| Lady's Guesses | Tea First | Milk First |
|---|---|---|
| Tea First | 3 | 1 |
| Milk First | 1 | 3 |
Making causal claims
How do we know \(X\) causes \(Y\)?
Can data help us make causal claims?
When is a causal claim not appropriate?
The lady tasting tea
A lady declares that by tasting a cup of tea made with milk she can discriminate whether the milk or the tea infusion was first added to the cup
- Is this a causal claim?
- How do you evaluate it?
An experiment
Suppose we have eight milk tea cups
4 milk first, 4 tea first
We arrange them in random order
Lady knows there are 4 of each, but not which ones
Results
Lady gets it right \(6/8\) times
What can we conclude?
Problem
How does “being able to discriminate” look like?
We do know how a person without the ability to discriminate milk/tea order looks like
This is our null hypothesis (\(H_0\))
Which lets us make probability statements about this hypothetical world of no effect
A person with no ability
| Count | Possible combinations | Total |
|---|---|---|
| 0 | xxxx | \(1 \times 1 = 1\) |
| 1 | xxxo, xxox, xoxx, oxxx | \(4 \times 4 = 16\) |
| 2 | xxoo, xoxo, xoox, oxox, ooxx, oxxo | \(6 \times 6 = 36\) |
| 3 | xooo, oxoo, ooxo, ooox | \(4 \times 4 = 16\) |
| 4 | oooo | \(1 \times 1 = 1\) |
- This is a symmetrical problem!
A person with no ability
| Count | Possible combinations | Total |
|---|---|---|
| 0 | xxxx | \(1 \times 1 = 1\) |
| 1 | xxxo, xxox, xoxx, oxxx | \(4 \times 4 = 16\) |
| 2 | xxoo, xoxo, xoox, oxox, ooxx, oxxo | \(6 \times 6 = 36\) |
| 3 | xooo, oxoo, ooxo, ooox | \(4 \times 4 = 16\) |
| 4 | oooo | \(1 \times 1 = 1\) |
A person with no ability
| Count | Possible combinations | Total |
|---|---|---|
| 0 | xxxx | \(1 \times 1 = 1\) |
| 1 | xxxo, xxox, xoxx, oxxx | \(4 \times 4 = 16\) |
| 2 | xxoo, xoxo, xoox, oxox, ooxx, oxxo | \(6 \times 6 = 36\) |
| 3 | xooo, oxoo, ooxo, ooox | \(4 \times 4 = 16\) |
| 4 | oooo | \(1 \times 1 = 1\) |
- A person guessing at random gets \(6/8\) cups right with probability \(\frac{16}{70} \approx 0.23\)
A person with no ability
| Count | Possible combinations | Total |
|---|---|---|
| 0 | xxxx | \(1 \times 1 = 1\) |
| 1 | xxxo, xxox, xoxx, oxxx | \(4 \times 4 = 16\) |
| 2 | xxoo, xoxo, xoox, oxox, ooxx, oxxo | \(6 \times 6 = 36\) |
| 3 | xooo, oxoo, ooxo, ooox | \(4 \times 4 = 16\) |
| 4 | oooo | \(1 \times 1 = 1\) |
- And at least \(6/8\) cups with \(\frac{16 + 1}{70} \approx 0.24\)
p-values
If the lady is not able to discriminate milk-tea order, the probability of observing \(6/8\) correct guesses or better is \(0.24\)
A p-value is the probability of observing a result equal or more extreme than what is originally observed…
… when the null hypothesis is true
Smaller p-values give more evidence against the null
Implying observed value is less likely to have emerged by chance
Rules of thumb
A convention in the social sciences is to claim that something with \(p < 0.05\) is statistically significant
Meaning we have enough evidence
Committing to a significance level \(\alpha\) implies accepting that sometimes we will get \(p < 0.05\) by chance
This is a false positive result
Types of error
Unobserved reality
|
||
|---|---|---|
| Decision | \(H_0\) true | \(H_0\) not true |
| Don't reject \(H_0\) | True negative | False negative (type II error) |
| Reject \(H_0\) | False positive (type I error) | True positive |
Hypothesis testing
We just computed p-values via permutation testing
Congenial with agnostic statistics because we do not need to assume anything beyond how the data was collected
Being agnostic helps us add credibility to our causal claims
Potential outcomes framework
Notation
\(D_i = \{0,1\}\) condition (0: control, 1: treatment)
\(Y_i(D_i)\) is the individual potential outcome
\[ Y_i = \begin{cases} Y_i(0) & : & D_i = 0 \\ Y_i(1) & : & D_i = 1 \end{cases} \]
Switching equation
\[ Y_i = Y_i(1) D_i + Y_i(0) (1 - D_i) \]
Toy example
| ID | \(Y_i(1)\) | \(Y_i(0)\) |
|---|---|---|
| 1 | 0 | 0 |
| 2 | 1 | 0 |
| 3 | 1 | 0 |
| 4 | 1 | 1 |
\(\tau_i = Y_i(1) - Y_i(0)\) is the unit-level treatment effect
Toy example
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 1 |
| 3 | 1 | 0 | 1 |
| 4 | 1 | 1 | 0 |
\(\tau_i = Y_i(1) - Y_i(0)\) is the unit-level treatment effect
\(E[\tau_i] = \frac{1}{n} \sum_{i=1}^n \tau_i\) is the average treatment effect (ATE)
Assumption
Stable Unit Treatment Value (SUTVA)
Potential outcomes are stable, meaning:
No unobserved multiple versions of the treatment: Treatment is the same for everyone
No interference: Unit \(i\)’s potential outcome is not affected by \(j\)’s potential outcome
Challenge
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 1 |
| 3 | 1 | 0 | 1 |
| 4 | 1 | 1 | 0 |
Challenge
Unobserved
|
|||
|---|---|---|---|
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) |
| 1 | 0 | 0 | 0 |
| 2 | 1 | 0 | 1 |
| 3 | 1 | 0 | 1 |
| 4 | 1 | 1 | 0 |
Assign condition \(D_i\) (0: control, 1: treatment)
Challenge
Unobserved
|
Observed
|
||||
|---|---|---|---|---|---|
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) | \(D_i\) | \(Y_i\) |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 0 | 1 | 1 | 1 |
| 4 | 1 | 1 | 0 | 0 | 1 |
To know \(E[\tau_i]\) we need \((Y_i(1) - Y_i(0))\)
But we only observe one at a time for each unit
Challenge
Unobserved
|
Observed
|
||||
|---|---|---|---|---|---|
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) | \(D_i\) | \(Y_i\) |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 0 | 1 | 1 | 1 |
| 4 | 1 | 1 | 0 | 0 | 1 |
This is the FUNDAMENTAL PROBLEM OF CAUSAL INFERENCE
Challenge
Unobserved
|
Observed
|
||||
|---|---|---|---|---|---|
| ID | \(Y_i(1)\) | \(Y_i(0)\) | \(\tau_i\) | \(D_i\) | \(Y_i\) |
| 1 | 0 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 1 | 0 | 0 |
| 3 | 1 | 0 | 1 | 1 | 1 |
| 4 | 1 | 1 | 0 | 0 | 1 |
How do we approximate \(E[\tau_i]\) with observed data?
