# 도구변수를 활용하여 인과효과를 추정하기

* 작성자: [최보경](https://www.facebook.com/pagebokyung/)
* 원문: [Simple example on using Instrumental Variables method for estimation](https://microsoft.github.io/dowhy/example_notebooks/dowhy-simple-iv-example.html)

## Loading the dataset

```python
import numpy as np
import pandas as pd
import patsy as ps

from statsmodels.sandbox.regression.gmm import IV2SLS
import os, sys
from dowhy import CausalModel
```

개인의 교육이 미래 소득에 주는 영향을 추정하기 위한, 가상의 데이터 셋을 생성합니다.

* 교란변수(unobserved confounder) : 개인의 능력(ability)
* 도구변수(instrumental variable) : 교육 보조(education\_voucher)

```python
n_points = 10000 # 데이터 개수
education_abilty = 1 # discrete
education_voucher = 2 # discrete
income_abilty = 2 # discrete
income_education = 4 # education 과 곱해지는 계수로, 찾고자 하는 인과효과 estimate 의 정답지


# confounder (unobserved)
ability = np.random.normal(0, 3, size=n_points)

# instrument
voucher = np.random.normal(2, 1, size=n_points)

# treatment
education = np.random.normal(5, 1, size=n_points) + education_abilty * ability +\
            education_voucher * voucher

# outcome
income = np.random.normal(10, 3, size=n_points) +\
         income_abilty * ability + income_education * education
```

```python
# 가상의 dataset (exclude confounder `ability` which we assume to be unobserved)
data = np.stack([education, income, voucher]).T
df = pd.DataFrame(data, columns = ['education', 'income', 'voucher'])
df.head()
```

|   | education | income    | voucher  |
| - | --------- | --------- | -------- |
| 0 | 5.638493  | 29.349797 | 1.974566 |
| 1 | 8.522389  | 46.239001 | 1.949956 |
| 2 | 12.840748 | 64.198096 | 2.253389 |
| 3 | 2.954071  | 15.591053 | 0.939508 |
| 4 | 10.117519 | 51.873490 | 1.779209 |

## Using DoWhy to estimate the causal effect of education on future income

**4가지 절차를 따릅니다.**

1. 문제를 인과 그래프로 모델링합니다. (model)
2. 관측된 변수로부터 인과추론이 될 수 있는지 확인합니다. (identify)
3. 인과효과를 추정합니다. (estimate)
4. 인과효과 추정치의 Robustness 를 확인합니다. (refute)

## step 1 : model

```python
model = CausalModel(
          data = df,
          treatment = 'education',
          outcome = 'income',
          common_causes = ['U'], # unobserved confounding 을 임의의 U 컬럼에 넣는 것
          instruments = ['voucher']
        )
model.view_model()

from IPython.display import Image, display
display(Image(filename="causal_model.png"))
```

![](/files/HRsF7iUbyq8crobpkxqK)

## step 2 : identify

```python
identified_estimand = model.identify_effect(proceed_when_unidentifiable = True)
print(identified_estimand)
```

```
Estimand type: nonparametric-ate

### Estimand : 1
Estimand name: backdoor1 (Default)
Estimand expression:
     d                           
────────────(Expectation(income))
d[education]                     
Estimand assumption 1, Unconfoundedness: If U→{education} and U→income then P(income|education,,U) = P(income|education,)

### Estimand : 2
Estimand name: iv
Estimand expression:
Expectation(Derivative(income, [voucher])*Derivative([education], [voucher])**
(-1))
Estimand assumption 1, As-if-random: If U→→income then ¬(U →→{voucher})
Estimand assumption 2, Exclusion: If we remove {voucher}→{education}, then ¬({voucher}→income)

### Estimand : 3
Estimand name: frontdoor
No such variable found!
```

**identify\_effect()**

Identified Estimand 를 리턴해주는 메소드로, Estimand 가 3가지 타입이 나옵니다.

Estimand 의 종류가 non-parametric ATE 일 때 `Backdoor / IV / Frontdoor` 와 같은 Identification methods 를 리턴합니다.

```python
# step 3 : estimate
# step 2 에서 Estimand 가 3가지가 있는데, 그중에서 IV 를 선택함. 

estimate = model.estimate_effect(identified_estimand, 
                                 method_name = 'iv.instrumental_variable',
                                 test_significance = True)

print(estimate)
```

```
Estimand type: nonparametric-ate

### Estimand : 1
Estimand name: backdoor
Estimand expression:
     d
────────────(Expectation(income))
d[education]
Estimand assumption 1, Unconfoundedness: If U→{education} and U→income then P(income|education,,U) = P(income|education,)

### Estimand : 2
Estimand name: iv
Estimand expression:
Expectation(Derivative(income, [voucher])*Derivative([education], [voucher])**
(-1))
Estimand assumption 1, As-if-random: If U→→income then ¬(U →→{voucher})
Estimand assumption 2, Exclusion: If we remove {voucher}→{education}, then ¬({voucher}→income)

### Estimand : 3
Estimand name: frontdoor
No such variable found!
```

