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# FIELLER procedure

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

### Options

`PRINT` = string token What to output (`value`); default `valu` Parameter estimates; default extracts these with `RKEEP` Variances and covariances; default extracts these with `RKEEP` Percentage points for limits; default 95, thus giving 95% confidence limits Whether to calculate relative potencies (`no`, `yes`); default `no` 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` Base of antilog transformation to be applied to value and limits, (`ten`, `e`); default `*` i.e. none 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 Positions of slope parameters in list of estimates; default last estimate Percentage doses; default 50, thus giving LD50 To store estimated values To store lower limits To store upper limits 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.

Procedure: `PROBITANALYSIS`.

Commands for: Regression analysis.

### Example

```CAPTION   'FIELLER example',\
!t('The example estimates relative potencies of analgesic drugs',\
'(data from Finney, 1971, Probit Analysis, 3rd Edition,',\
'p. 104).'); STYLE=meta,plain
UNITS     [NVALUES=14]
VALUES=3(1,2,3),5(4)] drug
1.5 103 19    3 120 53    6 123 83  1.5 60 14    3 110 54
6 100 81  .75  90 31  1.5  80 54    3  90 80    5 60 13
7.5  85 27   10  60 32   15  90 55   20  60 44  :
CAPTION   'Fit parallel probit lines.'
CALCULATE logdose = LOG10(dose)