decision theory. hw 9 homework 9 in homework 9, i asked you to find a meta- analysis and summarize...

Decision Theory

HW 9

Homework 9

In Homework 9, I asked you to find a meta-analysis and summarize it for me.

You didn’t have to read it, just read the abstract and summarize what the abstract said.

Importantly, I told you to write your summaries in your own words.

The Point

What was the point of the exercise?

Here’s what I was hoping: first, I hoped you’d all realize how much good research there is out there and how easily accessible it is. I hope in the future if someone says to you “I read that drinking wine causes cancer,” you know how to find out whether that’s true, with some simple searching and a few minutes of reading.

Use Your CT Skills

Second, I wanted you to put some of the skills we learned in action. I wanted you to realize which information was important (effect sizes! Statistical significance!), and include that in your summary. To learn how to effectively communicate scientific findings to others.

Learn Something

Third, I was hoping you might actually learn something– not just about whatever was in the meta-analysis you discovered, but maybe in trying to understand it, you might learn, for example, what a cohort study is, or what a case-control study is, or what a cross-sectional study is. Or maybe just what ‘epidemiology’ means.

Case-Control Studies

I can’t resist. A case-control study is one where you find a bunch of people who have a certain condition (say, high blood pressure) and you ask them, as well as some people who don’t have that condition, “have you been a smoker in the past 10 years?” or “how many eggs do you eat?” or something else about their lifestyle and experiences. (You have to know the question in advance, otherwise you’d be finding your hypotheses in your data.)

Case-Control Studies vs. RCTs

Case-controls aren’t the same as randomized controlled trials, because your subjects aren’t randomly selected, nor are they randomly sorted into controls vs. non-controls.

But they can often tell us something, or suggest further research, or be very informative when combined in meta-analyses.

Cohort Studies

Cohort studies are studies where you find a group that has a certain feature (“eats lots of eggs” or “watches TV 5 hours a day”) and then you follow them over a long period of time to see if they develop any conditions disproportionate to the general population. (Once again, you have to state in advance which conditions you’re looking for, to avoid finding your hypotheses in your data.)

Case-Control vs. Cohort Studies

Case-control studies are obviously a lot cheaper than cohort studies (you just have to ask a group of people some questions, rather than follow them for years), but they are subject to various reporting biases– for example, people are more likely to underestimate the number of eggs they’ve eaten in the past year if they’ve eaten fewer eggs recently, and to overestimate if they’ve eaten more eggs recently.

Cross Sectional and Longitudinal Studies

Cross-sectional studies look at a random selection of the population (so not like case-controls or cohort studies) and try to find correlations in the data, say, between smoking and heart disease. Longitudinal studies are like cross-sectional studies and cohort studies combined, in that they follow a cross-section over a long period of time, making repeated observations.

A lot of you re-stated what the abstracts said without understanding them at all, and without spending the 5 or 10 minutes it would take to figure out what they meant. Sometimes you didn’t understand things we already covered in class, that you could have reviewed in the slides online.

Effect Sizes

For example: effect sizes. Almost none of you reported any effect sizes. “According to the study, wine consumption significantly decreases your risk of heart disease,” you would say. How much does it decrease your risk? That would seem to be one of the most important questions about a finding wouldn’t it? If it decreases your risk by only a little bit, or by a huge amount, that would matter, right?

Effect Size Reported as Relative Risk

A lot of you discussed this study, because it was the first google result for “meta-analysis wine” and “wine” was my first suggested thing to google. But few of you reported the effect size, even though it was right there:

0.68 = 68% relative risk for wine drinkers relative to non-drinkers.

Not an Absolute Risk

But what does that mean, 68% relative risk ratio?

Some of you thought it meant wine-drinkers developed heart disease 68% of the time. If that were true, no one should ever, ever drink wine!

Of course, that’s not what it means.

Relative Risks

There’s a certain absolute risk (probability) that one who does not drink wine (not-W) will develop heart disease (H). There’s also an absolute risk that a wine drinker (W) will develop heart disease. The ratio of the second risk to the first is the relative risk of heart disease for wine drinkers relative to non-wine drinkers:

RR = P(H/ W) ÷ P(H/ not-W) = 68%

Arithmetic

RR = P(H/ W) ÷ P(H/ not-W) = 68%

A little bit of arithmetic gets us:

P(H/ W) = 68% x P(H/ not-W)

So the risk of heart disease for wine-drinkers is only 68% of the risk of heart disease for non-drinkers.

