Behavioral Data Mining. Lecture 19 Regression and Causal Effects

Transcription

1 Behavioral Data Mining Lecture 19 Regression and Causal Effects

2 Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression

3 Counterfactuals Counterfactual (N): Contrary to Fact What if the thermostat were set to 65? (sublime) What if kangaroos had no tails? (ridiculous)

4 Counterfactuals What if the thermostat were set to 65? Ceteris Paribus all other things being equal i.e. you didn t open the window. The intervention is clear. What if kangaroos had no tails? What is the intervention? How is all else kept equal?

5 Counterfactuals From the 2000 Presidential election in Florida: What if butterfly ballots had been replaced by standard ballots? - estimate the results of the election, which means modeling how each voter would have voted with the alternative ballot.

6 Potential Outcomes Counterfactual analysis is based on the notion of Potential Outcomes the idea that every unit could have received every possible treatment. i.e. the thermostat could have been set to any one of many temperature settings, or each Florida voter could have filled out one of two kinds of ballot. We only observe one treatment for each unit, but the model assumes that all the others are plausible outcomes in a complete dataset. The unobserved outcomes are counterfactual.

7 Treatments We will mostly focus on two-state treatments, and the states will conventionally be called Treatment and Control. The potential outcomes for individual i will be labeled y i 1 and y i 0 where the 1 superscript means treatment and the 0 superscript means control. We are typically interested in differences like y i 1 - y i 0 but only one alternative will have been observed. The other has to be imputed.

8 Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression

9 Regression Suppose a response variable Y depends only on a treatment variable D and a confounding variable S. We can fit a linear regression formula Y = αd + βs + γ representing the ideal geometry below (blue lines are contours of equal Y): D D = 1 D = 0 Strata by S slope - / S

10 Regression For the formula Y = αd + βs + γ, the average causal effect (Treatment Control), in fact the effect in every stratum, is This is true for any number of confounders S, since fixing S simplifies the formula to Y = αd + c D D = 1 Strata by S slope - / D = 0 S

11 Complications First of all, the dependence on D and S may not be perfectly linear. Regression models non-linear dependence on all variables, when we really want a best approximation for dependence on D. D D = 1 Strata by S D = 0 S

12 Complications Even given a true model which is perfectly linear, Y = αd + βs + γ, if there is an unmodeled confounder C which is correlated with D, the coefficient will be mis-estimated. D D = 1 D = 0 Strata by S slope - / S

13 Indicator Variables Rather than treating D and S as continous variables, we can encode them using indicator variables which are true iff each variable is equal to a specific value. i.e. rather than Y = αd + βs + γ, write Y = αd + β 2 S2 + β 3 S3 + γ Where D takes on values 0,1 and S2 = 1 iff S=2, S3 = 1 iff S=3. This allows the model to tolerate differences in offsets at S=2 and S=3. But it cannot model arbitrary non-linear dependences.

14 Potential Outcome Variables We use the notation Y 1, Y 0 to represent the potential outcomes corresponding to treatment and non-treatment. Note that these values do not depend on the actual assignment of a unit, which is given by D. Instead we have: or Y = Y 1 if D = 1 Y 0 if D = 0 Y = DY D Y 0 Which is to say that the actual assignment agrees with the potential outcome for that assignment.

15

16 Solution Note: We haven t yet made any causal estimates, we have only estimated actual and counterfactual expected values.

17 Fully-Saturated Models To model other dependencies, we have to add all interaction terms between D and S, i.e. Y = αd + β 2 S2 + β 3 S3 + γ 2 D. S2 + γ 3 D. S3 + δ This allows the model to model all possible potential outcomes and their average effects. Such a model is said to be fully saturated

18 Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression

19 Causal Effects Suppose we do want to estimate a causal effect Y = α + δd + ε as where is an intercept, and encodes random variation. If D is uncorrelated with, the standard regression coefficient works. If D is correlated with, the correlation is often treated as an omitted variable.

20 Omitted Variables The omitted variable model looks like this: Y = α + δd + Xβ + ε we have added a set of variables X and a new weight. Need to check that independent of D. The solution in a single calculation is: δ = Q T Q 1 Q T y (the standard matrix regression equation), where Q is the marix of observations of x, d. This assumes an appropriate structure for the causal graph:

21 Explanation next time Omitted Variables

22 Regression and Matching Matching offers more options for generating estimators than regression, but we can estimate some important ones. Lets start with an example, first without regression.

23 Regression and Matching Average treatment effect among the treated

24 Regression and Matching Average treatment effect among the treated

25 Regression and Matching Average treatment effect among the untreated

26 Regression and Matching Unconditional average treatment effect

27 Regression and Matching In the direct approach, we used weights for each difference that were marginal distributions for S. With regression however, we are implicitly using weights over the differences (derived from least squares) that are generally different. They are derived from within-stratum variances.

28 Weighted Regression Not too surprisingly, we can reproduce the earlier matching estimates by appropriately weighted the regression formula. It now looks like this: δ = Q T PQ 1 Q T Py Where P is a weight matrix, depending on the average treatment effect of interest. For the example above, P is a diagonal matrix. For the treatment effect among the treated: For treatment units, P ii = 1. For untreated units, P ii = p i /(1-p i ). p i is the estimated propensity score. Similar formulae apply for the other treatment effects.

29 Doubly Robust Estimators Regression estimates are generally biased because of variancederived weighting factors. On the other hand, propensity score models may also be biased if the propensity score estimate is wrong. Doubly-Robust method produce a correct method when either one of these estimators is accurate.

30 Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression