Causal Quartet

When statistics alone

do not tell the full story

Lucy D’Agostino McGowan

@LucyStats�

Department of Statistical Sciences

Wake Forest University

lucymcgowan.com/talk

TABLE OF CONTENTS

CAUSAL INFERENCE

CAUSAL QUARTET

02

SENSITIVITY ANALYSES

03

01

Quick Primer

​

What should we adjust for?

​

What if we haven’t measured everything?

​

3

Causal inference

4

Does X cause Y?

X

Y

Potential Outcomes

5

Potential Outcomes

6

Potential Outcomes

7

Split Decision: Life Stories

8

Vicky, CC BY 2.0 https://creativecommons.org/licenses/by/2.0, via Wikimedia Commons

Award-winning actor, rapper, and producer Ice-T unveils a compelling memoir of his early life robbing jewelry stores until he found fame and fortune—while a handful of bad choices sent his former crime partner down an incredibly different path.

Ice-T & Spike

9

r-causal.org

Ice-T, Spike,

& co

10

.

.

.

Fame & fortune

35 years in prison

Fame & fortune

One more heist

Abandons criminal life

Ice-T

35 years in prison

One more heist

Abandons criminal life

Spike

r-causal.org

.

.

.

.

.

.

Ice-T, Spike,

& co

11

Fame & fortune

35 years in prison

Fame & fortune

One more heist

Abandons criminal life

Ice-T

35 years in prison

One more heist

Abandons criminal life

Spike

r-causal.org

.

.

.

.

.

.

.

.

.

Ice-T, Spike,

& co

12

Fame & fortune

35 years in prison

Fame & fortune

One more heist

Abandons criminal life

Ice-T

35 years in prison

One more heist

Abandons criminal life

Spike

r-causal.org

.

.

.

.

.

.

.

.

.

Consistency

13

Consistency

14

the outcome we observe

Consistency

15

the outcome we observe

is exactly equal to

Consistency

16

the outcome we observe

is exactly equal to

the potential outcome under the observed exposure

Exchangeability

17

the potential

outcomes

Exchangeability

18

are independent

​

the potential

outcomes

Exchangeability

19

are independent

​

of the exposure assignment given some measured factors

​

the potential

outcomes

Exchangeability

20

“No unmeasured confounders”

• We have measured all Z
• We have adjusted for all Z

​

X

Y

Z

Exchangeability

21

“No unmeasured confounders”

• We have measured all Z
• We have adjusted for all Z

​

Abandoning crimnal life vs

one more heist

Fame & fortune vs 35 years in prison

Z

Positivity

22

Positivity

23

For all individuals within strata of Z there exists a non-zero probability of receiving every exposure level.

Exchangeability

24

“No unmeasured confounders”

✓ We have measured all Z

• We have adjusted for all Z

X

Y

Z

Causal Inference �is not a statistics problem

25

Causal Inference �is not a statistics problem

26

just

^

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

The problem

We have measured variables, what should we adjust for?

27

The problem

28

The problem

29

exposure

The problem

30

exposure

outcome

The problem

31

exposure

outcome

measured factor

Estimate the causal effect!

32

exposure

outcome

measured factor

A bit more info:

cor(x, z):

0.7

One unit increase in the exposure yields an average increase in the outcome of 1

The exposure and measured factor are positively correlated

33

To adjust or not adjust, that is the question

34

(1) Collider

35

(1) Collider

(2) Confounder

36

(1) Collider

(2) Confounder

(3) Mediator

37

(1) Collider

(2) Confounder

(4) M-Bias

(3) Mediator

38

(1) Collider

(2) Confounder

(4) M-Bias

Causal Quartet

(3) Mediator

39

40

41

42

43

44

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

(1) Collider

(2) Confounder

(4) M-Bias

(3) Mediator

45

X

Y

Z

U2

U1

X

Y

Z

X

Y

Z

(1) Collider

(2) Confounder

(4) M-Bias

(3) Mediator

46

X

Y

Z

U2

U1

X

Y

Z

X

Y

Z

(1) Collider

(4) M-Bias

(2) Confounder

(3) Mediator

47

X

Y

Z

U2

U1

X

Y

Z

(2) Confounder

(4) M-Bias

(1) Collider

(3) Mediator

48

X

Y

Z

U2

U1

X

Y

Z

(2) Confounder

(1) Collider

(4) M-Bias

(3) Mediator

49

X

Y

Z

X

Y

Z

50

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

51

52

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

^

53

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

^

54

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

^

55

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

^

56

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

^

The problem:

do we adjust for Z?

