Directed acyclic graphs (DAGs)

class: center, middle, inverse, title-slide

# Directed acyclic graphs (DAGs)
### 王　超辰 | Chaochen Wang
### 2019-08-26 17:00~18:00 @CSS

---

class: middle
# Objectives

- Introduction of using directed acyclic graphs (DAGs) to visualize the assumed structural relationships between variables.

- Identify collider using DAG helps us avoid spurious (fake) associations and collider bias.

???
Paradoxical associations between an outcome and exposure are common in epidemiological studies using observational data.

---

## Some Terminology

.pull-left[

```r
library(dagitty)
g <- dagitty('dag {
 V6 [pos="2,0"]
 V4 [pos="1,-1"]
 V5 [pos="1,1"]
 V2 [pos="-1,-1"]
 V3 [pos="-1,1"] 
 V1 [pos="-2,0"]

V1 -> V4
   V1 -> V5 -> V6 
   V1 -> V3 -> V4 
   V2 -> V3 -> V4 
   V2 -> V5 -> V6
}')
plot(g)
```
]

.pull-right[
![](img/daggitty.png)

- V2 is a **parent** of V5; 
- V5 is a **child** of V2;
- V6 is a **descendant** of V2 (and V5);
- V2 is an **ancestor** of V6 (and V5);
- V1 `\(\rightarrow\)` V3 `\(\leftarrow\)` V2 `\(\rightarrow\)` V5 is a **path**;
- V4 is a **collider** on the path: V1 `\(\rightarrow\)` V4 `\(\leftarrow\)` V3. 
]

---
class: middle

## Reminder of what is a confounder

.pull-left[
- Confounding arises from **common causes** of the exposure (A) and the outcome (Y).

- The confounder (W) in the figure wholly/partially accounts for the observed association of the exposure (A) on the outcome (Y).

- The presence of the confounder can lead to "confounding bias", and inaccurate estimates of the effects of A on Y.
]

.pull-right[

- This confounding bias will cause the crude odds ratio, different from the causal effect - conditional odds ratio.

<img src="img/confounder.png" style="display: block; margin: auto;" />
]

---
class: middle

## Statistical structure of confounding

.pull-left[

- The path `\(A \leftarrow W \rightarrow Y\)` is called a "back-door path".

- "back-door path": any path from A to Y that **starts with an arrow into A**.

- Confounding can be removed by conditioning on variables on the "back-door path" - **block the back-door path (via regression or stratification)**
]

.pull-right[

- To sufficiently **control for confounding**, we must identify a set of variables in the DAG that **block all open back-door paths** from the exposure (A) to the outcome (Y).

]

---
class: middle

### Confounding effect and regression adjustment (simulation demonstration)

```r
N <- 1000 # sample size
set.seed(777)

W <- rnorm(N) # confounder
A <- 0.5 * W + rnorm(N) # exposure
Y <- 0.3 * A + 0.4 * W + rnorm(N) # outcome

fit1 <- lm(Y ~ A) # crude model
fit2 <- lm(Y ~ A + W) # adjusted model 
```

???
- the confounder W is generated as a standard normal random variable with mean 0 and variance 1.

- The generation of A depends on the value of W plus an error term.

- Y is generated depending on both A and W plus an error term.

- both error terms follow independent standard normal distribution.

- this simulation assumes linear relationships between the variables.

- And the true causal effect of A on Y is 0.3.

---
class:middle

## Results from the crude and adjusted models (confounder effect)

```r
library(epiDisplay)
regress.display(fit1) ## crude coefficient
```

```
##  
##                   Coeff lower095ci upper095ci       Pr>|t|
## (Intercept) -0.06145865 -0.1266470  0.0037297 6.459891e-02
## A            0.47149852  0.4124036  0.5305934 1.425616e-49
```

```r
regress.display(fit2) ## adjusted coefficient
```

```
##  
##                   Coeff lower095ci   upper095ci       Pr>|t|
## (Intercept) -0.06023901 -0.1212212 0.0007432178 5.285274e-02
## A            0.28944793  0.2266320 0.3522639025 7.827229e-19
## W            0.42455863  0.3549939 0.4941233389 5.623482e-31
```

---
class: middle, inverse

## What if we condition on a collider?

---
class: middle

## Conditioning on a collider

- Suppose that V3 depends on both V1 and V2, while V1 and V2 are independent from each other:

- Then V1 and V2 will be **conditionally dependent (given V3)**, even though they are marginally independent.

