Quantile-Quantile Plot for Censored Lognormal Data
qq.lnorm produces a lognormal quantile-quantile (q-q) plot based on the product limit estimate (PLE) of the cumulative distribution function (CDF) F(x) for censored data. A line is added to the plot.
A data frame with the data(x) in column 1 and PLE in column 2
estimate of the log scale mean
estimate of log scale standard deviation
Additional parameters to plot
The PLE is used to determine the plotting positions on the horizontal
axis for the censored data version of a theoretical q-q plot for the lognormal distribution. Waller and Turnbull (1992)
provide a good overview of q-q plots and other graphical methods for
censored data. The lognormal q-q plot is obtained by plotting
detected values a[j](on log scale) versus H[p(j)] where H(p) is the
inverse of the distribution function of the standard normal
distribution. If the largest data value is not censored then the PLE
is 1 and H(1) is off scale. The "plotting positions" p[j] are
determined from the PLE of F(x) by multiplying each estimate by
n /(n+1), or by averaging adjacent values–see Meeker and Escobar
(1998, Chap 6)]. In complete data case without ties the first approach
is equivalent to replacing the sample CDF j / n with j / (n+1), and for
the second approach the plotting positions are equal to (j - .5) / n. If
the lognormal distribution is a close approximation to the empirical
distribution, the points on the plot will fall near a straight line.
An objective evaluation of this is obtained by calculating
square of the correlation coefficient associated with the plot.
A line is added to the plot based on the values of
If either of these is missing
sigma are estimated by
linear regression of log(y) on H[p(j)].
A list with components
The x coordinates of the points that were plotted
The y coordinates of the points that were plotted
The adjusted probabilities use to determine
The values of
Helsel and Cohen (1988) consider alternative procedures that can be used for calculating plotting positions for left censored data. Waller and Turnbull (1992) describe a modification of the Kaplan-Meier estimator that can be used for right censored data and note that for the purpose of assessing goodness of fit the choice of plotting positions makes little qualitative difference in the appearance of any particular plot. The two options in this function can be used for any type of censoring.
E. L. Frome
Fay, M. P. (1999), "Comparing Several Score Tests for Interval Censored Data," Statistics in Medicine, 1999; 18:273-85. (Corr: 1999, Vol 19, p.2681).
Frome, E. L. and Wambach, P. F. (2005), "Statistical Methods and Software for the Analysis of Occupational Exposure Data with Non-Detectable Values," ORNL/TM-2005/52,Oak Ridge National Laboratory, Oak Ridge, TN 37830. Available at: http://www.csm.ornl.gov/esh/aoed/ORNLTM2005-52.pdf
Hesel, D. R. and T. A. Cohn (1988), "Estimation of Descriptive Statistics for Multiply Censored Water Quality Data," Water Resources Research, 24, 1997-2004.
Meeker, W. Q. and L. A. Escobar (1998), Statistical Methods for Reliability Data, John Wiley and Sons, New York.
Ny, M. P. (2002), "A Modification of Peto's Nonparametric Estimation of Survival Curves for Interval-Censored Data," Biometrics, 58, 439-442.
Waller, L. A. and B. W. Turnbull (1992), "Probability Plotting with Censored Data," The American Statistician, 46(1), 5-12.
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