# invMillsRatio: Inverse Mill's Ratio of probit models In sampleSelection: Sample Selection Models

## Description

Calculates the 'Inverse Mill's Ratios' of univariate and bivariate probit models.

## Usage

 1 invMillsRatio( x, all = FALSE )

## Arguments

 x probit model estimated by probit, glm or vglm. all a logical value indicating whether the inverse Mill's Ratios should be calculated for all observations.

## Details

The formula to calculate the inverse Mill's ratios for univariate probit models is taken from Greene (2003, p. 785), whereas the formulas for bivariate probit models are derived in Henning and Henningsen (2005).

## Value

A data frame that contains the Inverse Mill's Ratios (IMR) and the delta values (see Greene, 2003, p. 784).

If a univariate probit estimation is provided, the variables IMR1 and IMR0 are the Inverse Mill's Ratios to correct for a sample selection bias of y = 1 and y = 0, respectively. Accordingly, 'delta1' and 'delta0' are the corresponding delta values.

If a bivariate probit estimation is provided, the variables IMRa1, IMRa0, IMRb1, and IMRb0 are the Inverse Mills Ratios to correct for a sample selection bias of y = 1 and y = 0 in equations 'a' and 'b', respectively. Accordingly, 'deltaa1', 'deltaa0', 'deltab1' and 'deltab0' are the corresponding delta values.

Arne Henningsen

## References

Greene, W. H. (2003) Econometric Analysis, Fifth Edition, Prentice Hall.

Henning, C.H.C.A and A. Henningsen (2005) Modeling Price Response of Farm Households in Imperfect Labor Markets in Poland: Incorporating Transaction Costs and Heterogeneity into a Farm Household Approach. Unpublished, University of Kiel, Germany.

## Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 ## Wooldridge( 2003 ): example 17.5, page 590 data(Mroz87) myProbit <- glm( lfp ~ nwifeinc + educ + exper + I( exper^2 ) + age + kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 ) Mroz87\$IMR <- invMillsRatio( myProbit )\$IMR1 myHeckit <- lm( log( wage ) ~ educ + exper + I( exper^2 ) + IMR, data = Mroz87\$lfp == 1, ] ) # using NO labor force participation as endogenous variable Mroz87\$nolfp <- 1 - Mroz87\$lfp myProbit2 <- glm( nolfp ~ nwifeinc + educ + exper + I( exper^2 ) + age + kids5 + kids618, family = binomial( link = "probit" ), data=Mroz87 ) all.equal( invMillsRatio( myProbit )\$IMR1, invMillsRatio( myProbit2 )\$IMR0 ) # should be true # example for bivariate probit library( "mvtnorm" ) library( "VGAM" ) nObs <- 1000 # error terms (trivariate normal) sigma <- symMatrix( c( 2, 0.7, 1.2, 1, 0.5, 1 ) ) myData <- as.data.frame( rmvnorm( nObs, c( 0, 0, 0 ), sigma ) ) names( myData ) <- c( "e0", "e1", "e2" ) # exogenous variables (indepently normal) myData\$x0 <- rnorm( nObs ) myData\$x1 <- rnorm( nObs ) myData\$x2 <- rnorm( nObs ) # endogenous variables myData\$y0 <- -1.5 + 0.8 * myData\$x1 + myData\$e0 myData\$y1 <- ( 0.3 + 0.4 * myData\$x1 + 0.3 * myData\$x2 + myData\$e1 ) > 0 myData\$y2 <- ( -0.1 + 0.6 * myData\$x1 + 0.7 * myData\$x2 + myData\$e2 ) > 0 # bivariate probit (using rhobit transformation) bProbit <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho, data = myData ) summary( bProbit ) # bivariate probit (NOT using rhobit transformation) bProbit2 <- vglm( cbind( y1, y2 ) ~ x1 + x2, family = binom2.rho( lrho = "identitylink" ), data = myData ) summary( bProbit2 ) # inverse Mills Ratios imr <- invMillsRatio( bProbit ) imr2 <- invMillsRatio( bProbit2 ) all.equal( imr, imr2, tolerance = .Machine\$double.eps ^ 0.25) # tests # E[ e0 | y1* > 0 & y2* > 0 ] mean( myData\$e0[ myData\$y1 & myData\$y2 ] ) mean( sigma[1,2] * imr\$IMR11a + sigma[1,3] * imr\$IMR11b, na.rm = TRUE ) # E[ e0 | y1* > 0 & y2* <= 0 ] mean( myData\$e0[ myData\$y1 & !myData\$y2 ] ) mean( sigma[1,2] * imr\$IMR10a + sigma[1,3] * imr\$IMR10b, na.rm = TRUE ) # E[ e0 | y1* <= 0 & y2* > 0 ] mean( myData\$e0[ !myData\$y1 & myData\$y2 ] ) mean( sigma[1,2] * imr\$IMR01a + sigma[1,3] * imr\$IMR01b, na.rm = TRUE ) # E[ e0 | y1* <= 0 & y2* <= 0 ] mean( myData\$e0[ !myData\$y1 & !myData\$y2 ] ) mean( sigma[1,2] * imr\$IMR00a + sigma[1,3] * imr\$IMR00b, na.rm = TRUE ) # E[ e0 | y1* > 0 ] mean( myData\$e0[ myData\$y1 ] ) mean( sigma[1,2] * imr\$IMR1X, na.rm = TRUE ) # E[ e0 | y1* <= 0 ] mean( myData\$e0[ !myData\$y1 ] ) mean( sigma[1,2] * imr\$IMR0X, na.rm = TRUE ) # E[ e0 | y2* > 0 ] mean( myData\$e0[ myData\$y2 ] ) mean( sigma[1,3] * imr\$IMRX1, na.rm = TRUE ) # E[ e0 | y2* <= 0 ] mean( myData\$e0[ !myData\$y2 ] ) mean( sigma[1,3] * imr\$IMRX0, na.rm = TRUE ) # estimation for y1* > 0 and y2* > 0 selection <- myData\$y1 & myData\$y2 # OLS estimation ols11 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols11 ) # heckman type estimation heckit11 <- lm( y0 ~ x1 + IMR11a + IMR11b, data = cbind( myData, imr ), subset = selection ) summary( heckit11 ) # estimation for y1* > 0 and y2* <= 0 selection <- myData\$y1 & !myData\$y2 # OLS estimation ols10 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols10 ) # heckman type estimation heckit10 <- lm( y0 ~ x1 + IMR10a + IMR10b, data = cbind( myData, imr ), subset = selection ) summary( heckit10 ) # estimation for y1* <= 0 and y2* > 0 selection <- !myData\$y1 & myData\$y2 # OLS estimation ols01 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols01 ) # heckman type estimation heckit01 <- lm( y0 ~ x1 + IMR01a + IMR01b, data = cbind( myData, imr ), subset = selection ) summary( heckit01 ) # estimation for y1* <= 0 and y2* <= 0 selection <- !myData\$y1 & !myData\$y2 # OLS estimation ols00 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols00 ) # heckman type estimation heckit00 <- lm( y0 ~ x1 + IMR00a + IMR00b, data = cbind( myData, imr ), subset = selection ) summary( heckit00 ) # estimation for y1* > 0 selection <- myData\$y1 # OLS estimation ols1X <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols1X ) # heckman type estimation heckit1X <- lm( y0 ~ x1 + IMR1X, data = cbind( myData, imr ), subset = selection ) summary( heckit1X ) # estimation for y1* <= 0 selection <- !myData\$y1 # OLS estimation ols0X <- lm( y0 ~ x1, data = myData, subset = selection ) summary( ols0X ) # heckman type estimation heckit0X <- lm( y0 ~ x1 + IMR0X, data = cbind( myData, imr ), subset = selection ) summary( heckit0X ) # estimation for y2* > 0 selection <- myData\$y2 # OLS estimation olsX1 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( olsX1 ) # heckman type estimation heckitX1 <- lm( y0 ~ x1 + IMRX1, data = cbind( myData, imr ), subset = selection ) summary( heckitX1 ) # estimation for y2* <= 0 selection <- !myData\$y2 # OLS estimation olsX0 <- lm( y0 ~ x1, data = myData, subset = selection ) summary( olsX0 ) # heckman type estimation heckitX0 <- lm( y0 ~ x1 + IMRX0, data = cbind( myData, imr ), subset = selection ) summary( heckitX0 )

### Example output

Please cite the 'maxLik' package as:
Henningsen, Arne and Toomet, Ott (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458. DOI 10.1007/s00180-010-0217-1.

If you have questions, suggestions, or comments regarding the 'maxLik' package, please use a forum or 'tracker' at maxLik's R-Forge site:
https://r-forge.r-project.org/projects/maxlik/
[1] TRUE

Attaching package: 'VGAM'

The following object is masked from 'package:sampleSelection':

probit

Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho,
data = myData)

Pearson residuals:
Min      1Q  Median     3Q   Max
probitlink(mu1) -3.099 -0.9009  0.4622 0.7685 2.346
probitlink(mu2) -3.168 -0.6602 -0.1993 0.6541 5.677
rhobitlink(rho) -7.651 -0.6032  0.2478 0.5139 4.140

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  0.28134    0.04178   6.734 1.66e-11 ***
(Intercept):2 -0.09840    0.04446  -2.213   0.0269 *
(Intercept):3  1.15844    0.12659   9.151  < 2e-16 ***
x1:1           0.35687    0.04369   8.168 3.13e-16 ***
x1:2           0.59222    0.04918  12.043  < 2e-16 ***
x2:1           0.29754    0.04298   6.923 4.43e-12 ***
x2:2           0.67683    0.05130  13.195  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-likelihood: -1095.843 on 2993 degrees of freedom

