Causal Inference: Confounding Variables and Omitted Variable Bias

The Scientific Method and Causal Inference

• The scientific method follows an iterative cycle designed to establish causal claims:

  • Theory: An overarching conceptual framework that provides an explanation for political or social phenomena.

  • Hypothesis: A concrete, falsifiable statement derived logically from the theory, stating an expected causal relationship between specific concepts.

  • Empirical Analysis: The systematic testing of hypotheses against observational or experimental data.

  • Evaluation of Hypothesis: Assessment of empirical results to determine whether evidence supports or rejects the hypothesis.

  • Evaluation of Theory: The cumulative results of hypothesis testing feed back to confirm, refine, alter, or falsify the broader theoretical model.

• Identifying causality empirically faces two primary obstacles:

  • Confounding Variables: Extraneous factors that jointly cause changes in both the explanatory variable and the outcome variable.

  • Reverse Causality: Situations in which the outcome variable exerts a reciprocal or backwards causal influence on the explanatory variable.

Confounding Variables and Causal Graphs

• In a basic causal setup, an empirical analysis seeks to evaluate whether an independent variable $X$ causally influences a dependent variable $Y$ ($X \rightarrow Y$).

• A third variable $Z$ is defined as a confounding variable (or confounder) if it satisfies two conditions simultaneously:

  • $Z$ causally affects $Y$ ($Z \rightarrow Y$).

  • $Z$ causally affects $X$ ($Z \rightarrow X$).

• Real-World Political Science Application:

  • Independent Variable ($X$): College Education.

  • Dependent Variable ($Y$): Political Participation.

  • Confounders ($Z$): Factors that determine whether an individual attends college while also determining their baseline rate of political participation.

  • Key confounding variables include family income, parents' level of education, and childhood neighbourhood environment.

Spurious Relationships

• A statistical correlation between $X$ and $Y$ in observational data is defined as a spurious relationship when that correlation disappears completely once the confounding variable $Z$ is controlled for or accounted for.

• Case Study: Elementary School Academic Achievement:

  • Research Goal: Determining what characteristics improve student academic achievement.

  • Sample Population: Elementary school children spanning Grade $4$, Grade $5$, and Grade $6$.

  • Dependent Variable ($Y$): Standardized achievement test scores in Math and English, measured on a scale from $0$ to $100$.

  • Candidate Independent Variable ($X$): Student physical height, measured in centimeters from $120\,cm$ to $180\,cm$.

• Bivariate Observation versus Controlled Reality:

  • A naive bivariate regression plot indicates a strong, positive relationship between student height and math test scores (indicated by a steeply sloped dashed blue line).

  • When data are disaggregated by grade level (Grade $4$, Grade $5$, and Grade $6$), the regression lines within every individual grade become completely horizontal (slopes equal to $0$).

  • Within Grade $4$, student test scores cluster around $20$ to $30$ across all heights ($125\,cm$ to $160\,cm$).

  • Within Grade $5$, student test scores cluster around $45$ to $55$ across all heights ($130\,cm$ to $170\,cm$).

  • Within Grade $6$, student test scores cluster around $70$ to $80$ across all heights ($145\,cm$ to $180\,cm$).

  • The initial correlation between height and test scores is entirely spurious; height has zero causal effect on academic achievement.

  • The confounding variable is the student's Grade ($Z$):

  • Advancing through grades causes an increase in student physical height ($Z \rightarrow X$).

  • Advancing through grades exposes students to more advanced instruction, causing higher test scores ($Z \rightarrow Y$).

Mathematical Framework of Omitted Variable Bias (OVB)

• When an empirical research design fails to control for a confounding variable $Z$, the estimated effect conflates the true causal relationship with systematic bias.

• Definitions of Structural Parameters:

  • $\beta$: The true causal effect of $X$ on $Y$.

  • $\beta > 0$: True positive causal relationship.

  • $\beta < 0$: True negative causal relationship.

  • $\beta = 0$: No causal relationship exists between $X$ and $Y$.

  • $\gamma$: The causal relationship between the confounder $Z$ and the dependent variable $Y$ ($Z \rightarrow Y$).

  • $\gamma > 0$: Positive association ($Z \uparrow \implies Y \uparrow$).

  • $\gamma < 0$: Negative association ($Z \uparrow \implies Y \downarrow$).