What Does That Mean?

What does that mean? Well, each of the studies in the meta-analysis is going to have different study groups with different rates of disease. But on average the rates are about 12% in the general population. If the absolute risk of heart disease in wine drinkers is 68% of the absolute risk in non-drinkers, and we assume the risk is 12% for non-drinkers then the risk for drinkers is 0.12 x 0.68 = 8%. So 4% less.

The Point

Effect sizes are important. Beyond the fact that the results were statistically significant (which is usually required for publication), effect sizes are the most important thing to know about. And they’re often reported as odds ratios and risk ratios, even in popular news articles. If you don’t know what they mean, you cannot be helped by what you read. You may even be harmed by it. That’s why I spend time in class on this.

Just three more issues. First, another important thing about a meta-analysis is whether it is systematic or not (did the authors look at *all* the studies on a particular topic). Since this is so important, you might consider noticing it, and putting such information in your summaries.

I didn’t in my summary, but my summary was unique, and it was another chance to learn something. Remember my summary?

My Summary

“This study was a meta-analysis of randomized controlled trials that looked at the effects of flavonoid-rich foods like chocolate, soy, and tea on the risks of heart attacks. It analyzed 133 studies, and found that chocolate, soy, and tea all reduced blood pressure and cholesterol, but no study actually measured the variable of interest, heart attack rates. The authors concluded that future studies should be longer, so they can actually measure heart attack rates.”

Main Finding

The study was about flavanoids and heart attacks. And the main finding was that there were no studies that actually looked at flavanoids and heart attacks. All the studies that claimed to look at those things actually looked at surrogate outcomes.

Surrogate Outcomes

A surrogate outcome is an outcome that’s (maybe, possibly) related to the outcome you want to consider.So for instance, if I wanted to find out who was going to get the best grade in class, I could actually measure all your grades and then calculate the final. Or I could measure your grade on just HW9 and use that as a “surrogate.”

Important Lesson

One important lesson is that studies that investigate surrogate outcomes tell you about surrogate outcomes, not about the outcomes you care about (like heart attacks).For example, a lot of people say things like “pomegranates help prevent cancer” because they increase a surrogate outcome, anti-oxidants. We’ve now discovered, unfortunately, that anti-oxidants cause cancer.

Using Your Own Words

The final issue is using your own words. Part of the reason I was so adamant about this is because I wanted you to try to understand the meta-analyses you read, so you could gain all the benefits (exercising skills, learning something new). It’s not so much that I care that you plagiarized, as I care that you didn’t learn anything when you wrote down “cohort studies” or “68% relative risk” without having any idea what that meant, or why anyone should care.

DECISION THEORY

Choices

Critical thinking, if you’ll recall, is not just about evaluating arguments and evidence in order to arrive at the truth.

It’s also about evaluating your goals, and your strategies for achieving them, so as to make the best (most rational) choices.

Decision Theory

Decision theory is a field of study that spans across economics, social science, philosophy, mathematics, and even business.

The goal of decision theory is to find general principles that can guide our choices in ways that are most likely to achieve what we want.

Not an Ethical Theory

Decision theory, however, is not a moral code. It explains how best to achieve what you want– even if what you want is becoming a murderer, becoming a millionaire, or becoming a monk.

The abstract problem in each of these cases is always the same: what decision is best for achieving my ends?

Could Be an Ethical Theory

Of course, some philosophers have thought that you can get an ethical theory out of the theory of rational choice (aka decision theory).

Will individuals who act rationally for their own benefit always form a just society? If the answer is yes, then decision theory has something to tell us about justice.

Problem Specification

Decision theoretic analysis begins with a problem specification: an analysis of a situation into acts, states, and outcomes.

Example

Suppose that I enter a dark room. There is a strong smell of gasoline in the room. I feel around on the walls for a light switch, but I cannot find one. I have matches in my pocket, and they would allow me to see, but lighting a match could cause an explosion. I have a decision to make: whether or not to light the match.

