10.3. Quantal Response Method
Parallel line models can be fitted using one of logit, probit, gompit (clolglog) or loglog link functions. Asymmetric dose structures and multiple test preparations are supported. To test the validity of assay a Weighted ANOVA table (including nonparallelism and nonlinearity tests) is displayed. Output includes estimates of effective dose (or lethal dose) for any userdefined percentile (including ED50, EC50 or LD50), the potency ratio and their confidence limits.
10.3.1. Quantal Response 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. Cases with a zero Subject will be considered missing.
As in other bioassay procedures, a [Dose] variable should also be selected. However, the choice of a [Preparation] variable is optional. If a [Preparation] variable is not selected, then an ED50 estimate can still be computed, fitting a single line (instead of parallel lines) on all data points.
Designs can be unbalanced, i.e. the number of replicates for each dosepreparation combination may be different, dose levels for standard and test preparations may be different, there can be more than one test preparation, but the first five characters of the standard preparation label should be “stand” or “refer” in any language (capitalisation is not significant). Otherwise the first preparation encountered in the [Preparation] column will be considered as the standard. For the two optional variables [Assay] and [Dilution], see sections 10.0.4. Multiple Assays with Combination and 10.0.2. Doses, Dilutions and Potency respectively.
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 change this value. If 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), 10 or 2based logarithm or leave it untransformed.
SpearmanKarber: When a nonnegative percentage is entered, ED50 and its confidence limits are also computed using the SpearmanKarber method. When this box contains a negative value, SpearmanKarber results are not reported.
Link Function: Select the model to be estimated; logit, probit, gompit (cloglog) or loglog.
Predictions (Interpolations): Predictions can be obtained for response or dose values using the estimated model parameters. For details see section 10.0.7. Prediction, Interpolation, Extrapolation.
10.3.2. Quantal Response Output Options
10.3.2.1. Regression Results
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. Let be the expected value for case j. Then:
Logit:
Probit:
Gompit (cloglog):
Loglog:
For further details see 7.2.5.1. Logit / Probit / Gompit Model Description.
A NewtonRaphson 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. Case (Diagnostic) Statistics
10.3.2.3. Effective Dose (or Lethal Dose)
By default, ED50 (or EC50 or LD50) values and their fiducial confidence limits are computed for all preparations. If a nonnegative percentage is entered in the SpearmanKarber box, ED50 values, their confidence limits and actual percentage trim used are also displayed for each preparation using this method. If the [Preparation] variable is not selected or 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 usersupplied 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:
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.
The trimmed SpearmanKarber (or Kaerber) and its confidence interval are computed as described in Hamilton at al (1977). If the consecutive response values are not monotonically increasing (or decreasing) their average is used. If the trim entered by the user has no solution, then the minimum trim estimated by the program is used. For each preparation, the percentage trim entered and the trim used by the program are displayed.
Effective Dose

Effective Dose 
Lower 95% 
Upper 95% 
Trim Entered 
Trim Used 
Standard S ED50 
2.3489 
1.9291 
2.8956 


SpearmanKarber 
2.3391 
1.8891 
2.8961 
0.00% 
9.09% 
Preparation T ED50 
2.0477 
1.6716 
2.5166 


SpearmanKarber 
1.9703 
1.5878 
2.4450 
0.00% 
9.09% 
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. Validity of Assay
Unlike Parallel Line Method and Slope Ratio Method, in Quantal Response Method the validity of assay is based on weighted sums of squares. This allows us to perform chisquare tests for various model parameters.
Nonlinearity test:
where Sxx, Syy and Sxy are sum of squares as defined in Finney, D. J. (1978) p. 372. The test statistic has (n – 4) degrees of freedom.
Nonparallelism test:
The test statistic has (m – 1) degrees of freedom.
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Pass/Fail 
Preparations 
0.066 
0.066 
1 
0.7969 
Pass 
Regression 
33.975 
33.975 
1 
0.0000 
Pass 
Nonparallelism 
0.001 
0.001 
1 
0.9743 
Pass 
Nonlinearity 
1.921 
1.921 
4 
0.7502 
Pass 
Standard S Nonlinearity 
0.851 
0.851 
2 
0.6533 
Pass 
Preparation T Nonlinearity 
1.070 
1.070 
2 
0.5856 
Pass 
Treatments 
35.964 
35.964 
7 
0.0000 
Pass 
Residual 





