17.2 Censoring
In the case of censoring, the values of the dependent variable are reported at a certain point if they are above or below a certain value.
If all data was reported at the correct value, the following following regression model could be executed:
##
## Call:
## lm(formula = yreal ~ x, data = censoring)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.60012 -0.59843 0.00981 0.78105 2.05397
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.25290 0.27739 -8.122 1.44e-10 ***
## x 0.52603 0.04888 10.762 2.17e-14 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.951 on 48 degrees of freedom
## Multiple R-squared: 0.707, Adjusted R-squared: 0.7009
## F-statistic: 115.8 on 1 and 48 DF, p-value: 2.169e-14Ignoring censoring leads to biased results:
##
## Call:
## lm(formula = y ~ x, data = censoring)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.94411 -0.51345 0.03089 0.37886 1.25169
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.54026 0.17443 -3.097 0.00326 **
## x 0.29534 0.03074 9.608 9.21e-13 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5981 on 48 degrees of freedom
## Multiple R-squared: 0.6579, Adjusted R-squared: 0.6508
## F-statistic: 92.32 on 1 and 48 DF, p-value: 9.212e-13Using the R package censReg) allows for the reduction of the bias:
##
## Call:
## censReg(formula = y ~ x, data = censoring)
##
## Observations:
## Total Left-censored Uncensored Right-censored
## 50 17 33 0
##
## Coefficients:
## Estimate Std. error t value Pr(> t)
## (Intercept) -1.41227 0.29576 -4.775 1.8e-06 ***
## x 0.41466 0.04717 8.791 < 2e-16 ***
## logSigma -0.33824 0.12631 -2.678 0.00741 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Newton-Raphson maximisation, 6 iterations
## Return code 1: gradient close to zero (gradtol)
## Log-likelihood: -44.496 on 3 Df