57

Causal Inference �is not a statistics problem

58

just

^

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

The Solution

59

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

The Solution

60

partial

^

The Solution

61

pre-exposure

partial

^

The Solution

62

time 0

​

partial

^

The Solution

63

follow-up

​

partial

^

The Solution

64

exposure: sodium intake

partial

^

The Solution

65

outcome: systolic blood pressure

partial

^

The Solution

66

measured factor: proteinuria

partial

^

(1) Collider

(2) Confounder

(4) M-Bias

(3) Mediator

67

X

Y

Z

U2

U1

X

Y

Z

X

Y

Z

(1) Collider

(2) Confounder

(4) M-Bias

(3) Mediator

68

X

Y

Z

U2

U1

X

Y

Z

X

Y

Z

sodium intake

systolic blood pressure

proteinuria

69

70

time

X

Y

Z

X

Y

Z

baseline

followup

71

time

baseline

followup

Systolic blood pressure at follow up

X

Y

Z

X

Y

Z

Sodium intake

at baseline

Proteinuria at follow up

72

time

baseline

followup

Systolic blood pressure at follow up

X

Y

Z

X

Y

Z

Sodium intake

at baseline

Proteinuria at follow up

True Causal Effect: 1

Estimated Causal Effect: 0.55

73

time

X

Y

Z

X

Y

Z

baseline

followup

74

time

X

Y

Z

X

Y

Z

baseline

followup

Sodium intake

at baseline

Systolic blood pressure at follow up

Proteinuria at baseline

75

time

X

Y

Z

X

Y

Z

baseline

followup

Sodium intake

at baseline

Systolic blood pressure at follow up

Proteinuria at baseline

True Causal Effect: 1

Estimated Causal Effect: 1

The Solution

76

y_followup ~ x_baseline + z_baseline

​

partial

^

The Solution

77

^

partial

^

The Solution

78

^

partial

^

The Solution

79

^

partial

^

(2) Confounder

(1) Collider

(4) M-Bias

(3) Mediator

80

X

Y

Z

X

Y

Z

(2) Confounder

(1) Collider

(4) M-Bias

(3) Mediator

81

X

Y

Z

X

Y

Z

(2) Confounder

(1) Collider

(4) M-Bias

(3) Mediator

82

X

Y

Z

X

Y

Z

On M-Bias

• The relationship between Z and the unmeasured confounders needs to be really large (Liu et al 2012)
• “To obsess about the possibility of [M-bias] generates bad practical advice in all but the most unusual circumstances” (Rubin 2009)
• There are (almost) no true zeros (Gelman 2011)
• Asymptotic theory shows that induction of M-bias is quite sensitive to various deviations from the exact M-Structure (Ding and Miratrix 2014)

83

84

devtools::install_github(

“r-causal/quartets”

)

install.packages(“quartets”)

D'Agostino McGowan L, Barrett M (2023). �Causal inference is not just a statistical problem. Preprint arXiv:2304.02683v3.

TABLE OF CONTENTS

CAUSAL INFERENCE

CAUSAL QUARTET

02

SENSITIVITY ANALYSES

03

01

Quick Primer

​

What should we adjust for?

​

Thinking about

unmeasured confounding

​

85

The problem

We might not have measured all of the important variables

86

​

Quantifying unmeasured confounding

87

Cornfield

Rosenbaum & Rubin

Lin, Psaty, & Kronmal

Cinelli & Hazlett

88

WHAT YOU NEED

EXPOSURE

​

OUTCOME

​

EFFECT

OUTCOME

​

UNMEASURED

CONFOUNDER

EFFECT

EXPOSURE

​

UNMEASURED

CONFOUNDER

EFFECT

89

90

tipr

D’Agostino McGowan, L. Sensitivity Analyses for Unmeasured Confounders. Curr Epidemiol Rep 9, 361–375 (2022). https://doi.org/10.1007/s40471-022-00308-6

TIPPING POINT

​

What will tip our confidence bound to cross the null?

