Hello everyone,

I am working with a sample of galaxies with multi-wavelength data and I am trying to find a relation between quantities that we measure based on different luminosities (say H-alpha, UV, IR). In this dataset I have undetected objects for both IR and H-alpha.

Now, I want to find the relation between something measured from Ha (like star formation rate) and the same thing measured from IR. I want to fit a linear fit to the logarithmic values. Since I have doubly censored data (both y and x – IR and Ha – are subject to censoring) the best way to calculate the fit parameters is to use Akritas–Thiel–Sen (ATS) line which is provided in cenkenfunction (NADA library).

I have a few questions in this regard:

1) When I run the cenken function for my data sometimes I get a NULL value for the intercept; why is that and how I can avoid it? Is it something that I should get worried about?

2) cenken function in NADA library only provides the slope and intercept and not the uncertainties in those parameters. Is there any package that provide line fit function for doubly censored data with the uncertainties?

3) Is there a way to force the intercept to be zero?

Thanks a lot for your help in advance. It’s very appreciated.

Irene,

There is a small research literature on doubly censored linear regression (below).  But you should check: `double censorship’ can either mean that both variables are right(left)-censored, or that the dependent variable is both right and left censored.  I do not know if any of these have been implemented in the R/CRAN system.

LINEAR REGRESSION WITH DOUBLY CENSORED DATA  BY CUN-HUI ZHANG1 AND XIN LI  Annals of Statistics 24, 2720-2743 1996

Regression M-estimators with doubly censored data   Minggao Gu and Jian-Jian Ren Ann. Statist. Volume 25, Number 6 (1997), 2638-2664.   [M-estimators are robust against outliers]

Regression M-estimators with non-i.i.d. doubly censored data  Jian-Jian Ren  Ann. Statist. Volume 31, Number 4 (2003), 1186-1219.

Quantile regression with doubly censored data Guixian Lin Xuming He, Stephen Portnoy Computational Statistics & Data Analysis Volume 56, Issue 4, 1 April 2012, Pages 797–812  [This gives regression lines for the median and other quantiles, good for large datasets with non-Gaussian scatter]

There is a trick that you might consider.  I am not sure that it gives correct confidence intervals, as it involves multiple hypothesis tests on a single dataset.  Let me know if you want advice from a statistician on this point.  Here goes:

1. Your data is of the form (x,y) where both x and y are left-censored
2. Transform your data using a trial linear slope, y’ = y – beta_0 x
3. Now apply a generalized Kendall’s tau test (e.g. cenken in CRAN’s NADA package) to determine with (say) 95% confidence that the data show no correlation.
4. Do this for a range of beta_0 values to obtain a range of beta with that give no correlation.  This gives an estimate of the confidence interval of beta, although it does not give a best-fit value for beta.