ATE decomposition
\[ E[\tau_i] \]
Substitute unit-level treatment effect
ATE decomposition
\[ \begin{align} E[\tau_i] & = E[Y_i(1) - Y_i(0)] \end{align} \]
Apply linearity of expectations
ATE decomposition
\[ \begin{align} E[\tau_i] & = E[Y_i(1) - Y_i(0)] \\ & = E[Y_i(1)] - E[Y_i(0)] \end{align} \]
We can only calculate
\[ E[Y_i(1) | D_i = 1] - E[Y_i(0) | D_i = 0] \]
We need to convince ourselves that
\[ E[Y_i(1)] - E[Y_i(0)] = E[Y_i(1) | D_i = 1] - E[Y_i(0) | D_i = 0] \]
ATE decomposition
\[ \begin{align} E[\tau_i] & = E[Y_i(1) - Y_i(0)] \\ & = E[Y_i(1)] - E[Y_i(0)] \end{align} \]
We can only calculate
\[ E[Y_i(1) | D_i = 1] - E[Y_i(0) | D_i = 0] \]
We need to convince ourselves that
\[ E[Y_i(1)] - E[Y_i(0)] = E[Y_i(1) \color{purple}{| D_i = 1}] - E[Y_i(0) \color{purple}{| D_i = 0}] \]
ATE decomposition
\[ \begin{align} E[\tau_i] & = E[Y_i(1) - Y_i(0)] \\ & = E[Y_i(1)] - E[Y_i(0)] \end{align} \]
We can only calculate
\[ E[Y_i(1) | D_i = 1] - E[Y_i(0) | D_i = 0] \]
We need to convince ourselves that \(D_i\) can be ignored
\[ E[Y_i(1)] - E[Y_i(0)] = E[Y_i(1) \color{purple}{| D_i = 1}] - E[Y_i(0) \color{purple}{| D_i = 0}] \]
Obstacle
\[ E[Y_i(1) | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Obstacle
\[ E[\color{purple}{Y_i(1)} | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Rewrite to include \(\tau_i\)
\[ = E[\color{purple}{\tau_i + Y_i(0)} | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Apply linearity of expectations again
\[ = E[\color{purple}{\tau_i} | D_i = 1] + E[\color{purple}{Y_i(0)} | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Obstacle
\[ E[\color{purple}{Y_i(1)} | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Rewrite to include \(\tau_i\)
\[ = E[\color{purple}{\tau_i + Y_i(0)} | D_i = 1] - E[Y_i(0) | D_i = 0] \]
Apply linearity of expectations again
\[ = \underbrace{E[\color{purple}{\tau_i} | D_i = 1]}_{\text{ATE on the treated (ATT)}} + \underbrace{E[\color{purple}{Y_i(0)} | D_i = 1] - E[Y_i(0) | D_i = 0]}_{\text{Selection bias}} \]
We want to eliminate selection bias
Strategy 1: Random assignment
If treatment \(D_i\) is randomly assigned
- Potential outcomes are independent from treatment: \((Y_i(0), Y_i(1)) \perp \!\!\! \perp D_i\)
- Positivity: \(0 < \text{Pr}[D_i = 1] < 1\)
- ATE \(E[\tau_i]\) is point-identified
- Estimate with difference in means between treatment and control: \(\widehat E_{DM}[\tau_i] = \widehat E[Y_i(1)] - \widehat E[Y_i(0)]\)
Randomization inference
Choose a test statistic and sharp null hypothesis
Calculate observed test statistic \(T^{obs}\)
Randomly shuffle treatment assignment
Calculate test statistic \(\widetilde T_1\)
Repeat \(K\) times to obtain distribution of test statistics \(\widetilde T = \{\widetilde T_1, \ldots, \widetilde T_K \}\)
p-value: Proportion in \(\widetilde T >= T^{obs}\)
Example
ID Y D
1 001 0.73452088 1
2 002 0.77074300 0
3 003 0.43603904 1
4 004 -0.08670346 0
5 005 0.68140405 0
6 006 1.97004908 0
7 007 0.03467949 0
8 008 -0.15014153 0
9 009 1.93785072 1
10 010 1.03405589 1
11 011 -0.63776244 0
12 012 -0.02706806 1
13 013 1.07300224 1
14 014 -1.93937350 1
15 015 0.02790501 1
16 016 -0.35149453 0
17 017 0.54079046 0
18 018 1.44996706 0
19 019 0.73112967 1
20 020 0.79690824 0
21 021 -0.27269854 0
22 022 -0.27808426 1
23 023 0.51189702 0
24 024 1.36787778 0
25 025 -0.53588199 0
26 026 -0.97449045 1
27 027 -0.30534497 1
28 028 0.78755569 0
29 029 1.29057502 0
30 030 -0.29850392 0
31 031 1.35695956 1
32 032 1.66123394 1
33 033 -0.10080816 1
34 034 2.22298971 1
35 035 0.37412061 0
36 036 0.96894977 0
37 037 -0.73311578 1
38 038 3.80851080 0
39 039 0.28927338 1
40 040 2.08703325 1
41 041 0.54731664 0
42 042 0.75394287 0
43 043 0.09015847 1
44 044 -0.90991412 0
45 045 -0.03775686 1
46 046 -0.01659609 0
47 047 1.40690978 1
48 048 -0.01060873 1
49 049 0.01867026 1
50 050 0.76318358 1
51 051 -0.79504822 0
52 052 -0.55555399 0
53 053 -1.33573394 0
54 054 0.32093037 0
55 055 -0.46741037 1
56 056 0.18678302 0
57 057 1.12459600 0
58 058 -0.07427236 1
59 059 -0.08427221 1
60 060 1.09717098 1
61 061 -0.63543347 0
62 062 0.33663997 0
63 063 0.36397851 0
64 064 -0.86762819 1
65 065 0.63749833 1
66 066 0.03136840 1
67 067 -0.50679316 0
68 068 1.45469951 1
69 069 1.05952630 0
70 070 -0.07452616 0
71 071 -0.58036503 1
72 072 -0.21894114 1
73 073 1.59581402 1
74 074 -0.32160300 0
75 075 -0.64248143 0
76 076 0.24132506 1
77 077 -1.54571192 0
78 078 1.42652245 1
79 079 0.13653512 1
80 080 -0.15238495 0
81 081 0.61609104 0
82 082 -0.19122112 1