91

92

D’Agostino McGowan, L. Sensitivity Analyses for Unmeasured Confounders. Curr Epidemiol Rep 9, 361–375 (2022). https://doi.org/10.1007/s40471-022-00308-6

93

{action}_{effect}_with_{what}

D’Agostino McGowan, L., (2022). tipr: An R package for sensitivity analyses for unmeasured confounders. Journal of Open Source Software, 7(77), 4495, https://doi.org/10.21105/joss.04495

94

{action}_{effect}_with_{what}

​

tip_rr_with_continous()

D’Agostino McGowan, L., (2022). tipr: An R package for sensitivity analyses for unmeasured confounders. Journal of Open Source Software, 7(77), 4495, https://doi.org/10.21105/joss.04495

95

{action}_{effect}_with_{what}

​

tip_rr_with_continous()

adjust_coef_with_binary()

D’Agostino McGowan, L., (2022). tipr: An R package for sensitivity analyses for unmeasured confounders. Journal of Open Source Software, 7(77), 4495, https://doi.org/10.21105/joss.04495

96

QUESTION

Cancer

VERSUS SULFONYLUREAS

Metformin

97

ANALYSIS

98

• New-user design
• Matched 42,217 new metformin users to 42,217 new sulfonylurea users
• Fit adjusted Cox proportional hazards model on the matched cohort

​

​

RESULTS

99

OUTCOME

METFORMIN� (ADJ HAZARD RATIO)

What if alcohol consumption is an unmeasured confounder?

100

What if heavy alcohol consumption is prevalent among 4% of Metformin users and 6% of Sulfonylurea users?

101

Meadows SO, Engel CC, Collins RL, Beckman RL, Cefalu M,

Hawes-Dawson J, et al. 2015 health related behaviors survey:

Substance use among US active-duty service members. RAND; 2018.

102

library(tipr)

adjust_hr_with_binary(

effect_observed = c(0.79, 0.87, 0.96),

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = 2)

​

What if we assume the effect of alcohol consumption on lung cancer after adjusting for other confounders is 2?

103

library(tipr)

adjust_hr_with_binary(

effect_observed = c(0.79, 0.87, 0.96),

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = 2)

​

What if we assume the effect of alcohol consumption on lung cancer after adjusting for other confounders is 2?

RESULTS

104

OUTCOME

METFORMIN� (ADJ HAZARD RATIO)

105

library(tipr)

adjust_hr_with_binary(

effect_observed = c(0.79, 0.87, 0.96),

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = 2)

​

What if we assume the effect of alcohol consumption on lung cancer after adjusting for other confounders is 2?

What if heavy alcohol consumption is prevalent among 4% of Metformin users and 6% of Sulfonylurea users?

106

Meadows SO, Engel CC, Collins RL, Beckman RL, Cefalu M,

Hawes-Dawson J, et al. 2015 health related behaviors survey:

Substance use among US active-duty service members. RAND; 2018.

107

library(tipr)

adjust_hr_with_binary(

effect_observed = c(0.79, 0.87, 0.96),

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = 2)

​

What if we assume the effect of alcohol consumption on lung cancer after adjusting for other confounders is 2?

108

library(tipr)

adjust_hr_with_binary(

effect_observed = c(0.79, 0.87, 0.96),

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = 2)

​

What if we assume the effect of alcohol consumption on lung cancer after adjusting for other confounders is 2?

109

# A tibble: 3 × 5

hr_adjusted hr_observed exposed_con…¹ unexp…² confo…³

<dbl> <dbl> <dbl> <dbl> <dbl>

1 0.805 0.79 0.04 0.06 2

2 0.887 0.87 0.04 0.06 2

3 0.978 0.96 0.04 0.06 2

# … with abbreviated variable names

# ¹​exposed_confounder_prev,

# ²​unexposed_confounder_prev,

# ³​confounder_outcome_effect

110

“If heavy alcohol consumption differed between groups, with 4% prevalence among metformin users and 6% among sulfonylureas users, and had an HR of 2 with lung cancer incidence the updated adjusted effect of metformin on lung cancer incidence would be an HR of 0.89 (95% CI: 0.81–0.98). Should an unmeasured confounder like this exist, our effect of metformin on lung cancer incidence would be attenuated and fall much closer to the null.

