10.3. Quantal Response Method

Logit, probit or gompit parallel line models can be fitted using a maximum likelihood procedure. Asymmetric dose structures and multiple test preparations are supported. The parallelism and linearity tests are performed. -.15pt'>), the potency ratio and their confidence limits.

10.3.1. Variable Selection

The first variable [Response] represents the number subjects responding positively (or negatively) to the test and the second [Subject] contains the total number of subjects in that group. Therefore, the following relation should hold for each case:

      0 ≤ Response ≤ Subject

If some cases do not conform to this, then the analysis will be aborted.

As in other bioassay procedures, a [Dose] and a [Preparation] variable should also be selected.

The next dialogue asks for the following convergence and model parameters.

Tolerance: This value is used to control the sensitivity of the maximum likelihood procedure employed. Under normal circumstances, you do not need to edit this value. If a convergence cannot be achieved, then larger values of this parameter can be tried by removing one or more zeros.

Maximum Number of Iterations: When convergence cannot be achieved with the default value of 100 function evaluations, a higher value can be tried.

Dose Transformation: It is possible to transform the dose variable by natural (default) or 10-based logarithm or leave it untransformed.

Logit / Probit / Gompit: Select the model to be estimated.

10.3.2. Output Options

10.3.2.1. Regression

The maximum likelihood model is constructed as a regression without a constant term (i.e. through the origin), with independent variables consisting of the transformed dose variable and a set of m dummy variables created from the preparations variable. When the convergence is achieved, the coefficient for the dose variable represents the estimated common slope and coefficients for the dummy variables represent the estimated intercept for each preparation.

The dependent variable is obtained from the response and subject variables. For the logit model:

for the probit model:

F_j is the cumulative normal probability at

where  is the expected logit or probit for case j, and for the gompit model:

F_j = 1-Exp(-Exp())

For further details see 7.2.5.1. Logit / Probit / Gompit Model Description.

A Newton-Raphson type maximum likelihood algorithm is employed to minimise the negative of the log likelihood function. The nature of this method implies that a solution (convergence) cannot always be achieved. In such cases, you are advised to edit the convergence parameters provided, in order to find the right levels for the particular problem at hand.

10.3.2.2. Validity of Assay

Three chi-square tests are performed:

1)    Pearson’s overall goodness of fit test:

     

where:

     

is the expected frequency for case j. The test statistic has (n – m -1) degrees of freedom.

2)    Non-linearity test:

     

where Sxx, Syy and Sxy are as defined in Finney, D. J. (1978) p. 372. The test statistic has (n – 4) degrees of freedom.

3)    Non-parallelism test:

     

The test statistic has (m – 1) degrees of freedom.

10.3.2.3. Effective Dose (or Lethal Dose)

By default, ED50 (or LD50) values and their confidence limits are computed for all preparations. If the [Preparation] variable contains only one value, then an ED50 estimate will still be calculated, fitting a single line (instead of parallel lines) on all data points. Let d be the user-supplied effective dose (or lethal dose) quantile. Then for the logit model compute:

     

and for the probit model:

      Y = Critical value of (1 - d) from inverse standard normal distribution.

The effective dose for preparation i is then found as:

     

where is the intercept for preparation i and  is the common slope.

To calculate the confidence limits of M_i first define:

     

where V_b is the variance of common slope and  is the critical value from normal distribution.

The confidence interval for potency ratio of each test preparation is defined as:

     

where:

     

and Vss, Vii and Vsi are the elements of covariance matrix of regression coefficients for standard and preparation i.

If you wish to compute other effective dose values then, on the Output Options Dialogue, click the [Opt] button situated to the left of the Effective Dose option. A further dialogue pops up asking for entry of a value between 0 and 1. The program will then output the effective dose and its confidence limits for this value, as well as its complementary value, for all preparations. For instance, if 0.9 is entered, ED10 and ED90 values will be computed and the output will look like as follows:

 

	Effective Dose	Lower 95%	Upper 95%
Standard ED10	4.4731	3.5712	5.2983
ED90	28.2233	23.6956	35.6538
Unknown ED10	6.6911	5.2925	8.0338
ED90	42.2176	34.4987	55.0306

 

10.3.2.4. Potency

The default method of iterative convergence used in calculating confidence intervals for potency has been changed as of this version of UNISTAT. The old method can still be invoked by entering the following line in [Options] section of the Documents\Unistat60\Unistat60.ini file:

   QuantalConfIntEP=0

The relative potency is for test preparation i is found as:

     

where and are the intercepts for test i and standard preparations and  is the common slope.

To calculate the confidence limits of M_i first define:

     

where V_b is the variance of common slope and  is the critical value from normal distribution.

First define:

     

     

The confidence interval for potency ratio of each test preparation is defined as:

     

where:

     

Note that M_i is the relative potency and M_iL and M_iU are the confidence limits for the relative potency. The estimated potency and its confidence interval are obtained by multiplying these relative values by the assigned potency supplied by the user for each test preparation separately.

