Causal InferencePSYC 573University of Southern CaliforniaMarch 29, 20221 / 35

Causation

Data are profoundly dumb about causal relationships

--- Pearl & Mackenzie (2018)

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Outline:

DAG
confounding
$d$ -separation
mediation

Materials based on chapters 5 and 6 of McElreath (2020)

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Causal Inference

Obtaining an estimate of the causal effect of one variable on another

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Causal Inference

Obtaining an estimate of the causal effect of one variable on another

an hour more exercise per day causes an increase in happiness by 0.1 to 0.2 points

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Causal Inference

Obtaining an estimate of the causal effect of one variable on another

an hour more exercise per day causes an increase in happiness by 0.1 to 0.2 points

Intervention: if I exercise one hour more, my happiness will increase by 0.1 to 0.2 points
Counterfactual: had I exercised one less hour, my happiness would have been 0.1 to 0.2 points less

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Directed Acyclic Graph

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Data from the 2009 American Community Survey (ACS)

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Data from the 2009 American Community Survey (ACS)

Does marriage cause divorce? (pay attention to the unit of analysis)

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Age at marriage?

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Directed Acyclic Graph (DAG)

Allows researchers to encode causal assumptions of the data

Based on knowledge of the data and the variables

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Directed Acyclic Graph (DAG)

Allows researchers to encode causal assumptions of the data

Based on knowledge of the data and the variables

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"Weak" assumptions

A may directly influence M
A may directly influence D
M may directly influence D

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"Weak" assumptions

A may directly influence M
A may directly influence D
M may directly influence D

"Strong" assumptions: things not shown in the graph

E.g., M does not directly influence A
E.g., A is the only relevant variable in the causal pathway M → D

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Basic Types of Junctions

Fork: A ← B → C

Chain/Pipe: A → B → C

Collider: A → B ← C

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Fork

aka Classic confounding

Confound: something that misleads us about a causal influence

M ← A → D

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Fork

aka Classic confounding

Confound: something that misleads us about a causal influence

M ← A → D

Assuming the DAG is correct,

the causal effect of M → D can be obtained by holding constant A
- stratifying by A; "controlling" for A

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$\begin{aligned} D_{i} & \sim N (μ_{i}, σ) \\ μ_{i} & = β_{0} + β_{1} A_{i} + β_{2} M_{i} \\ β_{0} & \sim N (0, 5) \\ β_{1} & \sim N (0, 1) \\ β_{2} & \sim N (0, 1) \\ σ & \sim t_{4}^{+} (0, 3) \end{aligned}$

library(brms)
m1 <- brm(Divorce ~ MedianAgeMarriage + Marriage,
  data = waffle_divorce,
  prior = prior(std_normal(), class = "b") +
    prior(normal(0, 5), class = "Intercept") +
    prior(student_t(4, 0, 3), class = "sigma"),
  seed = 941,
  iter = 4000
)

>#  Family: gaussian 
>#   Links: mu = identity; sigma = identity 
># Formula: Divorce ~ MedianAgeMarriage + Marriage 
>#    Data: waffle_divorce (Number of observations: 50) 
>#   Draws: 4 chains, each with iter = 4000; warmup = 2000; thin = 1;
>#          total post-warmup draws = 8000
># 
># Population-Level Effects: 
>#                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
># Intercept             3.49      0.77     1.96     4.99 1.00     5179     5008
># MedianAgeMarriage    -0.94      0.25    -1.42    -0.44 1.00     5605     5608
># Marriage             -0.04      0.08    -0.20     0.12 1.00     5198     4900
># 
># Family Specific Parameters: 
>#       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
># sigma     0.15      0.02     0.12     0.19 1.00     6071     5326
># 
># Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
># and Tail_ESS are effective sample size measures, and Rhat is the potential
># scale reduction factor on split chains (at convergence, Rhat = 1).

