A definition of causal effectWhat If: Chapter 1Elena Dudukina2020-10-181 / 20

INTRODUCTION

"No book can possibly provide a comprehensive description of all methodologies for causal inference across the sciences. The authors of any Causal Inference book will have to choose which aspects of causal inference methodology they want to emphasize."

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Why this book?"...to make causal inferences that are explicit about both the causal question and the assumptions underlying the data analysis"
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Why this book?"...to make causal inferences that are explicit about both the causal question and the assumptions underlying the data analysis"
"various data analysis approaches to estimate the causal effect of interest under a particular set of assumptions when data are collected on each individual in a population"
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NotationAA: treatment/exposure (1: treated, 0: untreated)
YY: outcome (1: death, 0: survival)
YaYa: counterfactual outcome under exposure AYa=1Ya=1: potential outcome under exposure level A=1
Ya=0Ya=0: potential outcome under exposure level A=0

Ya=0iYia=0: potential outcome for individual ii under A=0
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Counterfactual outcomes

Some distinguish between potential and counteractual outcomes. The outcome is potential before it has been observed. Once one outcome is observed it becomes factual, the unobserved potential outcome becomes counterfactual.

Hernán does not distinguish: " $Y^{a = 1}$ and $Y^{a = 0}$ are referred as potential outcomes or as counterfactual outcomes "

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Causal effect for an individual

Not what epidemiology does
"identifying individual causal effects is generally not possible"
When the potential outcome under one level of exposure is not the same as the outcome under another level of exposure $Y_{i}^{a = 1} \neq Y_{i}^{a = 0}$ , there's an effect of exposure A on individual's outcome
Examples:
- Zeus (individual causal effect): $Y_{Z e u s}^{a = 1} \neq Y_{Z e u s}^{a = 0}$

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Causal effect for an individual

Not what epidemiology does
"identifying individual causal effects is generally not possible"
When the potential outcome under one level of exposure is not the same as the outcome under another level of exposure $Y_{i}^{a = 1} \neq Y_{i}^{a = 0}$ , there's an effect of exposure A on individual's outcome
Examples:
- Zeus (individual causal effect): $Y_{Z e u s}^{a = 1} \neq Y_{Z e u s}^{a = 0}$
- Hera (no individual causal effect): $Y_{H e r a}^{a = 1} = Y_{H e r a}^{a = 0}$

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Consistency

One counterfactual outcome is an observed outcome!

For Zeus, who was treated (A=1), $Y^{a = 1}$ was Y=1

Consistency is an assumption, which allows us to shift from potential outcomes to observed outcomes

We always observe ONE of the potential outcomes:

Observed outcomes in treated
- $Y^{a = 1}$ = $Y$ among A=1
Observed outcomes in untreated
- $Y^{a = 0}$ = $Y$ among A=0

Also we assume no interference (individual's potential outcomes are independent)

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Average causal effectsIn a population of individuals
NeedOutcome YY
Exposure (action/intervention) levels to compare: AA (a=1 vs a=0)
Outcomes under exposure levels Ya=1Ya=1 and Ya=0Ya=0

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Greek gods dataset

Wonder data on Gods, for whom we observed both potential outcomes

Y_a0 is potential outcome under no treatment
Y_a1 is a potential outcome under treatment

Deterministic counterfactual outcomes (simplification)

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greek_gods

## # A tibble: 20 x 3
##    greek       Y_a0  Y_a1
##    <chr>      <dbl> <dbl>
##  1 Rheia          0     1
##  2 Kronos         1     0
##  3 Demeter        0     0
##  4 Hades          0     0
##  5 Hestia         0     0
##  6 Poseidon       1     0
##  7 Hera           0     0
##  8 Zeus           0     1
##  9 Artemis        1     1
## 10 Apollo         1     0
## 11 Leto           0     1
## 12 Ares           1     1
## 13 Athena         1     1
## 14 Hephaestus     0     1
## 15 Aphrodite      0     1
## 16 Cyclope        0     1
## 17 Persephone     1     1
## 18 Hermes         1     0
## 19 Hebe           1     0
## 20 Dionysus       1     0

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What are the risks of death under treatment levels?

