Near/Far Matching. Building a Stronger Instrument in an Observational Study of Perinatal Care for Premature Infants

Transcription

1 Near/Far Matching Building a Stronger Instrument in an Observational Study of Perinatal Care for Premature Infants Joint research: Mike Baiocchi, Dylan Small, Scott Lorch and Paul Rosenbaum

2 Classic set up

3 Simple Regression Y T

4 Simple Regression Y T

5 Simple Regression Y T ε

6 Regression with problems Y T ε U

7 Regression with serious problems Y T ε U

8 IV techniques the idea Y T ε Z U

9 IV techniques the idea Y T ε Z U

10 IV techniques the idea Y T ε Z U

11 IV techniques the idea Y T ε Z U

12 IV techniques the idea Y = BMI Exercise = T ε Z U = Lethargy

13 IV techniques the idea Y = BMI Exercise = T ε Free gym membership= Z U = Lethargy

14 An Encouragement Design

15 References Instrumental Variables: Angrist, Imbens, Rubin: Identifcation of causal effects using instrumental variables (with Discusssion). JASA 91, (1996) Encouragement Design: Holland, P.W.: Causal inference, path analysis, and recursive structural equations models. Sociological Methodology 18, (1988)

16 Note on Terminology Instrumental (Merriam-Webster): a) serving as a crucial means, agent, or tool b) of, relating to, or done with an instrument or tool

17 Note on Terminology Instrumental (Merriam-Webster): a) serving as a crucial means, agent, or tool b) of, relating to, or done with an instrument or tool

18 Note on Terminology Encouragement = Instrumental in selection

19 What s the difference: Strong vs. Weak

20 Example: Weak IV 1,000

21 Example: Weak IV 500 1,

22 Example: Weak IV ,

23 Example: Weak IV 250 1,

24 Example: Strong IV 250 1,

25 Example: Strong IV 250 1,

26 Example: Strong IV 400 1,

27 Example: Weak IV 250 1,

28 References Instrumental Variables: Angrist, Imbens, Rubin: Identifcation of causal effects using instrumental variables (with Discusssion). JASA 91, (1996) Encouragement Design: Holland, P.W.: Causal inference, path analysis, and recursive structural equations models. Sociological Methodology 18, (1988)

29 Key Take-Aways People worry about weak instruments. They are easily biased. They provide large (sometimes HUGE) confidence intervals. If you re not careful, you can get inappropriate confidence intervals. Angrist & Kreuger (1991) Does compulsory school attendance affect schooling and earnings? Near/far matching can create studies with stronger instruments. Near/far matching feels a lot like a randomized study with noncompliance.

30 Neonatal Intensive Care Units

31 Application: Regionalization Hospitals vary in their ability to care for premature infants. The American Academy of Pediatrics recognizes levels: 1, 2, 3A, 3B, 3C, 3D and Regional Centers. Regionalization of care refers to a policy that suggests or requires that high-risk mothers deliver at hospitals with greater levels of capabilities.

32 Application: Regionalization

33 Application: Regionalization H H

34 Application: Regionalization H H

35 Application: Regionalization H H

36 Application: Regionalization H H

37 Application: Regionalization H H

38 Application: Regionalization H H

39 Application: Regionalization H H

40 Application: Regionalization H H

41 The task at hand Regionalization is complex Focus on estimating the difference in death rates

42 The task at hand Regionalization is complex Focus on estimating the difference in death rates

43

44 Outcome Outcome

45 Outcome Outcome

46 The data Every baby delivered in a 10+ year period California Pennsylvania Missouri Mothers information ICD9 codes Delivery Post-delivery complications Some pre-delivery Some SES information Zip code of residence Birth/death certificates Census information PA and MO have zip code level CA will have block group

47 The data Every baby delivered in a 10+ year period California Pennsylvania Missouri Mothers information ICD9 codes Delivery Post-delivery complications Some pre-delivery Some SES information Zip code of residence Birth/death certificates Census information PA and MO have zip code level CA will have block group Pre-delivery Severity?

48 Summary of Problem Want to quantify effect of level of NICU on rate of death Observational data Selection bias Some selection variables are unobserved

49 Instrument: Excess Travel Time H H

50 Instrument: Excess Travel Time H H Excess Travel Time

51 Instrument: Excess Travel Time H H Excess Travel Time

52 Instrument: Excess Travel Time H H Excess Travel Time

53 Instrument: Excess Travel Time H H McClellan, McNeil & Newhouse; "Does more intensive treatment of acute myocardial infarction reduce mortality? JAMA. 272(11): , September 1994

54 Fewer Pairs at Greater Distances

55 Our method a quick sketch Use the idea of block design / pair matching to control observed variation. Use the idea of instrumental variables/encouragement to control unobserved variation.

