Inferência Causal
Abordagem de Neyman
Prof. Carlos Trucíos
ctrucios@unicamp.br
ctrucios@unicamp.br
Instituto de Matemática, Estatística e Computação Científica (IMECC),
Universidade Estadual de Campinas (UNICAMP).
Introdução
Introdução
Neyman
Neyman
- Neyman (1923) propõe um método para fazer inferência para o efeito causal médio (ACE: Average Causal Effect, também chamado de ATE: Average Treatment Effect).
- Diferente da abordagem de Fisher (p-valor), Neyman propõe um estimador pontual não viesado e, baseado na distribuição amostral do estimador pontual, intervalos de confiança.
- O teste de hipóteses é feito através da relação intervalo de confiança – teste de hipóteses.
Neyman
Considere um CRE com \(n\) unidades, em que \(n_1\) recebem o tratamento e \(n_0\) recebem o placebo. Para cada \(i\) (\(i = 1, \cdots, n.\)) temos os resultados potenciais \(Y_i(1)\) e \(Y_i(0)\), bem como o efeito causal individual \(\tau_i = Y_i(1) - Y_i(0)\). Então:
- Médias: \(\bar{Y}(1) = \displaystyle \sum_{i = 1}^n Y_i(1) \Big / n\) e \(\bar{Y}(0) = \displaystyle \sum_{i = 1}^n Y_i(0) \Big / n\).
- Variâncias: \(S^2(1) = \displaystyle \sum_{i = 1}^n (Y_i(1) - \bar{Y}(1))^2 \Big / n - 1\) e \(S^2(0) = \displaystyle \sum_{i = 1}^n (Y_i(0) - \bar{Y}(0))^2 \Big / n - 1\).
- Covariância: \(S(1, 0) = \displaystyle \sum_{i = 1}^n [Y_i(1) - \bar{Y}(1)][Y_i(0) - \bar{Y}(0)]\Big / n - 1\)
Neyman
Ademais, \[\tau = \displaystyle \sum_{i = 1}^n \tau_i \Big / n = \bar{Y}(1) - \bar{Y}(0) \quad e \quad S^2(\tau) = \sum_{i = 1}^n(\tau_i - \tau)^2 \Big / n - 1.\]
Lema
\[2 S(1, 0) = S^2(1) + S^2(0) - S^2(\tau).\]
Estamos interessados em estimar \(\tau\), baseados em \((Z_i, Y_i)_{i = 1}^n\) sob CRE.
Teorema de Neyman
Utilizando os resultados observados (\(Y_i\)), podemos calcular as médias e variâncias amostrais como:
\[\hat{\bar{Y}}(1) = \displaystyle \sum_{i = 1}^n Z_i Y_i \Big /n_1 \quad e \quad \hat{\bar{Y}}(0) = \displaystyle \sum_{i = 1}^n (1 - Z_i) Y_i \Big /n_0\]
\[\hat{S}^2(1) = \displaystyle \sum_{i = 1}^n Z_i (Y_i - \hat{\bar{Y}}(1))^2 \Big / n_1 - 1 \quad e \quad \hat{S}^2(0) = \displaystyle \sum_{i = 1}^n (1 - Z_i) (Y_i - \hat{\bar{Y}}(0))^2 \Big / n_0 - 1\]
Quais seriam as versões amostrais de \(S(1, 0)\) e \(S^2(\tau)\)?
Nenhuma!. Não existem versões amostrais de \(S(1, 0)\) nem \(S^2(\tau)\) pois os resultados potenciais \(Y_i(1)\) e \(Y_i(0)\) nunca são observados simultaneamente para cada unidade \(i\).
Teorema de Neyman
Teorema
Sob CRE:
- O estimador da diferença de médias, \(\hat{\tau} = \hat{\bar{Y}}(1) - \hat{\bar{Y}}(0)\) é não viesado para \(\tau\), ou seja, \(\mathbb{E}(\hat{\tau}) = \tau.\)
- \[\mathbb{V}(\hat{\tau}) = \dfrac{S^2(1)}{n_1} + \dfrac{S^2(0)}{n_0} - \dfrac{S^2(\tau)}{n} = \dfrac{n_0 S^2(1)}{n_1 n} + \dfrac{n_1 S^2(0)}{n_0 n} + 2\dfrac{S(1, 0)}{n}\]
- O estimador da variância, \(\hat{V} = \dfrac{\hat{S}^2(1)}{n_1} +\dfrac{\hat{S}^2(0)}{n_0}\) é conservador para estimar \(\mathbb{V}(\hat{\tau})\). Isto significa que \[\mathbb{E}(\hat{V}) - \mathbb{V}(\hat{\tau}) = \dfrac{S^2(\tau)}{n} \geq 0,\] com a igualdade acontecendo se \(\tau_i = \tau\) \(\forall i\).
Demostração: (no final da aula)
Teorema de Neyman
- Repare que apenas \(Z_1, \cdots, Z_n\) é aleatório.
