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代写接单- SOST 30172/70172: Quantitative Evaluation of Policies, Interventions and Experiments

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SOST 30172/70172: Quantitative Evaluation of Policies, Interventions and Experiments

06/03/2020

Introduction: glorifying higher education. . .

Meet Raymond and Bruno.

Introduction: glorifying higher education. . .

Raymond is capable; Bruno less so

Introduction: glorifying higher education. . .

University is good for Raymond and Bruno

Introduction: glorifying higher education. . .

Raymond is up for it; Bruno is not sure

Introduction: glorifying higher education. . .

Raymond v Bruno over-estimates the benefit of University

Introduction: thrashing physiotherapy. . .

Raymond is fit; Bruno, was recently injured. . .

Introduction: thrashing physiotherapy. . .

Physiotherapy is good for Raymond and Bruno,

Introduction: thrashing physiotherapy. . .

But Raymond does not need it; Bruno does. . .

Introduction: thrashing physiotherapy. . .

Raymond v Bruno, under-estimates the effect of physiotherapy

Introduction

You want to know if higher education, physiotherapy, etc. . . is good/bad for people (i.e. the effect of a policy or intervention)

To answer the question you need to know how Raymond would fare with AND without the policy. . .

. . . and you’d also need to know how Bruno had fared with AND without the policy.

However, you will only observe each person in one state of the world (either under the policy or in the absence of the policy).

Introduction

The assignment of a treatment (policy) on the people of a population will induce a behaviour

I Raymond goes to college and avoids physiotherapy I Bruno uses physiotherapy, but does not go to college

Data will reveal that behaviour.

The problem arises if people are assigned to (or choose) treatment depending on its perceived benefits

(e.g. physiotherapy; higher education).

Then, the behaviour that data can reveal does not fully (or at all) characterise what you need, which is

1. what had happened in the population if everyone had got the treatment

2. what had happened in the population if nobody had got the treatment

Introduction

This course is about understanding

I When the process of treatment assignment can induce behaviours in a population that reveal the true effect of the treatment

I If people self-select into treatment, we will try to understand if we can identify sub-populations for which true treatment effects are estimable

I If those sub-populations do not exist, then we will try to figure out assumptions under which we can claim that we might be able to estimate a treatment effect. . . and we will develop an understanding about how realistic those assumptions might be.

What we want to know. . .

To answer the preceding question we would want to know, for every individual in society,

1. What would happen to a person if she was subjected to the Active Treatment (e.g. higher education; physiotherapy).

2. What would happen to the same person if she was subjected to the Control Treatment. (e.g. higher education; physiotherapy)

Once we know 1 and 2 above, we know if the active treatment affects people (and how).

. . . are causal questions.

What are causal questions?

Questions about the effects caused by a treatment1. For a specific person we would ask:

I What would happen to that person under the active treatment?

I What would happen to that person under the control treatment?

I Would the person be better under the active or control treatments?

Causal effects are comparisons of potential outcomes under alternative treatments.

1‘Treatment’ must be broadly understood, not just as a medical innovation, but also as a policy, intervention, or form of manipulation.

Units

In a causal study, we will generally have a number of participants. These could be people, firms, countries, bacteria. . .

Rather than ‘participants’ we will talk about units.

Units have a ‘stamp date’: they are defined at a specific point in

time.

So, ‘Peter Smith’ is one unit on the 12th of January and a different unit on the 13th of January.

Units

We could refer to units by their name, but it is more efficient to number them i = 1,2,...,N

Here N is the total number of units in a study.

We can refer to a generic unit i and thus make statements that are true for each and every individual

Assignment (to Treatment/Control) Indicator

We use Zi to record a unit’s assignment to treatment. Zi = 1 if the unit received the active treatment; otherwise Zi = 0.

Causal studies might involve many different treatments; however in this course we restrict our attention to a two treatment situation:

I An active treatment (typically the new policy, innovation, technology. . . )

I A control treatment (a placebo, no action, the status quo)

Often we will just say treatment and control to refer to the active treatment and the control treatment respectively.

Potential Outcomes

Each unit i in a causal study has two potential outcomes:

I one if the unit receives the active treatment, denoted Yi(1) = Y1,i

I one if the unit receives the control treatment, denoted Yi(0) = Y0,i

In general, sinc Zi denotes assigment to treatment, we can refer to both potential outcomes at once, by writting Yi(Zi).

Cough. . .

Can a single nocturnal dose of buckwheat honey be effective to alleviate cough and improve sleep quality? Paul et al. [2007]

Cough is the reason for nearly 3% of all outpatient visits in the United States, more than any other symptom.

It most commonly occurs in conjunction with an upper respiratory tract infection.

At night, it is particularly bothersome because it disrupts sleep.

Despite the common occurrence of cough, there are no accepted therapies for this symptom.

Yet, consumers spend billions of dollars per year on medications for cough, particularly those with Dextromethorphan2

2Which is not supported as effective by the American Academy of Paediatrics or the American College of Chest Physicians.

Potential Outcomes

A Causal Effect is any comparison of a unit’s two potential outcomes, such as

τi = Yi(1) − Yi(0) τi′ = Yi(1)/Yi(0)

τ′′ = log ?Y (1)? − log ?Y (0)? and so on... (1) iii

The notation Yi(Zi) emphasises that unit i’s potential outcomes depend only on the unit’s assignment to treatment3.

3See discussion on SUTVA in a later lecture.

Potential Outcomes

The definition of treatment effects.

A first, minor, problem is that the unit treatment effect might vary across people.

Say we focus on τi = Yi(1) − Yi(0); then τi can change from person to person.

