FIELLER procedure

Calculates effective doses or relative potencies (P.W. Lane).


Options

PRINT = string
What to output (value); default valu

ESTIMATES = variate
Parameter estimates; default extracts these with RKEEP

VCOVARIANCE = symmetric matrix
Variances and covariances; default extracts these with RKEEP

%LIMIT = scalar
Percentage points for limits; default 95, thus giving 95% confidence limits

RELATIVE = string
Whether to calculate relative potencies (no, yes); default no

LINK = string
Which link function to assume when forming effective doses (probit, logit, complementaryloglog); default obtained using RKEEP, if the ESTIMATES or VARIANCES are obtained in that way, otherwise prob

LOGBASE = string
Base of antilog transformation to be applied to value and limits, (ten, e); default * i.e. none

DF = scalar
If this has a non-missing value, a t-distribution is used instead of a Normal distribution to calculate the confidence limits; default obtained using RKEEP if the ESTIMATES or VARIANCES are obtained in that way (setting DF to the number of residual d.f. when the dispersion factor is estimated, or a missing value when it is fixed), otherwise the default is a missing value


Parameters

TREATMENT = variates, scalars or texts
Positions of intercept parameters in list of estimates; default first estimate

SLOPE = variates, scalars or texts
Positions of slope parameters in list of estimates; default last estimate

%DOSE = variates or scalars
Percentage doses; default 50, thus giving LD50

VALUE = variates or scalars
To store estimated values

LOWER = variates or scalars
To store lower limits

UPPER = variates or scalars
To store upper limits

SE = variates or scalars
To store approximate s.e.s of values


Description

Quantal data from bioassay experiments can be analysed with the regression directives by fitting a generalized linear model with a binomial distribution and a probit, logit or complementary log-log link function. The coefficients estimated by the regression directives are the intercepts and slopes of the lines fitted on the transformed scale. However, you often need LD50s: that is, estimates of the median effective doses (or LD90s, and so on). These quantities are ratios of the estimated coefficients, and fiducial limits and approximate standard errors can be derived for their estimates using Fieller's theorem.

   By default, FIELLER assumes that parameter estimates and their covariances can be extracted using the RKEEP directive from a fit already done using the regression directives. If this is not the case, these quantities must be supplied by setting the ESTIMATES and VCOVARIANCE options.

   FIELLER can be used to estimate either effective doses of individual treatments (parallel or non-parallel regression lines) or relative potencies compared to a standard treatment (parallel regression lines only). For effective doses, you specify the percentage point with the option %DOSE: so the default, %DOSE=50, gives LD50s. The link function to be assumed in the calculations can be specified with the LINK option. If this is not specified, and the estimates or variances are being extracted using RKEEP, the LINK will be set (using the OMODEL option RKEEP) to the same link as in the fitted generalized linear model; otherwise the default is to use a probit link. For relative potencies, you should set option RELATIVE=yes; %DOSE and LINK are then not relevant. The DF parameter can be set to use a t-distribution instead of a Normal distribution when calculating the confidence limits; this would be relevant if the dispersion factor in the generalized linear model was being estimated instead of being fixed at 1 (i.e. if a heterogeneity factor is being used). If DF is unset and the estimates or variances are being extracted using RKEEP, FIELLER uses RKEEP to see whether the dispersion is fixed and, if so, sets DF to the number of residual d.f. in the analysis; otherwise the default setting for DF is a missing value, which indicates that a Normal distribution should be used.

   The SLOPE and TREATMENT parameters should be set to indicate the estimates of the slope and intercept parameters to be used. You can do this either by supplying a scalar or variate giving the position or positions in the list of estimates, or by giving a text containing their labels (as used in the tables of estimates printed by FIT &c). For effective doses, any model that includes a treatment effect should be fitted without an intercept (i.e. setting option FULL=yes in the TERMS statement and CONSTANT=omit in the FIT statement) so that the estimates produced by the regression directives are absolute intercepts rather than differences. For relative potencies, an intercept should be included, and the standard treatment represented as the first level of the treatments factor so that the estimates are differences of intercepts.

   The procedure prints the estimate with lower and upper fiducial limits. The range of the limits can be set by the %LIMIT option: the default is 95% limits. Printing can be turned off by setting PRINT=*. The results of the procedure can be stored in scalars or variates using the VALUE, LOWER and UPPER parameters; also an approximate standard error of the estimated value (before back transformation, if relevant) can be stored using parameter SE.

 

Options: PRINT, ESTIMATES, VCOVARIANCE, %LIMIT, RELATIVE, LINK, LOGBASE, DF.

Parameters: TREATMENT, SLOPE, %DOSE, VALUE, LOWER, UPPER, SE.


Method

The fiducial limits are calculated using Fieller's Theorem; see, for example,

Finney (1971, page 78).


Reference

Finney, D.J. (1971). Probit Analysis (third edition). Cambridge University Press, Cambridge.