83 083 0.91385643 1
84 084 -0.28182279 0
85 085 -0.85947519 0
86 086 -0.69182843 0
87 087 -0.47407936 0
88 088 0.58345214 1
89 089 1.65531353 1
90 090 -0.79963168 0
91 091 -0.52715729 1
92 092 0.39807853 1
93 093 0.41642051 1
94 094 -1.09120280 1
95 095 0.17952844 1
96 096 -1.83358910 0
97 097 1.15446368 1
98 098 0.44128714 1
99 099 1.47267069 0
100 100 0.06886880 0Create randomization distribution
replicate term estimate std.error statistic p.value
1 1 D 0.1153412268 0.1878193 0.6141073101 0.540573797
2 2 D -0.1826480966 0.1872737 -0.9753002555 0.332012531
3 3 D 0.2955517522 0.1857970 1.5907241794 0.114984414
4 4 D -0.0668328670 0.1880592 -0.3553820001 0.723071275
5 5 D 0.2622549322 0.1863063 1.4076545925 0.162397711
6 6 D 0.1611433858 0.1874750 0.8595459148 0.392172528
7 7 D 0.1017260419 0.1878996 0.5413850878 0.589571462
8 8 D -0.0279077043 0.1881593 -0.1483195916 0.882395417
9 9 D -0.1205967677 0.1877856 -0.6422043886 0.522313453
10 10 D 0.2249861257 0.1868029 1.2044036039 0.231334398
11 11 D 0.0111052940 0.1881770 0.0590151406 0.953066314
12 12 D -0.3259859333 0.1852768 -1.7594534571 0.081714278
13 13 D 0.1675767012 0.1874175 0.8941360314 0.373624078
14 14 D -0.1554821756 0.1875238 -0.8291330552 0.409057329
15 15 D -0.2242053566 0.1868125 -1.2001624229 0.232983652
16 16 D -0.1361690588 0.1876770 -0.7255501312 0.469933582
17 17 D -0.1545070202 0.1875320 -0.8238967521 0.412047883
18 18 D 0.2187984315 0.1868779 1.1708094455 0.244515455
19 19 D 0.1799442922 0.1873004 0.9607255307 0.339186460
20 20 D -0.0142672792 0.1881749 -0.0758192647 0.939722266
21 21 D -0.0023951847 0.1881802 -0.0127281426 0.989871318
22 22 D 0.1278327399 0.1877368 0.6809146571 0.497551454
23 23 D 0.0674356898 0.1880570 0.3585916815 0.720770705
24 24 D 0.0404156883 0.1881361 0.2148215669 0.830353599
25 25 D -0.3547861433 0.1847361 -1.9205023223 0.058301550
26 26 D 0.1320541191 0.1877070 0.7035120167 0.483407490
27 27 D 0.1677531492 0.1874158 0.8950851945 0.373065225
28 28 D 0.0666607318 0.1880599 0.3544655007 0.723813831
29 29 D 0.1726545544 0.1873704 0.9214611071 0.359200407
30 30 D -0.0793481629 0.1880096 -0.4220431583 0.673920037
31 31 D -0.1296072153 0.1877244 -0.6904122462 0.491583964
32 32 D 0.0533549863 0.1881032 0.2836474488 0.777290271
33 33 D -0.0375219099 0.1881422 -0.1994337788 0.842348788
34 34 D -0.2641900978 0.1862784 -1.4182539992 0.159299447
35 35 D -0.1221630750 0.1877753 -0.6505811126 0.516912825
36 36 D 0.0598267303 0.1880833 0.3180863354 0.751097054
37 37 D -0.0619667999 0.1880762 -0.3294770284 0.742511121
38 38 D 0.1682873113 0.1874110 0.8979587515 0.371484367
39 39 D -0.0278088028 0.1881594 -0.1477938476 0.882810423
40 40 D 0.0803134396 0.1880054 0.4271868498 0.670193955
41 41 D -0.3733916212 0.1843616 -2.0253224543 0.045745216
42 42 D -0.0841410631 0.1879883 -0.4475866333 0.655480471
43 43 D 0.3074203062 0.1856004 1.6563561851 0.101218395
44 44 D -0.1769068587 0.1873299 -0.9443597686 0.347347190
45 45 D -0.1642876295 0.1874472 -0.8764476426 0.383033209
46 46 D -0.0623282683 0.1880750 -0.3314011009 0.741078908
47 47 D -0.0613653298 0.1880783 -0.3262755251 0.744912490
48 48 D -0.0932679240 0.1879444 -0.4962527998 0.620830229
49 49 D -0.0539113345 0.1881016 -0.2866075911 0.775032449
50 50 D 0.0635588234 0.1880708 0.3379515468 0.736137859
51 51 D -0.1161411557 0.1878143 -0.6183829054 0.537827653
52 52 D 0.1248537843 0.1877573 0.6649744782 0.507727947
53 53 D -0.1517165512 0.1875553 -0.8089165188 0.420685306
54 54 D -0.2563627286 0.1863900 -1.3754105029 0.172140793
55 55 D -0.2824650274 0.1860046 -1.5185917411 0.132443367
56 56 D 0.0548065878 0.1880989 0.2913710949 0.771411820
57 57 D 0.2790049733 0.1860579 1.4995602140 0.137179645
58 58 D 0.0235251099 0.1881654 0.1250235893 0.900762201
59 59 D -0.2318957099 0.1867167 -1.2419656113 0.217261577
60 60 D 0.0905554253 0.1879579 0.4817856483 0.631046649
61 61 D -0.5035856373 0.1811743 -2.7795649665 0.006725205
62 62 D -0.2319067976 0.1867166 -1.2420259286 0.217311321
63 63 D -0.1669423055 0.1874232 -0.8907236412 0.375318661
64 64 D 0.2543662439 0.1864179 1.3644948595 0.175543018
65 65 D 0.2384262418 0.1866327 1.2775155920 0.204462124
66 66 D -0.3868883969 0.1840774 -2.1017705044 0.038205356
67 67 D -0.1041345613 0.1878861 -0.5542429198 0.580675694
68 68 D -0.2979368330 0.1857581 -1.6038967921 0.112105133
69 69 D -0.0563216820 0.1880944 -0.2994331388 0.765250326
70 70 D 0.2527161013 0.1864408 1.3554765049 0.178477362
71 71 D -0.0674049402 0.1880572 -0.3584279551 0.720811342
