Instrumental variables estimationWhat If: Chapter 16Elena Dudukina2022-03-211 / 17

16.1 The three instrumental conditions

RCT
- Z is the randomization assignment
- A is an indicator for receiving treatment
- Y is the outcome
- U is all factors (some unmeasured) that affect both the outcome and the adherence to the assigned treatment A
Need to ensure conditional exchangeability of the treated and the untreated
- Closing backdoor A ⬅ U ➡ Y not feasible when U or its components are unmeasured
Alternatively, can use non-backdoor approach
- Z is an instrumental variable (IV)
Z is a causal instrument

2 / 17

16.1 The three instrumental conditions

Z is associated with A
Z does not affect Y except through its potential effect on A
Z and A do not share causes

$U_{z}$ is a proxy instrument
- 16.2 $U_{z}$ is a common cause of A and Z
- 16.3 $U_{z}$ are associated due to conditioning on S

3 / 17

16.1 The three instrumental conditionsZ: the price of cigarettes
Check that Z and A are associatedPr[A=1|Z=1]−Pr[A=1|Z=0]>0Pr[A=1|Z=1]−Pr[A=1|Z=0]>0


data %>% group_by(highprice) %>% 
  count(qsmk) %>% 
  mutate(
    pct = n/sum(n) * 100
  ) %>% 
  filter(qsmk == 1, !is.na(highprice))

## # A tibble: 2 × 4
## # Groups:   highprice [2]
##   highprice qsmk      n   pct
##       <dbl> <fct> <int> <dbl>
## 1         0 1         8  19.5
## 2         1 1       370  25.8
4 / 17

16.1 The three instrumental conditionsZ is a weak instrument
Conditions 2 and 3 cannot be empirically verified
5 / 17

16.2 The usual IV estimandThe average causal effect of A on Y on the additive scale E[Ya=1]−E[Ya=0]E[Ya=1]−E[Ya=0] isE[Y|Z=1]−E[Y|Z=0]E[A|Z=1]−E[A|Z=0]E[Y|Z=1]−E[Y|Z=0]E[A|Z=1]−E[A|Z=0]
numerator of the IV estimand is the average causal effect of Z on Y: intention-to-treat effect
denominator is the average causal effect of Z on A: a measure of adherence to the assigned treatment A
the higher the noncompliance, the bigger the difference between the effect of Z on Y and the effect of A on Y

6 / 17

16.2 The usual IV estimand

IV estimand bypasses the need to adjust for the confounders
numerator model: $E [A | Z] = α_{0} + α_{1} Z$
denominator model: $E [Y | Z] = β_{0} + β_{1} Z$

# an instrument as weak if the F-statistic from the first-stage model is less than 10
# two-stage-least-squares regression
library(ivreg)
ivreg(wt82_71 ~ qsmk | highprice, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")

## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1     2.40      19.8     0.121   0.904    -36.5      41.3

# Chapter answer: 2.4 kg (-36.5; 41.3)

7 / 17

16.2 The usual IV estimandStrong parametric assumptions for two-stage-least-squares regression
Only valid as an estimator when the IV estimand can be interpreted as the average causal effect of treatment, whoch requires and additional assumption
8 / 17

16.3 A fourth identifying condition: homogeneityHomogeneityConstant effect of treatment A on outcome Y across all individuals (unrealistic)
Smoking cessation made everyone in the study population gain (or lose) the same weight
Equality of the average causal effect of A on Y across levels of Z in both the treated and in the untreated
confounders U are not additive effect modifiers (unrealistic)
Z-A association is constant across levels of the confounders U

Can IV methods validly estimate the average causal effect of treatment without homogeneity assumption?
9 / 17

16.4 An alternative fourth condition: monotonicity

Always-takers: $A^{z = 1} = 1$ and $A^{z = 0} = 1$
Never-takers: $A^{z = 1} = 0$ and $A^{z = 0} = 0$
Compliers: $A^{z = 1} = 1$ and $A^{z = 0} = 0$
Defiers: $A^{z = 1} = 0$ and $A^{z = 0} = 1$
"Monotonicity is not always a reasonable assumption in observational studies"
- No always-takers or defiers in RCTs by design

10 / 17

16.5 The three instrumental conditions revisited"Even in large samples, weak instruments introduce bias in the standard IV estimator and result in underestimation of its variance"
"The effect estimate is in the wrong place and the width of the confidence interval around it is too narrow"
Different IV definitions
Definition 2
Definition 3
Definition 4
Definition 5
data %<>% mutate(highprice2 = if_else(price82 >= 1.6, 1, 0),
          highprice3 = if_else(price82 >= 1.7, 1, 0),
          highprice4 = if_else(price82 >= 1.8, 1, 0),
          highprice5 = if_else(price82 >= 1.9, 1, 0)
         )
ivreg(wt82_71 ~ qsmk | highprice2, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")
## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1     41.3      165.     0.250   0.802    -282.      365.
ivreg(wt82_71 ~ qsmk | highprice3, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")
## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1    -40.9      188.    -0.218   0.828    -409.      327.
ivreg(wt82_71 ~ qsmk | highprice4, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")
## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1    -21.1      28.4    -0.742   0.458    -76.9      34.7
ivreg(wt82_71 ~ qsmk | highprice5, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")
## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1    -12.8      23.7    -0.541   0.588    -59.2      33.6
11 / 17

