SOCI 620: Quantitative methods 2

Agenda

Covariates for causal analysis

  1. Administrative
  2. Interacting variables in
    regression
  3. Causal analysis in
    regression
  4. Mediation, moderation,
    confounding, and collision
  5. Hands on:
    Building indicator
    variables in R

Slides are licensed under CC BY-NC-SA 4.0

Predicting
income:
interactions

Top-down photo of a cluster of four office desks, all facing in different directions. Three of the four desks are unoccupied.

Dummy variables


Indicator variable for male


Indicator variable for respondents at least 35 years old


Mean		 Std. dev
2.5%
97.5%

10.16 0.03 10.10 10.21

0.13 0.03 0.07 0.19

0.59 0.03 0.54 0.65

0.81 0.01 0.79 0.83

:
(men make about 14% more than women, on average)

:
(people at least 35 years old make about 81% more than those under 35, on average)

Interacting dummies


Interaction between both indicators


Mean		 Std. dev
2.5%
97.5%

10.20 0.03 10.13 10.27

0.05 0.05 -0.04 0.14

0.52 0.04 0.44 0.61

0.13 0.06 0.02 0.25

0.81 0.01 0.79 0.83
two crash test dummies sitting in the front seat of a car canoodling

Interacting dummies

Interacting dummies

Mean Exp(Mean)

10.20 26,853

0.05 1.05

0.52 1.69

0.13 1.14

Interpreting the interaction coefficient

The pay boost for being over 35 years old () is about 14% greater for men than for women.

OR

The income advantage for men over women () is about 14% greater for respondents over 35 years old.

Interacting continuous variables

Standardization:

Transforming a variable so that
and

Interacting continuous variables

Mean Exp(Mean)

10.58 39,549

0.40 1.49

0.20 1.22

-0.07 0.93

Interpreting the interaction coefficient

The extra income associated with increased union and professional dues () is reduced by about 7% for every standard-deviation increase in hours worked.

OR

The extra money made by working more hours () is reduced by about 7% for every standard-deviation increase in union dues paid.


Identifying cause & effect

Wile E Coyote stands on a stretched cord holding an anvil. He looks with panicked concern at Road Runner who stands next to him, sticking out their tongue.

Causal analysis

Causal question:

Does a change in one variable (X)
cause a change in another (Y)?

Regression only identifies statistical relationships, not causal relationships

To draw a “causal arrow” you need theory

Causal analysis

To establish a causal relationship you (usually) need

  1. Causal precedence
    A theoretical reason to believe that changes in X could affect Y (e.g. precedes Y in time)
  2. Statistical association
    An established statistical association between X and Y (e.g. a convincing coefficient estimate)
  3. No unaccounted-for confounders
    No other variables, observed or otherwise, that confound the association between X and Y

Confounding variables

A variable Z is a confounder of the relationship between X and Y if Z is a causal influence on both X and Y

Confounding variables

A variable Z is a confounder of the relationship between X and Y if Z is a causal influence on both X and Y



For example:
To establish a causal relationship between education and income, you need to account for race, which could affect both education and income

Types of covariates

Confounder

Z is a causal factor on both X and Y.

Must be “controlled for” to establish non-spurious relationship between X and Y.

E.g.:
Race confounds the relationship between education and income.

Mediator

Z is influenced by X and influences Y.

Including as covariate elaborates on relationship between X and Y.

E.g.:
Occupation mediates the relationship between gender and income.

Moderator

Z alters the relationship between X and Y.

Can be included as interaction variable to better describe the relationship between X and Y.

E.g.:
Marital status moderates the relationship between gender and income.

Collider

Z causally influenced by both X and Y.

Must not be “controlled for” when establishing relationship between X and Y.

E.g.:
Income is a collider for the relationship between gender and occupation.

Collider bias: an example

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Do college students who come from privileged backgrounds dedicate less effort to studying?

Collider bias: an example

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Do college students who come from privileged backgrounds dedicate less effort to studying?

Collider bias: an example

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Do college students who come from privileged backgrounds dedicate less effort to studying?

Collider bias: an example

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Do college students who come from privileged backgrounds dedicate less effort to studying?

Collider bias: an example

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Do college students who come from privileged backgrounds dedicate less effort to studying?

Image credit

Figures by Peter McMahan (source code)

Top-down photo of a cluster of four office desks, all facing in different directions. Three of the four desks are unoccupied.

Still from Severance (2022)

Wile E Coyote stands on a stretched cord holding an anvil. He looks with panicked concern at Road Runner who stands next to him, sticking out their tongue.

Road Runner and Wile E. Coyote © Warner Bros. Entertainment

A young woman seated in an auditorium looks perplexed as she takes notes in a spiral notebook.

Still from Gilmore Girls (2000)

Point out that the horizontal axis isn't from zero