Acts

The first aspect of the problem specification involves figuring out the different possible acts I can take.

Here, the relevant acts are: “light a match” and “don’t light a match.” There are other acts I could take (“scratch my nose”) but these acts aren’t relevant to the problem at hand.

Outcomes

We also need to know what the relevant outcomes are: what could result from taking one act or another act?

If I don’t light a match, then there will definitely not be an explosion. But if I do light a match, there are two possible outcomes: “there is an explosion” and “there is not an explosion.”

States

The outcome depends not just on my act, but on something I may not know about– the state of the world.

Either there is an explosive level of gasoline fumes in the air or there is not. If there is an explosive level, and I light a match, then there will be an explosion; otherwise I am safe.

Decision Table

State: Explosive gas

State: Non-explosive gas

Act: Light a match

Outcome: explosion

Outcome: no explosion

Act: Don’t light a match

Outcome: no explosion

Outcome: no explosion

Betting on Rugby

It’s not as easy as it might seem to draw up a decision table. Consider the following scenario: you have to choose whether to bet on the Hong Kong Dragons (rugby team) or the Cherry Blossoms (Japanese team). If you bet on the Dragons, you win $500 if they win, lose $200 if they lose; if you bet on the Cherry Blossoms, you win $300 if they win, and lose $200 if they lose.

Reasonable Table:

Dragons Win Cherry Blossoms Win

I bet on Dragons

+$500 -$200

I bet on Cherry Blossoms

-$200 +$300

Non-Obvious Bet

In this table, it’s not obvious who you should bet on. If you bet on the Dragons, it’s always possible that you could have done better by betting on the Cherry Blossoms. And if you bet on the Cherry Blossoms, it’s always possible that you could have done better by betting on the Dragons. You have to decide based on how likely you think one or the other team is to win.

Re-stating the Problem

However, I could describe the decision problem in a slightly different way.

There are two states that I really care about: either I win my bet or I lose my bet. Let’s draw the decision table up with those states instead of “Dragons win” and “Cherry Blossoms win.”

New Table:

I Win My Bet I Lose My Bet

I bet on Dragons

+$500 -$200

I bet on Cherry Blossoms

+$300 -$200

Obvious Bet

Now it seems like I should definitely bet on the Dragons, regardless of the odds that they will win. Consider this reasoning: If I bet on the Dragons and the state is “I win my bet” then I get $500, which is better than what I would have gotten from betting on the Cherry Blossoms. If the state is “I lose my bet” then I lose $200, but that’s the same amount I would have lost had I bet on the Cherry Blossoms.

The Dominance Principle

Betting on the Dragons is always better or equal to betting on the Cherry Blossoms, so clearly I should bet on the Dragons.

This is known as the dominance principle: if one action results in outcomes that are better than or equal to the outcomes of every other action, then you should prefer it.

What Has Gone Wrong?

What has gone wrong? If I analyze the problem one way, it’s unclear how to act, but if I analyze it a different way, it recommends betting on the Dragons.

The first thing to notice is that the second way of analyzing the problem can’t be right– it recommends the Dragons, no matter how bad they are!

The issue here is that “I win my bet” is not really one state, it’s two states: it’s the state where the Dragons won (when I bet on them) and it’s the state where the Cherry Blossoms won (when I bet on them). If you bet on the Cherry Blossoms and win, you would never say, “Darn! Since the state is ‘I won my bet’ I should have bet on the Dragons– then I would have $500!” You wouldn’t– because the Dragons didn’t win.

In general, the states should not depend upon the acts. Whether “I win my bet” is true depends upon who I bet on. In our reasonable table, the states were “Dragons win” and “Cherry Blossoms win”– these states are independent of which team you bet on (which act you choose).

Another Example

During the Cold War between the United States and the Soviet Union, many people argued that it was best for the U.S. to get rid of its nuclear weapons, even if the Soviet Union did not disarm.

The ‘Dove’ Argument

War with USSR No War with USSR

U.S. Disarms U.S. Becomes Communist

More Money for U.S.

U.S. Does Not Disarm

Everyone Dies What We Have Now

The ‘Dove’ Argument

Given this way of analyzing the decision problem, disarmament is clearly superior: if the U.S. disarms and there’s war, then it becomes communist… but that’s better than everyone dying in a nuclear holocaust. If the U.S. disarms and there’s no war, then the money used for nuclear programs can be used to improve society… and that’s better than spending a bunch of money on a nuclear program.