Number of Fisher scoring iterations: 3

No Hauck-Donner effect found in any of the estimates

Call:
vglm(formula = cbind(y1, y2) ~ x1 + x2, family = binom2.rho(lrho = "identitylink"),
data = myData)

Pearson residuals:
Min      1Q  Median     3Q   Max
probitlink(mu1) -3.086 -0.9034  0.4595 0.7653 2.346
probitlink(mu2) -3.273 -0.6617 -0.1966 0.6585 5.436
rho             -7.826 -0.5830  0.2583 0.5153 4.036

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept):1  0.28134    0.04178   6.734 1.66e-11 ***
(Intercept):2 -0.09840    0.04446  -2.213   0.0269 *
(Intercept):3  0.52210    0.04604  11.339  < 2e-16 ***
x1:1           0.35687    0.04369   8.168 3.13e-16 ***
x1:2           0.59222    0.04918  12.043  < 2e-16 ***
x2:1           0.29754    0.04298   6.923 4.43e-12 ***
x2:2           0.67683    0.05130  13.195  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log-likelihood: -1095.843 on 2993 degrees of freedom

Number of Fisher scoring iterations: 3

No Hauck-Donner effect found in any of the estimates

[1] TRUE
[1] 0.7880256
[1] 0.8411649
[1] -0.2699706
[1] -0.3589691
[1] 0.468903
[1] 0.4019226
[1] -0.7789105
[1] -0.8908078
[1] 0.4066549
[1] 0.4080471
[1] -0.5155529
[1] -0.618372
[1] 0.7308693
[1] 0.7616975
[1] -0.5709256
[1] -0.6736764

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.1136 -0.7843 -0.0275  0.7821  3.4793

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.56270    0.06376  -8.825  < 2e-16 ***
x1           0.46942    0.05975   7.856 4.04e-14 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.134 on 383 degrees of freedom
Multiple R-squared:  0.1388,	Adjusted R-squared:  0.1365
F-statistic: 61.72 on 1 and 383 DF,  p-value: 4.04e-14

Call:
lm(formula = y0 ~ x1 + IMR11a + IMR11b, data = cbind(myData,
imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-2.8721 -0.7544 -0.0240  0.6067  3.5395

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -1.9969     0.6878  -2.903  0.00391 **
x1            0.8282     0.1296   6.389 4.88e-10 ***
IMR11a        2.1404     2.2446   0.954  0.34090
IMR11b        1.0673     0.3525   3.028  0.00263 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.065 on 381 degrees of freedom
Multiple R-squared:  0.2434,	Adjusted R-squared:  0.2374
F-statistic: 40.85 on 3 and 381 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.5027 -0.6912  0.0488  0.7793  2.8668

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.83487    0.07539 -24.340  < 2e-16 ***
x1           0.37272    0.07992   4.663 5.46e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.096 on 215 degrees of freedom
Multiple R-squared:  0.09186,	Adjusted R-squared:  0.08763
F-statistic: 21.75 on 1 and 215 DF,  p-value: 5.461e-06

Call:
lm(formula = y0 ~ x1 + IMR10a + IMR10b, data = cbind(myData,
imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.5825 -0.7011  0.0936  0.7392  2.2169

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -3.2661     1.8446  -1.771 0.078053 .
x1            0.8804     0.1663   5.293 2.99e-07 ***
IMR10a        2.4208     1.6469   1.470 0.143057
IMR10b        1.0472     0.2947   3.554 0.000467 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.9822 on 213 degrees of freedom
Multiple R-squared:  0.2774,	Adjusted R-squared:  0.2673
F-statistic: 27.26 on 3 and 213 DF,  p-value: 5.822e-15

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.4618 -0.6188  0.0633  0.7342  2.9974

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -0.9861     0.1247  -7.906 1.07e-11 ***
x1            0.4616     0.1526   3.025  0.00332 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.128 on 82 degrees of freedom
Multiple R-squared:  0.1004,	Adjusted R-squared:  0.0894
F-statistic: 9.148 on 1 and 82 DF,  p-value: 0.003323

Call:
lm(formula = y0 ~ x1 + IMR01a + IMR01b, data = cbind(myData,
imr), subset = selection)

Residuals:
Min       1Q   Median       3Q      Max
-2.21368 -0.61448  0.02581  0.66745  2.18196

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   1.1408     5.7557   0.198  0.84339
x1            1.0279     0.4067   2.527  0.01346 *
IMR01a        2.1160     3.5098   0.603  0.54829
IMR01b        0.9031     0.3126   2.889  0.00497 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.019 on 80 degrees of freedom
Multiple R-squared:  0.2832,	Adjusted R-squared:  0.2563
F-statistic: 10.54 on 3 and 80 DF,  p-value: 6.489e-06

