10.4. FourParameter Logistic Model
This procedure features two implementations of the 4PL method; (1) according to European Pharmacopoeia (19972017) and (2) as described in United States Pharmacopoeia (2010) chapters <1032>, <1033>, <1034>, <81> and <111>. It is possible to estimate the Full and Reduced USP models including outlier detection, plate effects, equivalence tests, outlier detection and multiple potency estimates with confidence limits. 96well ELISA, as well as unbalanced designs are supported.
10.4.1. 4PL Variable Selection
The required data format is as described in section 10.0.1. Data Preparation (also see 10.0.2. Doses, Dilutions and Potency). Response data is stacked in a single column, a second column contains the dose level for each response and an optional categorical data column indicates which preparation a particular doseresponse pair belongs to. By default, all models are based on Completely Randomised Design.
The optional [Plate] variable is usually used to isolate the plate effect (or replicates). If all dose/treatment groups (or cells) have an equal number of replicates (or if the design is balanced) then including the plate effect is equivalent to analysing the data with Randomised Block Design.
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.
Predictions (interpolations) can be obtained for response or dose values using the estimated model parameters. For details see section 10.0.7. Prediction, Interpolation, Extrapolation.
To run an analysis, it is sufficient to select one [Data] and one [Dose] variable. In this case, a single sigmoid curve will be fitted to data. If a [Preparation] variable (which should be a numeric or string factor or categorical data variable) is also selected, then you will have the option of fitting a Full Model (also known as Unconstrained Model) where a separate sigmoid curve is fitted for each preparation or a Reduced Model (also known as Constrained Model) where parallel sigmoid curves are fitted for selected preparations.
The next dialogue asks for the following convergence and model options.
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 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.
Transform Response: It is possible to transform the response variable by one of e (natural), 10 or 2based logarithms or leave it untransformed (default).
Transform Dose: It is possible to transform the dose variable by one of natural (default), 10 or 2based logarithms or leave it untransformed.
Confidence Level for Regression Coefficients: While the general level of confidence (default 95%) is set on the Variable Selection Dialogue, confidence level for Regression Coefficients and Equivalence Tests can be set here separately. The default is 90%.
In the next dialogue, the model to be estimated is selected.
EP: Parallel sigmoid curves are fitted for all preparations. Validity tests based on weighted sums of squares and fiducial confidence limits of estimated potency are reported according to European Pharmacopoeia (19972017). See sections 10.3.2.5. Potency and 10.4.2.1. EP (European Pharmacopoeia).
Full Model USP: This is also known as the unrestricted, unconstrained or nonparallel model. A separate sigmoid curve is fitted for each preparation. See section 10.4.2.2. Full Model USP.
Reduced Model USP: Also known as the restricted, constrained or parallel model. Parallel sigmoid curves are fitted for all selected preparations.
If Reduced Model USP is selected, then a further dialogue will allow you to select the test preparations to include in the model. Goodness of Fit and Equivalence Tests options can be used to determine which preparations should be included.
USP Summary: UNISTAT implements Full and Reduced models as two separate procedures since each has a large number of diagnostic and equivalence tests. As of this version, we add a USP Summary option which combines the basic output from the two models in a single report. See section 10.4.2.4. USP Summary.
10.4.2. 4PL Output Options
The 4parameter logistic function is given as:
where:
A is upper asymptote,
D is lower asymptote,
B is Hill slope,
C is ED50 and
x is the random variable dose.
In the output, these parameters will be labelled by their above string literals. If this is not the terminology you are used to, you can change these by entering the following line:
4plLabels=i
C:\ProgramData\Unistat\Unistat10\Unistat10.ini
The parameter labels used will be:
for i = 0: Upper asymptote, Lower asymptote, Hill slope, ED50 (default),
for i = 1: A, D, B, EC50,
for i = 2: Top, Bottom, Hill, EC50.
10.4.2.1. EP (European Pharmacopoeia)
Parallel sigmoid curves are fitted for all preparations. European Pharmacopoeia approach is similar to Reduced Model USP in many respects, with two significant differences;
 EP reports fiducial limits of potency whereas USP reports confidence limits based on the standard error and tinterval and
 EP tests validity of assay using weighted sums of squares and chisquare tests whereas USP recommends either equivalence tests or Ftests based on the difference in residual sums of squares between the Full and Reduced models.
FourParameter Logistic Model according to European Pharmacopoeia has the following output options.
Validity of Data: A wide range of statistics on treatment groups are reported, including detection of outliers. This menu item is identical to the same one available under Parallel Line Method. For details see section 10.1.2.1. Validity of Data.
Regression Coefficients: The tstatistic is defined as:
and has a tdistribution with:
df = N – (3 + k)
degrees of freedom and N is the total number of observations used in the analysis, including all preparations.
Confidence intervals for regression coefficients are computed as:
where each coefficient’s standard error σ_{j} is the square root of the diagonal element of the covariance matrix:
where X is the Jacobian matrix. See Seber, G A. F. and Wild, C. J 2003.
Correlation Matrix of Regression Coefficients: This is a symmetric matrix with unity diagonal elements. It gives the correlation between regression coefficients and is obtained by dividing each element of (X’X)^{1} matrix by the square root of the diagonal elements corresponding to its row and column.
Case (Diagnostic) Statistics:
Prediction (interpolation) and outlier detection: See sections 10.0.6.2. ModelBased Outlier Detection and 10.0.7. Prediction, Interpolation, Extrapolation.
Confidence and Prediction Intervals: Approximate standard errors are calculated using the delta method (Van Der Vaart, 1998). Defining the column vector of partial derivatives with respect to each parameter as:
for each given level of dose calculate:
The standard error for actual Y (confidence) for that dose level is:
and the standard error for mean of Y (prediction) is:
The 95% intervals are then calculated as:
Validity of Assay: The following tests for model fit (or lack of fit) are provided.
Weighted ANOVA: Sum of squares are weighted as defined in Finney, D. J. (1978) p. 372.
Nonlinearity test:
Nonparallelism test:
with (m – 1) degrees of freedom.
Total sum of squares:
with N – 1 degrees of freedom.
Residual sum of squares:
where ybar is the mean response for each preparation and yhat is the overall mean of all responses.
Chisquare statistics are obtained by dividing SSQs with Rssqw / DoF(Rssqw).
Weighted Rsquared:
ANOVA of Regression: For each preparation the total sum of squares is:
with N – 1 degrees of freedom and the residual sum of squares is:
The unweighted Rsquared value is calculated as
Example
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Pass/Fail 
Preparations 
0.001 
0.530 
1 
0.4667 
Pass 
Regression 
9.432 
6600.317 
1 
0.0000 
Pass 
Nonparallelism 
0.000 
0.046 
1 
0.8306 
Pass 
Nonlinearity 
0.013 
8.894 
16 
0.9177 
Pass 
Standard S Nonlinearity 
0.008 
5.355 
8 
0.7191 
Pass 
Preparation T Nonlinearity 
0.005 
3.539 
8 
0.8961 
Pass 
Treatments 
9.445 
6609.786 
19 
0.0000 
Pass 
Residual 
0.029 

