Decile Plots

Here I provide two types of plots for each of p_hat_cnn , p_hat_covariate, and risk_pred_prob:

  1. Decile Plot A - The max value in a decile vs. the mean arrest outcome in that decile
  2. Decile Plot B - The mean arrest outcome at each decile index

Decile Plot 1 - p_hat_cnn

Decile Plot 2 - p_hat_covariate

Decile Plot 3 - risk_pred_prob

Baseline Regression

Table _03 - Model 03

Here we include the 18 raw skin-tone levels and do not account for the non-linearity in p_hat_cnn. This is our baseline.

Multihead(ResNet50)
Dependent variable:
Release Outcome
(1) (2) (3) (4) (5)
risk_pred_prob -1.074*** -1.077*** -1.072*** -0.742*** -0.682***
(-1.188, -0.960) (-1.192, -0.962) (-1.187, -0.956) (-0.855, -0.629) (-0.794, -0.569)
skin_tonenumber_f7ddc4 0.006 0.006 -0.021 -0.037
(-0.033, 0.045) (-0.034, 0.045) (-0.059, 0.017) (-0.075, 0.001)
age -0.0004 0.0003 0.001*
(-0.002, 0.001) (-0.001, 0.001) (0.00002, 0.002)
attractiveness -0.002 0.002 0.002
(-0.013, 0.009) (-0.009, 0.012) (-0.008, 0.013)
competence 0.002 -0.002 -0.003
(-0.010, 0.015) (-0.014, 0.010) (-0.015, 0.009)
dominance -0.002 0.003 0.005
(-0.011, 0.007) (-0.006, 0.011) (-0.004, 0.013)
trustworthiness 0.004 0.004 0.001
(-0.007, 0.015) (-0.007, 0.014) (-0.010, 0.012)
p_hat_covariates 1.085*** 1.003***
(1.015, 1.155) (0.932, 1.074)
p_hat_cnn 0.415***
(0.343, 0.487)
Constant 1.095*** 1.091*** 1.095*** 0.109* -0.171***
(1.059, 1.131) (1.043, 1.139) (1.018, 1.171) (0.012, 0.205) (-0.279, -0.063)
Observations 7,318 7,318 7,318 7,318 7,318
Adjusted R2 0.032 0.032 0.031 0.111 0.122
F Statistic 239.132*** (df = 1; 7316) 14.271*** (df = 18; 7299) 11.215*** (df = 23; 7294) 39.038*** (df = 24; 7293) 41.532*** (df = 25; 7292)
Note: p<0.1; p<0.05; p<0.01

Non-Linearity in p_hat_cnn

Average decile value for p_hat_cnn

Here I fixed the average decile values for p_hat_cnn and we now see the regression coefficient becoming significant.

Multihead(ResNet50)
Dependent variable:
Release Outcome
(1) (2) (3) (4)
risk_pred_prob -1.074*** -1.072*** -0.742*** -0.683***
(-1.188, -0.960) (-1.187, -0.956) (-0.855, -0.629) (-0.796, -0.571)
skin_tonenumber_f7ddc4 0.006 -0.021 -0.037
(-0.034, 0.045) (-0.059, 0.017) (-0.074, 0.001)
age -0.0004 0.0003 0.001
(-0.002, 0.001) (-0.001, 0.001) (-0.00005, 0.002)
attractiveness -0.002 0.002 0.002
(-0.013, 0.009) (-0.009, 0.012) (-0.008, 0.012)
competence 0.002 -0.002 -0.003
(-0.010, 0.015) (-0.014, 0.010) (-0.015, 0.009)
dominance -0.002 0.003 0.005
(-0.011, 0.007) (-0.006, 0.011) (-0.004, 0.013)
trustworthiness 0.004 0.004 0.001
(-0.007, 0.015) (-0.007, 0.014) (-0.009, 0.012)
p_hat_covariates 1.085*** 1.005***
(1.015, 1.155) (0.934, 1.075)
p_hat_cnn_decile_avr 0.412***
(0.339, 0.485)
Constant 1.095*** 1.095*** 0.109* -0.168**
(1.059, 1.131) (1.018, 1.171) (0.012, 0.205) (-0.276, -0.061)
Observations 7,318 7,318 7,318 7,318
Adjusted R2 0.032 0.031 0.111 0.121
F Statistic 239.132*** (df = 1; 7316) 11.215*** (df = 23; 7294) 39.038*** (df = 24; 7293) 41.372*** (df = 25; 7292)
Note: p<0.1; p<0.05; p<0.01

Direct coding of deciles

Here I include integers 1-10 for the corresponding decile that the observation is in.