16.5 The three instrumental conditions revisitedFor each of additional definitions of a high price, the 95% confidence interval around the estimate is very wide but is an underestimate of the true uncertainty
Stronger instrument that violates conditions 2 and 3 may be preferable to a more valid (less invalid) but weaker instrument
12 / 17

16.5 The three instrumental conditions revisited

Condition 2 is unverifiable from the data
- Absence of a direct effect of the instrument on the outcome (violation on fig 16.8)
- May be violated when a continuous or multilevel treatment is dichotomized
- "Even if condition 2 holds for the original treatment A, it does not have to hold for its dichotomized version A∗, because the path Z → A → Y represents a direct effect of the instrument Z that is not mediated through the treatment A∗ whose effect is being estimated in the IV analysis (16.9)

13 / 17

16.5 The three instrumental conditions revisited

Condition 3: confounding for the effect of the instrument on the outcome
- Unverifiable
- Adjust for known covariables
- Unverifiable assumption that there is no unmeasured confounding for the effect of Z on A within levels of the measured pre-instrument covariates V
- "Apply IV estimation repeatedly in each stratum of V, and pool the IV effect estimates under the assumption that the effect in the population (under homogeneity) or in the compliers (under monotonicity) is constant within levels of V"

14 / 17

16.5 The three instrumental conditions revisited

ivreg(wt82_71 ~ qsmk + sex + race + age + smokeintensity + smokeyrs + exercise + active + wt71 |  highprice5 +  sex + race + age + smokeintensity + smokeyrs + exercise + active + wt71, data = data) %>% 
  broom::tidy(., conf.int = T) %>% filter(term == "qsmk1")

## # A tibble: 1 × 7
##   term  estimate std.error statistic p.value conf.low conf.high
##   <chr>    <dbl>     <dbl>     <dbl>   <dbl>    <dbl>     <dbl>
## 1 qsmk1    -2.38      17.3    -0.138   0.891    -36.2      31.5

15 / 17

16.6 Instrumental variable estimation versus other methodsIV estimation requires modeling assumptions even if infinite data (on super-popultion) were available
Homogeneity condition is equal to setting to 0 the parameter corresponding to a product term in a structural mean modelEstimation cannot be nonparametric

Relatively minor violations of IV conditions "may result in large biases of unpredictable or counterintuitive direction"IV estimate may be more biased than an unadjusted estimate when IV conditions are violated

Hard to think about common causes of IV and the treatment
"Standard IV estimation is more restrictive than that for other methods" and "used to answer relatively simple causal questions"
16 / 17

References

Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC (v. 30mar21)
R ivreg package

17 / 17

16.1 The three instrumental conditions

RCT
- Z is the randomization assignment
- A is an indicator for receiving treatment
- Y is the outcome
- U is all factors (some unmeasured) that affect both the outcome and the adherence to the assigned treatment A
Need to ensure conditional exchangeability of the treated and the untreated
- Closing backdoor A ⬅ U ➡ Y not feasible when U or its components are unmeasured
Alternatively, can use non-backdoor approach
- Z is an instrumental variable (IV)
Z is a causal instrument

↑, ←, Pg Up, k	Go to previous slide
↓, →, Pg Dn, Space, j	Go to next slide
Home	Go to first slide
End	Go to last slide
Number + Return	Go to specific slide
b / m / f	Toggle blackout / mirrored / fullscreen mode
c	Clone slideshow
p	Toggle presenter mode
t	Restart the presentation timer
?, h	Toggle this help

Instrumental variables estimation

What If: Chapter 16

Elena Dudukina

2022-03-21

16.1 The three instrumental conditions

16.1 The three instrumental conditions

16.1 The three instrumental conditions

16.1 The three instrumental conditions

16.2 The usual IV estimand

16.2 The usual IV estimand

16.2 The usual IV estimand

16.3 A fourth identifying condition: homogeneity

16.4 An alternative fourth condition: monotonicity

16.5 The three instrumental conditions revisited

16.5 The three instrumental conditions revisited

16.5 The three instrumental conditions revisited

16.5 The three instrumental conditions revisited

16.5 The three instrumental conditions revisited

16.6 Instrumental variable estimation versus other methods

References

16.1 The three instrumental conditions

Help