  • $\alpha$: The causal relationship between the confounder $Z$ and the independent variable $X$ ($Z \rightarrow X$).

  • $\alpha > 0$: Positive association ($Z \uparrow \implies X \uparrow$).

  • $\alpha < 0$: Negative association ($Z \uparrow \implies X \downarrow$).

  • $\beta^*$: The apparent, unadjusted relationship between $X$ and $Y$ that appears in empirical analysis when $Z$ is omitted.

• The Fundamental Omitted Variable Bias Equation:   $\beta^* = \beta + \text{bias}$   $\text{bias} = \alpha \times \gamma$   $\beta^* = \beta + (\alpha \times \gamma)$

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Directional Scenarios of Spuriousness and Bias

• Mechanisms Producing Spurious Relationships ($\beta = 0$):

  • Case A ($\alpha > 0$, $\gamma > 0$):

  • When $Z$ rises, it causes $X$ to rise and $Y$ to rise.

  • Both $X$ and $Y$ move upward together due to $Z$.

  • Omitting $Z$ yields a positive spurious relationship: $\beta^* > 0$.

  • Case B ($\alpha < 0$, $\gamma < 0$):

  • When $Z$ rises, it causes both $X$ and $Y$ to decrease.

  • When $Z$ falls, it causes both $X$ and $Y$ to increase.

  • Because $X$ and $Y$ co-move in the same direction, omitting $Z$ yields a positive spurious relationship: $\beta^* > 0$.

  • Mathematically: $\text{bias} = (-)(-) = (+)$.

  • Case C ($\alpha > 0$, $\gamma < 0$ or $\alpha < 0$, $\gamma > 0$):

  • Increases in $Z$ push $X$ upward but pull $Y$ downward.

  • $X$ and $Y$ move in opposite directions as $Z$ fluctuates.

  • Omitting $Z$ yields a negative spurious relationship: $\beta^* < 0$.

  • Mathematically: $\text{bias} = (+)(-) = (-)$.

• Consequences When a True Relationship Exists ($\beta \neq 0$):

  • Omitting a confounder from empirical models produces three potential distortions:

  1. Overestimation: The estimated relationship is systematically inflated above its true value ($\beta^* > \beta$ for positive effects).

  2. Underestimation: The estimated relationship is attenuated or smaller in magnitude than the true value ($\beta^* < \beta$).

  3. Sign Reversal: The sign of the estimated relationship completely flips relative to the true effect (e.g., a beneficial causal factor appears harmful, or vice versa).

Case Study: Overestimation via Omitted Variable Bias

• Theoretical Model of Overestimation:

  • True positive causal effect: $\beta > 0$.

  • Confounder parameters: $\alpha > 0$ and $\gamma > 0$.

  • Bias structure: $\text{bias} = \alpha \times \gamma > 0$.

  • Resulting estimate: $\beta^* = \beta + \text{bias} \implies \beta^* > \beta$.

  • The naive analysis attributes the entire variation in $Y$ to $X$, inflating the perceived magnitude of the causal effect.

• Empirical Example: Homework Hours and Math Achievement:

  • Independent Variable ($X$): Number of hours spent weekly on homework (Math & English), spanning from $2$ to $10$ hours.

  • Dependent Variable ($Y$): Math test score ($0$ to $100$ points).

  • Confounding Variable ($Z$): School grade level (Grade $4$, Grade $5$, Grade $6$).

  • Causal paths:

  • Higher grades receive more assigned homework: $\alpha > 0$.

  • Students in higher grades attain higher math test scores due to advanced curriculum: $\gamma > 0$.

  • Within each grade level, extra hours of homework moderately improve math test scores: $\beta > 0$ (slopes of red lines within grades are positive).

  • Empirical consequence of omitting Grade ($Z$):

  • When Grade is omitted, the pooled regression line (dashed blue line) cuts across all three clusters.

  • The resulting estimate $\beta^*$ is substantially steeper than the actual within-grade causal slopes $\beta$.

  • The positive effect of homework on test scores is severely overestimated due to positive omitted variable bias ($\beta^* > \beta$).

Questions & Discussion

• Confounders of Education and Political Participation:

  • Question: Which one of the following is a potential confounding variable (or a confounder) for the relationship between College Education and Political Participation?