56 Our method: 1 st step Summarize discrepancies in subjects covariates We used Mahalanobis distance D M x 1, x 2 = (x 1 x 2 ) S 1 (x 1 x 2 )

57 Our method: 1 st step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn d ij = Mahalanobis distance between preemies i and j

58 Our method: 2 nd step Create a penalty for preemies with similar instrument values (e.g., calipers)

59 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn

60 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn

61 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn C z = c 11 c 12 c 13 c 1n c 21 c 22 c 31 c n1 c nn

62 Instrument: Excess Travel Time H H Selection is potentially biased!

63 Instrument: Excess Travel Time H H Selection is potentially biased!

64 Instrument: Excess Travel Time H H Selection largely due to the instrument!

65 Instrument: Excess Travel Time H H Selection largely due to the instrument!

66 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn C z = c 11 c 12 c 13 c 1n c 21 c 22 c 31 c n1 c nn

67 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn C z = c 11 c 12 c 13 c 1n c 21 c 22 c 31 c n1 c nn Diff Covariates + Diff Encouragement = Discrepancy Matrix

68 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn C z = c 11 c 12 c 13 c 1n c 21 c 22 c 31 c n1 c nn Diff Covariates + Diff Encouragement = Discrepancy Matrix (near) (far) (barrier to being paired)

69 Our method: 2 nd step D x = d 11 d 12 d 13 d 1n d 21 d 22 d 31 d n1 d nn C z = c 11 c 12 c 13 c 1n c 21 c 22 c 31 c n1 c nn D x + C z = D

70 Our method: 3 rd step Something has got to give: As we force separation in the instrument, it will be more difficult to find preemies with similar covariates. Allow some subjects to be removed from the study design by matching to sinks.

71 Our method: 3 rd step Let k=number of sinks. Then augment the matrix like so: D 0 0 D = n n discepancy matrix, after first two steps 0 = n k matrix, with all entries 0 = k k matrix, with entries

72 Two matched comparisons, one stronger and one weaker

73 The two matches Two matches 1) No sinks / no forced separation 2) 50% of babies matched to sinks / 25min separation

74 The two matches: Variables Excess Travel Time (i.e., Encouragement/Instrument) Pregnancy and birth variables Mother variables Mother s health insurance Mother s neighborhood Rare congenital anomalies Year Missing indicators

75 The two matches: Variables Excess Travel Time (i.e., Encouragement/Instrument) Pregnancy and birth variables Mother variables Mother s health insurance Mother s neighborhood Rare congenital anomalies Year Missing indicators In total: 45 covariates

76 The two matches 1. Weak Instrument a) No sinks (99,174 pairs) b) No forced separation 2. Strong Instrument a) 50% of babies matched to sinks (49,587 pairs) b) 25min separation

77 Weaker Instrument No sinks Number of pairs: 99,174 Stronger Instrument 50% of babies matched to sinks Number of pairs: 49,587 Variable Variable Type Encouraged Mean Unencouraged Mean Δ/sd Encouraged Mean Unencouraged Mean Δ/sd Excess travel time to hihg-level NICU (minutes) Birthweight (grams) Gestational age (weeks) Gestational diabetes, 1/0 Prenatal care (month) Singel birth, 1/0 Parity Mother's education (scale) Mother's age White, 1/0 Black, 1/0 Asian, 1/0 Other race, 1/0 Race missing, 1/0 Income ($1,000) Home value ($1,000) Has high school degree (fr) Has college degree (fr) Rent (fr) Below poverty (fr) Magnitude of encouragement Pregnancy and birth Mother Mother's neighborhood (zip code/census) , , , , , , , , , , , ,

78 Weaker Instrument No sinks Number of pairs: 99,174 Stronger Instrument 50% of babies matched to sinks Number of pairs: 49,587 Variable Variable Type Encouraged Mean Unencouraged Mean Δ/sd Encouraged Mean Unencouraged Mean Δ/sd Excess travel time to hihg-level NICU (minutes) Birthweight (grams) Gestational age (weeks) Gestational diabetes, 1/0 Prenatal care (month) Singel birth, 1/0 Parity Mother's education (scale) Mother's age White, 1/0 Black, 1/0 Asian, 1/0 Other race, 1/0 Race missing, 1/0 Income ($1,000) Home value ($1,000) Has high school degree (fr) Has college degree (fr) Rent (fr) Below poverty (fr) Magnitude of encouragement Pregnancy and birth Mother Mother's neighborhood (zip code/census) , , , , , , , , , , , ,