”

111

library(tipr)

sens <- adjust_hr_with_binary(

effect_observed = 0.96,

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06,

confounder_outcome_effect = seq(1.1, 3.5, by = 0.1))

​

112

library(ggplot2)

ggplot(sens, aes(x = confounder_outcome_effect,

y = hr_adjusted)) +

geom_point()

​

113

114

115

library(tipr)

tip_hr_with_binary(

effect_observed = 0.96,

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06)

​

116

library(tipr)

tip_hr_with_binary(

effect_observed = 0.96,

exposed_confounder_prev = .04,

unexposed_confounder_prev = .06)

​

117

# A tibble: 1 × 6

effect_adj…¹ effec…² expos…³ unexp…⁴ confo…⁵ n_unm…⁶

<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>

1 1 0.96 0.04 0.06 3.27 1

# … with abbreviated variable names ¹​effect_adjusted,

# ²​effect_observed, ³​exposed_confounder_prev,

# ⁴​unexposed_confounder_prev,

# ⁵​confounder_outcome_effect,

# ⁶​n_unmeasured_confounders

118

# A tibble: 1 × 6

effect_adj…¹ effec…² expos…³ unexp…⁴ confo…⁵ n_unm…⁶

<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>

1 1 0.96 0.04 0.06 3.27 1

# … with abbreviated variable names ¹​effect_adjusted,

# ²​effect_observed, ³​exposed_confounder_prev,

# ⁴​unexposed_confounder_prev,

# ⁵​confounder_outcome_effect,

# ⁶​n_unmeasured_confounders

119

“If heavy alcohol consumption differed between groups, with 4% prevalence among metformin users and 6% among sulfonylureas users, it would need to have an association with lung cancer incidence of 3.27 to tip this analysis at the 5% level, rendering the result inconclusive. Given that associations between lung cancer and alcohol consumption have not been reported to be this large, we may be able to rule this out as an impactful unmeasured confounder.

”

What is known about the unmeasured confounder?

Both exposure and outcome relationship is known

• adjust_* functions

120

D’Agostino McGowan, L. Sensitivity Analyses for Unmeasured Confounders. Curr Epidemiol Rep 9, 361–375 (2022). https://doi.org/10.1007/s40471-022-00308-6

What is known about the unmeasured confounder?

Both exposure and outcome relationship is known

Only one of the exposure/outcome relationships is known

• adjust_* functions

• adjust_* functions in an array
• tip_* functions

121

D’Agostino McGowan, L. Sensitivity Analyses for Unmeasured Confounders. Curr Epidemiol Rep 9, 361–375 (2022). https://doi.org/10.1007/s40471-022-00308-6

What is known about the unmeasured confounder?

Both exposure and outcome relationship is known

Only one of the exposure/outcome relationships is known

• adjust_* functions in an array
• tip_* functions in an array
• tip_coef_with_r2() ground in the measured confounders
• single number summaries: Robustness value r_value() & E-values e_value()

• adjust_* functions

• adjust_* functions in an array
• tip_* functions

Nothing

is known

122

D’Agostino McGowan, L. Sensitivity Analyses for Unmeasured Confounders. Curr Epidemiol Rep 9, 361–375 (2022). https://doi.org/10.1007/s40471-022-00308-6

123

devtools::install_github(

“r-causal/tipr”

)

install.packages(“tipr”)

D’Agostino McGowan, L., (2022). tipr: An R package for sensitivity analyses for unmeasured confounders. Journal of Open Source Software, 7(77), 4495, https://doi.org/10.21105/joss.04495

THANKS!

124

Lucy D’Agostino McGowan�

Department of Statistical Sciences

Wake Forest University

Images created with the assistance of DALL·E 2

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