20 


Total 
9.474 

39 


Rsquared 
0.996 



Pass 
ANOVA of Regression
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Pass/Fail 
Regression 
42.193 
4 
10.548 
8927.664 
0.0000 
Pass 
Error 
0.041 
35 
0.001 



Total 
42.235 
39 
1.083 



Rsquared 
0.999 




Pass 
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.
DoseResponse Plot: A doseresponse curve will be drawn for all preparations. Selecting Edit → DoseResponse Plot… (or doubleclicking on the graph area) will pop a dialogue where what is displayed on the plot can be controlled.
For each curve it is possible to draw lines for ED50, asymptotes, and their confidence limits with or without values. By default, confidence limits for actual Y values (i.e. the confidence interval) are drawn for each curve. If you wish to display the prediction interval instead, enter the following line:
4plPredictionInterval=1
C:\ProgramData\Unistat\Unistat10\Unistat10.ini
10.4.2.2. Full Model USP
A separate sigmoid curve is fitted for each preparation. Therefore, this model has 4k parameters, where k is the number of preparations including the standard (or reference). This is also known as the unrestricted, unconstrained or nonparallel model.
Validity of Data: A wide range of statistics on treatment groups are reported, including detection of outliers. This menu item is identical to the same one available under Parallel Line Method. For details see section 10.1.2.1. Validity of Data.
Regression Coefficients: The tstatistic is defined as:
and has a tdistribution with:
df = N – (4 * k)
degrees of freedom and N is the total number of observations used in the analysis, including all preparations. When a plate factor is selected, a Satterthwaite approximation of degrees of freedom is used as recommended by USP <1034>.
Confidence intervals for regression coefficients are computed as:
where each coefficient’s standard error σ_{j} is the square root of the diagonal element of the covariance matrix:
where X is the Jacobian matrix. See Seber, G A. F. and Wild, C. J 2003.
Correlation Matrix of Regression Coefficients: This is a symmetric matrix with unity diagonal elements. It gives the correlation between regression coefficients and is obtained by dividing each element of (X’X)^{1} matrix by the square root of the diagonal elements corresponding to its row and column.
Case (Diagnostic) Statistics:
Prediction (interpolation) and outlier detection: See sections 10.0.6.2. ModelBased Outlier Detection and 10.0.7. Prediction, Interpolation, Extrapolation.
Confidence and Prediction Intervals: Approximate standard errors are calculated using the delta method (Van Der Vaart, 1998). Defining the column vector of partial derivatives with respect to each parameter as:
for each given level of dose calculate:
The standard error for actual Y (confidence) for that dose level is:
and the standard error for mean of Y (prediction) is:
The 95% intervals are then calculated as:
Effective Dose: ED50 is readily estimated as a model parameter together with its standard error and 90% confidence limits. If you need to obtain other effective dose values for the currently estimated model (i.e. the socalled ED_{anything}), you can enter or edit the ED values from this dialogue, which pops up when you click the [Opt] button situated to the left of the Effective Dose option.
For a given percentage point P% the effective dose is calculated as:
Approximate standard errors are calculated using the delta method (Van Der Vaart, 1998). Defining the column vector of partial derivatives with respect to each parameter as:
for each given level of effective dose calculate:
The standard error for that effective dose is:
and the 90% intervals are then calculated as:
Goodness of Fit tests: The following measures of model fit (or lack of fit) are provided. The first option Validity of Assay is not available for the Full Model.
ANOVA of Regression: For each preparation the total sum of squares is:
with N – 1 degrees of freedom and the residual sum of squares is:
with 3 degrees of freedom (i.e. four parameters less one). Here ybar is the mean response for each preparation and yhat is the overall mean of all responses.
The Rsquared value is calculated as:
Measures of Variability: Let σ^{2} be the total variance of residuals. Then:
where the degrees of freedom is the same as in the Regression Coefficients output.
Upper limit of geometric standard deviation (GSD) is:
upper limit of the percent geometric coefficient of variation (%GCV) is:
and upper limit of the percent coefficient of variation (%CV) is calculated as:
If a [Plate] column is selected, the total variance is broken into within and between plate components and the onesided upper 95% limits are also computed for within plate standard deviation.