72 72 D -0.1670487340 0.1874223 -0.8912960940 0.374994788
73 73 D 0.0596402439 0.1880839 0.3170938070 0.751859597
74 74 D -0.0546999448 0.1880992 -0.2908036535 0.771816262
75 75 D 0.2048137968 0.1870396 1.0950291528 0.276193417
76 76 D -0.0801218863 0.1880062 -0.4261660881 0.670943519
77 77 D 0.3308682547 0.1851885 1.7866567616 0.077409327
78 78 D 0.4175200553 0.1833932 2.2766393095 0.025013324
79 79 D 0.0311804541 0.1881540 0.1657177190 0.868722740
80 80 D 0.1653957366 0.1874372 0.8824060238 0.379783223
81 81 D -0.1215843506 0.1877792 -0.6474858917 0.518880884
82 82 D 0.3851074139 0.1841155 2.0916623410 0.039141347
83 83 D 0.1135260783 0.1878306 0.6044066639 0.546993153
84 84 D -0.1218559597 0.1877774 -0.6489385287 0.518020776
85 85 D -0.1215504808 0.1877794 -0.6473047503 0.518948384
86 86 D -0.1600224776 0.1874848 -0.8535223382 0.395548806
87 87 D 0.0074699045 0.1881789 0.0396957684 0.968416616
88 88 D 0.2277930860 0.1867682 1.2196565828 0.225522598
89 89 D 0.2684913623 0.1862156 1.4418303287 0.152548597
90 90 D 0.3559145472 0.1847140 1.9268415429 0.056925270
91 91 D -0.0755612484 0.1880255 -0.4018669980 0.688664660
92 92 D -0.0311500673 0.1881541 -0.1655561746 0.868847307
93 93 D -0.3423288144 0.1849758 -1.8506680889 0.067373269
94 94 D 0.0748689655 0.1880283 0.3981791576 0.691370269
95 95 D 0.1451316428 0.1876084 0.7735880659 0.441113169
96 96 D 0.0873918104 0.1879732 0.4649163447 0.643050166
97 97 D 0.2053484400 0.1870336 1.0979227161 0.274931517
98 98 D -0.1291998977 0.1877273 -0.6882319790 0.492937991
99 99 D 0.0710205716 0.1880436 0.3776814634 0.706533612
100 100 D -0.0816945451 0.1879993 -0.4345469791 0.664850422
101 101 D -0.1311338243 0.1877136 -0.6985846891 0.486504532
102 102 D 0.2469548229 0.1865195 1.3240157922 0.188581000
103 103 D 0.2986198875 0.1857469 1.6076707752 0.111240994
104 104 D 0.0929681305 0.1879459 0.4946536924 0.621951956
105 105 D 0.1851593228 0.1872485 0.9888425235 0.325274211
106 106 D -0.3477073468 0.1848734 -1.8807861808 0.063050321
107 107 D 0.0984996912 0.1879171 0.5241655489 0.601351416
108 108 D -0.0523515012 0.1881061 -0.2783084332 0.781434966
109 109 D -0.2757815567 0.1861069 -1.4818448942 0.141596119
110 110 D -0.1894422457 0.1872048 -1.0119517342 0.314087538
111 111 D 0.2221667928 0.1868374 1.1890918956 0.237404397
112 112 D -0.1608331634 0.1874777 -0.8578787356 0.393447053
113 113 D -0.0997348392 0.1879105 -0.5307571581 0.596826640
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930 930 D 0.3953414818 0.1838940 2.1498333519 0.034188487
931 931 D -0.0906533658 0.1879574 -0.4823079613 0.630682670
932 932 D 0.1209587836 0.1877833 0.6441403533 0.521179662
933 933 D 0.1551751232 0.1875264 0.8274841984 0.409976032
934 934 D 0.3775654046 0.1842748 2.0489257938 0.043144113
935 935 D 0.0805858871 0.1880042 0.4286387094 0.669155265
936 936 D -0.0001591084 0.1881804 -0.0008455101 0.999327112
937 937 D 0.0965043981 0.1879277 0.5135187408 0.608748554
938 938 D -0.0519185103 0.1881073 -0.2760047917 0.783128431
939 939 D 0.0312188548 0.1881539 0.1659218682 0.868579376
940 940 D 0.0505260212 0.1881111 0.2685966344 0.788828090
941 941 D 0.1645808080 0.1874445 0.8780239950 0.382146048
942 942 D -0.1675343473 0.1874178 -0.8939082016 0.373563912
943 943 D 0.1803669101 0.1872963 0.9630032216 0.337930051
944 944 D -0.1159649102 0.1878154 -0.6174408490 0.538412148
945 945 D 0.1477183352 0.1875878 0.7874622588 0.432911728
946 946 D -0.0003544457 0.1881804 -0.0018835423 0.998501094
947 947 D 0.2627148548 0.1862997 1.4101732776 0.161837369
948 948 D 0.0131644365 0.1881757 0.0699582257 0.944382045
949 949 D 0.2223357512 0.1868353 1.1900092654 0.236951568
950 950 D -0.0048433990 0.1881797 -0.0257381533 0.979518601
951 951 D 0.0244222058 0.1881642 0.1297919867 0.896998323
952 952 D -0.1962438782 0.1871333 -1.0486848813 0.297168827
953 953 D 0.0597692882 0.1880835 0.3177806126 0.751379819
954 954 D -0.0754276834 0.1880261 -0.4011554747 0.689179248
955 955 D -0.1939764000 0.1871574 -1.0364343668 0.302720215
956 956 D 0.2692785872 0.1862040 1.4461479006 0.151333917
957 957 D 0.0577296755 0.1880900 0.3069258167 0.759563027
958 958 D 0.0368278373 0.1881436 0.1957432376 0.845225931
959 959 D -0.1843197442 0.1872570 -0.9843143272 0.327390066
960 960 D -0.0740925273 0.1880315 -0.3940432130 0.694428504
961 961 D 0.2523700149 0.1864456 1.3535854955 0.178988196
962 962 D -0.0695058196 0.1880493 -0.3696147871 0.712481503
963 963 D 0.2639323589 0.1862821 1.4168420208 0.160020694
