Download RM_Inferential_Stat_Intro

Introduction to
Inferential Statistics
Taking out the “loosey-goosey”
•
So far we’ve assessed relationships between variables two ways:
– Categorical variables: tables and proportions (percentages)
Higher rank 
more stress
Higher income 
fancier cars
– Continuous variables: scattergrams and simple correlation (r)
Higher income 
less crime
r = -.6
r2 = .36
Inferential statistics
•
•
•
•
Alas, results are often less extreme. Maybe 53 percent of officers are high stress and 47
percent are low stress. Maybe there are no faculty/staff cars valued 4 or 5. Maybe the r
between income and crime is only -.2 (r2 of .04). Can we still confirm the hypotheses?
Inferential statistics 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
Examples of inferential statistics (called “test statistics” because they “test” hypotheses)
– Independent and dependent variables are categorical
• Chi-Square (X2)
– Categorical dependent and continuous independent variables
• Difference between the means test (t statistic)
– Independent and dependent variables are continuous
• Regression (r2 and R2)
• b statistic - interpreted as unit change in DV for each unit change in IV
– Independent variables are nominal or continuous; dependent variable is nominal
• Logistic regression, generates uninterpretable “b” and exp(b) (a.k.a. odds ratio)
Requirements
– Must use probability sampling techniques (e.g., random sampling)
– “Parametric” inferential statistics, including r2, b and t
• Variables must be continuous and normally distributed in the population
– Non-parametric statistics
• Variables need not be normally distributed. We’ll cover one – Chi-Square (X2).
Some statistics used to test relationships
Procedure
Level of
Measurement
Statistic
Interpretation
Regression
All variables
continuous
r2, R2
Proportion of change in the dependent variable
accounted for by change in the independent
variable. R2 denotes cumulative effect of multiple
independent variables.
Unit change in the dependent variable caused by
a one-unit change in the independent variable
b
Logistic
regression
DV nominal &
dichotomous,
IV’s nominal or
continuous
b
exp(B)
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.
Range 0 to infinity; 1 denotes even odds, or no
relationship. Higher than 1 means positive
relationship, lower negative relationship. Use
percentage to describe likelihood of effect.
Chi-Square
All variables
categorical
(nominal or
ordinal)
X2
Reflects difference between Observed and
Expected frequencies. Use table to determine if
coefficient is sufficiently large to reject null
hypothesis
Difference
between
means
IV dichotomous, DV
continuous
t
Reflects magnitude of difference. Use table to
determine if coefficient is sufficiently large to
reject null hypothesis.
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
Error
variance
The big question
Once we remove the error
component, is enough “real”
relationship left to reject the
null hypothesis?
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 onehundred (< .05) that the relationship between variables was produced by chance.
– If the computer decides that the coefficient is sufficiently large it will award at least
one asterisk. The relationship between variables is “statistically significant” and the
null hypothesis (no relationship) is FALSE.
– If the coefficient is too small, no asterisk (*) is awarded. The association between
variables is deemed “non-significant” and the null hypothesis is TRUE. Working
hypotheses that depend on this relationship must be rejected.
•
For significant relationships, one to three asterisks usually appear next to the test
statistic’s coefficient (e.g., .25*, .36**, .41***). More asterisks = greater confidence that a
relationship is systematic – not the product of 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< .001)
•
Good
Better
Best
Instead of asterisks, sometimes the actual probability that a coefficient was produced by
chance are given, usually in a column labeled “p”.
– Again, significant relationships are denoted by p’s less than .05
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.)
Examples of tables from
articles, panels 1-12
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)
2
Hypothesis: Veteran status  less punitive police response to domestic violence
Method: Logistic regression Statistics: b and Odds Ratio (Exp b)
Fred Markowitz and Amy C.
Watson, “Police Response to
Domestic Violence Situations
Involving Veterans Exhibiting
Signs of Mental Illness,”
Criminology, (53:2, 2015)
3
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)
4
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)
5
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, SexRelated Victimization, and Jury
Decision Making,” Criminology
(45:4, 2007)
6
Hypothesis: Academic performance  Delinquency
Method: “Tobit” regression* Statistic: b
Richard B. Felson and Jeremy
Staff, “Explaining the
Academic PerformanceDelinquency Relationship,”
Criminology (44:2, 2006)
* Best when the DV for a large proportion of cases has a zero value
7
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)
8
Hypothesis: Father’s incarceration  Son’s delinquency
Method: Tobit regression Statistic: Random effect coefficient (S.E. in parentheses)
Michael E. Roettger and Raymond R. Swisher, “Associations of
Fathers’ History of Incarceration With Sons’ Delinquency and Arrest
Among Black, White and Hispanic Males in the United States,”
Criminology (49:4, 2011)
8
Hypothesis: Father’s incarceration  Son’s delinquency
Method: Logistic regression Statistic: Odds ratio (Standard Error in parentheses)
Michael E. Roettger and Raymond R. Swisher, “Associations of
Fathers’ History of Incarceration With Sons’ Delinquency and Arrest
Among Black, White and Hispanic Males in the United States,”
Criminology (49:4, 2011)
9
Hypothesis: Officer and driver race  Vehicle search
Method: Logistic regression Statistics: Odds ratio (Standard 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
10
Hypothesis: Offender race & gender  Use of intermediate sanctions
Method: Logistic regression Statistics: b and Exp b (odds ratio)
Brian D. Johnson and Stephanie M. Dipietro, “The Power of
Diversion: Intermediate Sanctions and Sentencing Disparity
Under Presumptive Guidelines,” Criminology (50:3, 2012)
11
Hypothesis: Race & ethnicity  Prosecution and sentencing outcomes
Method: Logistic regression Statistic: Odds ratio (Exp b)
Besiki L. Kutateladze, Nancy R. Andiloro, Brian D. Johnson and
Cassia C. Spohn, “Cumulative Disadvantage: Examining Racial and Ethic
Disparity in Prosecution and Sentencing,” Criminology (52:3, 2014)
12
Hypothesis: Marriage  Desistance from crime
Method: HLM (like logistic regression) Statistics: b (Coeff.) [Can compute log odds)
Bianca E. Bersani and Elaine Eggleston
Doherty, “When the Ties That Bind
Unwind: Examining the Enduring and
Situational Processes of Change Behind
the Marriage Effect,” Criminology (51:2,
2013)