Same Problem

But this overlooks a crucial fact: the probability that certain states will obtain depends on the acts in the table. If the U.S. disarms unilaterally, then it becomes much more likely that the USSR will attack and take over. We need a table where the states are independent of the acts chosen.

Amended Table

USSR Will Attack Anyways

USSR Will Attack Only If US Disarms

USSR Will Not Attack

U.S. Disarms U.S. Becomes Communist

U.S. Becomes Communist

More Money for U.S.

U.S. Does Not Disarm

Everyone Dies What We Have Now

What We Have Now

Here, the states are independent of the acts. We don’t have to know whether the US disarms or not in order to know “the USSR attacks anyway” or “the USSR only attacks if the US disarms.” We can know those things before the US disarms (or not) by asking Soviet politicians and policy makers. But once we make the table this way, we realize no act dominates the other.

Immediate Choices

When we make a decision table, we have to make choices about how to describe the acts and states. And obviously decision theory can’t help us with those choices. It can’t help us with a lot of choices: if a bear is chasing you, you don’t need or want a decision table to make a choice to run away. But for more difficult problems that require advance planning, decision theory can be useful.

Right Choice vs. Rational Choice

Decision theory is not concerned with making the right choice– it’s concerned with making the rational choice when you don’t know what the right choice is.The right choice is the choice that gets you the best outcome. If you are betting on sports, you typically won’t know what the right choice is (who will win). Sometimes the rational choice can still be the wrong choice.

Varieties of Decisions

Decision theorists classify decisions into three categories:

1. Decisions under certainty2. Decisions under risk3. Decisions under ignorance

Decisions under Certainty

Sometimes you know for certain what outcome an act will have. At a restaurant you are faced with a decision: do I order the soup or the salad? You know that if you order the soup, the outcome will be that you get soup, and if you order the salad, the outcome will be that you get salad. Any decision where all the acts are like this– you know their outcomes in advance– is a decision under certainty.

Hard Decisions

Just because a decision is made under certainty does not mean it is an easy or obvious decision.

Deciding which classes to take is a decision under certainty (often): you get what you sign up for. But should you take philosophy this semester or music? That can be a difficult decision.

Formal Approaches

We won’t be talking about decisions under certainty in this class.

There are formal approaches to these problems (for example, in the mathematics of linear programming), but that’s a little bit out of our range.

Decisions under Risk

Sometimes it is possible to assign determinate (or approximate) probabilities to various states.

For example, when I flip a fair coin, I know that the probability that it lands heads will be 0.5 and the probability that it lands tails will be 0.5. So if I bet on a fair coin (or an unfair coin, when I know its probabilities) that is a decision under risk.

Idealization

A lot of things are very difficult to assign exact probabilities to, but that doesn’t mean they shouldn’t be treated as decisions under risk.

Sometimes confronting a problem requires idealization. You don’t know exactly the probability that your plane flight will crash, but you can estimate it by considering the number of crashes divided by the number of flights.

Next Week

Next week we’ll talk about decisions under risk. They require a little bit more of a mathematical approach.

The general idea is to calculate which outcome we expect to happen given each of our acts, and then to choose the act that has the best expected outcome.

Decisions under Ignorance

A large number of decisions are decisions under ignorance (also known as a decision under uncertainty).

These are decisions where we cannot assign probabilities to some of the states, or where our estimates are too wide (“somewhere between a 20% and 80% chance”).

Example

For example, suppose you go on a first date with someone. What is the probability that you will wind up getting married to them and subsequently having children and grandchildren with them? You certainly can’t get an estimate by dividing your total number of marriages by your total number of first dates– you probably don’t have a very high number of either.

Example

So suppose your friend calls you and asks if you want to go see the new James Bond movie instead of going on this date. You really want grandchildren someday, but you can’t assign a probability to the state that you will have grandchildren with your date. This is a decision under ignorance.

Second Example

Suppose that Susan is starting her career in business. She has been very successful so far, and has made lots of contacts, and thinks that if she keeps at it, she will be able to become very rich.