- Assim, \(\mathbb{E}(\cdot)\) e \(\mathbb{V}(\cdot)\) são em \(Z_1, \cdots, Z_n\), que são permutações aleatórias de \(n_1\) 1s e \(n_0\) 0s.
- \(\hat{\tau}\) tem uma distribuição discreta sob \(\hat{\tau}^1, \cdots, \hat{\tau}^M\), com \(M = \binom{n}{n_1}\).
Quais são as diferenças entre o teste de Neyman e o de Fisher?
Fisher vs Neyman
Fisher vs Neyman
- FRT funciona para qualquer teste, já o teorema Neyman apenas fornece resultados para a diferença de médias.
- No FRT, \(\textbf{Y}\) é fixo, já na abordagem de Neyman \(\textbf{Y}(\textbf{z}^m)\) muda com mudanças de \(\textbf{z}^m\).
- No FRT, todos os \(T(\textbf{z}^m, \textbf{Y})\) são calculáveis, já na abordagem de Neyman, os \(\hat{\tau}^m\) são apenas valores hipotéticos. Isto, pois não todos os resultados potenciais são conhecidos (apenas conhecemos os resultados observados).
- Eles testam coisas diferentes.
Teorema de Neyman
Teorema
Seja \(n, n_1 \rightarrow \infty\). Se \(n_1/n \rightarrow k \in (0, 1)\), \(S^2(1), S^2(0), S(1, 0) < \infty\) e \[\max_{1\leq i \leq n} \{Y_i(1) - \bar{Y}(1)\}^2 \big / n \rightarrow 0, \quad \max_{1\leq i \leq n} \{Y_i(0) - \bar{Y}(0)\}^2 \big / n \rightarrow 0,\] então \[\dfrac{\hat{\tau} - \tau}{\sqrt{\mathbb{V}(\hat{\tau})}} \xrightarrow D N(0,1),\]
\[\hat{S}^2(1) \xrightarrow p S^2(1), \quad e \quad \hat{S}^2(0) \xrightarrow p S^2(0).\]
A prova do teorema é bastante técnica e está em Li e Peng (2017).
Teorema de Neyman
O teorema garante a normalidade e, ao mesmo tempo, que o estimador da variância, \(\hat{V} = \frac{\hat{S}^2(1)}{n_1} + \frac{\hat{S}^2(0)}{n_0}\), é maior (em probabilidade) do que \(\mathbb{V}(\hat{\tau}) = \frac{S^2(1)}{n_1} + \frac{S^2(0)}{n_0} - \frac{S^2(\tau)}{n}\), o que permite calcular intervalos de confiança conservadores (i.e, de maior amplitude): \[\hat{\tau} \pm z_{1-\alpha/2} \sqrt{\hat{V}}.\]
Pela relação intervalo de confiança-teste de hipóteses, podemos utilizar o intervalo para testar a hipótese nula fraca: \[H_{0N}: \tau = 0 \quad \text{ou equivalentemente} \quad \bar{Y}(1) = \bar{Y}(0)\]
Obs: \(H_{0F}\) e \(H_{0N}\) são conhecidas como hipótese nula forte e fraca, respectivamente.
Ilustração
Ilustração
Simulação
Consideremos uma amostra de tamanho 100 com 60 tratamentos e 40 placebos em que os resultados potenciais são gerados com efeito individual constante.
A Scientific Table é então
y0 y1
[1,] 5.531437384 6.531437
[2,] 3.530565789 4.530566
[3,] 3.082787789 4.082788
[4,] 2.742774038 3.742774
[5,] 2.554862255 3.554862
[6,] 2.518311752 3.518312
[7,] 2.376787709 3.376788
[8,] 2.336964819 3.336965
[9,] 2.247490449 3.247490
[10,] 2.189801099 3.189801
[11,] 2.069604626 3.069605
[12,] 1.958386866 2.958387
[13,] 1.896820472 2.896820
[14,] 1.871186074 2.871186
[15,] 1.702603813 2.702604
[16,] 1.643298497 2.643298
[17,] 1.611272905 2.611273
[18,] 1.568482604 2.568483
[19,] 1.526815575 2.526816
[20,] 1.496307607 2.496308
[21,] 1.463493142 2.463493
[22,] 1.450147361 2.450147
[23,] 1.373591086 2.373591
[24,] 1.358418817 2.358419
[25,] 1.327395254 2.327395
[26,] 1.268603743 2.268604
[27,] 1.261845243 2.261845
[28,] 1.233794988 2.233795
[29,] 1.232852669 2.232853
[30,] 1.228724181 2.228724
[31,] 1.194152401 2.194152
[32,] 1.178740088 2.178740
[33,] 1.154126273 2.154126