79 Application: Two matches Weaker Instrument No sinks Stronger Instrument 50% of babies matched to sinks Number of pairs: 99,174 Number of pairs: 49,587 Variable Variable Type Encouraged Mean Unencouraged Mean Δ/sd Encouraged Mean Unencouraged Mean Δ/sd Excess travel time to hihg-level NICU (minutes) High-level NICU, 1/0 Dead, 1/0 Magnitude of encouragement Delivery at a high-level NICU(Dij) Infant mortality (Rij)

80 Notation: Treatment Effects, Treatment Assignments

81 WARNING It s about to get mathy.

82 Notation Indices i denotes which pair There are I matched pairs, thus i = 1,, I. j denotes which subject within the matched pair Thus j = 1, 2. Covariates Observed: x ij Unobserved: u ij

83 Notation Matching on observed covariates x i1 =x i2 for all pairs i But it may be that u i1 u i2

84 Notation Instrument/Encouragement Z ij = 1, if subject j in the i th pair was encouraged Z ij =0, if subject j in the i th pair was unencouraged Note that, within a matched pair, Z i1 +Z i2 =1 Potential outcomes framework (Neyman 1923; Rubin 1974)

85 Encouragement Design Outcome (Y) Encouragement (Z) Treatment (T) Outcome (Y) Outcome (Y) Treatment (T) Outcome (Y)

86 Instrument Design Response (R) Instrument (Z) Dose (D) Response (R) Response (R) Dose (D) Response (R)

87 Notation Dose (d Tij,d Cij ) Can t observe: d Tij -d Cij Can observe: D ij =Z ij d Tij +(1-Z ij ) d Cij Response (r Tij,r Cij ) Can t observe: r Tij -r Cij Can observe: R ij =Z ij r Tij +(1-Z ij ) r Cij

88 Instrument Design Response (R) Instrument (Z) Dose (D) Response (R) Response (R) Dose (D) Response (R)

89 Instrument Design Response (R T ) Dose (D T ) Instrument (Z) Dose (D C ) Complier Response (R C )

90 Instrument Design Response (R T ) Dose (D T ) Instrument (Z) Response (R C ) Dose (D C ) Always-Taker

91 Instrument Design Dose (D T ) Response (R T ) Instrument (Z) Dose (D C ) Never-Taker Response (R C )

92 Instrument Design Dose (D T ) Response (R T ) Instrument (Z) Response (R C ) Dose (D C ) Defier

93 Instrument Design Dose (D T ) Response (R T ) Instrument (Z) Response (R C ) Dose (D C ) Defier

94 Instrument Design Response (R T ) Dose (D T ) Instrument (Z) Dose (D C ) Complier Response (R C )

95 Notation Let F = {(d Tij, d Cij, r Tij, r Cij, x ij, u ij ), i=1,,i,j=1,2} Let Z = Z 11, Z 12,, Z I2

96 Notation Let Z be the event that Z Ω Let Ω be the set containing the Ω = 2 I z of Z Then, in a randomized experiment: Pr(Z=z F, Z)=1/ Ω for each z Ω

97 Effect Ratios

98 The Effect Ratio λ = I i=1 I i=1 2 j =1 2 j =1 (r Tij r Cij ) (d Tij d Cij ) λ is parameter of the 2I subjects, fixed under F. λ is not directly observable. Under Fisher s sharp null hypothesis, H 0 : r Tij = r Cij, for all i, j, it follows that λ=0.

99 The Effect Ratio I 2 λ = I i=1 I i=1 2 j =1 2 j =1 (r Tij r Cij ) (d Tij d Cij ) i=1 j=1 r Tij r Cij is the effect of encouragement on response. I 2 i=1 j=1 d Tij d Cij is the effect of the encouragement on the dose. λ is the ratio of these effects. If λ = 1/100 then for every 100 discouraged by distance from delivering at a high level NICU there is one additional infant death.

100 Inference about an Effect Ratio in a Randomized Experiment

101 Inference Composite Null Consider H 0 λ : λ = λ0 This is a composite null, because no assumptions on distribution of F. Need to consider the supremum over null hypotheses of the probability of rejection.