Total 
35.964 

7 


Rsquared 
0.947 



**Fail** 
10.3.2.5. Potency
The relative potency 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:
First define:
The fiducial confidence interval for potency ratio of each test preparation is defined as:
where:
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 assumed 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:
If the data column [Dose] contains the actual dose levels administered in original dose units, we will obtain the estimated potency and its confidence limits in the same units. If, however, the [Dose] column contains unitless relative dose levels, then we may need to perform further calculations to obtain the estimated potency in original units. To do that you can enter assigned potency of the standard, assumed potency of each test preparation and predilutions for all preparations including the standard in a data column and select it as [Dilution] variable. UNISTAT will then calculate the estimated potency as described in section 10.0.2. Doses, Dilutions and Potency. Also see section 10.0.3. Potency Calculation Example.
10.3.2.6. Plot of Treatments
Response ratios are plotted on a logit, probit, gompit or loglog Yaxis (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.
10.3.2.7. DoseResponse Plot
A doseresponse curve will be drawn for each preparation. For each curve it is possible to display fiducial limits, ED50, and the fiducial limits for ED50.
Selecting Edit → DoseResponse Plot… (or doubleclicking on the graph area) will pop a dialogue where what is displayed on the plot can be controlled.
10.3.3. Quantal Response Examples
Example 1
Data is given in p. 4371 of European Pharmacopoeia (9^{th} Edition) Table 5.3.1.I.
Open BIOPHARMA9 and select Bioassay → Quantal Response Method. From the Variable Selection Dialogue select columns C35, C36, C37, L38 and L39 respectively as [Response], [Subject], [Dose], [Preparation] and [Dilution]. Click [Next], select Probit model and leave other entries unchanged.
Quantal Response Method
Valid Number of Cases: 8, 0 Omitted
Model selected: Logit
Regression Results

Coefficient 
Standard Error 
ZStatistic 
2Tail Probability 
Lower 90% 
Upper 90% 
Common Slope 
2.4011 
0.4170 
5.7582 
0.0000 
1.7152 
3.0869 
Intercept Standard S 
2.0504 
0.4086 
5.0176 
0.0000 
2.7226 
1.3783 
Intercept Preparation T 
1.7208 
0.3829 
4.4945 
0.0000 
2.3506 
1.0911 
Residual Variance = 
0.4781 
Degrees of Freedom = 
5 
Case (Diagnostic) Statistics

Response 
Subject 
Dose 
Preparation 
1 
0 
12 
1 
Standard S 
2 
3 
12 
1.6 
Standard S 
3 
6 
12 
2.5 
Standard S 
4 
10 
11 
4 
Standard S 
5 
0 
11 
1 
Preparation T 
** 6 
4 
12 
1.6 
Preparation T 
** 7 
8 
11 
2.5 
Preparation T 
8 
10 
11 
4 
Preparation T 

Estimated Response 
Residuals 
Standardised Residuals 
Probability 
1 
0.2419 
0.2419 
0.3499 
0.0202 
2 
2.1395 
0.8605 
1.2445 
0.1783 
3 
6.7138 
0.7138 
1.0323 
0.5595 
4 
9.8935 
0.1065 
0.1541 
0.8994 
5 
0.2218 
0.2218 
0.3207 
0.0202 
** 6 
2.1395 
1.8605 
2.6907 
0.1783 
** 7 
6.1543 
1.8457 
2.6692 
0.5595 
8 
9.8935 
0.1065 
0.1541 
0.8994 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Effective Dose

Effective Dose 
Lower 95% 
Upper 95% 
Trim Entered 
Trim Used 
Standard S ED50 
2.3489 
1.9291 
2.8956 


SpearmanKarber 
2.3391 
1.8891 
2.8961 
0.00% 
9.09% 
Preparation T ED50 
2.0477 
1.6716 
2.5166 