---
class: middle 
## Statistical structure of a collider (1)

.pull-left[

- imaging that rain (A) and a sprinkler (Y) are two reasons of a wet ground (C).

- If the ground is wet, and it is not raining, then the sprinkler must be on.

]

.pull-right[

- Now the arrows are **towards C from A and Y**

- If we condition on C (using regression or stratification), we will create **collider bias**.

- If we ignore the colliding structure, we may conclude that *rain has a negative effect on the sprinkler* even we know that this is not true. 
]

???
assume that the sprinkler is on a daily timer, and not related to the weather.

---
class: middle 
## Statistical structure of a collider (2)

- Conditioning on a collider will introduce an association between Y and A (opens the back-door paths) even if they were unrelated.

```r
N <- 1000 # sample size
set.seed(777)

A <- rnorm(N) # exposure
Y <- 0.3 * A + rnorm(N) # outcome
C <- 1.2 * A + 0.9 * Y + rnorm(N) # collider

fit3 <- lm(Y ~ A) # crude model
fit4 <- lm(Y ~ A + C) # adjusted model
```

???
- A was simulated as a standard normally distributed variable. 
- Y equals the value of A plus an error term.
- C is generated depending on both A and Y plus error. 
- the true causal effect of A on Y is 0.3

---
class: middle

## Results from the crude and adjusted models (collider effect)

```r
regress.display(fit3) ## crude coefficient
```

```
##  
##                  Coeff  lower095ci upper095ci       Pr>|t|
## (Intercept) 0.01026968 -0.05003051 0.07056988 7.382948e-01
## A           0.32588676  0.26534718 0.38642635 8.478273e-25
```

```r
regress.display(fit4) ## adjusted coefficient
```

```
##  
##                   Coeff   lower095ci  upper095ci        Pr>|t|
## (Intercept)  0.03534516 -0.009980569  0.08067088  1.262735e-01
## A           -0.41615560 -0.485538292 -0.34677290  4.878072e-30
## C            0.49066896  0.456016177  0.52532175 2.701447e-126
```
 
???
- unlike adjusting for confounders, now the crude coefficient is the true causal effect between A and Y.

- whereas the regression adjusting for collider is substantially biased.

---
class: middle

## What did we learn?

- If we ignore the collider structure of the variables, and just focus on the model predictive performance. Someone might erroneously choose the model controls the collider and report the **completely wrong (the opposite direction) association between A and Y**.

- Conditioning on the collider creates a paradox/collider bias.

- DAG can help us understand the biological mechanisms in clinical epidemiological settings - **choose the correct confounders**.

---
class: middle

## Why is it important?

- The following graph shows another, more complex collider structure usually known as M-bias.

- The collider (C) is the effect of a common cause (W1) of the exposure (A) and common cause (W2) of the outcome (Y).

- There is only one back-door path, and it is already blocked by the collider (C) -- thus we do not need to adjust for anything.

- But some could consider C to be a classical confounder as it is associated with both exposure A (via `\(A \leftarrow W \rightarrow C\)`), and with the outcome Y (via `\(C \leftarrow W2 \rightarrow Y\)`) and is not in the causal pathway between A and Y.

---
class:middle

## Motivating example

- Evidence shows that exceeding the recommendations for 24h dietary sodium intake is associated with increased levels of systolic blood pressure (SBP).

- With advancing age, the kidney function declines.

- Likewise, age is associated with changes in the blood pressures (SBP).

- Therefore, age is a common cause of both high SBP and impaired sodium homeostasis. - **Age is a confounder** for assocation between sodium intake (SOD) and SBP

- However, high levels of 24-h excretion of urinary protein (PRO) are caused by high SBP and increased sodium intake (SOD). - **PRO is a collider** between SOD and SBP.

---
class:middle 
## DAG for the relationship of the exposure, outcome, confounder and collider

We are interested in estimating the effect of sodium intake (SOD) on SBP.

---
class:middle 
## Data generation (simulation demo)

```r
generateData <- function(n, seed){
 set.seed(seed)
 Age_years <- rnorm(n, 65, 5)
 Sodium <- Age_years/18 + rnorm(n)
 sbp <- 1.05 * Sodium + 2.00 * Age_years + rnorm(n)
 hypertension <- ifelse(sbp > 140, 1, 0)
 PRO <- 2.00 * sbp + 2.80 * Sodium + rnorm(n)
 data.frame(sbp, hypertension, Sodium, Age_years, PRO)
}

ObsData <- generateData(n = 1000, seed = 777)

head(ObsData)
```

```
##        sbp hypertension   Sodium Age_years      PRO
## 1 139.7215            0 3.373645  67.44893 289.2140
## 2 130.4990            0 3.194070  63.00729 268.1802
## 3 141.0323            1 4.064678  67.55418 290.9316
## 4 127.5514            0 2.313437  63.00594 263.2170
## 5 150.7820            1 4.090570  73.19343 313.2848
## 6 141.4549            1 3.887545  68.10637 293.4832
```