102 Models have incredible power Models force you to clarify your thinking. By writing down a model, you take a stand on how the world works. If the model is correct, there are very powerful mathematical machines you can deploy to get precise answers.

103 Models have incredible power If the model is wrong, then the power (in the statistical sense) of your inference is not credible.

104 Inference Test Statistic Consider the following test statistic T λ 0 = 1 I I 2 Z ij (R ij λ 0 D ij ) 2 (1 Z ij )(R ij λ 0 D ij ) = 1 I i=1 I i=1 j =1 V i (λ 0 ) j =1

105 Inference Unobserved to Observed Note that in T λ 0 If Z ij = 1, then R ij λ 0 D ij = r Tij λ 0 d Tij, and If Z ij = 0, then R ij λ 0 D ij = r Cij λ 0 d Cij Thus we can write: 2 2 V i λ 0 = j=1 Z ij r Tij λ 0 d Tij j=1 1 Z ij r Cij λ 0 d Cij For the variance of the test statistic S 2 λ 0 = 1 I 2 V I I 1 i=1 i λ 0 T λ 0

106 Inference t-stat! Then, for each k>0, lim sup I Pr T I λ S I λ k F I, Z I Φ( k) lim sup I Pr T I λ S I λ k F I, Z I Φ k.

107 Application to the Study of Perinatal Care

108 Pop death rate: ~1.90% Application: Estimating λ Inference about the effect ratio, λ, under the assumption of random assignment of excess travel time within pairs matched for covariates Weaker Instrument Stronger Instrument 99,174 Pairs of Two Babies 49,587 Pairs of Two Babies Point Estimate 0.92% 0.90% 95% CI (0.36%, 1.48%) (0.57%, 1.23%) Length of 95% CI 1.12% 0.66%

109 Pop death rate: ~1.90% Application: Estimating λ Inference about the effect ratio, λ, under the assumption of random assignment of excess travel time within pairs matched for covariates Weaker Instrument Stronger Instrument 99,174 Pairs of Two Babies 49,587 Pairs of Two Babies Point Estimate 0.92% 0.90% 95% CI (0.36%, 1.48%) (0.57%, 1.23%) Length of 95% CI 1.12% 0.66% Bound, Jaeger & Baker. (1995), Problems With Instrumental Variables Estimation When the Correlation Between the Instruments and the Endogenous Explanatory Variable Is Weak, JASA, 90,

110 General Method: Quantifying Departures from Random Assignment

111 Sensitivity Analysis: Framework Up to this point we have assumed Pr Z = z F, Z = 1/ Ω for each z Ω. Now we will allow Pr Z = z F = π ij. Consider matched pair i, then 1 Γ π ij 1 π ij π ij 1 π ij Γ for all i, j, j with x ij = x ij.

112 Sensitivity Analysis: Numerical Examples Gamma of 1.25 Doubling of the odds of death. Doubling of the odds of treatment. Gamma of 1.08 Doubling of the odds of death Increase of 25% of the odds of treatment

113 Application to the Study of Regionalization of Perinatal Care

114 Pop death rate: ~1.90% Application: Sensitivity Analysis Inference about the effect ratio, λ with sensitivity analysis. Weaker Instrument Stronger Instrument 99,174 Pairs of Two Babies 49,587 Pairs of Two Babies Point Estimate 0.92% 0.90% 95% CI (0.36%, 1.48%) (0.57%, 1.23%) Length of 95% CI 1.12% 0.66% Sensitivity (Γ) 1.07 >1.22

115 Pop death rate: ~1.90% Application: Sensitivity Analysis Inference about the effect ratio, λ with sensitivity analysis. Weaker Instrument Stronger Instrument 99,174 Pairs of Two Babies 49,587 Pairs of Two Babies Point Estimate 0.92% 0.90% 95% CI (0.36%, 1.48%) (0.57%, 1.23%) Length of 95% CI 1.12% 0.66% Sensitivity (Γ) 1.07 >1.22 Small & Rosenbaum (2008), War and Wages: The Strength of Instrumental Variables and Their Sensitivity to Unobserved Biases, JASA, 103,