964 964 D -0.1295452377 0.1877248 -0.6900804894 0.491824752
965 965 D -0.1163097194 0.1878132 -0.6192839173 0.537220818
966 966 D -0.1194344938 0.1877932 -0.6359893532 0.526266899
967 967 D -0.1818610577 0.1872815 -0.9710571170 0.333974991
968 968 D 0.2699424451 0.1861942 1.4497894882 0.150394826
969 969 D 0.0776819375 0.1880167 0.4131651065 0.680394951
970 970 D 0.1243453088 0.1877607 0.6622541797 0.509385099
971 971 D -0.1700123354 0.1873951 -0.9072401625 0.366577972
972 972 D 0.3378920046 0.1850590 1.8258607013 0.071217264
973 973 D -0.0543077219 0.1881004 -0.2887166822 0.773408882
974 974 D -0.0730340845 0.1880357 -0.3884054122 0.698576817
975 975 D 0.1000386096 0.1879088 0.5323783962 0.595800831
976 976 D 0.0098591257 0.1881777 0.0523926247 0.958331952
977 977 D 0.2722372411 0.1861601 1.4623820239 0.146843089
978 978 D 0.0905166497 0.1879581 0.4815788603 0.631181789
979 979 D 0.0357434323 0.1881457 0.1899773735 0.849751598
980 980 D 0.1145542049 0.1878243 0.6099010368 0.543359456
981 981 D -0.0755271118 0.1880257 -0.4016851460 0.688804464
982 982 D 0.3042643656 0.1856534 1.6388837073 0.104652038
983 983 D 0.3512700620 0.1848047 1.9007640093 0.060348489
984 984 D -0.3619354086 0.1845945 -1.9607047573 0.052760702
985 985 D -0.0643086052 0.1880682 -0.3419429752 0.733129923
986 986 D -0.1248011166 0.1877576 -0.6646927040 0.507869302
987 987 D 0.1221635701 0.1877753 0.6505837605 0.516848331
988 988 D -0.3229263433 0.1853315 -1.7424257492 0.084918592
989 989 D 0.1608528964 0.1874776 0.8579847817 0.393028671
990 990 D 0.3947600648 0.1839068 2.1465228910 0.034445357
991 991 D 0.0949213739 0.1879359 0.5050730497 0.614642742
992 992 D -0.0145640428 0.1881746 -0.0773964230 0.938473668
993 993 D 0.2461230992 0.1865308 1.3194772590 0.190160515
994 994 D -0.2246358607 0.1868072 -1.2025008644 0.232085504
995 995 D 0.0287986601 0.1881579 0.1530558213 0.878676812
996 996 D -0.0843542311 0.1879874 -0.4487229027 0.654628605
997 997 D -0.1017955399 0.1878992 -0.5417560632 0.589288478
998 998 D 0.0079708015 0.1881787 0.0423576288 0.966305745
999 999 D 0.1881300175 0.1872183 1.0048696697 0.317508581
1000 1000 D 0.0571169562 0.1880919 0.3036651493 0.762048710
conf.low conf.high df outcome
1 -0.257401024 0.488083478 97.56246 Y
2 -0.554674724 0.189378531 90.45119 Y
3 -0.073288075 0.664391579 95.27674 Y
4 -0.440055266 0.306389531 97.47826 Y
5 -0.107464262 0.631974126 97.99182 Y
6 -0.210973317 0.533260089 96.37068 Y
7 -0.271537380 0.474989463 90.56870 Y
8 -0.401305227 0.345489818 97.96644 Y
9 -0.493480127 0.252286592 93.41292 Y
10 -0.145718265 0.595690516 97.99895 Y
11 -0.362562959 0.384773547 93.26586 Y
12 -0.693798673 0.041826806 95.16538 Y
13 -0.204739627 0.539893030 90.37705 Y
14 -0.527647679 0.216683328 97.35870 Y
15 -0.594951460 0.146540747 97.52113 Y
16 -0.508850592 0.236512474 93.14713 Y
17 -0.526772479 0.217758439 95.65075 Y
18 -0.152056350 0.589653213 97.96544 Y
19 -0.192007205 0.551895790 92.81085 Y
20 -0.387839713 0.359305154 95.03726 Y
21 -0.375969435 0.371179066 95.21183 Y
22 -0.244783597 0.500449077 96.77883 Y
23 -0.306356704 0.441228084 86.85719 Y
24 -0.332942221 0.413773597 97.83283 Y
25 -0.722333946 0.012761660 81.27898 Y
26 -0.240451711 0.504559949 97.84388 Y
27 -0.204444957 0.539951255 92.48379 Y
28 -0.306919712 0.440241176 90.59252 Y
29 -0.199439220 0.544748328 92.74930 Y
30 -0.452457652 0.293761327 97.77840 Y
31 -0.502188916 0.242974485 96.98437 Y
32 -0.320018098 0.426728071 96.17703 Y
33 -0.411021229 0.335977409 95.20102 Y
34 -0.633869087 0.105488891 97.67129 Y
35 -0.495022521 0.250696371 93.47787 Y
36 -0.313423825 0.433077285 97.88772 Y
37 -0.435283081 0.311349481 96.24445 Y
38 -0.203784596 0.540359218 94.72147 Y
39 -0.401225064 0.345607458 97.58099 Y
40 -0.292844998 0.453471878 96.59542 Y
41 -0.739572029 -0.007211213 91.57912 Y
42 -0.457392280 0.289110153 94.07766 Y
43 -0.061431591 0.676272203 87.82609 Y
44 -0.548735064 0.194921347 96.38157 Y
45 -0.536494857 0.207919598 93.48983 Y
46 -0.435753520 0.311096984 94.04070 Y
47 -0.434606175 0.311875516 97.88083 Y
48 -0.466249781 0.279713933 97.74096 Y
49 -0.427305494 0.319482825 95.68842 Y
50 -0.309760388 0.436878035 95.96795 Y
51 -0.489080676 0.256798365 93.42708 Y
52 -0.248041145 0.497748713 92.13334 Y
53 -0.524301240 0.220868138 90.47374 Y
54 -0.626248642 0.113523185 97.97721 Y
55 -0.652100589 0.087170534 88.16199 Y
56 -0.318685783 0.428298958 93.66225 Y
57 -0.090561513 0.648571460 91.26777 Y
58 -0.349905192 0.396955412 97.53505 Y
59 -0.602504569 0.138713149 96.42495 Y