Equivalence Tests: These tests are used to decide which test preparations are to be included in the Reduced Model as recommended by USP <1034>.
Mean Difference from Standard: For each model parameter other than ED50 (i.e. A, D and B), the difference between the standard and each test preparation and their confidence limits are tabulated. For instance, for the upper asymptote A:
The 90% intervals are then calculated as:
The criteria against which an accept or reject decision is to be made (i.e. the socalled goalposts) are not included here. These criteria must be based on the specific properties of the bioassay.
Difference Plots: Three plots for parameters A, D and B are displayed with error bars. Each plot will have k – 1 points, one for each test preparation.
Outlier Plot: Standardised residuals are plotted against a randomized Xaxis. If a [Plate] factor has been selected, the residuals are marked by their plate membership, otherwise by preparation.
Homogeneity of Variance Plot: Studentised residuals are plotted against estimated (fitted) response values. Residuals are marked by their preparation membership. If the horizontal spread of points indicates a bias, the response variable may need to be transformed to improve the homogeneity of variance.
DoseResponse Plot: A doseresponse curve will be drawn for each preparation. Selecting Edit → DoseResponse Plot… (or doubleclicking on the graph area) will pop a dialogue where what is displayed on the plot can be controlled.
For each curve it is possible to draw lines for ED50, asymptotes, and their confidence limits with or without values By default, confidence limits for actual Y values (i.e. the confidence interval) are drawn for each curve. If you wish to display the prediction interval instead, enter the following line:
4plPredictionInterval=1
C:\ProgramData\Unistat\Unistat10\Unistat10.ini
Here are a few examples of what can be achieved:
10.4.2.3. Reduced Model USP
This is also known as the restricted, constrained or parallel model. USP <1034> recommends that the best subset of preparations to be included in Reduced Model should be selected using Equivalence Tests as described in Full Model output above.
This model assumes that all preparations have the same A, D and B coefficients and only differ in parameter C (i.e. ED50). Therefore, the Reduced Model has 3 + k’ parameters to estimate; A, D, B, C_{j} j = 1, …, k’, where k’ is the number of preparations selected, including the standard.
Validity of Data: A wide range of statistics on treatment groups are reported, including detection of outliers. This menu item is identical to the same one available under Parallel Line Method. For details see section 10.1.2.1. Validity of Data.
Regression Coefficients: The tstatistic is defined as in Full Model, except for the degrees of freedom, which is for Reduced Model:
df = N’ – (3 + k)
where N’ is the total number of observations used in Reduced Model.
Correlation Matrix of Regression Coefficients: Correlations between all 3 + k’ parameters are displayed.
Case (Diagnostic) Statistics: These are as described in Full Model.
Effective Dose: You can obtain effective dose values other than 50% and their confidence limits as described in Full Model.
Goodness of Fit tests: The first option Validity of Assay is based on the description given by Gottschalk, P. G. and Dunn, J. R. (2005). The remaining two options, ANOVA of Regression and Measures of Variability are as in Full Model.
Validity of Assay: This option is only available for USP Reduced Model and parallelism and linearity tests are available only when it is run immediately after a Full Model. This is because the parallelism and linearity tests here depend on the difference in residual sum of squares for Full and Reduced models. The USP Summary option performs the two models consecutively and therefore parallelism and linearity tests are always available under this output option. See section 10.4.2.4. USP Summary.
The sum of squares for the nonparallelism test is the difference between residual SSQ of the Reduced model minus sum of all residual SSQs of the Full model:
with:
df = (4 * k’) – (3 + k’)
degrees of freedom, where k’ is the number of preparations selected in Reduced Model.
The nonlinearity test is defined as the difference between weighted nonlinearity SSQ and sum of all residual SSQs of the Full model. For details see Gottschalk, P. G. and Dunn, J. R. (2005).
Example
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Pass/Fail 
Preparations 
0.231 
1 
0.231 
161.363 
0.0000 
Pass 
Regression 
41.963 
3 
13.988 
9788.536 
0.0000 
Pass 
Nonparallelism 
0.002 
3 
0.001 
0.489 
0.6937 
Pass 
Nonlinearity 
0.011 
12 
0.001 
0.619 
0.8022 
Pass 
Treatments 
42.206 
19 
2.221 
1554.519 
0.0000 
Pass 
Residual 
0.029 
20 
0.001 