[34,] 1.136876241 2.136876
[35,] 1.129138145 2.129138
[36,] 1.091129190 2.091129
[37,] 1.086578287 2.086578
[38,] 1.049756979 2.049757
[39,] 1.034405646 2.034406
[40,] 1.013648407 2.013648
[41,] 0.972700005 1.972700
[42,] 0.947718882 1.947719
[43,] 0.938918367 1.938918
[44,] 0.930975215 1.930975
[45,] 0.912622493 1.912622
[46,] 0.906122053 1.906122
[47,] 0.898091600 1.898092
[48,] 0.890358968 1.890359
[49,] 0.881581185 1.881581
[50,] 0.855943255 1.855943
[51,] 0.832073746 1.832074
[52,] 0.800477650 1.800478
[53,] 0.786604233 1.786604
[54,] 0.772637062 1.772637
[55,] 0.769510785 1.769511
[56,] 0.767649268 1.767649
[57,] 0.705733005 1.705733
[58,] 0.630076722 1.630077
[59,] 0.619247754 1.619248
[60,] 0.608887798 1.608888
[61,] 0.605445167 1.605445
[62,] 0.601326153 1.601326
[63,] 0.581004639 1.581005
[64,] 0.580513903 1.580514
[65,] 0.554804620 1.554805
[66,] 0.541912851 1.541913
[67,] 0.458092664 1.458093
[68,] 0.446501564 1.446502
[69,] 0.432933287 1.432933
[70,] 0.414536910 1.414537
[71,] 0.411949086 1.411949
[72,] 0.406352747 1.406353
[73,] 0.402976422 1.402976
[74,] 0.330015591 1.330016
[75,] 0.328348341 1.328348
[76,] 0.327567771 1.327568
[77,] 0.321524446 1.321524
[78,] 0.320680385 1.320680
[79,] 0.317501161 1.317501
[80,] 0.310050851 1.310051
[81,] 0.309874094 1.309874
[82,] 0.304518342 1.304518
[83,] 0.303330507 1.303331
[84,] 0.281418712 1.281419
[85,] 0.268290976 1.268291
[86,] 0.255402436 1.255402
[87,] 0.228566642 1.228567
[88,] 0.192877303 1.192877
[89,] 0.185665682 1.185666
[90,] 0.185283957 1.185284
[91,] 0.163040638 1.163041
[92,] 0.162535875 1.162536
[93,] 0.125687529 1.125688
[94,] 0.099078962 1.099079
[95,] 0.080654525 1.080655
[96,] 0.073148464 1.073148
[97,] 0.057259259 1.057259
[98,] 0.022616780 1.022617
[99,] 0.019628326 1.019628
[100,] 0.006032503 1.006033Ilustração
Simulação
Escolhemos 1 amostra e calculamos: \(\hat{\tau}\), \(\hat{V}\), intervalo de confiança e \(\mathbb{V}(\hat{\tau})\).
[1] 0.8674413[1] 0.03907339Code
[1] 0.4800083 1.2548742[1] 0.03125819Ilustração
Simulação
E se gerarmos não um, mas vários CRE?
Code
tau_hat_p <- c()
V_hat_p <- c()
lim_sup <- c()
lim_inf <- c()
est <- c()
for (i in 1:10^4) {
z_permut <- sample(z)
tau_hat_p[i] <- mean(y1[z_permut == 1]) - mean(y0[z_permut == 0])
V_hat_p[i] <- var(y1[z_permut == 1]) / n1 + var(y0[z_permut == 0]) / n0
lim_sup[i] <- tau_hat_p[i] + 1.96*sqrt(V_hat_p[i])
lim_inf[i] <- tau_hat_p[i] - 1.96*sqrt(V_hat_p[i])
est[i] <- (tau_hat_p[i] - tau) / sqrt(V_hat_p[i])
}
mean(V_hat_p)[1] 0.03126173Ilustração
Ilustração
Simulação
Repetiremos o experimento mas mudando os resultados potenciais:
Code
[1] 0.00575214Code
tau_hat_p <- c()
V_hat_p <- c()
lim_sup <- c()
lim_inf <- c()
cobertura <- c()
for (i in 1:10^4) {
z_permut <- sample(z)
tau_hat_p[i] <- mean(y1[z_permut == 1]) - mean(y0[z_permut == 0])
V_hat_p[i] <- var(y1[z_permut == 1]) / n1 + var(y0[z_permut == 0]) / n0
lim_sup[i] <- tau_hat_p[i] + 1.96*sqrt(V_hat_p[i])
lim_inf[i] <- tau_hat_p[i] - 1.96*sqrt(V_hat_p[i])
cobertura[i] <- ifelse(lim_inf[i] < tau && tau < lim_sup[i], 1, 0)
est[i] <- (tau_hat_p[i] - tau) / sqrt(V_hat_p[i])
}
c(mean(V_hat_p), mean(cobertura))[1] 0.03123609 1.00000000O que implica fazermos “y0 = sort(y0, decreasing = FALSE)”?