60 -0.282502880 0.463613731 96.71514 Y
61 -0.863908765 -0.143262510 83.39433 Y
62 -0.602628120 0.138814524 94.17082 Y
63 -0.539005062 0.205120451 95.38574 Y
64 -0.115582966 0.624315454 97.80989 Y
65 -0.131976820 0.608829304 97.23435 Y
66 -0.752303419 -0.021473375 95.50193 Y
67 -0.476990354 0.268721231 97.96073 Y
68 -0.666783229 0.070909563 93.62386 Y
69 -0.429637580 0.316994216 96.98606 Y
70 -0.117412232 0.622844434 95.06809 Y
71 -0.440703168 0.305893287 95.84444 Y
72 -0.539070405 0.204972937 96.17396 Y
73 -0.313693636 0.432974124 96.19657 Y
74 -0.427978528 0.318578638 97.96278 Y
75 -0.166369877 0.575997470 97.79537 Y
76 -0.453324639 0.293080866 95.73258 Y
77 -0.037114244 0.698850753 88.70385 Y
78 0.053519231 0.781520879 96.65965 Y
79 -0.342235823 0.404596731 97.35835 Y
80 -0.206711168 0.537502641 95.06011 Y
81 -0.494378415 0.251209714 94.89505 Y
82 0.019583546 0.750631282 94.83046 Y
83 -0.259294816 0.486346972 96.40401 Y
84 -0.494877997 0.251166078 90.54252 Y
85 -0.494193249 0.251092287 97.98502 Y
86 -0.532298870 0.212253915 93.59699 Y
87 -0.365975565 0.380915374 97.77709 Y
88 -0.142842871 0.598429043 97.98926 Y
89 -0.101062425 0.638045149 97.68481 Y
90 -0.010690459 0.722519553 97.02282 Y
91 -0.448732703 0.297610207 97.14718 Y
92 -0.404535930 0.342235796 97.99489 Y
93 -0.709625961 0.024968332 93.54376 Y
94 -0.298299595 0.448037526 97.32303 Y
95 -0.227363265 0.517626550 94.11685 Y
96 -0.285759184 0.460542805 95.45574 Y
97 -0.165818891 0.576515771 97.88952 Y
98 -0.501753923 0.243354127 97.67687 Y
99 -0.302429814 0.444470957 92.37838 Y
100 -0.454791730 0.291402639 97.61037 Y
101 -0.503753657 0.241486008 95.77150 Y
102 -0.123192062 0.617101708 97.89591 Y
103 -0.070155967 0.667395742 94.57045 Y
104 -0.280005415 0.465941676 97.97823 Y
105 -0.186618375 0.556937021 94.16855 Y
106 -0.714706367 0.019291674 95.42524 Y
107 -0.274433456 0.471432838 97.62932 Y
108 -0.426224487 0.321521485 87.13496 Y
109 -0.645113505 0.093550391 97.81214 Y
110 -0.561008438 0.182123947 96.66136 Y
111 -0.148812503 0.593146089 93.81893 Y
112 -0.533757174 0.212090848 82.42707 Y
113 -0.472788066 0.273318388 94.92637 Y
114 -0.236338982 0.508547163 97.93150 Y
115 -0.321580895 0.425646577 91.68534 Y
116 -0.222211057 0.522424126 96.05871 Y
117 -0.553476619 0.189918503 97.12489 Y
118 -0.300019022 0.446341258 97.36522 Y
119 -0.055962016 0.680976848 91.56523 Y
120 -0.416623986 0.330212998 96.32593 Y
121 -0.342466821 0.404421431 96.79862 Y
122 -0.049819330 0.686202779 96.71792 Y
123 -0.713052106 0.020953286 96.64206 Y
124 -0.471919963 0.274080958 96.15451 Y
125 -0.402944592 0.343956708 96.76388 Y
126 -0.466073939 0.280518390 91.67763 Y
127 -0.212571044 0.531615552 97.43115 Y
128 -0.589257462 0.153730448 86.51538 Y
129 -0.503430602 0.242293601 91.28765 Y
130 -0.104683882 0.634911538 94.70416 Y
131 -0.431711801 0.314809858 97.85599 Y
132 -0.427197168 0.319543238 96.18022 Y
133 -0.289457511 0.457096004 93.64186 Y
134 -0.085240278 0.652985257 97.23883 Y
135 -0.270506067 0.475642879 93.90946 Y
136 -0.173651011 0.569040187 97.65185 Y
137 -0.264397227 0.481667107 93.39566 Y
138 -0.146357952 0.595158481 97.16949 Y
139 0.076785174 0.802690979 96.54811 Y
140 -0.320378040 0.426200477 97.95618 Y
141 -0.514695412 0.230550445 92.57918 Y
142 -0.445669708 0.300719899 97.17312 Y
143 -0.488188888 0.257259572 97.90736 Y
144 -0.460362605 0.286317182 91.89961 Y
145 -0.107199726 0.632358451 96.54741 Y
146 -0.537841532 0.206631300 92.49132 Y
147 -0.465064583 0.280932187 97.64635 Y
148 -0.627300705 0.112464112 97.43031 Y
149 -0.407565158 0.339222443 97.60678 Y
150 -0.430066910 0.316690070 95.67103 Y
151 -0.535941075 0.208079229 97.55892 Y
152 -0.388894514 0.358353630 93.99848 Y
153 -0.161359574 0.580865161 96.97026 Y
154 -0.436115825 0.310730154 94.03698 Y
155 -0.319180106 0.427651597 95.21879 Y
156 -0.220701658 0.523878896 96.12291 Y
157 -0.430110837 0.316434013 97.82274 Y
158 -0.451244687 0.295180064 95.89680 Y
159 -0.061752335 0.674956362 97.80744 Y
160 -0.393528342 0.354250946 88.99958 Y
161 -0.276178876 0.470081660 93.99871 Y
162 -0.464255407 0.281959385 95.60470 Y
163 -0.047265807 0.689360010 89.05240 Y
164 -0.283978236 0.462848147 90.15748 Y
165 -0.364965143 0.381947766 97.52980 Y
166 -0.405056449 0.341962437 95.44109 Y
167 -0.085809945 0.652620687 95.43078 Y
168 -0.649174838 0.089320612 96.94975 Y
169 -0.487017037 0.258469931 97.81291 Y
170 -0.561587108 0.182163009 90.61633 Y
171 -0.645456022 0.094330861 87.34750 Y
172 -0.200379059 0.543327811 97.97913 Y
