Causality

Principles of Correlation and Causation

• Correlation occurs when the attributes of one variable vary in relation to the attributes of another variable (they covary).

• Any two variables that change together can be correlated; correlation does not inherently mean causation.

• Variables can change together due to:

  • Random chance.

  • Some spurious factor.

• Two correlated variables might not have any true underlying causal relationship with each other.

• A statistically significant correlation is only one criterion required to establish a causal relationship between variables.

• Example of a spurious correlation (Spreading Love and Margarine, tylervigen.com 2024):

  • Fake AI-generated paper title: "Spreading Love and Margarine: An Examination of the Butter-Splitter Correlation in Maine" (Paper URL: https://tylervigen.com/spurious/research-papers/5920_spreading-love-and-margarine-an-examination-of-the-butter-splitter-correlation-in-maine.pdf).

  • Core correlation: Per capita consumption of margarine in the United States correlates with the divorce rate in Maine.

  • Per capita margarine consumption values (Source: US Department of Agriculture): $8.2$, $7.1$, $4.8$, $5.0$, $4.78$, $4.55$, $4.32$, $3.7$.

  • Divorce rate in Maine data source: CDC National Vital Statistics.

  • Timeframe: $2000-2009$

  • Statistical metrics: $r = 0.993$, $r^2 = 0.985$, $p < 0.01$ (URL: tylervigen.com/spurious/correlation/5920).

Spurious, Intervening, and Causal Relationships

• Variable Definitions:

  • Independent Variable: The variable hypothesized to cause or influence changes in another variable.

  • Dependent Variable: The variable that changes as a result of the independent variable.

  • Spurious Variable: A third variable that independently exerts an effect on both the independent variable and the dependent variable.

• Spurious Relationship vs. Nonspurious Relationship:

  • Spurious Relationship: The apparent relationship between an independent variable and a dependent variable is actually driven by a third (spurious) variable.

  • Nonspurious Relationship: A genuine relationship where the causal order flows directly from the independent variable ($X$) to the dependent variable ($Y$), denoted as $X \rightarrow Y$.

• Spurious Variables vs. Intervening Variables:

  • Spurious variables have a separate, independent effect on both the independent variable and the dependent variable.

  • Intervening variables come between the independent variable and the dependent variable in temporal sequence.

  • Types of Intervening Variables:

    • Mediating Variables (Mediator): Help explain the mechanism through which the independent variable influences the dependent variable.

    • Moderating Variables (Moderator): Affect the direction or strength of the relationship between the independent variable and the dependent variable.

Criteria for Nomothetic Causality

• Three mandatory criteria exist for establishing nomothetic causality:

  1. Correlation: Attributes of one variable must be associated with attributes of another variable; they must covary.

  2. Time Order: The independent variable must precede the dependent variable in temporal sequence.

  3. Nonspuriousness: The covariation between the independent variable and dependent variable cannot be driven or explained by some third variable.

Causal Claims, Policy, and Social Consequences

• Case Study: "Broken Windows" Policing:

  • In $1982$, James Q. Wilson and George L. Kelling published "Broken Windows: The Police and Neighborhood Safety."

  • Causal Claim: Signs of physical or social disorder (small crimes) directly lead to further disorder and escalation into major offenses (serious crimes).

  • Policy Implication: Target minor offenses very aggressively to prevent bigger crimes from occurring, utilizing harsh punishments for minor infractions.

  • Media Reference: "BRIC TV: Broken Windows Policing" (URL: https://youtu.be/K5whwqJhZh8).

• Empirical Evaluation in New York City (NYC):

  • Hypothesized Relationship: Small crimes $\rightarrow$ Big crimes.

  • NYC Policing Statistics ($1990s$):

    • $1993 - 1996$: Arrest rates increased by $50\%$.

    • $1994 - 1998$: Misdemeanor arrests rose by $40{,}000$ each year.

  • NYC Crime Decreases ($1990 - 2009$, Fritsch 2016):

    • $82\%$ decrease in homicides.

    • $77\%$ decrease in rapes.

    • $84\%$ decline in robberies.

Statistical Significance and Benchmarks in Sociology

• Fundamentals of Statistical Significance:

  • Correlations, maximum likelihoods, and other statistical measurements are generally not reported in formal research unless they are found to be statistically significant.

  • Statistical procedures are employed to explicitly test null hypotheses.

  • A statistically significant finding means that the observed results are unlikely to be caused by random chance.

  • Statistical significance measures the probability of finding a relationship in sample data when no actual relationship exists in the broader population.

• Standard Benchmarks for Statistical Significance in Sociology:

  • Statistical significance is represented by a $p$-value, which indicates the probability that a result would occur by random chance.

  • $*p \le .05$: The result would happen by random chance $5$ times out of $100$.

  • $**p \le .01$: The result would happen by random chance $1$ time out of $100$.

  • $***p \le .001$: The result would happen by random chance $1$ time out of 1000$.\n * Results with smaller p-values are least likely to happen by random chance, making them the most statistically significant and the most likely to be genuinely driven by the independent variable.\n* Sociological Literature Example Citation:\n * Lerch, J. C., Frank, D. J., & Schofer, E. (2024$). The Social Foundations of Academic Freedom: Heterogeneous Institutions in World Society,$1960$to$2022$. *American Sociological Review*,$89$(1),$88-125$$.

  • DOI: https://doi.org/10.1177/00031224231214000

  • SAGE Journal URL: https://journals.sagepub.com/doi/10.1177/00031224231214000

Null Hypothesis Testing and Type I / Type II Errors

• Error Matrix for Null Hypothesis Decisions:

  • Type I Error:

    • Claim: A relationship exists between variables.

    • Reality: No relationship exists (the Null Hypothesis is True).

    • Definition: Falsely rejecting the Null Hypothesis.

  • Type II Error:

    • Claim: No relationship exists (accepting/failing to reject the Null Hypothesis).

    • Reality: A relationship does exist.

    • Definition: Falsely accepting (or failing to reject) the Null Hypothesis.