???
- The simulation assumes linear relationships between the variables; 
- The true causal effect of sodium intake on SBP is 1.05; 
- The coefficients for the association of PRO with SBP ad sodium intake are 2.0, and 2.8

---
class: middle

## Linear regressions using the `ObsData`

- Model 0: `\(\text{SBP} = \beta_0 + \beta_1\text{SOD} + \varepsilon\)` ;

- Model 1: `\(\text{SBP} = \beta_0 + \beta_1\text{SOD} + \beta_2 \text{Age} + \varepsilon\)` ;

- Model 2: `\(\text{SBP} = \beta_0 + \beta_1\text{SOD} + \beta_2 \text{Age} + \beta_3\text{PRO} + \varepsilon\)`

```r
# Model fit
fit0 <- lm(sbp ~ Sodium, data = ObsData)
fit1 <- lm(sbp ~ Sodium + Age_years, data = ObsData)
fit2 <- lm(sbp ~ Sodium + Age_years + PRO, data = ObsData)
```

???
- We fit three different linear regression models 
- to evaluate the effect of sodium intake on SBP
- Model 0 is the unadjusted model 
- Model 1 adjusted for age only 
- Model 2 adjusted for age and the collider (PRO).

---
class: middle
### Results from the crude, and age-adjusted models for the association between SOD and SBP

```r
regress.display(fit0, decimal = 3)
```

```
## Linear regression predicting sbp
## 
## Coeff.(95%CI) P(t-test) P(F-test)
## Sodium (cont. var.) 3.96 (3.375,4.545) < 0.001 < 0.001 
## 
## No. of observations = 1000
```

```r
regress.display(fit1, decimal = 3)
```

```
## Linear regression predicting sbp
## 
## adj. coeff.(95%CI) P(t-test) P(F-test)
## Sodium (cont. var.) 1.039 (0.977,1.102) < 0.001 < 0.001 
## 
## Age_years (cont. var.) 2.004 (1.992,2.017) < 0.001 < 0.001 
## 
## No. of observations = 1000
```

???

- The crude model showed biased association between sodium intake and SBP caused by confounding effect from age.

- The age-adjusted model showed the causal effect that is close to the true effect.

---
class: middle 
### Results from the age and PRO(collider)-adjusted models for the association between SOD and SBP

```r
regress.display(fit2, decimal = 3)
```

```
## Linear regression predicting sbp
## 
## adj. coeff.(95%CI) P(t-test) P(F-test)
## Sodium (cont. var.) -0.902 (-0.973,-0.831) < 0.001 < 0.001 
## 
## Age_years (cont. var.) 0.417 (0.364,0.47) < 0.001 < 0.001 
## 
## PRO (cont. var.) 0.396 (0.383,0.409) < 0.001 < 0.001 
## 
## No. of observations = 1000
```

???

- The collider model suggested a negative relationship between sodium intake and SBP.
- -0.902 suggested that for one unit increase in sodium take, the expected SBP **decrease** by 0.9 mmHg.
- This is not only wrong, it is completely wrong.

---
class: middle 
## How about conditioning colliders in a logistic regression model?

- Model 0: `\(\text{logit}(\text{Hypertension}) = \beta_0 + \beta_1\text{SOD} + \varepsilon\)` ;

- Model 1: `\(\text{logit}(\text{Hypertension}) = \beta_0 + \beta_1\text{SOD} + \beta_2 \text{Age} + \varepsilon\)` ;

- Model 2: `\(\text{logit}(\text{Hypertension}) = \beta_0 + \beta_1\text{SOD} + \beta_2 \text{Age} + \beta_3\text{PRO} + \varepsilon\)`

---
class: middle

### Logistic regression model1 -- crude model:

```r
fit3 <- glm(hypertension ~ Sodium, family = binomial(link = "logit"), data = ObsData)
# logistic.display(fit3, decimal = 3)
library(broom)
library(tidyverse)
tidy(fit3, exponentiate = TRUE, conf.int = TRUE) %>% 
 mutate_if(is.numeric, round, digits = 4)
```

```
## # A tibble: 2 x 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0.0218 0.332 -11.5 0 0.0112 0.0411
## 2 Sodium 2.15 0.0829 9.24 0 1.83 2.54
```