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Posterior predictive checks

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Pedicting an Intervention

What would happen to the divorce rate if we encourage more people to get married, so that marriage rate increases by 1 per 10 adults?

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Pedicting an Intervention

What would happen to the divorce rate if we encourage more people to get married, so that marriage rate increases by 1 per 10 adults?

Based on our DAG, this should not change the median marriage age

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Pedicting an Intervention

What would happen to the divorce rate if we encourage more people to get married, so that marriage rate increases by 1 per 10 adults?

Based on our DAG, this should not change the median marriage age

Marriage	MedianAgeMarriage	Estimate	Est.Error	Q2.5	Q97.5
2	2.5	1.07	0.034	0.999	1.14
3	2.5	1.03	0.068	0.894	1.16

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Randomization15 / 35

Framing ExperimentX: exposure to a negatively framed news story about immigrants
Y: anti-immigration political action
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Framing Experiment

X: exposure to a negatively framed news story about immigrants
Y: anti-immigration political action

No Randomization

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Potential confound:

Location
Usual outlet/source to acquire information

Framing Experiment

X: exposure to a negatively framed news story about immigrants
Y: anti-immigration political action

No Randomization

Randomization

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Potential confound:

Location
Usual outlet/source to acquire information

Back-Door Criterion

The causal effect of X → Y can be obtained by blocking all the backdoor paths that do not involve descendants of X

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Back-Door Criterion

The causal effect of X → Y can be obtained by blocking all the backdoor paths that do not involve descendants of X

Randomization: (when done successfully) eliminates all paths entering X
Conditioning (holding constant)

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Dagitty

library(dagitty)
dag4 <- dagitty("dag{
  X -> Y; W1 -> X; U -> W2; W2 -> X; W1 -> Y; U -> Y
}")
latents(dag4) <- "U"
adjustmentSets(dag4, exposure = "X", outcome = "Y",
               effect = "direct")

># { W1, W2 }

impliedConditionalIndependencies(dag4)

># W1 _||_ W2

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Post-Treatment Bias19 / 35

Data for Framing Experiment

cong_mesg: binary variable indicating whether or not the participant agreed to send a letter about immigration policy to his or her member of Congress
emo: post-test anxiety about increased immigration (0-9)
tone: framing of news story (0 = positive, 1 = negative)

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Results
 
    No adjustment 
    Adjusting for feeling 
  
    b_Intercept 
    −0.81 [−1.18, −0.45] 
    −2.01 [−2.60, −1.40] 
  
    b_tone 
    0.22 [−0.29, 0.74] 
    −0.14 [−0.71, 0.42] 
  
    b_emo 
     
    0.32 [0.21, 0.43] 
  
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Negative framing: emphasizing costs Positive framing: emphasizing benefits

Results

	No adjustment	Adjusting for feeling
b_Intercept	−0.81 [−1.18, −0.45]	−2.01 [−2.60, −1.40]
b_tone	0.22 [−0.29, 0.74]	−0.14 [−0.71, 0.42]
b_emo		0.32 [0.21, 0.43]

Which one estimates the causal effect?

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Negative framing: emphasizing costs Positive framing: emphasizing benefits

Mediation

In the DAG, E is a post-treatment variable potentially influenced by T

E is a potential mediator

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Mediation

In the DAG, E is a post-treatment variable potentially influenced by T

E is a potential mediator

A mediator is very different from a confounder

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Mediation Analysis

$\begin{aligned} {emo}_{i} & \sim N (μ_{i}^{e}, σ) \\ μ_{i}^{e} & = β_{0}^{e} + β_{1} {tone}_{i} \\ {cong_mesg}_{i} & \sim B e r n (μ_{i}^{c}, σ^{c}) \\ l o g i t (μ_{i}^{c}) & = η_{i} \\ η_{i} & = β_{0}^{c} + β_{2} {tone}_{i} + β_{3} {emo}_{i} \\ β_{0}^{e}, β_{0}^{c} & \sim N (0, 5) \\ β_{1}, β_{2}, β_{3} & \sim N (0, 1) \\ σ & \sim t_{4}^{+} (0, 3) \end{aligned}$

m_med <- brm(
  # Two equations for two outcomes
  bf(cong_mesg ~ tone + emo) +
    bf(emo ~ tone) +
    set_rescor(FALSE),
  data = framing,
  seed = 1338,
  iter = 4000,
  family = list(bernoulli("logit"), gaussian("identity"))
)