  • A: Family income

  • B: Parents' education

  • C: Neighbourhood

  • D: All of the above

  • Answer: D (All of the above).

  • Explanation: Family income, parents' level of education, and neighbourhood conditions simultaneously impact a person's probability of attending college and their baseline political socialization and propensity to participate in civic activities.

• Plausibility of Causality in Student Height:

  • Question: Do you think the relationship between student height and math test scores is a causal relationship?

  • A: Yes

  • B: No

  • Answer: B (No).

  • Explanation: There is no plausible biological or cognitive mechanism whereby greater physical height increases mathematical ability. The observed correlation is purely a byproduct of age and grade-level progression.

• Spurious Sign under Two Positive Relationships:

  • Question: Suppose there is no relationship between $X$ and $Y$ ($\beta = 0$). What would be the sign of the spurious relationship ($\beta^*$) if we omit $Z$, given $\alpha > 0$ and $\gamma > 0$?

  • A: $\beta^* > 0$ (Positive)

  • B: $\beta^* < 0$ (Negative)

  • C: $\beta^* = 0$ (No relationship)

  • Answer: A ($\beta^* > 0$: Positive).

  • Explanation: Since $Z$ increases both $X$ and $Y$, omitting $Z$ makes $X$ and $Y$ appear to increase together, producing a positive spurious parameter: $\beta^* = 0 + (\alpha \times \gamma) > 0$.

• Spurious Sign under Two Negative Relationships:

  • Question: Suppose there is no relationship between $X$ and $Y$ ($\beta = 0$). What would be the sign of the spurious relationship ($\beta^*$) if we omit $Z$, given $\alpha < 0$ and $\gamma < 0$?

  • A: $\beta^* > 0$ (Positive)

  • B: $\beta^* < 0$ (Negative)

  • C: $\beta^* = 0$ (No relationship)

  • Answer: A ($\beta^* > 0$: Positive).

  • Explanation: As $Z$ rises, both $X$ and $Y$ fall; as $Z$ falls, both $X$ and $Y$ rise. Because they move in the exact same direction, their correlation is positive. The product of two negative coefficients is positive ($(-)\times(-) = (+)$), yielding $\beta^* > 0$.

• Spurious Sign under Diverging Relationships:

  • Question: Suppose there is no relationship between $X$ and $Y$ ($\beta = 0$). What would be the sign of the spurious relationship ($\beta^*$) if we omit $Z$, given $\alpha > 0$ and $\gamma < 0$?

  • A: $\beta^* > 0$ (Positive)

  • B: $\beta^* < 0$ (Negative)

  • C: $\beta^* = 0$ (No relationship)

  • Answer: B ($\beta^* < 0$: Negative).

  • Explanation: An increase in $Z$ increases $X$ but decreases $Y$. As a consequence, higher values of $X$ are matched with lower values of $Y$, generating a negative spurious slope: $\beta^* = 0 + (\alpha \times \gamma) < 0$.

• Magnitude of Estimate in Positive Confounding:

  • Question: Suppose there is a positive relationship between $X$ and $Y$ ($\beta > 0$), and $Z$ has positive relationships with both $X$ ($\alpha > 0$) and $Y$ ($\gamma > 0$). If we omit $Z$, would the relationship between $X$ and $Y$ ($\beta^*$) be greater or smaller than its true relationship ($\beta$)?

  • A: $\beta^* > \beta$

  • B: $\beta^* < \beta$

  • Answer: A ($\beta^* > \beta$).

  • Explanation: With $\text{bias} = \alpha \times \gamma > 0$, omitting $Z$ adds positive bias to the true parameter: $\beta^* = \beta + \text{bias}$, leading directly to an overestimation ($\beta^* > \beta$).

(Note: While the theoretical framework, variable relationships, and mathematical omitted variable bias model are grounded in the lecture notes and Source 1, the specific operationalization details, data collection design, and reverse causality assessment extend adjacent to and beyond what the notes explicitly cover to fulfill the tutorial prompt.)

1. Dependent and Independent Variables

1. Phenomenon of Interest:
      Political participation, specifically democratic and civic engagement among citizens.

2. Potential Causes of Political Participation:

   • College education (attainment of higher education)

   • Parents' level of education (family educational background)

   • Household income / family wealth

   • Childhood neighborhood environment

   • Political interest and youth civic socialization

3. Selected Independent Variable ($X$):
      College education.

2. Causal Theory and Hypothesis

1. Causal Theory and Mechanism:

   • (a) Expected Direction of Effect: Attaining a college education increases an individual's level of political participation ($X \rightarrow Y$). As an individual's educational attainment rises, their rate and intensity of political engagement increase.