---
class: middle

### Logistic regression model2 -- age adjusted:

```r
fit4 <- glm(hypertension ~ Sodium + Age_years, family = binomial(link = "logit"), data = ObsData)
# logistic.display(fit4, decimal = 3)
tidy(fit4, exponentiate = TRUE, conf.int = TRUE) %>% 
 mutate_if(is.numeric, round, digits = 4)
```

```
## # A tibble: 3 x 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0 34.2 -7.69 0 0 0 
## 2 Sodium 7.95 0.384 5.40 0 4.00 18.2
## 3 Age_years 42.6 0.489 7.67 0 18.3 126.
```

???

- Age adjusted OR for one-unit increase of Sodium intake on being hypertensive was 7.94.

---
class: middle

### Logistic regression model3 -- collider adjusted:

```r
fit5 <- glm(hypertension ~ Sodium + Age_years + PRO, family = binomial(link = "logit"), data = ObsData)
# logistic.display(fit5, decimal = 3)
tidy(fit5, exponentiate = TRUE, conf.int = TRUE)%>% 
 mutate_if(is.numeric, round, digits =4)
```

```
## # A tibble: 4 x 7
## term estimate std.error statistic p.value conf.low conf.high
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 (Intercept) 0 119. -5.13 0 0 0 
## 2 Sodium 0.0179 1.27 -3.17 0.0015 0.001 0.160
## 3 Age_years 3.43 0.771 1.60 0.110 0.811 18.0 
## 4 PRO 6.43 0.424 4.39 0 3.25 17.9
```

???
- Collider adjusted model gave protective effect again.

- And one unit increase in sodium intake would decrease the risk of hypertension by 98%!!!

---
class: middle

### Logistic regression models -- odds ratios:

One unit increase in SOD intake would **decrease the risk of hypertension by 98%!!!**

---
class: middle

## What did we learn?

- The decision whether to include or exclude the variable in a regression model is based on whether the purpose of the study is prediction or explanation/causation.

- Adding a collider to a regression model is **not advised when we are intereted in estimation of causal effects**, because this will open back-door path.

- However, if prediction is the purpose of the model, inclusion of colliders may be advisable if it can reduce the model's prediction error.

- Most epidemiological researches are trying to understand/explain **how the world works (causation)**. We should be aware of such kind of collider bias.

---
class: middle

# Exercise -- (1)

A case-control study is conducted in order to address whether exposure to a particular non-steroidal anti-inflammatory drug during the first trimester of pregancy causes a congenital defect `\((D)\)` arising in the second trimester:

- `\(D = 1\)` for cases, `\(D = 0\)` for controls;

- `\(E^*\)` denotes the use of the drug during the first trimester as **self-reported 1 month postpartum**;

- `\(E\)` denotes the use of the drug during the first trimester as **recorded in accurate medical records**.

Draw an appropriate causal diagram for this problem, involving `\(D\)`, `\(E\)` and `\(E^*\)`

---
class: middle

# Exercise -- (1)

Here are the results of various analyses exploring (crude and adjusted) associations between `\(E\)`, `\(E^*\)`, and `\(D\)`:

- Crude OR (odds ratio) for association between `\(E\)` and `\(D\)`: `\(1.73\)`;

- OR for association between `\(E\)` and `\(D\)`, adjusted for `\(E^*\)`: `\(1.06\)`;

- Crude OR (odds ratio) for association between `\(E^*\)` and `\(D\)`: `\(2.00\)`;

- OR for association between `\(E^*\)` and `\(D\)`, adjusted for `\(E\)`: `\(1.52\)`.

**What is the most appropriate analysis to answer the question of interest?**

---
class: middle

# Exercise -- (1)

.pull-left[- The case-control study setting with recall bias:

- The causal effect of interest is from `\(E\)` to `\(D\)`;

- The exposure reported by the mother `\(E^*\)` might be affected by the true exposure as well as the disease outcome.
]

.pull-right[

- The arrow from `\(D\)` to `\(E^*\)` represents **the recall bias**;

- Because conditional on `\(E\)`, there is an association between `\(E^*\)` and `\(D\)` (OR = 1.52), therefore, **recall bias is present**.

- If there is no recall bias, ie. no arrow from `\(D\)` to `\(E^*\)`, conditioning on E will give us a conditional odds ratio of 1 `\(\neq\)` 1.52.