The approximate variance of M_i is:

     

Weights are computed after the estimated potency and its confidence interval are found:

     

and % Precision is:

     

10.3.2.5. Plot of Treatments

Response ratios are plotted on a logit, probit or gompit Y-axis (see Scale Type), versus log of dose, according to the model selected. A line of best fit is also drawn for each preparation. If you want to edit the properties of the graph, you can send it to Graphics Editor by clicking on the [Opt] button situated to the left of the plot option. The Edit → Data Series dialogue on the graphics window menu provides you with necessary controls to edit all aspects of the plot.

10.3.3. Examples

Example 1

Data is given in European Pharmacopoeia (2008), Table 5.3.1.-I on p. 589.

Open BIOPHARMA6 and select Bioassay → Quantal Response Method. From the Variable Selection Dialogue select columns C30 to C33 respectively as [Response], [Subject], [Dose] and [Preparation]. Click [Next], select Probit model and leave other entries unchanged. On the Output Options Dialogue. Click the [Opt] button situated to the left of the Potency option. Enter 140 as the assigned potency for the unknown. Click [Back] to get back to output options, click [All] to perform all tests in one go and then click [Finish].

Quantal Response Method

Model selected: Probit

Regression

	Coefficient	Standard Error
Common Slope	2.4011	0.4170
Intercept Standard S	-2.0504	0.4086
Intercept Preparation T	-1.7208	0.3829

 

Validity of Assay

	Chi-Square	DoF	Probability
Goodness of Fit	1.9225	5	0.8598
Non-linearity	1.9215	4	0.7502
Non-parallelism	0.0010	1	0.9743

 

Effective Dose

	Effective Dose	Lower 95%	Upper 95%
Standard S ED50	2.3489	1.9291	2.8956
Preparation T ED50	2.0477	1.6716	2.5166

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	140.0000	160.5974	120.9660	215.1559

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0170	0.0017	75.3225

 

G =	0.1131
C =	1.1275

 

 

Table 5.3.2.-I. also gives the results for logit and gompit methods. Click on the [Last Procedure Dialogue] button on the Output Medium Toolbar. This will display the Output Options Dialogue again. Click [Back], select the Logit model, click [Next] and select only the Potency output option.

Quantal Response Method

Model selected: Logit

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	140.0000	162.8590	121.1311	221.1056

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0172	0.0015	74.3779

 

G =	0.1455
C =	1.1703

 

Select the Gompit model and repeat the analysis.

Quantal Response Method

Model selected: Gompit

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	140.0000	158.3126	118.7082	213.2961

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0174	0.0017	74.9834

 

G =	0.1179
C =	1.1336

 

Example 2

Data is given in European Pharmacopoeia (2008), Table 5.3.3.-I on p. 591.

Open BIOPHARMA6 and select Bioassay → Quantal Response Method. From the Variable Selection Dialogue select columns C34 to C36, Response, Subject, LogDose respectively as [Response], [Subject], [Dose] and C38 Preparation as [Preparation]. Click [Next], select Probit model and select the dose transformation None, since the data is already logged base 10. On the Output Options Dialogue click [Finish]. The following output is obtained:

Quantal Response Method

Model selected: Probit

Regression

	Coefficient	Standard Error
Common Slope	-1.4880	0.3063
Intercept 1	-7.9314	1.6586

 

Validity of Assay

	Chi-Square	DoF	Probability
Goodness of Fit	2.7112	8	0.9512
Non-linearity	2.7112	8	0.9512

 

Effective Dose

	Effective Dose	Lower 95%	Upper 95%
1 ED50	-5.3302	-5.6568	-5.0022

 

 

Remember that the dose data were already logged base 10. Applying the back-transformation (again base 10):

      -M_T + Log(1000/50)

and reversing the limits we obtain:

 

	Effective Dose	Lower 95%	Upper 95%
1 ED50	6.6313	6.3033	6.9578

 

Example 3

Table 18.2.1. on p. 376 from Finney, D. J. (1978) gives data for an unbalanced assay with one test preparation. Finney gives the results in Table 18.3.1. on p. 380.

Open BIOFINNEY and select Bioassay → Quantal Response Method. From the Variable Selection Dialogue select columns C20 to C23 respectively as [Response], [Subject], [Dose] and [Preparation]. Click [Next], select Probit model and leave other entries unchanged. On the Output Options Dialogue click the [Opt] button situated to the left of the Potency option, enter 20, click [Back] and then click [Finish] to obtain the following output.

Quantal Response Method

Model selected: Probit

Regression

	Coefficient	Standard Error
Common Slope	1.3914	0.1234
Intercept Standard	-3.3660	0.3061
Intercept Unknown	-3.9263	0.3520

 

Validity of Assay

	Chi-Square	DoF	Probability
Goodness of Fit	5.7033	11	0.8924
Non-linearity	5.4208	10	0.8614
Non-parallelism	0.2824	1	0.5951

 

Effective Dose

	Effective Dose	Lower 95%	Upper 95%
Standard ED50	11.2359	10.0319	12.6068
Unknown ED50	16.8071	14.5334	19.5569

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Unknown	20.0000	13.3704	11.0678	16.0812

 

Test Preparation	Variance	Weight	% Precision
Unknown	0.0086	0.6113	82.7786

 

G =	0.0295
C =	1.0304