Ilustração
Ilustração
Simulação
Repetiremos o experimento mas mudando os resultados potenciais:
[1] 0.01574629Code
tau_hat_p <- c()
V_hat_p <- c()
lim_sup <- c()
lim_inf <- c()
cobertura <- c()
for (i in 1:10^4) {
z_permut <- sample(z)
tau_hat_p[i] <- mean(y1[z_permut == 1]) - mean(y0[z_permut == 0])
V_hat_p[i] <- var(y1[z_permut == 1]) / n1 + var(y0[z_permut == 0]) / n0
lim_sup[i] <- tau_hat_p[i] + 1.96*sqrt(V_hat_p[i])
lim_inf[i] <- tau_hat_p[i] - 1.96*sqrt(V_hat_p[i])
cobertura[i] <- ifelse(lim_inf[i] < tau && tau < lim_sup[i], 1, 0)
est[i] <- (tau_hat_p[i] - tau) / sqrt(V_hat_p[i])
}
c(mean(V_hat_p), mean(cobertura))[1] 0.03122516 0.99350000O que implica fazermos “y0 = sample(y0)”?
Ilustração
Ilustração
- Quando \(\tau_i = \tau\), \(\mathbb{V}(\hat{\tau}) = \mathbb{E}(\hat{V})\). Nas nossas simulações os resultados são bem próximos.
- Em todos os outros casos temos que \(\mathbb{E}(\hat{V}) > \mathbb{V}(\hat{\tau})\).
- Note que como estamos utilizamos MC = 10^4 e não todas as possíveis permutações, os resultados não são exatos mas sim aproximados.
Ilustração
Simulação
Code
set.seed(123)
eps <- rbinom(n, 1, 0.4)
y0 <- (1 - eps) * rexp(n) + eps*rcauchy(n)
tau <- 1
y1 <- y0 + tau
tau_hat_p <- c()
V_hat_p <- c()
lim_sup <- c()
lim_inf <- c()
cobertura <- c()
est <- c()
for (i in 1:10^4) {
z_permut <- sample(z)
tau_hat_p[i] <- mean(y1[z_permut == 1]) - mean(y0[z_permut == 0])
V_hat_p[i] <- var(y1[z_permut == 1]) / n1 + var(y0[z_permut == 0]) / n0
lim_sup[i] <- tau_hat_p[i] + 1.96*sqrt(V_hat_p[i])
lim_inf[i] <- tau_hat_p[i] - 1.96*sqrt(V_hat_p[i])
cobertura[i] <- ifelse(lim_inf[i] < tau && tau < lim_sup[i], 1, 0)
est[i] <- (tau_hat_p[i] - tau) / sqrt(V_hat_p[i])
}Ilustração
::: {.callout-note} ### Simulação
Ilustração
Simulação
lim_inf tau lim_sup
1 -2.3125940236 1 2.462131
2 -2.6242117937 1 1.858665
3 -0.7010915757 1 2.555248
4 -0.5280264732 1 2.717127
5 -0.3420326567 1 2.914151
6 -0.1773311099 1 3.077994
7 0.0168086246 1 3.624109
8 -0.4358171698 1 2.817504
9 -0.5772159830 1 2.668433
10 0.1250953916 1 3.365354
11 0.5072443055 1 4.098246
12 -1.9395897579 1 2.858510
13 -2.1317645140 1 2.653403
14 -0.0851779737 1 3.156221
15 -0.3955806289 1 2.845080
16 0.4448547697 1 4.057552
17 -2.4666477227 1 2.004497
18 -0.2821605489 1 2.972503
19 0.2642077209 1 3.893325
20 -2.0503707101 1 2.730265
21 -0.3109196788 1 2.932356
22 0.6623007697 1 4.242760
23 -0.2234031745 1 3.054162
24 -0.2176093639 1 3.032076
25 -0.4021756153 1 2.851615
26 -1.7485937626 1 3.029218
27 -1.8528201912 1 2.947404
28 -2.6500558151 1 1.841206
29 -2.9479762394 1 1.495257
30 -0.3109272058 1 2.946649
31 -0.4799688036 1 2.757914
32 -2.6228942070 1 1.858439
33 -2.5123570608 1 1.967980
34 0.2066221903 1 3.819700
35 -0.1028372927 1 3.150706
36 -0.3413204675 1 2.919592
37 -0.3333261333 1 2.916257
38 -2.6712269830 1 1.800780
39 -2.7003826357 1 1.773272
40 -0.3733445700 1 2.862686
41 -0.4651727443 1 2.781354
42 -1.7497430414 1 3.041975
43 -2.6507604872 1 1.811084
44 -1.7389185388 1 3.069674
45 -1.7690734075 1 3.019032
46 0.8987270500 1 4.477390
47 -0.6358514055 1 2.600204
48 0.4652658654 1 4.051407
49 0.7233174257 1 4.289506
50 -2.8060354011 1 1.650323
51 0.1047499129 1 3.708567
52 -1.6926855072 1 3.103478
53 -1.6551082224 1 3.152423
54 -0.3453934405 1 2.945730
55 -0.0660743095 1 3.572443
56 -1.8102432048 1 2.978086
57 0.1958534586 1 3.804073
58 0.6337586963 1 4.231145
59 -0.5501012528 1 2.694929
60 -1.4271082956 1 3.379706
61 -0.1976828336 1 3.039239
62 -2.0646502440 1 2.714740