Total 
42.235 
39 
1.083 



Rsquared 
0.999 




Pass 
Potency: In transformed (logged) scale, the relative potency for test preparation j is found as:
and its confidence limits as:
where the variance of relative potency for test preparation j is:
The relative potency and its confidence limits are then transferred (antilogged) back to the original scale.
Weights (which are needed when various potencies are to be combined) are computed as:
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.
DoseResponse Plot: This has the same options as the doseresponse plot described for the Full Model in the previous section.
10.4.2.4. USP Summary
As we have seen in previous sections, USP recommends equivalence tests in order to assess validity of assays which require running Full and Reduced models consecutively. Classic parallelism and linearity tests are also based on the difference in residual sums of squares between the Full and Reduced models. Under this menu item we provide a procedure combining the two models in a single run, generating a combined report. Details are as in sections 10.4.2.2. Full Model USP and 10.4.2.3. Reduced Model USP.
Note that this dialogue does not have an [Opt] button for DoseResponse Plot option. This is because the output has two plots for Full and Reduced models. If you wish to edit these grapsh, you will need to run Full and Reduced models separately.
10.4.3. 4PL Examples
Example 1
Data is given in Table 5.4.1.I. on p. 4373 of European Pharmacopoeia (9^{th} Edition).
Open BIOPHARMA9 and select Bioassay → FourParameter Logistic Model. From the Variable Selection Dialogue select columns C44, C45, L46 and L47 respectively as [Data], [Dose], [Preparation] and [Dilution]. Click [Next] and leave all entries on the Convergence Criteria dialogue unchanged. On the next two dialogues select EP: Fit parallel sigmoids and check Preparation T. On Output Options Dialogue check only the Regression Results and Potency options.
Click [Finish] to obtain the following output. Note that the potency reported here is relative potency.
FourParameter Logistic Model
Valid Number of Cases: 40, 0 Omitted
Model selected: EP
Regression Results

