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 The first condition is necessary because the regression line is estimated from
the partial mortalities. The second condition, called goodness-of-fit, can be
tested by the ?2 statistic, which is a measure of the distance of the data points
from the regression line. A low ?2 indicates a good fit. By convention, the fit is
considered adequate if the P-value for ?2 is >0.05 (in other words, goodness-of-
fit is rejected if P#0.05). If the P-value is not provided, the goodness-of-fit ?2
should be compared against tabled values with k - 2 df, where k is the number of
partial mortalities. If there are only two partial mortalities (k = 2), then there are
0 df, and the goodness-of-fit cannot be tested (i.e., a line between two points is
always a perfect fit). When there are only two partial mortalities, the LC50 is
below) with mortality expressed on a probit scale. Goodness-of-fit can also be
assessed by eye, if the data are plotted on log-probit paper, or if the computer
program provides a plot.
The SAS probit procedure (PROC PROBIT) provides a goodness-of-fit ?2
and its associated P-value if the LACKFIT option is specified. Model-predicted
mortalities can also be plotted against observed mortalities to assess model fit.
The INVERSECL option provides an estimate of LC50 as well as other effects
concentrations ranging from LC1 to LC99.
Logistic Method. The Logistic method is similar to the Probit method
except that mortalities are converted to logits rather than probits. A logit is log
[M/(100 - M)], where M is percent mortality. The LC50 is derived from a
regression of logits on log concentration. As with the Probit method, the
Logistic method can be used whenever there are two or more partial mortalities,
and the data fit the regression line. SAS PROC PROBIT can calculate LC50
using the Logistic method by specifying the D=LOGISTIC option in the model
statement.
Trimmed Spearman-Karber (TSK) Method. The TSK method is a
nonparametric method that can be calculated by hand using the procedure in
Gelber et al. (1985). The calculations can be tedious, especially for processing
large numbers of tests, and computer programs are usually used. The method is
labelled "trimmed" because extreme values (mortality much higher or lower than
50 percent) are "trimmed" or removed prior to calculation of the LC50. Thus, the
LC50 is calculated using points near 50 percent mortality, which may produce a
more robust estimate. The TSK method can be used in many cases where the
Probit method is unsuitable. Access to appropriate computer programs and
difficulties in deciding what values to trim are probably the major factors
limiting widespread use of the TSK method. Investigators with access to reliable
programs should not hesitate to use the TSK method whenever there are two or
more partial mortalities. Information concerning TSK computer programs may
be obtained from the USEPA Environmental Research Laboratories in Athens,
GA, or Duluth, MN, or CSC/USEPA, Cincinnati, OH.
used when:
There are 0 or 1 partial mortalities.
L22
Appendix L
Statistical Methods
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