116 What changes when an instrument is strengthened?

117 What changes? 1. Smaller study looks less like population Fewer black mothers Fewer renters That is, less urban

118 Weaker Instrument No sinks Number of pairs: 99,174 Stronger Instrument 50% of babies matched to sinks Number of pairs: 49,587 Variable Variable Type Encouraged Mean Unencouraged Mean Δ/sd Encouraged Mean Unencouraged Mean Δ/sd Excess travel time to hihg-level NICU (minutes) Birthweight (grams) Gestational age (weeks) Gestational diabetes, 1/0 Prenatal care (month) Singel birth, 1/0 Parity Mother's education (scale) Mother's age White, 1/0 Black, 1/0 Asian, 1/0 Other race, 1/0 Race missing, 1/0 Income ($1,000) Home value ($1,000) Has high school degree (fr) Has college degree (fr) Rent (fr) Below poverty (fr) Magnitude of encouragement Pregnancy and birth Mother Mother's neighborhood (zip code/census) , , , , , , , , , , , ,

119 Weaker Instrument No sinks Number of pairs: 99,174 Stronger Instrument 50% of babies matched to sinks Number of pairs: 49,587 Variable Variable Type Encouraged Mean Unencouraged Mean Δ/sd Encouraged Mean Unencouraged Mean Δ/sd Excess travel time to hihg-level NICU (minutes) Birthweight (grams) Gestational age (weeks) Gestational diabetes, 1/0 Prenatal care (month) Singel birth, 1/0 Parity Mother's education (scale) Mother's age White, 1/0 Black, 1/0 Asian, 1/0 Other race, 1/0 Race missing, 1/0 Income ($1,000) Home value ($1,000) Has high school degree (fr) Has college degree (fr) Rent (fr) Below poverty (fr) Magnitude of encouragement Pregnancy and birth Mother Mother's neighborhood (zip code/census) , , , , , , , , , , , ,

120 What changes? 1. Smaller study looks less like population Fewer black mothers Fewer renters That is, less urban 2. Compliers change Larger study: 14 minutes difference Smaller study: 34 minutes difference

121 Why is it OK to throw out data?

122 Stronger instrument by design In some sense we already know this tradeoff: Larger studies with poor design and low compliance Vs. Smaller studies with good design and high levels of compliance

123 Stronger Instruments by Design

124 Our method Advantages Other researchers worry about weak instruments. Now we do something about it! More compliance leads to better estimates Less bias Tighter confidence intervals Sensitivity analysis Easier to explain Mimics an experimental design Avoids MLE (i.e., parametric assumptions)

125 The End More questions?

126 The End Baiocchi, Small, Lorch & Rosenbaum: Building a Stronger Instrument in an Observational Study of Perinatal Care for Premature Infants JASA. Dec 2010, Vol. 105, No. 492:

127 Pop death rate: ~1.90% Application: Estimating λ Inference about the effect ratio, λ, under the assumption of random assignment of excess travel time within pairs matched for covariates

128 Visualizing our method

129 Encouragement Our method two criteria for matching Distance in Covariates

130 Encouragement Our method near in covariates Distance in Covariates

131 Encouragement Our method far in covariates Distance in Covariates

132 Encouragement Our method far in covariates Distance in Covariates

133 Encouragement Our method good pair Distance in Covariates

134 Encouragement Our method ok pair Distance in Covariates

135 Encouragement Our method strength of instrument Distance in Covariates

136 Encouragement Our method strength of instrument Weak IV Distance in Covariates

137 Encouragement Our method strength of instrument Medium IV Distance in Covariates

138 Encouragement Our method strength of instrument Strong IV Distance in Covariates

139 Encouragement Our method good pair Strong IV Distance in Covariates

140 Encouragement Our method near in instrument Strong IV Distance in Covariates

141 Encouragement Our method near in instrument Strong IV Distance in Covariates

142 Encouragement Our method near in instrument Strong IV Distance in Covariates

143 Encouragement Our method near in instrument Possible Bias Strong IV Distance in Covariates

144 Encouragement Our method near in instrument Strong IV Distance in Covariates

145 Encouragement Our method far in instrument Strong IV Distance in Covariates

146 Encouragement Our method far in instrument Strong IV Distance in Covariates

147 Encouragement Our method far in instrument Stronger Encouragement Strong IV Distance in Covariates

148 Encouragement Find the experiment Strong IV Distance in Covariates

149 Encouragement Find the experiment Strong IV Distance in Covariates

150 The framework Treatment (T) Encouragement (Z) Treatment (T)

151 The framework Treatment (T) Encouragement (Z) Treatment (T)

152 The framework Response (R) D A Treatment (T) Encouragement (Z) Treatment (T) Response (R) D A

153 The framework Response (R) D A Treatment (T) Encouragement (Z) Treatment (T) Response (R) D A

154 The framework Response (R) D A Encouragement (Z) Treatment (T) Treatment (T) Response (R) Response (R) Response (R) D A D A D A