   • (b) Causal Mechanism: Higher education provides individuals with cognitive and communication skills—such as information processing, critical reading, and public speaking—which lower the informational and bureaucratic costs of understanding politics. Furthermore, college campuses serve as environments of civic socialization, connecting students to politically active peers, student organizations, and civic networks that build personal political efficacy (the belief that one's participation can influence political outcomes).

2. Derivation of Causal Hypothesis:

   • (a) Unit of Analysis: The individual citizen.

   • (b) Data Used: Cross-sectional survey data collected from a nationally representative random sample of voting-age adults (e.g., the Canadian Election Study or American National Election Studies).

   • (c) Operationalization of the Dependent Variable ($Y$): Voter turnout in the most recent national election, quantified as a binary variable where $Y = 1$ if the respondent voted and $Y = 0$ if the respondent did not vote.

   • (d) Operationalization of the Independent Variable ($X$): College degree completion, quantified as a binary indicator where $X = 1$ if the individual completed a bachelor's degree or higher, and $X = 0$ if the individual did not complete a bachelor's degree.

   • (e) Operationalized Causal Hypothesis: Individuals who have completed a college degree ($X = 1$) are significantly more likely to vote in national elections ($Y = 1$) than individuals who have not completed a college degree ($X = 0$).

3. Confounding Variables

1. Identification of Confounder ($Z$):
      Parents' level of education is a major confounding variable because it simultaneously causes changes in both the explanatory variable and the outcome variable:

   • Effect on Independent Variable ($Z \rightarrow X$): Highly educated parents generally possess higher socioeconomic status, place a greater premium on academic achievement, and provide greater resources to ensure their children attend and graduate from college ($\beta_{ZX} = \text{path } \rightarrow \text{positive}$ or $\bar{\alpha} > 0$).

   • Effect on Dependent Variable ($Z \rightarrow Y$): Highly educated parents also socialize their children into civic life from a young age through frequent discussions about politics at home, exposure to the news, and direct role-modeling of voting behavior, raising their baseline propensity to participate politically regardless of the child's own degree attainment ($\bar{\gamma} > 0$).

2. Analysis of Bias from Omitting $Z$ (Diagram-Based Method):

   • Structural parameters from the lecture model:

     • Let $\beta$ be the true causal effect of college education ($X$) on political participation ($Y$), where $\beta > 0$.

     • Let $\alpha$ be the relationship between the confounder ($Z$) and the independent variable ($X$), where $\alpha > 0$.

     • Let $\gamma$ be the relationship between the confounder ($Z$) and the dependent variable ($Y$), where $\gamma > 0$.

     • Let $\beta^*$ be the estimated relationship between $X$ and $Y$ when $Z$ is omitted.

   • The omitted variable bias equation is:      $\text{bias} = \alpha \times \gamma$$\beta^* = \beta + \text{bias}$$\beta^* = \beta + (\alpha \times \gamma)$

   • Direction of bias:      $\text{bias} = (+) \times (+) > 0$$\beta^* > \beta$

   • Substantive Consequence: Omitting parents' education causes positive omitted variable bias, resulting in an overestimation of the causal effect of college education. The naive analysis mistakenly credits college education with civic engagement that was actually generated by parental socialization, making college appear more impactful than it truly is.

4. Reverse Causality

• Assessment: There is little plausible possibility of reverse causality in this empirical setup.

• Explanation: Reverse causality requires the dependent variable to exert a backwards causal influence on the independent variable ($Y \rightarrow X$). In this design, clear temporal ordering rules out reverse causation: college degree attainment ($X$) occurs in early adulthood (typically between ages $18$ and $22$), whereas adult political participation ($Y$) takes place in elections held after graduation (e.g., ages $25$, $30$, or older). An individual's decision to cast a ballot in an election at age $30$ cannot travel backward in time to cause their graduation from college a decade earlier. While early political interest in high school could lead someone to choose a political science major, that represents prior civic disposition (another confounding variable $Z$), not adult voting behavior ($Y$) causing past college completion ($X$).