DAGs II

POL51

Juan F. Tellez

University of California, Davis

September 30, 2024

Plan for today

Controlling for confounds

Intuition

Limitations

Where are we so far?

Want to estimate the effect of X on Y

Elemental confounds get in our way

DAGs + ggdag() to model causal process

figure out which variables to control for and which to avoid

Do waffles cause divorce?

Remember, we have strong reason to believe the South is confounding the relationship between Waffle Houses and divorce rates:

Speak about true effects and null effects

A problem we share with Lincoln

We need to control (for) the South

It has a bad influence on divorce, waffle house locations, and the integrity of the union

But how do we do control (for) the South? And what does that even mean?

We’ve already done it

One way to adjust/control for backdoor paths is with multiple regression:

In general: \(Y = \alpha + \beta_1X_1 + \beta_2X_2 + \dots\)

In this case: \(Y = \alpha + \beta_1Waffles + \beta_2South\)

In multiple regression, coefficients (\(\beta_i\)) are different: they describe the relationship between X1 and Y, after adjusting for the X2, X3, X4, etc.

What does it mean to control?

\(Y = \alpha + \beta_1Waffles + \beta_2South\)

Three ways of thinking about \(\color{red}{\beta_1}\) here:

• The relationship between Waffles and Divorce, controlling for the South

• The relationship between Waffles and Divorce that cannot be explained by the South

• The relationship between Waffles and Divorce, comparing among similar states (South vs. South, North vs. North)

Comparing models

Let’s fit two models of the effect of GDP on life expectancy, one with controls and one without:

no_controls = lm(lifeExp ~ gdpPercap, data = gap)
controls = lm(lifeExp ~ gdpPercap + pop + year, data = gap)

Population and year might be forks here: they affect both economic activity and life expectancy

Note

I’ve re-scaled gdp (10,000s) and population (millions)

The regression table

We can compare models using a regression table;

many different functions, we’ll use huxreg() from the huxtable package

huxreg("No controls" = no_controls, "With controls" = controls)

Note

Format: huxreg("label for model 1" = model_1, "label for model 2" = model_2, ...)

The regression table

huxreg("No controls" = no_controls, "With controls" = controls)

	No controls	With controls
(Intercept)	53.956 ***	-411.450 ***
	(0.315)	(27.666)
gdpPercap	7.649 ***	6.729 ***
	(0.258)	(0.244)
pop		0.006 **
		(0.002)
year		0.235 ***
		(0.014)
N	1704	1704
R2	0.341	0.440
logLik	-6422.205	-6282.869
AIC	12850.409	12575.737
*** p < 0.001; ** p < 0.01; * p < 0.05.

Comparison

• Model 1 has no controls: just the relationship between GDP and life expectancy

• Model 2 controls/adjusts for: population and year

• the effect of GDP per capita on life expectancy changes with controls

• The estimate is smaller

	No controls	With controls
(Intercept)	53.956 ***	-411.450 ***
	(0.315)	(27.666)
gdpPercap	7.649 ***	6.729 ***
	(0.258)	(0.244)
pop		0.006 **
		(0.002)
year		0.235 ***
		(0.014)
N	1704	1704
R2	0.341	0.440
logLik	-6422.205	-6282.869
AIC	12850.409	12575.737
*** p < 0.001; ** p < 0.01; * p < 0.05.

Interpretation

No controls: every additional 10k of GDP = 7.6 years more life expectancy

With controls: after adjusting for population and year, every additional 10k of GDP = 6.7 years more of life expectancy

	No controls	With controls
(Intercept)	53.956 ***	-411.450 ***
	(0.315)	(27.666)
gdpPercap	7.649 ***	6.729 ***
	(0.258)	(0.244)
pop		0.006 **
		(0.002)
year		0.235 ***
		(0.014)
nobs	1704	1704
*** p < 0.001; ** p < 0.01; * p < 0.05.