Since we magically observed both potential outcomes we can contrast the risk in the whole population under A=1 and A=0 directly:

greek_gods %>% 
  summarise_if(is.numeric, mean) %>% 
  kableExtra::kable(format = "html")

Y_a0	Y_a1
0.5	0.5

20 gods
50% (N=10) dies if treated and 50% dies if untreated

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Average causal effect

is inequality of counterfactual risks on the populational level:

$P r [Y^{a = 1} = 1] \neq P r [Y^{a = 0} = 1]$

If $P r [Y^{a = 1} = 1]$ = $P r [Y^{a = 0} = 1]$ --> no average causal effect (null effect)

Greek gods
- Absolute scale: $P r [Y^{a = 1} = 1]$ - $P r [Y^{a = 0} = 1]$ = 0.5-0.5 = 0
- Multiplicative scale: $P r [Y^{a = 1} = 1]$ / $P r [Y^{a = 0} = 1]$ = 0.5/0.5 = 1

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Average causal effects vs. not individual causal effects

# no individual causal effect
greek_gods %>%
  filter(Y_a0 == Y_a1)

## # A tibble: 8 x 3
##   greek       Y_a0  Y_a1
##   <chr>      <dbl> <dbl>
## 1 Demeter        0     0
## 2 Hades          0     0
## 3 Hestia         0     0
## 4 Hera           0     0
## 5 Artemis        1     1
## 6 Ares           1     1
## 7 Athena         1     1
## 8 Persephone     1     1

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# individual causal effect
greek_gods %>%
  filter(Y_a0 != Y_a1) %>% 
  arrange(Y_a1)

## # A tibble: 12 x 3
##    greek       Y_a0  Y_a1
##    <chr>      <dbl> <dbl>
##  1 Kronos         1     0
##  2 Poseidon       1     0
##  3 Apollo         1     0
##  4 Hermes         1     0
##  5 Hebe           1     0
##  6 Dionysus       1     0
##  7 Rheia          0     1
##  8 Zeus           0     1
##  9 Leto           0     1
## 10 Hephaestus     0     1
## 11 Aphrodite      0     1
## 12 Cyclope        0     1

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Measures of causal effectRisk differencePr[Ya=1=1]Pr[Ya=1=1] - Pr[Ya=0=1]Pr[Ya=0=1]

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Measures of causal effect

Risk difference
- $P r [Y^{a = 1} = 1]$ - $P r [Y^{a = 0} = 1]$
Risk ratio
- $P r [Y^{a = 1} = 1]$ / $P r [Y^{a = 0} = 1]$

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Measures of causal effect

Risk difference
- $P r [Y^{a = 1} = 1]$ - $P r [Y^{a = 0} = 1]$
Risk ratio
- $P r [Y^{a = 1} = 1]$ / $P r [Y^{a = 0} = 1]$
Odds ratio
- $\frac{P r [Y^{a = 1} = 1] / P r [Y^{a = 1} = 0]}{P r [Y^{a = 0} = 1] / P r [Y^{a = 0} = 0]}$

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Random error

Sampling variability: "working with samples prevents one from obtaining the exact proportion of individuals in the population who had the outcome under certain treatment value, e.g., the probability of death under no treatment cannot be directly computed"
- Follows the law of big numbers
Non-deterministic (stochastic) counterfactuals Shift from certainly having a 100% chance of dying under treatment or surviving under no treatment to "90% chance of dying if treated, and a 10% chance of dying if untreated".
- Does not follow the law of big numbers