## Step 3: Estimate

```
#Choose the second estimand: using IV
estimate = model.estimate_effect(identified_estimand,
        method_name="iv.instrumental_variable", test_significance=True)

print(estimate)
```

```
*** Causal Estimate ***

## Identified estimand
Estimand type: nonparametric-ate

### Estimand : 1
Estimand name: iv
Estimand expression:
Expectation(Derivative(income, [voucher])*Derivative([education], [voucher])**
(-1))
Estimand assumption 1, As-if-random: If U→→income then ¬(U →→{voucher})
Estimand assumption 2, Exclusion: If we remove {voucher}→{education}, then ¬({voucher}→income)

## Realized estimand
Realized estimand: Wald Estimator
Realized estimand type: nonparametric-ate
Estimand expression:

Expectation(Derivative(income, voucher))⋅Expectation(Derivative(education, vou

      -1
cher))
Estimand assumption 1, As-if-random: If U→→income then ¬(U →→{voucher})
Estimand assumption 2, Exclusion: If we remove {voucher}→{education}, then ¬({voucher}→income)
Estimand assumption 3, treatment_effect_homogeneity: Each unit's treatment ['education'] is affected in the same way by common causes of ['education'] and income
Estimand assumption 4, outcome_effect_homogeneity: Each unit's outcome income is affected in the same way by common causes of ['education'] and income

Target units: ate

## Estimate
Mean value: 3.984270555650861
p-value: [0, 0.001]
```

**해석 방법** 교육(education)을 한 단위 높일 때, 소득(income)이 4.01 포인트 가량 높아진다는 의미를 가지는 Estimate (추정치)를 보입니다.

추정된 인과효과는, p-value 가 0 \~ 0.001 사이로 통계적으로 유의합니다.

그 다음으로는, Placebo refutation test 를 통해서 이 추정치의 강건성을 검사합니다. 이 테스트에서는, 독립확률변수(X)가 도구변수와의 상관관계를 유지하면서, Treatment를 대체하도록 합니다. 따라서 진실된 인과효과는 0이 됩니다.

이 Estimator 가 올바르게 0에 가까운 값을 리턴하는지를 확인합니다.

## step 4 : refute

```
ref = model.refute_estimate(identified_estimand, 
                            estimate,
                            method_name = "placebo_treatment_refuter", 
                            placebo_type = "permute")

print(ref)
```

```
Refute: Use a Placebo Treatment
Estimated effect:3.984270555650861
New effect:-0.004644847508369445
p value:0.43999999999999995
```

## 2SLS와 비교

2-stage linear square regression 동일한 로직이기 때문에 기존 패키지의 coefficient, p-value 와 동일한 결과를 보입니다.

```
income_vec, endog = ps.dmatrices("income ~ education", data=df)
exog = ps.dmatrix("voucher", data=df)
```

```
m = IV2SLS(income_vec, endog, exog).fit()
m.summary()
```

| Dep. Variable:    | income           | R-squared:          | 0.890     |
| ----------------- | ---------------- | ------------------- | --------- |
| Model:            | IV2SLS           | Adj. R-squared:     | 0.890     |
| Method:           | Two Stage        | F-statistic:        | 1.373e+04 |
|                   | Least Squares    | Prob (F-statistic): | 0.00      |
| Date:             | Sun, 19 Dec 2021 |                     |           |
| Time:             | 09:48:50         |                     |           |
| No. Observations: | 10000            |                     |           |
| Df Residuals:     | 9998             |                     |           |
| Df Model:         | 1                |                     |           |

|           | coef    | std err | t       | P>\|t\| | \[0.025 | 0.975] |
| --------- | ------- | ------- | ------- | ------- | ------- | ------ |
| Intercept | 10.2193 | 0.314   | 32.568  | 0.000   | 9.604   | 10.834 |
| education | 3.9779  | 0.034   | 117.195 | 0.000   | 3.911   | 4.044  |

| Omnibus:       | 3.175 | Durbin-Watson:    | 1.953 |
| -------------- | ----- | ----------------- | ----- |
| Prob(Omnibus): | 0.204 | Jarque-Bera (JB): | 3.137 |
| Skew:          | 0.040 | Prob(JB):         | 0.208 |
| Kurtosis:      | 3.032 | Cond. No.         | 25.7  |


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