Multihead(ResNet50)
Dependent variable:
Release Outcome
(1) (2) (3) (4)
risk_pred_prob -1.074*** -1.072*** -0.742*** -0.685***
(-1.188, -0.960) (-1.187, -0.956) (-0.855, -0.629) (-0.798, -0.573)
skin_tonenumber_f7ddc4 0.006 -0.021 -0.036
(-0.034, 0.045) (-0.059, 0.017) (-0.074, 0.002)
age -0.0004 0.0003 0.001
(-0.002, 0.001) (-0.001, 0.001) (-0.0001, 0.002)
attractiveness -0.002 0.002 0.002
(-0.013, 0.009) (-0.009, 0.012) (-0.008, 0.012)
competence 0.002 -0.002 -0.003
(-0.010, 0.015) (-0.014, 0.010) (-0.015, 0.009)
dominance -0.002 0.003 0.005
(-0.011, 0.007) (-0.006, 0.011) (-0.004, 0.013)
trustworthiness 0.004 0.004 0.001
(-0.007, 0.015) (-0.007, 0.014) (-0.010, 0.012)
p_hat_covariates 1.085*** 1.004***
(1.015, 1.155) (0.933, 1.074)
p_hat_cnn_decile 0.016***
(0.013, 0.018)
Constant 1.095*** 1.095*** 0.109* 0.056
(1.059, 1.131) (1.018, 1.171) (0.012, 0.205) (-0.040, 0.153)
Observations 7,318 7,318 7,318 7,318
Adjusted R2 0.032 0.031 0.111 0.121
F Statistic 239.132*** (df = 1; 7316) 11.215*** (df = 23; 7294) 39.038*** (df = 24; 7293) 41.239*** (df = 25; 7292)
Note: p<0.1; p<0.05; p<0.01

Higher order

We now include two higher order terms of p_hat_cnn, none of which become significant.

Multihead(ResNet50)
Dependent variable:
Release Outcome
(1) (2) (3) (4) (5) (6)
risk_pred_prob -1.074*** -1.072*** -0.742*** -0.682*** -0.681*** -0.680***
(-1.188, -0.960) (-1.187, -0.956) (-0.855, -0.629) (-0.794, -0.569) (-0.794, -0.568) (-0.793, -0.568)
skin_tonenumber_f7ddc4 0.006 -0.021 -0.037 -0.037 -0.036
(-0.034, 0.045) (-0.059, 0.017) (-0.075, 0.001) (-0.075, 0.001) (-0.074, 0.002)
age -0.0004 0.0003 0.001* 0.001* 0.001*
(-0.002, 0.001) (-0.001, 0.001) (0.00002, 0.002) (0.00001, 0.002) (0.00004, 0.002)
attractiveness -0.002 0.002 0.002 0.002 0.002
(-0.013, 0.009) (-0.009, 0.012) (-0.008, 0.013) (-0.008, 0.013) (-0.008, 0.013)
competence 0.002 -0.002 -0.003 -0.003 -0.003
(-0.010, 0.015) (-0.014, 0.010) (-0.015, 0.009) (-0.015, 0.009) (-0.015, 0.009)
dominance -0.002 0.003 0.005 0.005 0.005
(-0.011, 0.007) (-0.006, 0.011) (-0.004, 0.013) (-0.004, 0.013) (-0.004, 0.013)
trustworthiness 0.004 0.004 0.001 0.001 0.001
(-0.007, 0.015) (-0.007, 0.014) (-0.010, 0.012) (-0.010, 0.012) (-0.010, 0.012)
p_hat_covariates 1.085*** 1.003*** 1.002*** 1.001***
(1.015, 1.155) (0.932, 1.074) (0.931, 1.073) (0.930, 1.072)
p_hat_cnn 0.415*** 0.271 2.524
(0.343, 0.487) (-0.436, 0.979) (-1.938, 6.985)
I(p_hat_cnn2) 0.099 -3.157
(-0.387, 0.585) (-9.543, 3.230)
I(p_hat_cnn3) 1.534
(-1.466, 4.533)
Constant 1.095*** 1.095*** 0.109* -0.171*** -0.120 -0.627
(1.059, 1.131) (1.018, 1.171) (0.012, 0.205) (-0.279, -0.063) (-0.394, 0.154) (-1.657, 0.402)
Observations 7,318 7,318 7,318 7,318 7,318 7,318
Adjusted R2 0.032 0.031 0.111 0.122 0.122 0.121
F Statistic 239.132*** (df = 1; 7316) 11.215*** (df = 23; 7294) 39.038*** (df = 24; 7293) 41.532*** (df = 25; 7292) 39.934*** (df = 26; 7291) 38.479*** (df = 27; 7290)
Note: p<0.1; p<0.05; p<0.01