SpearmanKarber 
1.9703 
1.5878 
2.4450 
0.00% 
9.09% 
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Pass/Fail 
Preparations 
0.066 
0.066 
1 
0.7969 
Pass 
Regression 
33.975 
33.975 
1 
0.0000 
Pass 
Nonparallelism 
0.001 
0.001 
1 
0.9743 
Pass 
Nonlinearity 
1.921 
1.921 
4 
0.7502 
Pass 
Standard S Nonlinearity 
0.851 
0.851 
2 
0.6533 
Pass 
Preparation T Nonlinearity 
1.070 
1.070 
2 
0.5856 
Pass 
Treatments 
35.964 
35.964 
7 
0.0000 
Pass 
Residual 





Total 
35.964 

7 


Rsquared 
0.947 



**Fail** 
Potency
Assigned potency of Standard S: 132 IU/vial
Assumed potency of Preparation T: 140 IU/vial

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
160.5974 
120.9660 
215.1559 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
114.71% 
86.40% 
153.68% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
75.32% 
133.97% 
G = 
0.1131 
C = 
1.1275 
Table 5.3.2.I. also gives the results for logit and gompit methods. Click on the [Last 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
Valid Number of Cases: 8, 0 Omitted
Model selected: Logit
Potency
Assigned potency of Standard S: 132 IU/vial
Assumed potency of Preparation T: 140 IU/vial

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
162.8590 
121.1311 
221.1056 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
116.33% 
86.52% 
157.93% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
74.38% 
135.77% 
Select the Gompit model and repeat the analysis.
Quantal Response Method
Valid Number of Cases: 8, 0 Omitted
Model selected: Logit
Potency
Assigned potency of Standard S: 132 IU/vial
Assumed potency of Preparation T: 140 IU/vial

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
158.3126 
118.7082 
213.2961 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
113.08% 
84.79% 
152.35% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
74.98% 
134.73% 
Select the LogLog model and repeat the analysis.
Quantal Response Method
Valid Number of Cases: 8, 0 Omitted
Model selected: Logit
Potency
Assigned potency of Standard S: 132 IU/vial
Assumed potency of Preparation T: 140 IU/vial

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
159.1364 
120.0325 
211.3075 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
113.67% 
85.74% 
150.93% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
75.43% 
132.78% 
Example 2
Data is given in p. 4373 of European Pharmacopoeia (9^{th} Edition) Table 5.3.3.I.
Open BIOPHARMA9 and select Bioassay → Quantal Response Method. From the Variable Selection Dialogue select columns C40 to C42, Response, Subject, LogDose respectively as [Response], [Subject], [Dose]. You do not need to select a [Preparation] variable, though you can also select C43 Preparation as [Preparation] as it contains only one value. 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
Valid Number of Cases: 10, 0 Omitted
Model selected: Probit
Regression Results

Coefficient 
Standard Error 
ZStatistic 
2Tail Probability 
Lower 95% 
Upper 95% 
Common Slope 
1.4880 
0.3063 
4.8580 
0.0000 
2.0883 
0.8877 
Intercept 1 
7.9314 
1.6586 
4.7820 
0.0000 
11.1822 
4.6806 
Effective Dose

Effective Dose 
Lower 95% 
Upper 95% 
1 ED50 
5.3302 
5.6568 
5.0022 
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Pass/Fail 
Preparations 
0.000 
0.000 
0 
* 
* 
Regression 
23.337 
23.337 
1 
0.0000 
Pass 
Nonparallelism 
0.000 
0.000 
0 
* 
* 
Nonlinearity 
2.711 
2.711 
8 
0.9512 
Pass 
Treatments 
26.049 
26.049 
9 
0.0020 
Pass 
Residual 





Total 
26.049 

9 


Rsquared 
0.896 




Remember that the dose data were already logged base 10. Applying the backtransformation (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].
Standard and test preparations are equipotent. The standard has an assigned potency of 20 IU/vial and the test preparation has an assumed potency of 20 IU/vial. Create a new column of data as follows and select it as [Dilution] variable.
20 IU/vial
1
20 IU/vial
1
For further information see section 10.0.2. Doses, Dilutions and Potency.
Click [Next], select Probit model and leave other entries unchanged.
Quantal Response Method
Valid Number of Cases: 14, 0 Omitted
Model selected: Probit
Regression Results