63 -2.7802337322 1 1.669823
64 -0.3595384384 1 2.918993
65 0.5704051684 1 4.150743
66 -2.7542991585 1 1.712118
67 -0.6094635784 1 2.633835
68 0.3052981784 1 3.908642
69 -0.3873920656 1 2.872384
70 0.2947881470 1 3.929062
71 0.3354991619 1 3.946830
72 -0.2065778287 1 3.036562
73 0.6295266793 1 4.215042
74 0.1978947708 1 3.793144
75 -2.7022795170 1 1.781908
76 0.8876916344 1 4.460966
77 -2.0294604847 1 2.754008
78 -1.7965558070 1 2.982167
79 -0.1825118640 1 3.057730
80 -2.3400475324 1 2.141853
81 0.7972758941 1 4.383837
82 -2.6248992918 1 1.851342
83 -2.2391587284 1 2.534556
84 0.4875895517 1 4.102409
85 -1.6894818347 1 3.095156
86 -2.6679338628 1 1.803140
87 0.6714271089 1 4.254788
88 -0.5000203285 1 2.753786
89 -0.2865649886 1 2.947400
90 0.1270778386 1 3.369613
91 -2.7092373745 1 1.745094
92 -0.0846349656 1 3.164825
93 -0.1795793984 1 3.073924
94 0.7695847529 1 4.344575
95 -2.8200014689 1 1.631473
96 -1.9342858308 1 2.848428
97 -0.2954830823 1 2.949759
98 0.8345030447 1 4.410989
99 -0.2789869608 1 2.982013
100 0.2917082727 1 3.889307
101 -2.4511093767 1 2.020767
102 -2.9276221668 1 1.525662
103 -2.0046616174 1 2.775127
104 0.2845827431 1 3.877459
105 0.4644896402 1 4.052189
106 0.5160322514 1 4.122795
107 -0.2791849922 1 2.991555
108 -0.2144026450 1 3.041908
109 0.4877057401 1 4.096182
110 -0.0144068435 1 3.239545
111 -2.7664312969 1 1.698009
112 0.0014101773 1 3.238937
113 -0.4428523685 1 2.817539
114 -0.1318369465 1 3.135731
115 -0.5034555032 1 2.767194
116 0.4276436817 1 4.013587
117 -2.8017914790 1 1.672067
118 -0.1275102412 1 3.108797
119 -0.6601597059 1 2.591799
120 -1.6123565946 1 3.170439
121 0.2008926091 1 3.798422
122 -2.9085576837 1 1.544381
123 0.3684066836 1 3.967858
124 -0.6703722012 1 2.575430
125 -2.4362566572 1 2.035293
126 -0.4392321328 1 2.805676
127 -2.6115850667 1 1.865581
128 0.3218951545 1 3.915901
129 0.6289053400 1 4.223916
130 -0.2811042338 1 2.988744
131 -2.1799191928 1 2.602400
132 -2.8035098003 1 1.642327
133 -0.4633459053 1 2.796293
134 -1.8734838181 1 2.922796
135 0.2402660691 1 3.849344
136 -2.4628864552 1 2.030142
137 -0.2848575740 1 2.952196
138 -2.6970341993 1 1.762612
139 -0.3634104069 1 2.878096
140 0.8356754274 1 4.407406
141 0.3736931300 1 3.969653
142 -0.4029425202 1 2.847435
143 0.3601208541 1 3.950422
144 -1.6390827022 1 3.156964
145 -1.4055227198 1 3.395473
146 0.4515151456 1 4.057834
147 -1.8616132205 1 2.936971
148 -0.1343403689 1 3.131885
149 0.3625148636 1 3.972237
150 -0.8945075804 1 2.366551
151 -2.5691332057 1 1.917495
152 -0.0008299218 1 3.618124
153 -3.1198141023 1 1.305819
154 -0.2795657340 1 2.968623
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1912 -2.6271981073 1 1.844969
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1923 0.6468558176 1 4.229485
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1927 -0.4505372651 1 2.809500
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1929 -0.0743498353 1 3.183568
1930 -2.7386552188 1 1.723221
1931 -0.1196654667 1 3.518834
1932 -2.8654985132 1 1.593965
1933 -2.0047568020 1 2.786642
1934 0.3526869090 1 3.945459
1935 -1.8083067339 1 3.000623
1936 -2.4168970133 1 2.069044
1937 -2.3496186583 1 2.424020
1938 -1.9450261812 1 2.846924
1939 -2.4429526538 1 2.332611
1940 -0.0671333731 1 3.162186
1941 -2.7948003388 1 1.651372
1942 0.1973593820 1 3.444120
1943 -1.7940419896 1 2.997091
1944 -2.6262227014 1 1.848689
1945 -2.6179859636 1 1.849902
1946 0.5366001703 1 4.125102
1947 -0.6846370727 1 2.563657
1948 -0.5263685071 1 2.737096