In other words: .shout[comparing countries with similar population levels and in the same year], every additional 10k of GDP = 6.7 years more of life expectancy

Another example: 🚽 and 💰

How much does having an additional bathroom boost a house’s value?

price	bedrooms	bathrooms	sqft_living	waterfront
379900	3	2	3110	FALSE
998000	4	2	2910	FALSE
530000	2	2	1785	FALSE
530000	3	2	2030	FALSE
396000	3	2	2340	FALSE
348000	2	2	1270	FALSE
452000	2	2	1740	FALSE
201000	3	1	900	FALSE

Another example: 🚽 and 💰

A huge effect:

no_controls = lm(price ~ bathrooms, data = house_prices)
huxreg("No controls" = no_controls)

	No controls
(Intercept)	10708.309
	(6210.669)
bathrooms	250326.516 ***
	(2759.528)
N	21613
R2	0.276
logLik	-304117.741
AIC	608241.481
*** p < 0.001; ** p < 0.01; * p < 0.05.

The problem

We are comparing houses with more and fewer bathrooms. But houses with more bathrooms tend to be larger! So house size is confounding the relationship between 🚽 and 💰

Controlling for size

What happens if we control for how large a house is?

• The effect goes very close to zero (by house price standards), and is even negative

controls = lm(price ~ bathrooms + sqft_living, data = house_prices)
huxreg("No controls" = no_controls, "Controls" = controls)

	No controls	Controls
(Intercept)	10708.309	-39456.614 ***
	(6210.669)	(5223.129)
bathrooms	250326.516 ***	-5164.600
	(2759.528)	(3519.452)
sqft_living		283.892 ***
		(2.951)
N	21613	21613
R2	0.276	0.493
logLik	-304117.741	-300266.206
AIC	608241.481	600540.413
*** p < 0.001; ** p < 0.01; * p < 0.05.

Does this actually work? Back to the waffles

These are examples with real data, where we can’t know for sure if our controls are doing what we think

Only way to know for sure is with made-up data, where we know the effects ex ante:

fake = tibble(south = sample(c(0, 1), size = 50, replace = TRUE), 
              waffle = rnorm(n = 50, mean = 20, sd = 4) + 10 * south,
              divorce = rnorm(n = 50, mean = 20, sd = 2) + 8 * south) 

What do we know?

fake = tibble(south = sample(c(0, 1), size = 50, replace = TRUE), 
              waffle = rnorm(n = 50, mean = 20, sd = 4) + 10*south,
              divorce = rnorm(n = 50, mean = 20, sd = 2) + 8*south) 

We know that waffles have 0 effect on divorce

We know that the south has an effect of 10 on divorce

We know that the south has an effect of 8 on waffles

Controlling for the South

Fit a naive model without controls, and the correct one controlling for the South:

naive_waffles = lm(divorce ~ waffle, data = fake)
control_waffles = lm(divorce ~ waffle + south, data = fake)

The results

Perfect! Naive model is confounded; but controlling for the South, we get pretty close to the truth (0 effect)

	Naive model	Control South
(Intercept)	7.479 ***	15.774 ***
	(1.671)	(1.787)
waffle	0.648 ***	0.191 *
	(0.064)	(0.086)
south		6.285 ***
		(0.980)
nobs	50	50
*** p < 0.001; ** p < 0.01; * p < 0.05.

What’s going on?

In our made-up world, if we control for the South we can get back the uncounfounded estimate of Divorce ~ Waffles

But what’s lm() doing under-the-hood that makes this possible?

What’s going on?

• lm() is estimating \(South \rightarrow Divorce\) and \(South \rightarrow Waffles\)

• it is then subtracting out or removing the effect of South on Divorce and Waffles

• what’s left is the relationship between Waffles and Divorce, adjusting for the influence of the South on each

Visualizing controlling for the South

This is the confounded relationship between waffles and divorce (zoomed out)

Add the south

We can see what we already know: states in the South tend to have more divorce, and more waffles

Effect of south on divorce

\(South \rightarrow Divorce = 10\) How much higher, on average, divorce is in the South than the North

Remove effect of South on divorce

Regression subtracts out the effect of the South on divorce

Next: effect of South on waffles

\(South \rightarrow Waffles = 8\) How many more, on average, Waffle Houses there are in the South than the North

Subtract out the effect of south on waffles

Regression subtracts out the effect of the South on waffles

What’s left over?