Coefficient 
Standard Error 
tStatistic 
2Tail Probability 
Lower 90% 
Upper 90% 
Upper asymptote 
3.1960 
0.0327 
97.7429 
0.0000 
3.1407 
3.2512 
Lower asymptote 
0.1455 
0.0143 
10.2042 
0.0000 
0.1214 
0.1695 
Hill slope 
1.1244 
0.0294 
38.2340 
0.0000 
1.0747 
1.1741 
Standard S ED50 
0.1348 
0.0037 
36.4278 
0.0000 
0.1285 
0.1410 
Preparation T ED50 
0.0924 
0.0026 
35.7697 
0.0000 
0.0880 
0.0967 
Total Residual Variance = 
0.0012 
Degrees of Freedom = 
35 
Weighted ANOVA

Sum of Squares 
ChiSquare 
DoF 
Probability 
Preparations 
0.001 
0.530 
1 
0.4667 
Regression 
9.432 
6600.317 
1 
0.0000 
Nonparallelism 
0.000 
0.046 
1 
0.8306 
Nonlinearity 
0.013 
8.894 
16 
0.9177 
Standard S Nonlinearity 
0.008 
5.355 
8 
0.7191 
Preparation T Nonlinearity 
0.005 
3.539 
8 
0.8961 
Treatments 
9.445 
6609.786 
19 
0.0000 
Residual 
0.029 

20 

Total 
9.474 

39 

Rsquared 
0.996 



ANOVA of Regression
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Regression 
42.193 
4 
10.548 
8927.664 
0.0000 
Error 
0.041 
35 
0.001 


Total 
42.235 
39 
1.083 


Rsquared 
0.999 




Potency
Assigned potency of Standard S: 0.4 IU/ml

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
0.5835 
0.5568 
0.6116 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
145.89% 
139.20% 
152.90% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
95.42% 
104.81% 
Example 2
Open the file 4PL and select Bioassay → FourParameter Logistic Model. From the Variable Selection Dialogue select columns C1 to C4 respectively as [Data], [Dose], [Preparation] and [Plate]. Click [Next], enter 1 for Transform Response, 3 for Transform Dose and leave other entries unchanged. On the next dialogue select Full Model: Fit a separate sigmoid on each preparation. Leave Output Options Dialogue unchanged and then click [Finish].
FourParameter Logistic Model
Valid Number of Cases: 384, 0 Omitted
Model selected: Full Model USP
Full Model USP
Regression Results

Coefficient 
Standard Error 
tStatistic 
2Tail Prob 
Lower 90% 
Upper 90% 
Standard A 
14851.5573 
721.5338 
20.5833 
0.0000 
13377.0205 
16326.0941 
D 
161.5165 
7.8716 
20.5189 
0.0000 
145.4300 
177.6030 
B 
1.4833 
0.0782 
18.9697 
0.0000 
1.6431 
1.3235 
EC50 
0.0111 
0.0004 
25.8819 
0.0000 
0.0102 
0.0120 
Test A A 
14945.5751 
577.3904 
25.8847 
0.0000 
13765.6117 
16125.5385 
D 
158.2746 
10.9899 
14.4018 
0.0000 
135.8155 
180.7337 
B 
1.5004 
0.0828 
18.1187 
0.0000 
1.6696 
1.3312 
EC50 
0.0316 
0.0013 
24.2711 
0.0000 
0.0289 
0.0343 
Test B A 
14027.4183 
838.3262 
16.7326 
0.0000 
12314.2027 
15740.6339 
D 
262.6613 
10.8566 
24.1937 
0.0000 
240.4746 
284.8480 
B 
1.4996 
0.0917 
16.3570 
0.0000 
1.6869 
1.3122 
EC50 
0.0056 
0.0003 
22.2848 
0.0000 
0.0051 
0.0061 
Test C A 
14995.9650 
578.6668 
25.9147 
0.0000 
13813.3931 
16178.5369 
D 
159.0249 
7.1096 
22.3675 
0.0000 
144.4955 
173.5542 
B 
2.0018 
0.1060 
18.8896 
0.0000 
2.2184 
1.7852 
EC50 
0.0156 
0.0005 
33.5103 
0.0000 
0.0147 
0.0166 
Total Residual Variance = 
0.0451 
Degrees of Freedom = 
368 
Satterthwaite DoF = 
4.7019 
Correlation Matrix of Regression Coefficients

