Survival Function Estimation from Current Status Data

1. Survival Function Estimation from Current Status Data SWB JSM San Diego 2012

2. Objective: Compare Baseline and Endline U5 • U5 = P[Life  5 years] • Added objective: Compare survivor function estimates • P[Life > t|Baseline] vs. P[Life > t|Endline], t =1,2,…,60+ months • Baseline = 2007; Endline = 2010

3. Data were from Surveys • IRC surveyed two Afghan refugee camps in Pakistan in 2007 and one after humanitarian aid in 2010 • IRC = International Rescue Committee, www.rescue.org • Before 2007 = “baseline” and after 2010 = “endline,” • 2008 earthquake prevented endline survey of other refugee camp

4. Data Details • Many questions: intrusive, detailed • Data included survey dates, live birth dates within last 5 years, deaths, and current status (dead or alive) • Pdf files showed questionnaires • Excel spreadsheets contained data relevant to objective(s)

5. Data Problems • If exact dates were unknown, month or season were given, including “monsoon” • Some ages at deaths were inadmissable; deaths would have occurred after survey dates • Baseline and endline data were in different formats • Some instructions seemed to have been misunderstood; e.g., baseline survey date

6. Some Endline Grouped Age-at-Death Data

7. Sufficient Current Status Data • Not all data are shown. This is endline data

8. Nonparametric Max. Likelihood from Current Status Data • ln L = S dj ln(Fj) + S (sjdj) ln(1Fj) • Fj = P[Life  j]; dj and sj are current status deaths from the both surveys and all live births for j = 1,2,…,60+ • Use Excel Solver to maximize log likelihood as function of pk  0, Fj = Spk, k = 1,2,…,j • Also used least squares fit to current status npmle

9. Kaplan-Meier Estimate from Admissible Ages at Deaths • Data arranged in form of Nevada table • Used workbook described at http://home.comcast.net/~pstlarry/KMUsrMan.htm • Includes Greenwood’s estimate of standard deviation of survival function

10. CDF Estimates (Current Status)

11. CDF Estimates from Current Status Data

12. Std. Error Est. is Greenwood’s standard deviation See Banerjee and Wellner for confidence intervals from current status data Birth, Deaths, U5, and Std. Err.

13. Conclusions • Estimates from current status and admissible ages at deaths agreed tolerably • U5 = ~10% baseline and ~4% endline • Pakistan U5 is 8.7% (Wikipedia) • Infant mortality (1 year) is almost 4% baseline AND endline • Pakistan infant mortality is 6.7% (World Bank, World Development measures)

14. References • Bannerjee, Moulinath and Jon A. Wellner “Confidence Intervals for Current Status Data,” Scand. J. of Statist., Vol. 32, pp. 405-424, 2005 • George, L. L. “Kaplan-Meier Reliability Estimation Spreadsheet,” ASQ Reliability Review, vol. 25, no. 2, http://home.comcast.net/~pstlarry/KMUsrMan.htm, June 2005 • ibid.“What Price Required Data?” http:// home.comcast.net/~pstlarry /NwsRev2.doc pp. 3-4, 2000 • Greenwood, M. “The Natural Duration of Cancer,” Reports on Public Health and Medical Subjects 33, His Majesty’s Stationery Office, pp 1-26, 1926 • Miller, Rupert “What Price Kaplan-Meier?” Biometrics, vol. 39, pp. 1077-1081, 1983 • ibid. Beyond ANOVA, Basics of Applied Statistics, Chapman and Hall, New York, 1997 • “Survival Analysis,” http://en.wikipedia.org/wiki/Survival_analysis Nov. 2, 2011