The true effect of waffles on divorce \(\approx\) 0

The other confounds

The perplexing pipe

Remember, with a perplexing pipe, controlling for Z blocks the effect of X on Y:

Simulation

Let’s make up some data to show this: every unit of foreign aid increases corruption by 8; every unit of corruption increases the number of protest by 4

fake_pipe = tibble(aid = rnorm(n = 200, mean = 10), 
                   corruption = rnorm(n = 200, mean = 10) + 8 * aid, 
                   protest = rnorm(n = 200, mean = 10) + 4 * corruption)

What is the true effect of aid on protest? Tricky since the effect runs through corruption

For every unit of aid, corruption increases by 8; and for every unit of corruption, protest increases by 4…

The effect of aid on protest is \(4 \times 8 = 32\)

The data

aid	corruption	protest
9.23	82.95	342.13
7.18	65.69	275.31
9.59	84.85	349.96
9.37	84.27	346.49
9.81	89.66	368.28
8.97	80.65	332.55

Bad controls

Remember, with a pipe controlling for Z (corruption) is a bad idea

Let’s fit two models, where one makes the mistake of controlling for corruption

right_model = lm(protest ~ aid, data = fake_pipe)
bad_control = lm(protest ~ aid + corruption, data = fake_pipe)

Bad controls

Notice how the model that mistakenly controls for Z tells you that X basically has no effect on Y (wrong)

	Correct model	Bad control
(Intercept)	49.803 ***	9.783 ***
	(3.122)	(0.901)
aid	31.970 ***	-0.817
	(0.312)	(0.507)
corruption		4.094 ***
		(0.063)
nobs	200	200
*** p < 0.001; ** p < 0.01; * p < 0.05.

The exploding collider

Remember, with an exploding collider, controlling for M creates strange correlations between X and Y:

Simulation

Let’s make up some data to show this:

fake_collider = tibble(x = rnorm(n = 100, mean = 10), 
                   y = rnorm(n = 100, mean = 10),
                   m = rnorm(n = 100, mean = 10) + 8 * x + 4 * y)

• X has an effect of 8 on M

• Y has an effect of 4 on M

• X has no effect on Y

The data

x	y	m
9.094432	8.115710	114.9313
10.043012	10.645053	133.7738
10.225254	10.171595	131.9025
9.454381	10.527610	128.8469
10.024405	11.665213	135.4701
9.594685	9.837076	126.1522

Bad controls

What’s the true effect of X on Y? it’s zero

Remember, with a collider controlling for M is a bad idea

Let’s fit two models, where one makes the mistake of controlling for M

right_model = lm(y ~ x, data = fake_collider)
collided_model = lm(y ~ x + m, data = fake_collider)

Bad controls

Notice how the model that mistakenly controls for M tells you that X has a strong, negative effect on Y (wrong)

	Correct model	Collided!
(Intercept)	8.697 ***	-1.331 ***
	(0.989)	(0.356)
x	0.138	-1.957 ***
	(0.099)	(0.059)
m		0.238 ***
		(0.006)
nobs	100	100
*** p < 0.001; ** p < 0.01; * p < 0.05.

Colliding as sample selection

Most of the time when we see a collider, it’s because we’re looking at a weird sample of the population we’re interested in

Examples: the non-relationship between height and scoring, among NBA players; the (alleged) negative correlation between how surprising and reliable findings are, among published research

Hiring at Google

Imagine Google wants to hire the best of the best, and they have two criteria: interpersonal skills, and technical skills

Say Google can measure how socially and technically skilled someone is (0-100)

fake_google = tibble(social_skills = rnorm(n = 200, mean = 50, sd = 10), 
                     tech_skills = rnorm(n = 200, mean = 50, sd = 10))

The two are causally unrelated: one does not affect the other; improving someone’s social skills would not hurt their technical skills

The data

social_skills	tech_skills
37.87	38.46
27.31	54.83
56.66	52.63
43.63	32.23
48.43	42.49
44.41	47.49
44.91	25.93
42.47	22.64