A 
D 
B 
EC50 
Standard A 
1.0000 
0.2339 
0.5689 
0.4288 
D 
0.2339 
1.0000 
0.5698 
0.4309 
B 
0.5689 
0.5698 
1.0000 
0.0015 
EC50 
0.4288 
0.4309 
0.0015 
1.0000 
Test A A 
1.0000 
0.2417 
0.4974 
0.2356 
D 
0.2417 
1.0000 
0.6673 
0.6290 
B 
0.4974 
0.6673 
1.0000 
0.2818 
EC50 
0.2356 
0.6290 
0.2818 
1.0000 
Test B A 
1.0000 
0.2347 
0.6275 
0.5546 
D 
0.2347 
1.0000 
0.5172 
0.3068 
B 
0.6275 
0.5172 
1.0000 
0.1771 
EC50 
0.5546 
0.3068 
0.1771 
1.0000 
Test C A 
1.0000 
0.1421 
0.4344 
0.3558 
D 
0.1421 
1.0000 
0.4831 
0.4325 
B 
0.4344 
0.4831 
1.0000 
0.0532 
EC50 
0.3558 
0.4325 
0.0532 
1.0000 
Case (Diagnostic) Statistics

LogE (Response) 
Dose 
Preparation 
Plate 
Estimated Response 
95% lb Actual Y 
95% lb Mean of Y 
** 1 
4.5962 
0.5000 
Standard 
Plate 1 
5.1005 
0.5007 
4.1345 
2 
4.9461 
0.5000 
Standard 
Plate 1 
5.1005 
0.5007 
4.1345 
3 
5.2660 
0.5000 
Standard 
Plate 2 
5.1005 
0.5007 
4.1345 
4 
5.1606 
0.5000 
Standard 
Plate 2 
5.1005 
0.5007 
4.1345 
5 
5.2318 
0.5000 
Standard 
Plate 3 
5.1005 
0.5007 
4.1345 
6 
5.0652 
0.5000 
Standard 
Plate 3 
5.1005 
0.5007 
4.1345 
7 
5.2122 
0.5000 
Standard 
Plate 4 
5.1005 
0.5007 
4.1345 
8 
5.3453 
0.5000 
Standard 
Plate 4 
5.1005 
0.5007 
4.1345 
9 
4.7638 
0.2500 
Standard 
Plate 1 
5.1288 
0.5449 
4.2419 
… 
… 
… 
… 
… 
… 
… 
… 

95% ub Mean of Y 
95% ub Actual Y 
Residuals 
Standardised Residuals 
Studentised Residuals 
** 1 
6.0666 
9.7004 
0.5043 
2.3756 
0.5164 
2 
6.0666 
9.7004 
0.1544 
0.7273 
0.1581 
3 
6.0666 
9.7004 
0.1655 
0.7794 
0.1694 
4 
6.0666 
9.7004 
0.0601 
0.2829 
0.0615 
5 
6.0666 
9.7004 
0.1312 
0.6182 
0.1344 
6 
6.0666 
9.7004 
0.0353 
0.1662 
0.0361 
7 
6.0666 
9.7004 
0.1117 
0.5262 
0.1144 
8 
6.0666 
9.7004 
0.2447 
1.1529 
0.2506 
9 
6.0158 
9.7127 
0.3650 
1.7196 
0.3723 
… 
… 
… 
… 
… 
… 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Effective Dose