>#  Family: MV(bernoulli, gaussian) 
>#   Links: mu = logit
>#          mu = identity; sigma = identity 
># Formula: cong_mesg ~ tone + emo 
>#          emo ~ tone 
>#    Data: framing (Number of observations: 265) 
>#   Draws: 4 chains, each with iter = 4000; warmup = 2000; thin = 1;
>#          total post-warmup draws = 8000
># 
># Population-Level Effects: 
>#                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
># congmesg_Intercept    -2.01      0.30    -2.60    -1.42 1.00     9742     6632
># emo_Intercept          3.40      0.24     2.93     3.86 1.00    10684     6756
># congmesg_tone         -0.15      0.29    -0.73     0.41 1.00     9449     6097
># congmesg_emo           0.32      0.06     0.21     0.43 1.00     9514     6710
># emo_tone               1.14      0.33     0.47     1.79 1.00    10417     5856
># 
># Family Specific Parameters: 
>#           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
># sigma_emo     2.73      0.12     2.51     2.98 1.00    10496     6553
># 
># Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
># and Tail_ESS are effective sample size measures, and Rhat is the potential
># scale reduction factor on split chains (at convergence, Rhat = 1).

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Direct Effect

Causal effect when holding mediator at a specific level

cond_df <- data.frame(tone = c(0, 1, 0, 1),
                      emo = c(0, 0, 9, 9))
cond_df %>%
  bind_cols(
    fitted(m_med, newdata = cond_df)[ , , "congmesg"]
  ) %>%
  knitr::kable()

tone	emo	Estimate	Est.Error	Q2.5	Q97.5
0	0	0.122	0.032	0.069	0.195
1	0	0.108	0.033	0.054	0.183
0	9	0.699	0.071	0.549	0.826
1	9	0.669	0.063	0.539	0.786

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Indirect Effect

Change in $Y$ of the control group if their mediator level changes to what the treatment group would have obtained

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Indirect Effect

Change in $Y$ of the control group if their mediator level changes to what the treatment group would have obtained

Quick Demo using posterior means¹

T = 0, E(M) = 3.39
T = 1, E(M) = 3.39 + 1.14 = 4.53

[1]: Fully Bayesian analyses in the note

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Indirect Effect

Change in $Y$ of the control group if their mediator level changes to what the treatment group would have obtained

Quick Demo using posterior means¹

T = 0, E(M) = 3.39
T = 1, E(M) = 3.39 + 1.14 = 4.53

[1]: Fully Bayesian analyses in the note

tone	emo	Estimate	Est.Error	Q2.5	Q97.5
0	3.39	0.286	0.042	0.208	0.372
0	4.53	0.365	0.048	0.275	0.462

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Potential Confounding

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Potential Confounding

Maybe age is related to both emo and cong_mesg?

m_med2 <- brm(
  # Two equations for two outcomes
  bf(cong_mesg ~ tone + emo + age) +
    bf(emo ~ tone + age) +
    set_rescor(FALSE),
  data = framing,
  seed = 1338,
  iter = 4000,
  family = list(bernoulli("logit"),
                gaussian("identity"))
)

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Unobserved Confounding

Can be incorporated by assigning priors to the unobserved confounding paths

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Collider Bias28 / 35

E.g., Is the most newsworthy research the least trustworthy?