Simulate the hiring process

Now imagine that they add up the two skills to see a person’s overall quality:

fake_google %>% 
  mutate(total_score = social_skills + tech_skills)

social_skills	tech_skills	total_score
54.2	53.6	108
61.5	39.6	101
55.1	59.8	115
67.3	49.3	117
57.9	58.6	116
45.7	36.2	81.9
36.8	69.5	106
40.1	33.1	73.2
52.6	47.4	100
59	46.4	105
46	47	93.1
47.5	39.3	86.8
26.8	55.3	82
53.2	60	113
56.7	52.6	109
41.8	59.5	101
44.7	38.5	83.1
58	50	108
46.5	50.8	97.3
41.4	42.8	84.2
46.2	44.2	90.4
55.8	53.4	109
39.6	38.6	78.2
48.1	45.6	93.7
52.4	48.2	101
54.5	50.9	105
53.2	46.4	99.6
56.1	43	99.1
53.1	59.2	112
63.9	50.4	114
34.6	47.8	82.3
40.6	44.4	85
52.6	37.2	89.7
63.4	49.2	113
35.4	47.7	83.2
55.7	26.6	82.3
47.7	42.5	90.2
29.6	52.8	82.4
56.2	57.3	114
58.9	57.1	116
51.4	40	91.4
40.5	48.6	89.1
40.5	64.9	105
34.3	52.5	86.8
50.1	36.3	86.4
35.2	78	113
42.7	68.4	111
53.7	59.8	113
64.4	48.1	113
63.9	39.8	104
43.1	43.7	86.8
37.9	38.5	76.3
45.8	58	104
58.9	43.2	102
50.2	59.4	110
55.6	51.9	108
43.5	54.5	98
52.6	41.1	93.7
31.1	38.4	69.5
48.8	58.3	107
41.6	45.2	86.8
51.5	34.1	85.5
69.6	62.2	132
38.4	66.4	105
43.9	51.3	95.2
64.3	56.8	121
53	58.6	112
51.7	38.4	90.1
49.7	54.8	104
52.2	54.9	107
47.2	43.5	90.7
57	55.4	112
50.9	38.7	89.6
49.9	34.9	84.8
49	55.6	105
30.8	53.1	83.9
63.4	62	125
48.5	52.6	101
42.5	47.8	90.4
57.5	64.3	122
51.7	45.8	97.6
62.9	64.2	127
54.8	52.1	107
53.5	68.6	122
63.4	48.8	112
66.4	57.2	124
49.7	52.6	102
35.2	53.4	88.6
49.3	56.6	106
40.4	34.2	74.6
48.8	41.8	90.6
46	53.1	99.1
43.4	62	105
59.7	56.2	116
52.6	56	109
37	70.6	108
29.9	46.9	76.8
45.3	47	92.3
46.9	44.2	91.1
46.1	45.9	92
56.1	42.6	98.7
62.3	43.8	106
44.6	35.6	80.2
44.9	25.9	70.8
56.2	43.9	100
43.6	32.2	75.9
49.7	56.1	106
60.8	53.7	115
54.1	46.8	101
59.2	57.2	116
52	54	106
61.2	51.7	113
35.5	48.1	83.6
61.7	57.7	119
51.8	40.3	92.1
42	26.6	68.6
44.4	47.5	91.9
43.1	49.7	92.7
43.4	56.8	100
48.3	59.5	108
42.6	42.8	85.4
49.8	51.9	102
56	39.6	95.5
48.6	53.1	102
63.5	36.8	100
62.9	26.1	89
52.6	41.5	94.1
27.3	54.8	82.1
37.8	48.5	86.3
43.8	48	91.8
58.8	47.1	106
36.1	46.3	82.4
59.6	58.9	118
63.1	37.9	101
54.4	65.4	120
62	46	108
43.5	39.6	83.1
42.3	58.6	101
41.2	46.3	87.4
42.8	52.5	95.3
58	45.4	103
49.3	35	84.3
60.9	49	110
56	50.4	106
58.1	38.4	96.5
57.8	48	106
52.2	62.8	115
48.7	53.6	102
37.9	57.4	95.2
55.8	42	97.8
51	61.5	112
62.8	48.5	111
43.6	38	81.6
35.3	53.9	89.2
40.3	46.4	86.7
26.3	45.6	71.9
52.1	35.6	87.7
44.8	50.8	95.6
53.6	41.6	95.2
57.4	39.8	97.2
52	60.4	112
48.4	42.5	90.9
41.9	43.1	85.1
49.6	23	72.7
58.2	41.8	100
58.4	53.8	112
53.5	47.5	101
48.9	32.6	81.5
59.5	58	117
54.6	50.6	105
38.7	61.9	101
48.8	47.2	96
61.8	55.8	118
47.2	69.2	116
48.1	72.7	121
39.1	43.4	82.5
66.2	52.5	119
45	33.7	78.7
36.2	59.1	95.3
49.3	44.7	94
58.7	62.6	121
53.5	37.8	91.3
44	58.1	102
36.8	49.2	86
44.6	57.6	102
42.6	29	71.6
68.9	38.9	108
42.5	22.6	65.1
56	51.9	108
26.5	59.3	85.8
43.3	56.6	100
44.4	57.4	102
35.9	46.3	82.2
51.7	46.3	98.1
57.3	54.2	111
50.8	52.4	103
45.9	51.8	97.7
44.6	49.1	93.7
53.6	51	105
53.2	55.8	109