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Causation vs association

greek_gods_obs

## # A tibble: 20 x 5
##    greek          A Y_obs  Y_a0  Y_a1
##    <chr>      <dbl> <dbl> <dbl> <dbl>
##  1 Rheia          0     0     0    NA
##  2 Kronos         0     1     1    NA
##  3 Demeter        0     0     0    NA
##  4 Hades          0     0     0    NA
##  5 Hestia         1     0    NA     0
##  6 Poseidon       1     0    NA     0
##  7 Hera           1     0    NA     0
##  8 Zeus           1     1    NA     1
##  9 Artemis        0     1     1    NA
## 10 Apollo         0     1     1    NA
## 11 Leto           0     0     0    NA
## 12 Ares           1     1    NA     1
## 13 Athena         1     1    NA     1
## 14 Hephaestus     1     1    NA     1
## 15 Aphrodite      1     1    NA     1
## 16 Cyclope        1     1    NA     1
## 17 Persephone     1     1    NA     1
## 18 Hermes         1     0    NA     0
## 19 Hebe           1     0    NA     0
## 20 Dionysus       1     0    NA     0

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greek_gods_obs %>% 
  group_by(A) %>% 
  count(Y_obs) %>% 
  mutate(
    denominator = sum(n),
    Pr_Y_obs = round(n / denominator, 1)
  ) %>% 
  filter(Y_obs == 1)

## # A tibble: 2 x 5
## # Groups:   A [2]
##       A Y_obs     n denominator Pr_Y_obs
##   <dbl> <dbl> <int>       <int>    <dbl>
## 1     0     1     3           7      0.4
## 2     1     1     7          13      0.5

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Observed (associational) risk difference, risk ratio, and odds ratio:

Risk difference
- $P r [Y = 1 | A = 1]$ - $P r [Y = 1 | A = 0]$

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Observed (associational) risk difference, risk ratio, and odds ratio:

Risk difference
- $P r [Y = 1 | A = 1]$ - $P r [Y = 1 | A = 0]$
Risk ratio
- $P r [Y = 1 | A = 1]$ / $P r [Y = 1 | A = 0]$

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Observed (associational) risk difference, risk ratio, and odds ratio:

Risk difference
- $P r [Y = 1 | A = 1]$ - $P r [Y = 1 | A = 0]$
Risk ratio
- $P r [Y = 1 | A = 1]$ / $P r [Y = 1 | A = 0]$
Odds ratio
- $\frac{P r [Y = 1 | A = 1] / P r [Y = 0 | A = 1]}{P r [Y = 1 | A = 0] / P r [Y = 0 | A = 0]}$

19 / 20

Observed (associational) risk difference, risk ratio, and odds ratio:

Risk difference
- $P r [Y = 1 | A = 1]$ - $P r [Y = 1 | A = 0]$
Risk ratio
- $P r [Y = 1 | A = 1]$ / $P r [Y = 1 | A = 0]$
Odds ratio
- $\frac{P r [Y = 1 | A = 1] / P r [Y = 0 | A = 1]}{P r [Y = 1 | A = 0] / P r [Y = 0 | A = 0]}$
What is (observed; associational) vs. What if (counterfactual; causal)

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Conceptualizaton of causation as comparison of counterfactual entities20 / 20

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A definition of causal effect

What If: Chapter 1

Elena Dudukina

2020-10-18

INTRODUCTION

"No book can possibly provide a comprehensive description of all methodologies for causal inference across the sciences. The authors of any Causal Inference book will have to choose which aspects of causal inference methodology they want to emphasize."

Why this book?

"...to make causal inferences that are explicit about both the causal question and the assumptions underlying the data analysis"

Why this book?

"...to make causal inferences that are explicit about both the causal question and the assumptions underlying the data analysis"

"various data analysis approaches to estimate the causal effect of interest under a particular set of assumptions when data are collected on each individual in a population"

Notation

Counterfactual outcomes

Causal effect for an individual

Causal effect for an individual

Consistency

One counterfactual outcome is an observed outcome!

Average causal effects

Greek gods dataset

What are the risks of death under treatment levels?

Average causal effect

is inequality of counterfactual risks on the populational level:

Average causal effects vs. not individual causal effects

Measures of causal effect

Measures of causal effect

Measures of causal effect

Random error

Causation vs association

Conceptualizaton of causation as comparison of counterfactual entities

INTRODUCTION

"No book can possibly provide a comprehensive description of all methodologies for causal inference across the sciences. The authors of any Causal Inference book will have to choose which aspects of causal inference methodology they want to emphasize."

Help