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Conditioning on a collider creates spurious associationsnice person → date ← good-looking person
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Conditioning on a collider creates spurious associations

nice person → date ← good-looking person
impulsivity → high-risk youth ← delinquency

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Conditioning on a collider creates spurious associations

nice person → date ← good-looking person
impulsivity → high-risk youth ← delinquency
healthcare worker → COVID-19 testing ← COVID-19 severity²

[2]: See https://www.nature.com/articles/s41467-020-19478-2

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Conditioning on a collider creates spurious associations

nice person → date ← good-looking person
impulsivity → high-risk youth ← delinquency
healthcare worker → COVID-19 testing ← COVID-19 severity²

[2]: See https://www.nature.com/articles/s41467-020-19478-2

standardized test → admission ← research skills

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Conditioning on a collider creates spurious associations

nice person → date ← good-looking person
impulsivity → high-risk youth ← delinquency
healthcare worker → COVID-19 testing ← COVID-19 severity²

[2]: See https://www.nature.com/articles/s41467-020-19478-2

standardized test → admission ← research skills
maternal smoking → birth weight → birth defect ← mortality

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Final Example31 / 35

Student Admissions at UC Berkeley (1973)
 
    Dept 
    App_Male 
    Admit_Male 
    Percent_Male 
    App_Female 
    Admit_Female 
    Percent_Female 
  


    A 


1 


41 
  

    B 


0 


00 
  

    C 


9 


06 
  

    D 


1 


93 
  

    E 


7 


92 
  

    F 


9 


04 
  

    Total 


5 


35 
  



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Dept	App_Male	Admit_Male	Percent_Male	App_Female	Admit_Female	Percent_Female
A	825	512	62.1	108	89	82.41
B	560	353	63.0	25	17	68.00
C	325	120	36.9	593	202	34.06
D	417	138	33.1	375	131	34.93
E	191	53	27.7	393	94	23.92
F	373	22	5.9	341	24	7.04
Total	2691	1198	44.5	1835	557	30.35

Causal Thinking

What do we mean by the causal effect of gender?

What do we mean by gender bias?

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Instrumental Variables

See more in the note

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RemarksCausal inference requires causal assumptionsYou need a DAG

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RemarksCausal inference requires causal assumptionsYou need a DAG

Blindly adjusting for covariates does not give better resultspost-treatment bias, collider bias, etc

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RemarksCausal inference requires causal assumptionsYou need a DAG

Blindly adjusting for covariates does not give better resultspost-treatment bias, collider bias, etc

Think carefully about what causal quantity is of interestE.g., direct, indirect, total

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RemarksCausal inference requires causal assumptionsYou need a DAG

Blindly adjusting for covariates does not give better resultspost-treatment bias, collider bias, etc

Think carefully about what causal quantity is of interestE.g., direct, indirect, total

Causal inferences are possible with both experimental and non-experimental data
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Causal Inference

PSYC 573

University of Southern California

March 29, 2022

Causation

Causal Inference

Causal Inference

Causal Inference

Directed Acyclic Graph

Directed Acyclic Graph (DAG)

Directed Acyclic Graph (DAG)

Basic Types of Junctions

Fork

Fork

Posterior predictive checks

Pedicting an Intervention

Pedicting an Intervention

Pedicting an Intervention

Randomization

Framing Experiment

Framing Experiment

Framing Experiment

Back-Door Criterion

Back-Door Criterion

Dagitty

Post-Treatment Bias

Data for Framing Experiment

Results

Results

Mediation

Mediation

Mediation Analysis

Direct Effect

Indirect Effect

Indirect Effect

Indirect Effect

Potential Confounding

Potential Confounding

Unobserved Confounding

Collider Bias

Conditioning on a collider creates spurious associations

Conditioning on a collider creates spurious associations

Conditioning on a collider creates spurious associations

Conditioning on a collider creates spurious associations

Conditioning on a collider creates spurious associations

Final Example

Student Admissions at UC Berkeley (1973)

Causal Thinking

Instrumental Variables

Remarks

Remarks

Remarks

Remarks

Causation

Help