1949 -2.4794050789 1 2.012446
1950 -1.7385302306 1 3.062790
1951 -1.7097512798 1 3.071057
1952 -2.0225789012 1 2.758943
1953 -1.8006284389 1 2.983988
1954 -0.1059521308 1 3.150019
1955 -0.2339470638 1 3.001681
1956 -0.5836713245 1 2.673638
1957 -0.2450075982 1 3.001733
1958 0.5757035922 1 4.162191
1959 0.5210386242 1 4.123039
1960 1.0929755606 1 4.657506
1961 0.4995946771 1 4.089385
1962 -1.8526442868 1 2.938256
1963 0.3220631852 1 3.942637
1964 0.5943425569 1 4.181467
1965 0.3422689105 1 3.948006
1966 -0.1312620685 1 3.109617
1967 0.2680755182 1 3.885409
1968 -2.7019036949 1 1.756173
1969 -2.9024729073 1 1.540562
1970 -0.6927238129 1 2.554496
1971 -0.2105376218 1 3.056196
1972 -0.8130093585 1 2.452662
1973 -0.0244047409 1 3.228327
1974 -0.0384432987 1 3.584021
1975 -0.3818386412 1 2.863414
1976 0.5422302438 1 4.139822
1977 -2.7333030807 1 1.741052
1978 0.6405745019 1 4.211831
1979 -2.0656287042 1 2.701775
1980 -0.0448476381 1 3.206919
1981 -0.5105551535 1 2.756784
1982 -0.3427954890 1 2.897713
1983 -0.4141858167 1 2.837233
1984 -3.2110994949 1 1.208296
1985 -2.1249147789 1 2.649237
1986 -2.3280109715 1 2.441926
1987 -0.2543237299 1 2.980830
1988 0.0899545969 1 3.335372
1989 -1.7635356320 1 3.027198
1990 0.6547418761 1 4.231582
1991 -0.1452686932 1 3.104714
1992 -2.2402299674 1 2.537650
1993 -2.7826234308 1 1.680675
1994 -0.3839533012 1 2.880822
1995 0.0958832691 1 3.710453
1996 -0.4088695752 1 2.835764
1997 -0.0429503392 1 3.215584
1998 0.4907195834 1 4.099509
1999 -0.0342626473 1 3.213816
2000 -1.7826865465 1 3.008812
2001 -1.8373048064 1 2.949368
2002 -2.6349932601 1 1.838481
2003 -0.0571936783 1 3.185214
2004 -2.7571133708 1 1.710086
2005 -2.4292859735 1 2.049756
2006 -2.5051674580 1 1.976654
2007 -2.7265087639 1 1.733176
2008 -0.2348608111 1 3.031435
2009 -0.4839647545 1 2.757873
2010 0.3718953888 1 3.959082
2011 -2.8559992451 1 1.600373
2012 0.3011546913 1 3.555948
2013 -0.3554882135 1 2.927946
2014 0.4055086136 1 4.023416
2015 -2.6422721754 1 1.813056
2016 -2.6905995556 1 1.777731
2017 0.1593030668 1 3.768958
2018 -0.1664671438 1 3.079522
2019 -0.5952380035 1 2.646390
2020 -2.4474584870 1 2.022219
2021 -0.1765285095 1 3.101900
2022 0.0348957958 1 3.275067
2023 -1.9989550463 1 2.786998
2024 -2.2029258938 1 2.306332
2025 -2.7352991681 1 1.721145
2026 -2.7988584823 1 1.659765
2027 -2.7058851283 1 1.757312
2028 -0.1264971422 1 3.104590
2029 -0.6042635913 1 2.655510
2030 -1.7200254103 1 3.077254
2031 0.1800448965 1 3.799767
2032 -0.1049052951 1 3.146904
2033 0.4115216036 1 4.000280
2034 -2.9510310650 1 1.483079
2035 -0.1589615510 1 3.086203
2036 -2.6072465256 1 1.879400
2037 0.6075955149 1 4.203536
2038 -2.6423240130 1 1.819825
2039 0.5468516494 1 4.151664
2040 -2.6913945219 1 1.772259
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2046 0.7275820519 1 4.311082
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2053 0.1675662176 1 3.783496
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2060 0.4486622750 1 4.045469
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2086 0.2666236051 1 3.500999
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2089 0.6064682399 1 4.198956
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2092 0.6950846869 1 4.284387
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2095 -0.1166712949 1 3.126722
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2100 -0.2690357063 1 2.988221
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2103 0.0661931363 1 3.315728