- Conditioning on `\(E^*\)` changes the estimate of association between `\(E\)` and `\(D\)` to null (1.06 `\(\approx\)` 1.00) `\(\Rightarrow\)` `\(E^*\)` is the collider.

]

---
class: middle

# Exercise -- (2)

A nutritional epidemiologist comes to discuss an analysis of some data that she has collected. She is interested in whether or not eating dark green vegetables protects against stomach cancer. She also collected the following baseline information from the participants (left), but she also believes that there are some unobserved variables (right) that she was not able to collect:

.pull-left[
**Observed variables**

1. Age
2. Alcohol intake
3. Smoking status
]

.pull-right[
**Unobserved variables**

1. Allergies
2. Risky behaviours
]

---
class: middle

### Following causal (DAG) diagram is drawn by this nutritional epidemiologist:

???

- exposure: dark green vegetable intake 
- outcome: stomach cancer
- observed variables: age, smoking, alochol intake
- unobserved variables: allergies, and risky behaviour (circled)
- She believes this is a plausible reprenstation of the relationship between these variables.

---
class: middle

## Let's use the convenient DAG web application

[http://www.dagitty.net/dags.html](http://www.dagitty.net/dags.html)

Johannes Textor, Benito van der Zander, Mark K. Gilthorpe, Maciej Liskiewicz, George T.H. Ellison. 
Robust causal inference using directed acyclic graphs: the R package 'dagitty'.
International Journal of Epidemiology 45(6):1887-1894, 2016.

---
class: middle

### We will get the following DAG graph

???
- To denote that a variable is unobserved, we can move the mouse pointer on that variable and press U. 
- Under the assumption that this diagram is correct, can you find out the minimal sufficient set of confounders?

---
class: middle, inverse

## What do you think about the consequences of adjusting for the following variables?

--
### Smoking status

--
### Alcohol intake

???
- to see the effect of adjusting on a variable, move the mouse over that variable and press A.

- There are no sufficient adjustment sets that include alochol.

- This is because the path via Allergies and Risky behaviour will be opened by conditioning on alcohol (M-bias)

- **This is also an example shows that conditioning on a baseline covariate can induce (rather than reduce) bias!!**

---
class: middle

## Look at some of the results of various analyses of the data

The following odds ratios (OR) are for 1 unit increase in the exposure:

- OR for association between green vegetable intake and stomach cancer, **adjusted for age: 1.04**

- OR for the same association **adjusted for age and alcohol intake: 0.66**

- OR for the same association **adjusted for age and smoking: 1.03**

- OR for the same association **adjusted for age, alcohol and smoking: 0.72**

_Can you explain the changes of the above results?_

---
class: middle

### Discussion on the different results shown in the previous slide

- From the discussion so far we conclude that adjustment for age or age + smoking will estimate the true causal effect of green vegetable intake on stomach cancer.

- However, if we adjust for alcohol intake, the exposure showed a protective effect (OR = 0.66 or 0.72).

- Suppose that: 
    - certain allergies leads to both low alcohol consumption and low green vegetable intake
    - risky behaviours cause high alcohol consumption and smoking
    - Conditioning on alcohol intake will induce a positive association between allergies and risky behaviour.

???

If someone is a heavy drinker, we also know that he/she is allergic to alcohol (so drinking alcohol is a risky behaviour). And somehow this allergy is related with lower intake of darkgreen vegetable. Then given the level of alcohol drinking (within strata of alcohol intake), a low vegetable intake, associated with allergies will related with risky behaviour, and will turn out to be associated with stomach cancer.

---
class:middle, inverse

## Your friend wants to report the results adjusted for alcohol because they showed that vegetable intake is protective against stomach cancer.

--
## do you agree?

--
## you better say NO.

--
## because this is a fake association caused by *collider bias* (controlling for alcohol intake)

---
class: middle, inverse

## Finnally,

--
## Do you agree with your friend's DAG graph?

???

- If there is an arrow from alcohol to stomach cancer, then this would suggest we should collect data on allergies or about risky behaviour.

- Without this DAG graph, such kind of observation may not be so apparent.

---
class: middle

## Thanks to the following paper

Luque-Fernandez, M. A. et al. Educational Note: Paradoxical collider effect in the analysis of non-communicable disease epidemiological data: a reproducible illustration and web application. International Journal of Epidemiology 48, 640–653 (2019).

## Thanks to [Dr. Karla Diaz-Ordaz](https://www.lshtm.ac.uk/aboutus/people/diaz-ordaz.karla)

---
class: middle, center

# The end

## slide address: https://wangcc.me/DAG-CSS