# qq.lnorm: Quantile-Quantile Plot for Censored Lognormal Data In STAND: Statistical Analysis of Non-Detects

## Description

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.

## Usage

 `1` ```qq.lnorm(pl, mu, sigma, aveple = TRUE,...) ```

## Arguments

 `pl` A data frame with the data(x) in column 1 and PLE in column 2 `mu` estimate of the log scale mean `sigma` estimate of log scale standard deviation `aveple` if `TRUE`, calculate plotting positions by averaging `...` Additional parameters to plot

## Details

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 valuesâ€“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 `Rsq` the square of the correlation coefficient associated with the plot.

A line is added to the plot based on the values of `mu` and `sigma`. If either of these is missing `mu` and `sigma` are estimated by linear regression of log(y) on H[p(j)].

## Value

A list with components

 `x` The x coordinates of the points that were plotted `y` The y coordinates of the points that were plotted `pp` The adjusted probabilities use to determine `x` `par` The values of `mu`, `sigma`, and `Rsq`

## Note

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

## References

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.

`plekm`, `plend`, `pleicf`

## Examples

 ```1 2 3 4 5 6``` ```data(SESdata) # use SESdata data set Example 1 from ORNLTM-2005/52 pnd<- plend(SESdata) qq.lnorm(pnd) # lognormal q-q plot based on PLE Ia <- "Q-Q plot For SESdata " qqout <- qq.lnorm(pnd,main=Ia) # lognormal q-q plot based on PLE qqout ```

### Example output

```Loading required package: survival
\$x
[1] -1.30015343 -1.13097761 -0.80754104 -0.60017878 -0.46049454 -0.32929135
[7] -0.24500622 -0.16242937 -0.08094729  0.04044051  0.20354423  0.37228936
[13]  0.50593365  0.60017878  0.70009021  0.80754104  0.98916863  1.21123213
[19]  1.40074506  1.66069761  2.14119812

\$y
[1] 0.015 0.025 0.040 0.045 0.050 0.070 0.075 0.095 0.100 0.125 0.145 0.150
[13] 0.165 0.270 0.290 0.345 0.395 0.420 0.495 0.840 1.140

\$pp
[1] 0.09677419 0.12903226 0.20967742 0.27419355 0.32258065 0.37096774
[7] 0.40322581 0.43548387 0.46774194 0.51612903 0.58064516 0.64516129
[13] 0.69354839 0.72580645 0.75806452 0.79032258 0.83870968 0.88709677
[19] 0.91935484 0.95161290 0.98387097

\$par
mu     sigma       Rsq
-2.278651  1.246852  0.983000
```

STAND documentation built on May 2, 2019, 3:39 p.m.