Dose 
Standard Error 
Lower 90% 
Upper 90% 
Standard ED10 
0.0025 
0.0002 
0.0021 
0.0030 
ED20 
0.0044 
0.0003 
0.0038 
0.0049 
ED50 
0.0111 
0.0004 
0.0102 
0.0120 
ED80 
0.0283 
0.0018 
0.0247 
0.0319 
ED90 
0.0489 
0.0043 
0.0402 
0.0576 
Test A ED10 
0.0073 
0.0006 
0.0061 
0.0085 
ED20 
0.0125 
0.0007 
0.0111 
0.0140 
ED50 
0.0316 
0.0013 
0.0289 
0.0343 
ED80 
0.0796 
0.0059 
0.0676 
0.0916 
ED90 
0.1367 
0.0137 
0.1086 
0.1647 
Test B ED10 
0.0013 
0.0001 
0.0010 
0.0016 
ED20 
0.0022 
0.0002 
0.0019 
0.0026 
ED50 
0.0056 
0.0003 
0.0051 
0.0061 
ED80 
0.0141 
0.0009 
0.0122 
0.0160 
ED90 
0.0242 
0.0022 
0.0196 
0.0287 
Test C ED10 
0.0052 
0.0003 
0.0045 
0.0059 
ED20 
0.0078 
0.0004 
0.0071 
0.0085 
ED50 
0.0156 
0.0005 
0.0147 
0.0166 
ED80 
0.0312 
0.0015 
0.0281 
0.0343 
ED90 
0.0468 
0.0031 
0.0404 
0.0531 
ANOVA of Regression
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Standard Regression 
332.193 
3 
110.731 
2665.488 
0.0000 
Error 
3.822 
92 
0.042 


Total 
336.015 
95 
3.537 


Rsquared 
0.989 




Test A Regression 
307.441 
3 
102.480 
3200.335 
0.0000 
Error 
2.946 
92 
0.032 


Total 
310.387 
95 
3.267 


Rsquared 
0.991 




Test B Regression 
248.561 
3 
82.854 
2184.435 
0.0000 
Error 
3.489 
92 
0.038 


Total 
252.050 
95 
2.653 


Rsquared 
0.986 




Test C Regression 
373.567 
3 
124.522 
3189.062 
0.0000 
Error 
3.592 
92 
0.039 


Total 
377.159 
95 
3.970 


Rsquared 
0.990 




Measures of Variability

Standard Deviation 
Upper 95% Standard Deviation 
Upper 95% GSD 
Upper 95% %GCV 
Upper 95% %CV 
Within Plate 
0.0958 
0.1019 
1.1072 
10.7249 
10.2144 
Between Plate 
0.1894 




Total 
0.2123 
0.5621 
1.7543 
75.4291 
60.9521 
Mean Difference from Standard

Mean Difference 
Lower 90% 
Upper 90% 
A Test A 
94.0178 
1794.5193 
1982.5549 
Test B 
824.1390 
3084.5298 
1436.2518 
Test C 
144.4077 
1745.7603 
2034.5757 
D Test A 
3.2419 
30.8677 
24.3840 
Test B 
101.1448 
73.7400 
128.5496 
Test C 
2.4916 
24.1683 
19.1850 
B Test A 
0.0171 
0.2157 
0.2498 
Test B 
0.0163 
0.2300 
0.2625 
Test C 
0.5185 
0.2493 
0.7876 
Click the Last Dialogue button to obtain the Output Options Dialogue and click [Back]. Select Reduced Model: Fit parallel sigmoids on selected preparations. On the next dialogue check only Test A, as it is the only test preparation meeting the selection criteria. Leave output options unchanged and then click [Finish].
FourParameter Logistic Model
Valid Number of Cases: 192, 192 Omitted
Model selected: Reduced Model USP
Regression Results

Coefficient 
Standard Error 
tStatistic 
2Tail Prob 
Lower 90% 
Upper 90% 
A 
14901.5591 
444.0137 
33.5610 
0.0000 
13997.7665 
15805.3517 
D 
160.4999 
6.2979 
25.4846 
0.0000 
147.6804 
173.3193 
B 
1.4941 
0.0551 
27.1126 
0.0000 
1.6063 
1.3820 
Standard EC50 
0.0111 
0.0004 
29.9367 
0.0000 
0.0104 
0.0119 
Test A EC50 
0.0314 
0.0010 
30.0153 
0.0000 
0.0293 
0.0336 
Total Residual Variance = 
0.0440 
Degrees of Freedom = 
187 
Satterthwaite DoF = 
4.7821 
Correlation Matrix of Regression Coefficients

