holcomb, “Understanding the Influence of Victim Gender in Death Penalty Cases: The Importance of Victim Race, Sex-Related Victimization, and Jury Decision Making,” Criminology (45:4, 2007)

14. 5
Hypothesis: Strains of imprisonment  Recidivism Method: Logistic regression Statistics: B and Exp B (odds-ratio)
Shelley Johnson Listwan, Christopher J. Sullivan, Robert Agnew, Francis T. Cullen and Mark Colvin, “The Pains of Imprisonment Revisited: The Impact of Strain on Inmate Recidivism,” Justice Quarterly (30:1, 2013)

15. 6
Hypothesis: Officer and driver race  Vehicle search Method: Logistic regression Statistics: Odds ratio (Exp B) (Stand. Error in parentheses)
Jeff Rojek, Richard Rosenfeld and Scott Decker, “Policing Race: The Racial Stratification of Searches in Police Traffic Stops,” Criminology (50:4, 2012