Inverse Mill's Ratio - A Comment

[Emad Shehata]

Inverse Mills' Ratio - Comment

- It is imporyant to read this article
Why are there so many formulas for the inverse of Mills’ ratio?
http://www.stata.com/support/faqs/st...e-mills-ratio/

- If model is nonselection, the natural choice for the nonselection hazard is [PDF/(1-CDF)]
as in: IMR1 - IMR2 - IMR3 - IMR4

- If model is selection, the natural choice for nonselection hazard is [PDF / CDF]
as in: IMR5 - IMR6 [Stata’s heckman command]

the next example will explain different methods to compute Inverse Mills' Ratio from probit regression

clear all
input y x
0 1.9
0 2.4
0 1.1
0 0.5
0 1.2
1 3.6
1 9.1
1 8.1
1 9.8
1 3.3
0 6.6
0 6.1
end

* PDF = Probability Density Function
* CDF = Cumulative  Density Function
* pdf = normalden(xb) = normd(xb)
* cdf = normal(xb)    = normprob(xb)

 probit y x

 predict cdf1 , pr

 predict xb , xb

 gen double pdf1 = exp(-(xb^2/2))/sqrt(2*_pi)
 gen double pdf = normalden(xb)
 gen double cdf = normal(xb)

* Inverse Mills' Ratio from [predict probit postestimation]
 predict IMR1 , score

 gen double IMR2 = pdf/cdf
 replace IMR2 = pdf/(cdf-1) if y == 0

 gen double IMR3 = cond(y == 1, pdf/cdf, pdf/(cdf-1))

 gen double IMR4 = y*(pdf/cdf) + (1-y)*pdf/(cdf-1)

 gen double IMR5 = exp(-(xb^2/2))/(sqrt(2*_pi)*normal(xb))

 gen double IMR6 = normalden(xb)/normal(xb)

 format y %10.0f
 format xb pdf pdf1 cdf cdf1 IMR1 IMR2 IMR3 IMR4 IMR5 IMR6 %10.4f
 list y xb pdf pdf1 cdf cdf1
 list y IMR1 IMR2 IMR3 IMR4 IMR5 IMR6

list y xb pdf pdf1 cdf cdf1

     +-------------------------------------------------+
     | y        xb      pdf     pdf1      cdf     cdf1 |
     |-------------------------------------------------|
  1. | 0   -1.0656   0.2261   0.2261   0.1433   0.1433 |
  2. | 0   -0.9097   0.2638   0.2638   0.1815   0.1815 |
  3. | 0   -1.3151   0.1680   0.1680   0.0942   0.0942 |
  4. | 0   -1.5022   0.1291   0.1291   0.0665   0.0665 |
  5. | 0   -1.2839   0.1750   0.1750   0.0996   0.0996 |
     |-------------------------------------------------|
  6. | 1   -0.5355   0.3456   0.3456   0.2961   0.2961 |
  7. | 1    1.1796   0.1990   0.1990   0.8809   0.8809 |
  8. | 1    0.8677   0.2738   0.2738   0.8072   0.8072 |
  9. | 1    1.3978   0.1502   0.1502   0.9189   0.9189 |
 10. | 1   -0.6291   0.3273   0.3273   0.2646   0.2646 |
     |-------------------------------------------------|
 11. | 0    0.4000   0.3683   0.3683   0.6554   0.6554 |
 12. | 0    0.2441   0.3872   0.3872   0.5964   0.5964 |
     +-------------------------------------------------+

. list y IMR1 IMR2 IMR3 IMR4 IMR5 IMR6

     +-------------------------------------------------------------+
     | y      IMR1      IMR2      IMR3      IMR4     IMR5     IMR6 |
     |-------------------------------------------------------------|
  1. | 0   -0.2639   -0.2639   -0.2639   -0.2639   1.5780   1.5780 |
  2. | 0   -0.3222   -0.3222   -0.3222   -0.3222   1.4533   1.4533 |
  3. | 0   -0.1855   -0.1855   -0.1855   -0.1855   1.7829   1.7829 |
  4. | 0   -0.1383   -0.1383   -0.1383   -0.1383   1.9406   1.9406 |
  5. | 0   -0.1943   -0.1943   -0.1943   -0.1943   1.7570   1.7570 |
     |-------------------------------------------------------------|
  6. | 1    1.1672    1.1672    1.1672    1.1672   1.1672   1.1672 |
  7. | 1    0.2259    0.2259    0.2259    0.2259   0.2259   0.2259 |
  8. | 1    0.3392    0.3392    0.3392    0.3392   0.3392   0.3392 |
  9. | 1    0.1634    0.1634    0.1634    0.1634   0.1634   0.1634 |
 10. | 1    1.2368    1.2368    1.2368    1.2368   1.2368   1.2368 |
     |-------------------------------------------------------------|
 11. | 0   -1.0687   -1.0687   -1.0687   -1.0687   0.5619   0.5619 |
 12. | 0   -0.9595   -0.9595   -0.9595   -0.9595   0.6493   0.6493 |
     +-------------------------------------------------------------+

My best regards

[Nick Cox]

On spelling and punctuation only, see http://www.stata.com/statalist/archi.../msg00433.html

I copy the substantive contents here.

st: Mills' ratio

---

From	"Nick Cox"
To	<[email protected]>
Subject	st: Mills' ratio
Date	Sun, 13 Mar 2005 17:26:00 -0000

---

The dreaded "mill's ratio" made another appearance recently. Other mutations appear intermittently.

The correct naming is easy to determine.

This ratio is named for John P. Mills whose tabulation appears in

Table of the Ratio: Area to Bounding Ordinate, for Any Portion of Normal Curve. Biometrika 18: 395--400 (1926)

Thus allowing for differences of opinion on what should be punctuated and how, three (and only three) variations appear acceptable.

Mills ratio

Mills' ratio

Mills's ratio

This is a hard name to get right even for people for whom English or American is their first language.