173 -0.322143385 0.424461651 97.88393 Y
174 -0.299311305 0.447426073 93.47186 Y
175 -0.398035200 0.348960424 96.09832 Y
176 -0.210556526 0.533647463 96.52781 Y
177 -0.383276528 0.363618728 97.68617 Y
178 -0.334602126 0.412216372 96.91076 Y
179 -0.477478012 0.268254314 97.63113 Y
180 -0.230431685 0.514645869 94.17582 Y
181 -0.386338598 0.360560223 97.57108 Y
182 -0.435010365 0.311541235 97.10333 Y
183 -0.601587603 0.139546162 97.75261 Y
184 -0.565877114 0.177282337 94.37028 Y
185 -0.409041814 0.339040338 85.72842 Y
186 -0.590841377 0.150880507 97.24690 Y
187 -0.702822135 0.033290663 85.09257 Y
188 -0.332094572 0.415122154 92.76955 Y
189 -0.224360049 0.520220045 97.34137 Y
190 -0.397854468 0.349004382 97.50500 Y
191 -0.352558788 0.395882218 83.56458 Y
192 -0.400294176 0.346642703 96.56574 Y
193 -0.370528651 0.376462846 96.78931 Y
194 -0.533260256 0.211563786 90.80471 Y
195 -0.721027395 0.012218955 97.88215 Y
196 -0.252434693 0.493262023 94.13226 Y
197 -0.533640013 0.210426111 97.93533 Y
198 -0.489228928 0.256434907 95.46477 Y
199 -0.326742564 0.419900377 97.99179 Y
200 -0.367241485 0.381384928 82.51251 Y
201 -0.434640204 0.311877928 97.49832 Y
202 -0.580525113 0.162037902 93.81811 Y
203 -0.411275340 0.336073682 91.84908 Y
204 -0.309823257 0.436742027 96.72400 Y
205 -0.171694351 0.570994521 96.80218 Y
206 -0.226366978 0.518662766 93.42443 Y
207 -0.456168530 0.290792301 90.07577 Y
208 -0.253010740 0.492322698 97.99949 Y
209 -0.398980326 0.348069201 95.50822 Y
210 -0.334324606 0.412392099 97.94735 Y
211 -0.561772570 0.181239332 97.55874 Y
212 -0.512501299 0.232408353 96.47168 Y
213 -0.526497660 0.217840283 97.68891 Y
214 -0.397620726 0.350043150 89.83666 Y
215 -0.741502833 -0.009485446 93.19903 Y
216 -0.269035387 0.476747166 97.28550 Y
217 -0.445992042 0.300367427 97.43053 Y
218 -0.614392546 0.127807389 82.66221 Y
219 -0.378028357 0.369361684 92.86611 Y
220 -0.142852055 0.598549078 96.61359 Y
221 0.161889810 0.879307132 94.84163 Y
222 -0.187690079 0.555795758 95.28894 Y
223 -0.108047600 0.631424687 97.97437 Y
224 -0.059093659 0.677570914 96.41699 Y
225 -0.399306540 0.347837885 94.55685 Y
226 -0.319206383 0.427356998 97.97431 Y
227 -0.371614342 0.376741336 84.60542 Y
228 -0.257691666 0.487803735 97.52272 Y
229 -0.099199720 0.639774500 97.98211 Y
230 -0.046167606 0.689495481 97.94034 Y
231 -0.190005913 0.553704567 93.93893 Y
232 -0.502724759 0.242422348 96.99519 Y
233 -0.139830929 0.601426831 96.57932 Y
234 -0.194015985 0.549694769 95.47151 Y
235 -0.392582747 0.354807543 92.52241 Y
236 -0.443856035 0.302482025 97.99715 Y
237 -0.552188113 0.191341429 96.28987 Y
238 -0.234637774 0.511116123 89.04386 Y
239 -0.312237513 0.434358197 96.74183 Y
240 -0.542609241 0.201185145 97.35795 Y
241 -0.174769581 0.568549607 91.80659 Y
242 -0.552207384 0.191217432 97.33883 Y
243 -0.361039744 0.385819400 97.99998 Y
244 -0.549939495 0.193566377 97.42698 Y
245 -0.265794850 0.479896034 97.48320 Y
246 -0.424005926 0.323034024 93.60281 Y
247 -0.446805466 0.299555825 97.27880 Y
248 -0.472097913 0.273791623 97.24318 Y
249 -0.494350660 0.250981027 97.46537 Y
250 -0.184611099 0.558685702 95.98412 Y
251 -0.571479380 0.171251034 96.17164 Y
252 -0.494630252 0.250896842 95.42656 Y
253 -0.492181257 0.255036349 81.74920 Y
254 -0.501773368 0.243427507 96.72388 Y
255 -0.506419678 0.239005054 93.24276 Y
256 -0.349835321 0.397121652 96.53715 Y
257 -0.438387498 0.308168812 96.58232 Y
258 -0.144215108 0.597128436 97.92782 Y
259 -0.198966233 0.544699470 97.86710 Y
260 -0.745275078 -0.013992410 97.14885 Y
261 -0.582268989 0.159876128 97.11315 Y
262 -0.376141908 0.371556352 90.06627 Y
263 -0.609859650 0.131101155 95.15390 Y
264 -0.558831916 0.184408442 96.48635 Y
265 -0.421878380 0.324937925 96.00970 Y
266 -0.583220134 0.158810932 97.82186 Y
267 -0.378979591 0.368143976 95.43409 Y
268 -0.410082534 0.337046084 94.00851 Y
269 -0.188546219 0.554714210 97.99468 Y
270 -0.311945321 0.434659041 96.61347 Y
271 -0.208546281 0.535940820 92.99703 Y
272 -0.072593848 0.665677092 89.12609 Y
273 -0.645849906 0.093033229 95.16980 Y
274 -0.227365850 0.517695835 93.41472 Y
275 -0.825815345 -0.101578118 89.41795 Y
276 -0.327590946 0.419655867 92.09143 Y
277 -0.391825560 0.355265779 95.42828 Y
278 -0.455648262 0.290956810 93.40129 Y
279 -0.376133128 0.370792652 97.46871 Y
280 -0.017488325 0.716458826 94.57283 Y
281 -0.340285718 0.406619719 96.46893 Y
282 -0.070538518 0.667229374 92.64844 Y
283 -0.248807954 0.496894679 93.15679 Y
284 -0.486645663 0.258927222 97.02596 Y
