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: " Ya=1 and Ya=0 are referred as potential outcomes or as counterfactual outcomes "
When the potential outcome under one level of exposure is not the same as the outcome under another level of exposure Ya=1i≠Ya=0i, there's an effect of exposure A on individual's outcome
Examples:
When the potential outcome under one level of exposure is not the same as the outcome under another level of exposure Ya=1i≠Ya=0i, there's an effect of exposure A on individual's outcome
Examples:
For Zeus, who was treated (A=1), Ya=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:
Also we assume no interference (individual's potential outcomes are independent)
Wonder data on Gods, for whom we observed both potential outcomes
Y_a0
is potential outcome under no treatmentY_a1
is a potential outcome under treatmentDeterministic counterfactual outcomes (simplification)
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
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 |
Pr[Ya=1=1]≠Pr[Ya=0=1]
If Pr[Ya=1=1] = Pr[Ya=0=1] --> no average causal effect (null effect)
# no individual causal effectgreek_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
# individual causal effectgreek_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
Risk difference
Risk ratio
Risk difference
Risk ratio
Odds ratio
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"
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".
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
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
Observed (associational) risk difference, risk ratio, and odds ratio:
Observed (associational) risk difference, risk ratio, and odds ratio:
Risk difference
Risk ratio
Observed (associational) risk difference, risk ratio, and odds ratio:
Risk difference
Risk ratio
Odds ratio
Observed (associational) risk difference, risk ratio, and odds ratio:
Risk difference
Risk ratio
Odds ratio
What is (observed; associational) vs. What if (counterfactual; causal)
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