This means that, in the population, τi might have a distribution. What we do, in these situations, is to focus on the Average

Treatment Effect,

ATE = E [Yi(1) − Yi(0)] = E [Yi(1)] − E [Yi(0)]

The definition of treatment effects.

What these definitions of treatment effects imply is that, since we want to estimate AT E = E [Yi(1) − Yi(0)], which is a population characteristic, we need:

I The full population distribution/density of Yi(1), or at least E (Yi (1))

I The full population distribution/density of Yi(0), or at least E (Yi (0))

Identification (the major problem).

Why are causal questions difficult?

Because a person receives only one treatment, not both.

We cannot see how a person had fared under both the active and control treatments. We never see the causal effect for any one person.

Identification (the major problem).

Given a dataset, we only observe one potential outcome per unit: Yi(1) if a unit is assigned to treatment, Zi = 1, and Yi(0) otherwise -Zi = 0 or 1 − Zi = 1). That is, we observe4

Yi =Zi ·Yi(1)+(1−Zi)·Yi(0) (2)

Therefore we cannot estimate a unit’s casual effect

(e.g. τi = Yi(1) − Yi(0)). We say that units’ causal effects are not

identified.

The imperfect solution to not observing both Yi(1) and Yi(0) is to collect data from many units (some in the active treatment group, some in the control treatment group), and see what can be done with these data. . .

4ThisfollowsbecauseifZi =1,then1−Zi =0andYi =Yi(1); conversely if Zi = 0, 1−Zi = 1 and Yi = Yi(0)

Identification (the major problem).

To compute causal effects such as AT E = E [Yi(1) − Yi(0)] exactly, we need to have the population distribution of Yi(1) and Yi(0), or at least their expected values.

Data only reveal some bits of those distributions:

I People who received the treatment contribute partial information about the distribution of Yi(1)

I People who received the control treatment contribute partial information about the distribution of Yi(0)

To what extent are these bits of information sufficient to accurately approximate the true distributions of Yi(1) and Yi(0) in the population? This is the central question of causal inference.

Identification (the major problem).

So, once a data set is available, what are these data capable of revealing about Yi(1) and Yi(0)?

It will depend on

I The nature of the treatment

I The availability of the treatment to people

I The extent to which the researcher controls the treatment I The extent to which a person can get hold of the treatment

on the basis of, say, the perceived benefits of that treatment I The extent to which a unit’s treatment affects other units’

treatment.

We face a problem of Identification (what do data reveal); not of statistical inference/methods (neither the amount of data nor the sophistication of the statistical model are at the core of causal inference).

An implicit assumption: SUTVA.

The notation Yi(Zi) is not at all naive. It makes two implicit assertions:

1. There is only one version of the treatment.

2. A unit’s potential outcomes depend only on the unit’s

assignment to treatment.

In practice it is systematically assumed that both conditions hold.

However, neither of these assumptions need to be true in practice.

An implicit assumption: SUTVA.

Stable Unit Treatment Value Assumption (SUTVA):

1. No interference among units The treatment applied to one unit does not effect the outcome for another unit.

2. No hidden variation of treatment There is only a single version of each treatment level.

An implicit assumption: SUTVA.

SUTVA: No interference

This was a main concern in early work in agricultural experiments. In those settings, “guard rows” were used to separate experimental plots and avoid contamination

(interference).

This can be a strong assumption in observational studies. For instance, in studies of immunisation, where inoculation of vaccines to substantial sub-populations can generate positive externalities for the non-vaccinated)

An implicit assumption: SUTVA.

How does interference complicate things up?

Suppose you want to study the effect of promotions on performance at work. A firm has two types of worker: half are Normans and half are Psychos. The latter are awful characters and their promotion would depress Normans if they are not promoted as well.

The potential outcomes for each type of worker are: Norman: YN(1) and YN(0) − W · ZP

Psycho: YP (1) and YP (0)

Here ZP is the treatment assigned to a Psycho (ZP = 1 if Psycho is promoted, otherwise ZP = 0). W is the loss of productivity that a Norman would experience if a Psycho gets promoted and Norman does not.

In both cases, assume Y (1) > Y (0).

An implicit assumption: SUTVA.

How does interference complicate things up?

The true average treatment effect of promotion is

AT E = 12 E hYN (1) − YN (0) + YP (1) − YP (0)i > 0

But because of the violation of SUTVA, data will tend to reveal not YN(0), but YN(0)−W ·ZP, so in the best case scenario, data will tend to approximate this parameter instead:

ATEFalse =1hYN(1)−YN(0)+YP(1)−YP(0)i 2

+ 12 W · Z P

which over-estimates the effect of promotions.

An implicit assumption: SUTVA.

SUTVA No hidden variation of the treatments Again, this can be problematic in practice.

I For instance, if testing a new pill, the concentration in all the pills must be the same.

I If studying retirement, some individuals might be partially retired, and variation in part time work might be substantial.

This component of SUTVA is very often not explored in research (a mistake).

An implicit assumption: SUTVA.

SUTVA is imposed on the population by default

It can be, as we have seen, a strong assumption.

And, yet, it does absolutely nothing to help us to identify causal effects.

Bibliography I

Ian M Paul, Jessica Beiler, Amyee McMonagle, Michele L Shaffer, Laura Duda, and Cheston M Berlin. Effect of honey, dextromethorphan, and no treatment on nocturnal cough and sleep quality for coughing children and their parents. Archives of pediatrics & adolescent medicine, 161:1140–1146, December 2007. ISSN 1538-3628. doi: 10.1001/archpedi.161.12.1140.