285 -0.697657749 0.037433063 97.62654 Y
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915 0.044402083 0.773577385 92.69315 Y
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920 -0.549236484 0.194242083 97.99109 Y
921 -0.352225584 0.394864559 95.31142 Y
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924 -0.602367015 0.138901502 95.99072 Y
925 -0.675985417 0.060968180 94.67214 Y
926 0.101105177 0.824932826 93.95592 Y
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929 0.023407628 0.754268304 93.53602 Y
930 0.030117793 0.760565171 92.10777 Y
931 -0.463735489 0.282428757 96.21077 Y
932 -0.252282686 0.494200253 86.96891 Y
933 -0.216971624 0.527321870 97.85836 Y
934 0.011874818 0.743255992 97.93078 Y
935 -0.292643409 0.453815183 95.12399 Y
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937 -0.276449076 0.469457872 97.64263 Y
938 -0.425234222 0.321397202 97.52055 Y
939 -0.342428041 0.404865750 92.80321 Y
940 -0.322967972 0.424020014 94.09857 Y
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944 -0.488796227 0.256866406 95.58493 Y
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948 -0.360676962 0.387005835 90.04292 Y
949 -0.148487691 0.593159193 96.85382 Y
950 -0.378286357 0.368599559 97.86603 Y
951 -0.349006195 0.397850606 97.52645 Y
952 -0.568085003 0.175597246 88.80564 Y
953 -0.313838267 0.433376843 90.94944 Y
954 -0.448560560 0.297705193 97.97312 Y
955 -0.565694797 0.177741997 91.85957 Y
956 -0.100248169 0.638805344 97.76944 Y
957 -0.315614693 0.431074044 96.22866 Y
958 -0.336646505 0.410302179 95.75392 Y
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960 -0.447361815 0.299176760 95.40408 Y
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963 -0.106207194 0.634071911 88.98404 Y
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966 -0.492107279 0.253238292 97.93197 Y
967 -0.553640889 0.189918774 95.41794 Y
968 -0.099676930 0.639561820 95.45889 Y
969 -0.295465552 0.450829427 97.28082 Y
970 -0.248326815 0.497017432 96.60778 Y
971 -0.542044064 0.202019393 94.89174 Y
972 -0.029805066 0.705589075 89.19848 Y
973 -0.427593493 0.318978049 97.86031 Y
974 -0.446286173 0.300218004 95.91606 Y
975 -0.273372100 0.473449319 88.31621 Y
976 -0.363988178 0.383706429 90.01099 Y
977 -0.097201562 0.641676044 97.78684 Y
978 -0.282493111 0.463526411 97.72707 Y
979 -0.338004025 0.409490890 90.65664 Y
980 -0.258244602 0.487353012 96.59656 Y
981 -0.448733048 0.297678825 96.44619 Y
982 -0.064457993 0.672986724 92.02546 Y
983 -0.015588673 0.718128797 95.50293 Y
984 -0.728270706 0.004399889 97.71351 Y
985 -0.437548870 0.308931660 97.47735 Y
986 -0.497582821 0.247980587 94.29962 Y
987 -0.250503270 0.494830410 97.31388 Y
988 -0.691220599 0.045367913 88.22544 Y
989 -0.211266576 0.532972369 96.41722 Y
990 0.029536166 0.759983964 92.58593 Y
991 -0.278036530 0.467879278 97.89154 Y
992 -0.388218149 0.359090063 93.44413 Y
993 -0.124156667 0.616402865 95.61295 Y
994 -0.595378556 0.146106834 97.37339 Y
995 -0.344711190 0.402308510 95.61100 Y
996 -0.457439189 0.288730727 97.36995 Y
997 -0.474950097 0.271359017 92.54924 Y
998 -0.365787031 0.381728634 91.65427 Y
999 -0.183538396 0.559798431 95.13806 Y
1000 -0.316304512 0.430538425 94.76730 YVisualize
Visualize
Visualize
Visualize
Normal-approximation inference
Compare with:
Normal approximation compares test statistic to t-distribution
Based on “weak” null hypothesis \(H_0: E[\tau_i] = 0\)
vs. randomization inference sharp null \(H_0 : \tau_i = Y_i(1) - Y_i(0) = 0 \forall i\)
Strategy 2: Ignorability
What if we cannot assume random assignment?
Strategy 2: Ignorability
We can still conduct valid causal inference if we believe treatment assignment is strongly ignorable (conditional on covariates)
Requirements:
- Conditional independence: \((Y_i(0), Y_i(1)) \perp \!\!\! \perp D_i | \mathbf{X}_i\)
- Positivity: There exists \(\epsilon > 0\) such that \[\epsilon < Pr[D_i = 1 | \mathbf{X}_i] < 1 - \epsilon\]
Strategy 2: Ignorability
If strong ignorability holds:
- ATE \(E[\tau_i]\) is point-identified
- Conditional ATE (CATE) \(E[\tau_i | \mathbf{X}_i]\) is point-identified
Estimation strategies (more in the lab):
- Conditioning on covariates (regression)
- Propensity score matching
- IPW estimation
- Doubly robust estimation
Next week: Causal Inference II
General topic: Satisfying ignorability by design.
Regression discontinuity
Difference-in-differences
Focus: How do they satisfy ignorability?
Just for reference: Methodological critiques
CHECK UPDATED SYLLABUS ON GITHUB!