Simulating the hiring process

Now imagine that Google only hires people who are in the top 15% of quality (in this case that’s 112.8 or higher)

fake_google %>% 
  mutate(total_score = social_skills + tech_skills) %>% 
  mutate(hired = case_when(total_score >= quantile(total_score, .85) ~ "yes", 
                           total_score < quantile(total_score, .85) ~ "no"))

social_skills	tech_skills	total_score	hired
54.2	53.6	108	no
61.5	39.6	101	no
55.1	59.8	115	yes
67.3	49.3	117	yes
57.9	58.6	116	yes
45.7	36.2	81.9	no
36.8	69.5	106	no
40.1	33.1	73.2	no
52.6	47.4	100	no
59	46.4	105	no
46	47	93.1	no
47.5	39.3	86.8	no
26.8	55.3	82	no
53.2	60	113	yes
56.7	52.6	109	no
41.8	59.5	101	no
44.7	38.5	83.1	no
58	50	108	no
46.5	50.8	97.3	no
41.4	42.8	84.2	no
46.2	44.2	90.4	no
55.8	53.4	109	no
39.6	38.6	78.2	no
48.1	45.6	93.7	no
52.4	48.2	101	no
54.5	50.9	105	no
53.2	46.4	99.6	no
56.1	43	99.1	no
53.1	59.2	112	no
63.9	50.4	114	yes
34.6	47.8	82.3	no
40.6	44.4	85	no
52.6	37.2	89.7	no
63.4	49.2	113	no
35.4	47.7	83.2	no
55.7	26.6	82.3	no
47.7	42.5	90.2	no
29.6	52.8	82.4	no
56.2	57.3	114	yes
58.9	57.1	116	yes
51.4	40	91.4	no
40.5	48.6	89.1	no
40.5	64.9	105	no
34.3	52.5	86.8	no
50.1	36.3	86.4	no
35.2	78	113	yes
42.7	68.4	111	no
53.7	59.8	113	yes
64.4	48.1	113	no
63.9	39.8	104	no
43.1	43.7	86.8	no
37.9	38.5	76.3	no
45.8	58	104	no
58.9	43.2	102	no
50.2	59.4	110	no
55.6	51.9	108	no
43.5	54.5	98	no
52.6	41.1	93.7	no
31.1	38.4	69.5	no
48.8	58.3	107	no
41.6	45.2	86.8	no
51.5	34.1	85.5	no
69.6	62.2	132	yes
38.4	66.4	105	no
43.9	51.3	95.2	no
64.3	56.8	121	yes
53	58.6	112	no
51.7	38.4	90.1	no
49.7	54.8	104	no
52.2	54.9	107	no
47.2	43.5	90.7	no
57	55.4	112	no
50.9	38.7	89.6	no
49.9	34.9	84.8	no
49	55.6	105	no
30.8	53.1	83.9	no
63.4	62	125	yes
48.5	52.6	101	no
42.5	47.8	90.4	no
57.5	64.3	122	yes
51.7	45.8	97.6	no
62.9	64.2	127	yes
54.8	52.1	107	no
53.5	68.6	122	yes
63.4	48.8	112	no
66.4	57.2	124	yes
49.7	52.6	102	no
35.2	53.4	88.6	no
49.3	56.6	106	no
40.4	34.2	74.6	no
48.8	41.8	90.6	no
46	53.1	99.1	no
43.4	62	105	no
59.7	56.2	116	yes
52.6	56	109	no
37	70.6	108	no
29.9	46.9	76.8	no
45.3	47	92.3	no
46.9	44.2	91.1	no
46.1	45.9	92	no
56.1	42.6	98.7	no
62.3	43.8	106	no
44.6	35.6	80.2	no
44.9	25.9	70.8	no
56.2	43.9	100	no
43.6	32.2	75.9	no
49.7	56.1	106	no
60.8	53.7	115	yes
54.1	46.8	101	no
59.2	57.2	116	yes
52	54	106	no
61.2	51.7	113	yes
35.5	48.1	83.6	no
61.7	57.7	119	yes
51.8	40.3	92.1	no
42	26.6	68.6	no
44.4	47.5	91.9	no
43.1	49.7	92.7	no