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.3059 -0.7136 -0.0389  0.7179  3.2808

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.38799    0.06894 -34.639  < 2e-16 ***
x1           0.50504    0.07066   7.148 6.28e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.13 on 312 degrees of freedom
Multiple R-squared:  0.1407,	Adjusted R-squared:  0.138
F-statistic: 51.09 on 1 and 312 DF,  p-value: 6.28e-12

Call:
lm(formula = y0 ~ x1 + IMR00a + IMR00b, data = cbind(myData,
imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.5236 -0.7402 -0.0121  0.7445  3.0029

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -1.6417     0.7507  -2.187 0.029509 *
x1            0.7906     0.1267   6.240 1.44e-09 ***
IMR00a        0.1775     1.1859   0.150 0.881131
IMR00b        1.5404     0.4502   3.421 0.000707 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.072 on 310 degrees of freedom
Multiple R-squared:  0.2317,	Adjusted R-squared:  0.2243
F-statistic: 31.17 on 3 and 310 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-4.5384 -0.8422  0.0159  0.8606  3.8369

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.05005    0.05292  -19.84   <2e-16 ***
x1           0.61499    0.05168   11.90   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.264 on 600 degrees of freedom
Multiple R-squared:  0.191,	Adjusted R-squared:  0.1896
F-statistic: 141.6 on 1 and 600 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1 + IMR1X, data = cbind(myData, imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-4.5180 -0.8633  0.0390  0.8578  3.8874

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.62505    0.20925  -7.766 3.53e-14 ***
x1           0.77235    0.07558  10.220  < 2e-16 ***
IMR1X        0.92323    0.32519   2.839  0.00468 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.257 on 599 degrees of freedom
Multiple R-squared:  0.2017,	Adjusted R-squared:  0.199
F-statistic: 75.68 on 2 and 599 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.5320 -0.8754 -0.0490  0.8899  4.1030

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.05778    0.06565 -31.344   <2e-16 ***
x1           0.63987    0.06951   9.205   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.258 on 396 degrees of freedom
Multiple R-squared:  0.1763,	Adjusted R-squared:  0.1742
F-statistic: 84.74 on 1 and 396 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1 + IMR0X, data = cbind(myData, imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.6358 -0.8835 -0.0711  0.9058  3.8633

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  -1.2215     0.3177  -3.844 0.000141 ***
x1            0.8415     0.1019   8.260 2.22e-15 ***
IMR0X         0.8865     0.3296   2.689 0.007463 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.248 on 395 degrees of freedom
Multiple R-squared:  0.1911,	Adjusted R-squared:  0.187
F-statistic: 46.65 on 2 and 395 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.8057 -0.7642 -0.0296  0.7640  3.5448

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.64705    0.05710 -11.332   <2e-16 ***
x1           0.49052    0.05561   8.821   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.142 on 467 degrees of freedom
Multiple R-squared:  0.1428,	Adjusted R-squared:  0.141
F-statistic: 77.82 on 1 and 467 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1 + IMRX1, data = cbind(myData, imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.0754 -0.7093 -0.0606  0.6356  3.6354

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.50842    0.11934 -12.640  < 2e-16 ***
x1           0.76903    0.06251  12.303  < 2e-16 ***
IMRX1        1.18395    0.14660   8.076 5.76e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.07 on 466 degrees of freedom
Multiple R-squared:  0.2481,	Adjusted R-squared:  0.2448
F-statistic: 76.87 on 2 and 466 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1, data = myData, subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.5498 -0.7580 -0.0424  0.8063  3.1909

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.15926    0.05227 -41.307   <2e-16 ***
x1           0.48537    0.05431   8.938   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.152 on 529 degrees of freedom
Multiple R-squared:  0.1312,	Adjusted R-squared:  0.1295
F-statistic: 79.88 on 1 and 529 DF,  p-value: < 2.2e-16

Call:
lm(formula = y0 ~ x1 + IMRX0, data = cbind(myData, imr), subset = selection)

Residuals:
Min      1Q  Median      3Q     Max
-3.8148 -0.7703  0.0468  0.7909  2.7010

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.34764    0.11035 -12.212  < 2e-16 ***
x1           0.78002    0.06249  12.483  < 2e-16 ***
IMRX0        1.29836    0.15796   8.219 1.59e-15 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.086 on 528 degrees of freedom
Multiple R-squared:  0.2297,	Adjusted R-squared:  0.2268
F-statistic: 78.74 on 2 and 528 DF,  p-value: < 2.2e-16

sampleSelection documentation built on Jan. 13, 2021, 7:49 p.m.