Susan’s Choice

Susan is recently married, and she wants to have children some day. She is faced with the following decision: should she have children now and postpone her career, or should she keep on the successful track she’s on and have children later? She knows whatever she does now she’ll be good at, but she doesn’t know what she’ll be good at later.

Decision under Ignorance

Susan has read a lot of parenting books and studies done on parenting late in life, so she is pretty confident that she can assign a probability to being a good parent later vs. now. However, she has no idea whether she will be able to resume a successful business career after staying home for years to raise her children. She cannot assign that state a probability, so this is a decision under ignorance.

Note

[Note: I’m not saying that women have to stay home when they have children. Susan just happens to be conservative, and she thinks that women should not work when their children are young. As decision theorists, we don’t judge what other people’s goals are, we just help them make the best decisions to achieve those goals.]

Acts

The acts here are clear:

A1: Susan has children now, and a career later.

A2: Susan has a career now, and children later.

Relevant States

S1: In 7 years, Susan will be able to be both a good mother and a good businesswoman.S2: In 7 years, she will be able to be a good mother, but not a good businesswoman.S3: In 7 years, she will not be able to be a good mother, but she will be able to be a good businesswoman.S4: In 7 years, she will neither be able to be a good mother nor a good businesswoman.

Outcomes

The outcome for each act-state pair is pretty clear here. For example, for A2 and S1:A2: Susan has a career now, and children later.S1: In 7 years, she will be able to be a good mother but not able to be a good businesswoman.O1: She is a good mother later and a good businesswoman now.

It will be easier if we can summarize things as follows:

M = Susan is a good mother.─M = Susan is not a good mother.B = Susan is a good businesswoman.─B = Susan is not a good businesswoman.

Susan’s Decision Table

S1 S2 S3 S4A1:

children now

M & B M & ─B M & B M & ─B

A2: career now

M & B M & B ─M & B ─M & B

Preferences

How is Susan going to make this decision. Well, first, a decision theorist needs to know her preferences. Which outcomes does she prefer to which other outcomes.

For example, clearly Susan prefers M & B to ─M & B: she prefers being a good mother and having a good career to just being a bad mother and having a good career.

Preferences

But what about what happens in S4: if she chooses children now, she’ll be a good mother, but not a good businesswoman (M & ─B). If she chooses a career now, she’ll be a good businesswoman, but not a good mother (─M & B). Which of these two does she prefer, being only a good mother or only a good businesswoman?

Indifference

Susan might have a definite preference, but we also allow her to be indifferent: she might think that (M & ─B) and (─M & B) are equally good outcomes, and she would be equally happy with either of them.

Rational Preferences

We’re still not in a position to help Susan. In order to help her, the decision theorist requires that her preferences be rational. Rationality here has nothing to do with which things she prefers. It’s OK if she prefers ─M & ─B to M & B. That would be strange, but it’s not irrational.

Rational Preferences

Rational preferences have the following features:

1. They are connected.2. They are (appropriately) asymmetrical.3. They are transitive.

Connected

Your preferences for outcomes O1, O2, O3,… are connected if, for any two of those outcomes X and Y:

Either you prefer X to YOr you prefer Y to XOr you are indifferent between X and Y

Asymmetrical

Additionally, rational preferences have certain asymmetries. If X and Y are outcomes, then:

You can’t both prefer X to Y and prefer Y to X.You can’t both prefer X to Y and be indifferent between X and Y.

Note

We can combine connectedness and asymmetry and get:One and only one of the following is true for any two outcomes X and Y:

You prefer X to Y.You prefer Y to XYou are indifferent between X and Y.

Transitivity

Rational preferences are also transitive, which means:If you think X is better than Y, and you think Y is better than Z, then you think X is better than Z too.If you think X is better than Y, and you think Y and Z are the same (you’re indifferent), then you think X is better than Z.

Transitivity Continued

If you think X is better than Y, and you think X and Z are equal (you’re indifferent), then you think Z is better than Y.

If you think X and Y are equal, and Y and Z are equal, then you think X and Z are equal.

Intransitivity Problems

What’s wrong with intransitive preferences? Well suppose your preferences are intransitive in the following way:

• You prefer A to B• You prefer B to C• You prefer C to A