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9899 -0.0527680365 1 3.199415
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9903 -0.3862644326 1 2.856625
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9907 -2.7721911198 1 1.700384
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9910 -0.4970715194 1 2.779121
9911 -0.0725725221 1 3.193639
9912 0.7150541916 1 4.301403
9913 0.6966349144 1 4.294087
9914 -2.4740273061 1 1.993714
9915 -2.5627153514 1 1.922346
9916 -0.1311221905 1 3.108408
9917 -0.0910162421 1 3.180885
9918 0.2976792120 1 3.897047
9919 0.2677021503 1 3.880227
9920 0.2284188290 1 3.480938
9921 -2.7162274180 1 1.748145
9922 -2.9825590594 1 1.450715
9923 -0.4693255809 1 2.777451
9924 -0.4107853689 1 2.834768
9925 0.4848115939 1 4.065642
9926 -0.3523947050 1 2.911669
9927 -2.2598101034 1 2.226568
9928 -2.4811073039 1 1.997098
9929 -2.6817071536 1 1.802820
9930 -1.9430840695 1 2.827205
9931 0.3596525039 1 3.945459
9932 -2.5652360862 1 1.917411
9933 0.8602263900 1 4.440117
9934 -0.5463611991 1 2.720780
9935 0.8021467767 1 4.394789
9936 -0.1750236380 1 3.075759
9937 -1.7037525131 1 3.077844
9938 0.4830302683 1 4.078026
9939 0.5284284073 1 4.101613
9940 -0.2887813483 1 2.992548
9941 -0.3452262022 1 2.909133
9942 -0.3629911530 1 2.884898
9943 -2.0123814412 1 2.782054
9944 -2.6598889585 1 1.808895
9945 -0.0002253230 1 3.604935
9946 -2.7554497102 1 1.699740
9947 -2.8566202482 1 1.594958
9948 0.3314702162 1 3.927835
9949 0.0368299141 1 3.644587
9950 0.1493396027 1 3.753041
9951 0.2253564824 1 3.835820
9952 0.5304397921 1 4.144706
9953 0.1571734430 1 3.771624
9954 -0.1662656003 1 3.085298
9955 -0.3864518566 1 2.860892
9956 -0.0433250992 1 3.212464
9957 0.7681575653 1 4.352000
9958 0.8718743570 1 4.453651
9959 0.8793787169 1 4.441911
9960 -2.5265774533 1 1.965370
9961 0.1016910932 1 3.347292
9962 -0.2359977176 1 3.005963
9963 -2.6406815590 1 1.836356
9964 -2.3514826621 1 2.147401
9965 -0.3700352021 1 2.877147
9966 -1.8494142663 1 2.924586
9967 -2.8846452788 1 1.562214
9968 -1.6298044445 1 3.178439
9969 -2.3658641367 1 2.404440
9970 0.6966068176 1 4.280777
9971 -0.1388453454 1 3.100718
9972 0.3063148980 1 3.898710
9973 0.0830399911 1 3.338334
9974 0.5868445358 1 4.187903
9975 0.7510849524 1 4.341515
9976 -0.1154854555 1 3.129957
9977 -2.0087568767 1 2.777198
9978 -0.3803837155 1 2.853865
9979 -2.8163010221 1 1.640168
9980 -1.7896067531 1 2.988980
9981 -0.3261934810 1 2.910405
9982 0.3698323993 1 3.992969
9983 -0.1445816201 1 3.094574
9984 -2.0253505702 1 2.761431
9985 -0.1807268406 1 3.064367
9986 -1.6938602528 1 3.092944
9987 -0.2425542146 1 3.007077
9988 0.4743182441 1 4.075302
9989 -0.1091470814 1 3.146750
9990 -2.2217109464 1 2.283357
9991 -2.3795820364 1 2.114696
9992 -0.3491155542 1 2.890191
9993 -0.1173523287 1 3.140284
9994 -2.8450439926 1 1.605647
9995 -1.8711860824 1 2.915942
9996 0.5795263586 1 4.183061
9997 -2.5639536886 1 1.919245
9998 -0.4685165688 1 2.798797
9999 -2.8053965008 1 1.645460
10000 -2.5720408136 1 1.895579Aplicação
Aplicação
- Suponha que estamos em um CRE.
- Temos \(Y_i\) (os resultados observados) e \(Z_i\) (o vetor de indicador de tratamento).
- O objetivo é determinar se o uso do tratamento causa algum efeito médio em \(Y_i\).
- Seu colega, quase que imediatamente, resolve ajustar um modelo de regressão do tipo \[Y_i = \beta_0 + \beta_1 Z_i + \epsilon_i.\]
Aplicação
- Regressão é um dos métodos mais conhecidos por usuários treinados (e destreinados) e é frequentemente utilizado para fazer inferência acerca do ACE.