43.4	56.8	100	no
48.3	59.5	108	no
42.6	42.8	85.4	no
49.8	51.9	102	no
56	39.6	95.5	no
48.6	53.1	102	no
63.5	36.8	100	no
62.9	26.1	89	no
52.6	41.5	94.1	no
27.3	54.8	82.1	no
37.8	48.5	86.3	no
43.8	48	91.8	no
58.8	47.1	106	no
36.1	46.3	82.4	no
59.6	58.9	118	yes
63.1	37.9	101	no
54.4	65.4	120	yes
62	46	108	no
43.5	39.6	83.1	no
42.3	58.6	101	no
41.2	46.3	87.4	no
42.8	52.5	95.3	no
58	45.4	103	no
49.3	35	84.3	no
60.9	49	110	no
56	50.4	106	no
58.1	38.4	96.5	no
57.8	48	106	no
52.2	62.8	115	yes
48.7	53.6	102	no
37.9	57.4	95.2	no
55.8	42	97.8	no
51	61.5	112	no
62.8	48.5	111	no
43.6	38	81.6	no
35.3	53.9	89.2	no
40.3	46.4	86.7	no
26.3	45.6	71.9	no
52.1	35.6	87.7	no
44.8	50.8	95.6	no
53.6	41.6	95.2	no
57.4	39.8	97.2	no
52	60.4	112	no
48.4	42.5	90.9	no
41.9	43.1	85.1	no
49.6	23	72.7	no
58.2	41.8	100	no
58.4	53.8	112	no
53.5	47.5	101	no
48.9	32.6	81.5	no
59.5	58	117	yes
54.6	50.6	105	no
38.7	61.9	101	no
48.8	47.2	96	no
61.8	55.8	118	yes
47.2	69.2	116	yes
48.1	72.7	121	yes
39.1	43.4	82.5	no
66.2	52.5	119	yes
45	33.7	78.7	no
36.2	59.1	95.3	no
49.3	44.7	94	no
58.7	62.6	121	yes
53.5	37.8	91.3	no
44	58.1	102	no
36.8	49.2	86	no
44.6	57.6	102	no
42.6	29	71.6	no
68.9	38.9	108	no
42.5	22.6	65.1	no
56	51.9	108	no
26.5	59.3	85.8	no
43.3	56.6	100	no
44.4	57.4	102	no
35.9	46.3	82.2	no
51.7	46.3	98.1	no
57.3	54.2	111	no
50.8	52.4	103	no
45.9	51.8	97.7	no
44.6	49.1	93.7	no
53.6	51	105	no
53.2	55.8	109	no

General population

No relationship between social and technical skills among all job candidates

Collided!

If we only look at Google workers we see a trade-off between social and technical skills:

Limitations

It’s cool that we can control for a confound, or avoid colliders/pipes and get back the truth

But there are big limitations we must keep in mind when evaluating research:

• We need to know what to control for (confident in our DAG)

• We need to have data on the controls (e.g., data on Z)

• We need our data to measure the variable well (e.g., # of homicides a good proxy for crime?)

Stuff that’s hard to measure

Ability is a likely fork for the effect of Education on Earnings; but how do you measure ability?

🚨 Your turn: pipes, colliders 🚨

Using the templates in the class script:

• Make a realistic pipe scenario

• Use models to show that everything goes wrong when you mistakenly control for the pipe

• Make a realistic fork scenario

• Use models to show that everything goes wrong when you fail to control for the fork

10:00