A 
D 
B 
Standard EC50 
Test A EC50 
A 
1.0000 
0.2272 
0.5248 
0.2760 
0.2792 
D 
0.2272 
1.0000 
0.6047 
0.4291 
0.4232 
B 
0.5248 
0.6047 
1.0000 
0.1069 
0.1030 
Standard EC50 
0.2760 
0.4291 
0.1069 
1.0000 
0.3294 
Test A EC50 
0.2792 
0.4232 
0.1030 
0.3294 
1.0000 
Case (Diagnostic) Statistics

LogE(Response) 
Dose 
Preparation 
Plate 
Estimated Response 
95% lb Actual Y 
95% lb Mean of Y 
** 1 
4.5962 
0.5000 
Standard 
Plate 1 
5.0936 
1.9426 
4.5478 
2 
4.9461 
0.5000 
Standard 
Plate 1 
5.0936 
1.9426 
4.5478 
3 
5.2660 
0.5000 
Standard 
Plate 2 
5.0936 
1.9426 
4.5478 
4 
5.1606 
0.5000 
Standard 
Plate 2 
5.0936 
1.9426 
4.5478 
5 
5.2318 
0.5000 
Standard 
Plate 3 
5.0936 
1.9426 
4.5478 
… 
… 
… 
… 
… 


… 

95% ub Mean of Y 
95% ub Actual Y 
Residuals 
Standardised Residuals 
Studentised Residuals 
** 1 
5.6395 
8.2446 
0.4974 
2.3705 
0.5053 
2 
5.6395 
8.2446 
0.1475 
0.7030 
0.1498 
3 
5.6395 
8.2446 
0.1723 
0.8214 
0.1751 
4 
5.6395 
8.2446 
0.0669 
0.3190 
0.0680 
5 
5.6395 
8.2446 
0.1381 
0.6583 
0.1403 
… 
… 
… 
… 
… 
… 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Effective Dose

Dose 
Standard Error 
Lower 90% 
Upper 90% 
Standard ED10 
0.0026 
0.0002 
0.0022 
0.0029 
ED20 
0.0044 
0.0002 
0.0040 
0.0048 
ED50 
0.0111 
0.0004 
0.0104 
0.0119 
ED80 
0.0281 
0.0014 
0.0253 
0.0310 
ED90 
0.0484 
0.0032 
0.0419 
0.0550 
Test A ED10 
0.0072 
0.0004 
0.0063 
0.0081 
ED20 
0.0124 
0.0006 
0.0113 
0.0136 
ED50 
0.0314 
0.0010 
0.0293 
0.0336 
ED80 
0.0795 
0.0040 
0.0714 
0.0877 
ED90 
0.1368 
0.0091 
0.1183 
0.1554 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
25.144 
3 
8.381 
245.449 
0.0000 
Regression 
1266.637 
3 
422.212 
12364.460 
0.0000 
Nonparallelism 
0.013 
3 
0.004 
0.124 
0.9460 
Nonlinearity 
0.000 
38 
0.000 
0.000 
1.0000 
Treatments 
1287.181 
47 
27.387 
802.021 
0.0000 
Plate 
5.019 
3 
1.673 
48.993 
0.0000 
Residual 
11.371 
333 
0.034 


Total 
1300.756 
383 
3.396 


Rsquared 
0.991 




ANOVA of Regression
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Regression 
654.740 
4 
163.685 
4514.212 
0.0000 
Error 
6.781 
187 
0.036 


Total 
661.520 
191 
3.463 


Rsquared 
0.990 




Measures of Variability

Standard Deviation 
Upper 95% Standard Deviation 
Upper 95% GSD 
Upper 95% %GCV 
Upper 95% %CV 
Within Plate 
0.0968 
0.1058 
1.1116 
11.1636 
10.6130 
Between Plate 
0.1862 




Total 
0.2098 
0.5535 
1.7393 
73.9327 
59.8725 
Potency

Estimated Potency 
Lower 95% 
Upper 95% 
Test A 
2.8258 
2.5551 
3.1251 

Relative Potency 
Lower 95% 
Upper 95% 
Test A 
282.58% 
255.51% 
312.51% 

Percent CI 
Lower 95% 
Upper 95% 
Test A 
100.00% 
90.42% 
110.59% 