- O procediemnto padrão consiste em ajusta a regressão por MQO e utilizar \(\hat{\beta}\) como o estimador do ACE (\(\tau\)).
- De fato, pode-se provar que \(\hat{\beta} = \hat{\tau}\).
- Contudo, a variância do estimador obtido por MQO é \[\hat{V}_{MQO} = \dfrac{n(n_1 -1)}{(n-2)n_1n_0}\hat{S}^2(1) + \dfrac{n(n_0 -1)}{(n-2)n_1n_0}\hat{S}^2(0) \approx \dfrac{\hat{S}^2(1)}{n_0} + \dfrac{\hat{S}^2(0)}{n_1}\]
- Mas \(\hat{V} \neq \hat{V}_{MQO}\) 😞
Aplicação
- Não podemos utilizar \(\hat{V}_{MQO}\) 😢.
- Felizmente, o estimador robusto da variância EHW (Eicker-Huber-White) é bastante próximo de \(\hat{V}\) 🏄, \[\hat{V}_{EHW} = \dfrac{\hat{S}^2(1) (n_1 - 1)}{n_1 n_1} +\dfrac{\hat{S}^2(0) (n_0 - 1)}{n_0 n_0} \approx \dfrac{\hat{S}^2(1)}{n_1} + \dfrac{\hat{S}^2(0)}{n_0}.\]
- Ademais, o estimador HC2 que é uma variante do EHW é exatamente igual a \(\hat{V}\) 😄.
hccm do pacote car do R calcula tanto o estimador EHW quanto o HC2!.
Aplicação
Utilizaremos o dataset lalonde do pacote Matching para mostrar como utilizar o método de Neyman, bem como comparar com o modelo de regressão.
Podemos, então calcular facilmente \(\hat{\tau}\) e \(\hat{V}\).
Aplicação
Vejamos o que acontece se ajustarmos um modelo de regressão por MQO.
Contudo, lembre-se que não devemos utilizar \(\hat{V}_{MQO}\), mas podemos utilizar os estimadores EHW ou HC2.
Para mais detalhes de estimadores robustos da matriz de covariância, ver aqui
Aplicação
\[H_{0N}: \tau = 0\] Estatística de teste:
- \(\frac{\hat{\tau}-\tau}{\sqrt{\hat{V}}} = \frac{1794.3431}{670.9967} = 2.674146 \quad , \quad \frac{\hat{\tau}-\tau}{\sqrt{\hat{V}_{MQO}}} = \frac{1794.3431}{632.8536} = 2.835321\)
- \(\frac{\hat{\tau}-\tau}{\sqrt{\hat{V}_{EHW}}} = \frac{1794.3431}{669.3155} = 2.680863 \quad , \quad \frac{\hat{\tau}-\tau}{\sqrt{\hat{V}_{HC2}}} = \frac{1794.3431}{670.9967} = 2.674146\)
Mesmo a um nível de significância de 0.5% (o que não é tão inusual na área de saúde), se utilizarmos regressão por MQO rejeitariamos \(H_{0N}\). A mesma conclusão não é obtida se utilizamor as correções EHW ou mesmo HC2 🤣.
Demostração
Demostração do Teorema de Neyman
Teorema
Sob CRE:
- O estimador da diferença de médias, \(\hat{\tau} = \hat{\bar{Y}}(1) - \hat{\bar{Y}}(0)\) é não viesado para \(\tau\), ou seja, \(\mathbb{E}(\hat{\tau}) = \tau.\)
- \[\mathbb{V}(\hat{\tau}) = \dfrac{S^2(1)}{n_1} + \dfrac{S^2(0)}{n_0} - \dfrac{S^2(\tau)}{n} = \dfrac{n_0 S^2(1)}{n_1 n} + \dfrac{n_1 S^2(0)}{n_0 n} + 2\dfrac{S(1, 0)}{n}\]
- O estimador da variância, \(\hat{V} = \dfrac{\hat{S}^2(1)}{n_1} +\dfrac{\hat{S}^2(0)}{n_0}\) é conservador para estimar \(\mathbb{V}(\hat{\tau})\). Isto significa que \[\mathbb{E}(\hat{V}) - \mathbb{V}(\hat{\tau}) = \dfrac{S^2(\tau)}{n} \geq 0,\] com a igualdade acontecendo se \(\tau_i = \tau\) \(\forall i\).
Referências
- Peng Ding (2023). A First Course in Causal Inference. Capítulo 4
- Li, X., & Ding, P. (2017). General forms of finite population central limit theorems with applications to causal inference. Journal of the American Statistical Association, 112(520), 1759-1769.
- Long, J. S., & Ervin, L. H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54(3), 217-224.
- https://www.causalconversations.com/post/randomization-based-inference/
Carlos Trucíos (IMECC/UNICAMP) | ME920/MI628 - Inferência Causal | ctruciosm.github.io