155 Clarifying who we re talking about Compliance Class Encouragement (Z) Treatment (T) 1 1 Complier Never Taker Always Taker Defier 0 1

156 Clarifying who we re talking about Compliance Class Encouragement (Z) Treatment (T) 1 1 Complier Never Taker Always Taker Defier 0 1

157 Clarifying who we re talking about Compliance Class Encouragement (Z) Treatment (T) 1 1 Complier Never Taker Always Taker Defier 0 1

158 Building intuition for implementation

159 Variable Type High NICU Low NICU sd Δ/sd Mortality Outcome 2.26% 1.25% 13.33% 0.08 Difference in Travel Time Instrument % attending high level NICU Treatment 100.0% 0.0% 49.7% 2.01 Birth weight 2, , Preemie covariates Gestational age GI 0.9% 0.6% 8.7% 0.04 GU 0.9% 0.8% 9.0% 0.01 CNS 0.9% 0.4% 8.3% 0.05 Pulmonary 0.8% 0.7% 8.8% 0.01 % of preemies with type of Cardio 1.4% 0.7% 10.5% 0.06 congenital disorders Skeletal 0.7% 0.9% 9.0% Skin 0.0% 0.0% 0.0% 0.00 Chromosomes 0.4% 0.3% 6.3% 0.02 Other_Anomaly 0.8% 0.1% 7.0% 0.09 Gestational_DiabetesM 4.9% 4.3% 21.0% 0.03 Mother's education Insurance - Fee for service 24.0% 24.5% 42.8% Insurance - HMO 32.3% 27.8% 46.0% 0.10 Insurance - Government 23.5% 24.2% 42.6% Insurance - Other Mother covariates 16.8% 21.4% 39.1% Uninsured 2.2% 1.6% 13.7% 0.04 Prenatal care Single birth (y/n) 79.0% 86.1% 38.3% Parity Mother's age Median income 41, , , Median home value 97, , , % completed high school 79.9% 80.0% 9.7% Census level covariates % completed college 22.2% 19.4% 13.1% 0.21 % renting 31.4% 27.9% 12.8% 0.28 % below poverty line 13.4% 11.8% 9.9% 0.16

160 Pre-matching Variable Type High NICU Low NICU sd Δ/sd Mortality Outcome 2.26% 1.25% 13.33% 0.08 Difference in Travel Time Instrument % attending high level NICU Treatment 100.0% 0.0% 49.7% 2.01 Birth weight 2, , Preemie covariates Gestational age

161 Covariates across the instrument 1st Quartile 2nd Quartile 3rd Quartile 4th Quartile max(δ/sd) Mortality 1.93% 2.08% 1.47% 1.74% 0.05 Difference in Travel Time (3.19) % attending high level NICU 81.1% 69.8% 49.9% 21.6% 1.20 Birth weight 2, , , , Gestational age

162 Post-matching Matched Pairs 49,587 Variable Type Encouraged Mean Unencouraged Mean Mortality Outcome 1.54% 1.94% 12.86% Difference in Travel Time Instrument % attending high level NICU Treatment 68.6% 25.4% 49.7% 0.87 Birth weight 2, , Preemie covariates Gestational age sd Δ/sd

163 Result Point Estimate 95% CI Length of 95% CI Sensitivity (Γ) Weaker Instrument Stronger Instrument 99,174 Pairs 49,587 Pairs of Two Babies of Two Babies 0.92% 0.90% (0.36%, 1.48%) (0.57%, 1.23%) 1.12% 0.66% 1.07 >1.22

164 Final thoughts Near/Far deals with binary outcomes 2SLS can lead to logical absurdities when outcomes are binary Increase the strength of the instrument Stronger instruments lead to more robust results Sensitivity analysis for this method is available This technique has the potential to be a Philosopher s stone

165 Why not use 2SLS? Linear probability models (LPMs) have trouble when your parameter values are up against the edge of parameter space

166 Quality of Care Trouble with time Treatment effect High level NICU Low level NICU Time

167 Quality of Care Trouble with time Treatment effect Treatment effect High level NICU Low level NICU Time

168 Modeling hospital choice

169 Instrument: Probabilities Empirical distributions M

170 Instrument: Probabilities Empirical distributions?

171 Instrument: Probabilities Near Neighbors?

172 Instrument: Probabilities Conditional logistic model/bayesian hierarchical modeling H H M

173 Instrument: Probabilities Conditional logistic model/bayesian hierarchical modeling H H M

174