Coefficient 
Standard Error 
ZStatistic 
2Tail Probability 
Lower 90% 
Upper 90% 
Common Slope 
1.3914 
0.1234 
11.2782 
0.0000 
1.1885 
1.5944 
Intercept Standard 
3.3660 
0.3061 
10.9960 
0.0000 
3.8695 
2.8625 
Intercept Unknown 
3.9263 
0.3520 
11.1554 
0.0000 
4.5053 
3.3474 
Residual Variance = 
2.3831 
Degrees of Freedom = 
11 
Case (Diagnostic) Statistics

Response 
Subject 
Dose 
Preparation 
1 
0 
33 
3.4 
Standard 
2 
5 
32 
5.2 
Standard 
3 
11 
38 
7 
Standard 
4 
14 
37 
8.5 
Standard 
5 
18 
40 
10.5 
Standard 
6 
21 
37 
13 
Standard 
7 
23 
31 
18 
Standard 
8 
30 
37 
21 
Standard 
9 
27 
30 
28 
Standard 
** 10 
2 
40 
6.5 
Unknown 
11 
10 
30 
10 
Unknown 
** 12 
18 
40 
14 
Unknown 
** 13 
21 
35 
21.5 
Unknown 
** 14 
27 
37 
29 
Unknown 

Estimated Response 
Residuals 
Standardised Residuals 
Probability 
1 
1.5884 
1.5884 
1.0289 
0.0481 
2 
4.5393 
0.4607 
0.2984 
0.1419 
3 
9.6950 
1.3050 
0.8453 
0.2551 
4 
12.9095 
1.0905 
0.7064 
0.3489 
5 
18.4982 
0.4982 
0.3227 
0.4625 
6 
21.4748 
0.4748 
0.3076 
0.5804 
7 
23.0640 
0.0640 
0.0414 
0.7440 
8 
29.8926 
0.1074 
0.0696 
0.8079 
9 
26.9414 
0.0586 
0.0380 
0.8980 
** 10 
8.9266 
6.9266 
4.4869 
0.2232 
11 
13.0679 
3.0679 
1.9873 
0.4356 
** 12 
24.8085 
6.8085 
4.4104 
0.6202 
** 13 
28.5854 
7.5854 
4.9136 
0.8167 
** 14 
33.5394 
6.5394 
4.2361 
0.9065 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Effective Dose

Effective Dose 
Lower 95% 
Upper 95% 
Trim Entered 
Trim Used 
Standard ED50 
11.2359 
10.0319 
12.6068 


SpearmanKarber 
11.0745 
9.8289 
12.4780 
0.00% 
10.00% 
Unknown ED50 
16.8071 
14.5334 
19.5569 


SpearmanKarber 
16.0704 
12.9779 
19.8997 
0.00% 
27.03% 
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Pass/Fail 
Preparations 
0.594 
0.594 
1 
0.4407 
Pass 
Regression 
130.268 
130.268 
1 
0.0000 
Pass 
Nonparallelism 
0.282 
0.282 
1 
0.5951 
Pass 
Nonlinearity 
5.421 
5.421 
10 
0.8614 
Pass 
Standard Nonlinearity 
2.058 
2.058 
7 
0.9565 
Pass 
Unknown Nonlinearity 
3.363 
3.363 
3 
0.3390 
Pass 
Treatments 
136.566 
136.566 
13 
0.0000 
Pass 
Residual 





Total 
136.566 

13 


Rsquared 
0.958 



Pass 
Potency
Assigned potency of Standard: 20 IU/vial
Assumed potency of Unknown: 20 IU/vial

Estimated Potency 
Lower 95% 
Upper 95% 
Unknown 
13.3704 
11.0678 
16.0812 

Relative Potency 
Lower 95% 
Upper 95% 
Unknown 
66.85% 
55.34% 
80.41% 

Percent CI 
Lower 95% 
Upper 95% 
Unknown 
100.00% 
82.78% 
120.28% 