10.1. Parallel Line Method
Balanced, symmetric or unbalanced assays can be analysed. The analysis is based on a regression of the response variable against the natural logarithm of the dose variable. A separate line is fitted on each preparation, subject to a constraint that they should be parallel. An assay is said to be balanced when:
1) there is an equal number of cases in each treatment group,
2) there is an equal number of dose groups for each preparation and
3) successive dose levels are the same for all preparations.
An assay fulfilling the first two conditions but having different dose levels for different preparations (yet having the same ratio of successive dose levels) will be called symmetric. Assays not fulfilling one or more of these conditions will be called asymmetric or unbalanced.
For validity tests, the following Analysis of Variance (ANOVA) options are available:
1) Completely randomised design
2) Randomised block design
3) Latin square design
4) Twin and triple crossover designs
The unbalanced assays can only be analysed using the Completely Randomised Design option. All other options require symmetric or balanced assays. In most cases, the program will detect whether an assay is unbalanced, symmetric or balanced and apply the relevant algorithm automatically.
Note that the example data sets given in European Pharmacopoeia (19972017) Parallel Line Method are all balanced.
10.1.1. Parallel Line Variable Selection
Once the data is arranged as described above (see section 10.0.1. Data Preparation), select Bioassay → Parallel Line Method from UNISTAT menus. A Variable Selection Dialogue will pop up.
Data columns available for selection are listed on the left. Variables are referred to by their column numbers, which are prefixed by a single letter representing the type of data. For instance, in the above example C1, C2 and C4 are numeric columns, whereas L3 means that column three contains Long Strings. Columns containing Short Strings (up to 8 characters) are prefixed by (S). Other data types that will probably not be used in bioassays are date (D) and time (T). If Column Labels have been entered, they will also appear in the list next to the column numbers.
You will need to assign tasks to variables by sending them to boxes on the right. To do this, highlight the variable on the left list and click on the desired task button (i.e. one of the command buttons in the middle of the dialogue). Likewise, you can deselect an already selected variable by highlighting it on the right list first and then clicking its task button.
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. It is compulsory to select at least three variables [Data], [Dose] and [Preparation], which have bold button fonts. There are three optional variables [Plate], [Assay] and [Dilution]. A plate effect, that will be subtracted from the residual sum of squares, can be calculated if required. For the latter two options see sections 10.0.4. Multiple Assays with Combination and 10.0.2. Doses, Dilutions and Potency respectively. Note that if the design is balanced, the plate effect will generate the same results as Randomised Block Design.
The frame Select Data Type (at the top) displays options for the type of Analysis of Variance to be performed. The number of variables to be selected is different for these types of analyses. When the second option Randomised Block Design is selected, four compulsory variables will need to be selected.
The third and fourth options Latin Square Design and Crossover Design require selection of five variables.
When all variables are selected, click the [Next] button to proceed to Output Options Dialogue.
10.1.2. Parallel Line Output Options
Output options that have further options under them (i.e. further dialogues and windows to display) have an [Opt] button placed to the left of their check boxes. When you click [Finish] without clicking on an [Opt] button first, the program will generate output with default selections. If you want to change the defaults, you can click on the [Opt] button to display further dialogues for this particular output option. Then you can either obtain this particular output option on its own by clicking [Finish], or click [Back] to display the Output Options Dialogue again and output all selected options together.
Data
This output option (which is available for all bioassay analysis methods supported here) will enable the user to include a printout of data used in the analysis as part of the output. This will be useful in fulfilling reporting and data integrity requirements.
Data

Response 
Dose 
Preparations 
1 
161 
1 
Standard S 
2 
151 
1 
Preparation T 
3 
162 
1.5 
Preparation T 
4 
194 
2.25 
Standard S 
5 
176 
1.5 
Standard S 
6 
193 
2.25 
Preparation T 
7 
160 
1 
Preparation T 
8 
192 
2.25 
Preparation T 
9 
195 
2.25 
Standard S 
10 
184 
1.5 
Standard S 
11 
181 
1.5 
Preparation T 
12 
166 
1 
Standard S 
13 
178 
1.5 
Preparation T 
14 
150 
1 
Standard S 
15 
174 
1.5 
Standard S 
16 
199 
2.25 
Preparation T 
17 
201 
2.25 
Standard S 
18 
161 
1 
Preparation T 
19 
187 
2.25 
Standard S 
20 
172 
1.5 
Standard S 
21 
161 
1 
Standard S 
22 
160 
1 
Preparation T 
23 
202 
2.25 
Preparation T 
24 
186 
1.5 
Preparation T 
25 
171 
1.5 
Standard S 
26 
170 
1.5 
Preparation T 
27 
193 
2.25 
Preparation T 
28 
163 
1 
Standard S 
29 
154 
1 
Preparation T 
30 
198 
2.25 
Standard S 
31 
194 
2.25 
Preparation T 
32 
192 
2.25 
Standard S 
33 
151 
1 
Preparation T 
34 
171 
1.5 
Preparation T 
35 
151 
1 
Standard S 
36 
182 
1.5 
Standard S 
10.1.2.1. Validity of Data
One of the basic assumptions of Parallel Line Method is that for each treatment group (i.e. a unique dosepreparation combination) observations are normally distributed. Normality, homogeneity of variance and outliers of treatment groups can be detected and tested using the following descriptive statistics, tests and plotting procedures provided under this comprehensive menu option.
Note that under each test type we provide multiple alternatives. There are four normality tests, five homogeneity of variance tests and three outlier tests. In most cases you will need to use only one of these tests from each type. You can simply uncheck the tests you do not need and UNISTAT will remember your choices in future session.
10.1.2.1.1. Summary Statistics
The selected statistics will be displayed for each unique dosepreparation group.
For more information see section 5.1.1. Summary Statistics.
Example
Summary Statistics
Data variable: Response
Subsample selected by: Dose x Preparations
Quantile Method: Simple Average

Valid Cases 
Mean 
Median 
Variance 
1 x Standard S 
6 
158.6667 
161.0000 
43.4667 
1 x Preparation T 
6 
156.1667 
157.0000 
22.1667 
1.5 x Standard S 
6 
176.5000 
175.0000 
28.7000 
1.5 x Preparation T 
6 
174.6667 
174.5000 
75.0667 
2.25 x Standard S 
6 
194.5000 
194.5000 
23.5000 
2.25 x Preparation T 
6 
195.5000 
193.5000 
16.3000 
10.1.2.1.2. Normality Tests
Pass/Fail test results will be displayed by comparing the calculated pvalue to the alpha value provided in the above dialogue. Smaller pvalues indicate nonnormality.
For more information see section 6.3.3. Normality Tests.
Example
ShapiroWilk Normality Test
Alpha = 0.05
Dose x Preparations 
Test Statistic 
Probability 
Pass/Fail 
1 x Standard S 
0.8581 
0.1827 
Pass 
1 x Preparation T 
0.8118 
0.0749 
Pass 
1.5 x Standard S 
0.8965 
0.3536 
Pass 
1.5 x Preparation T 
0.9757 
0.9284 
Pass 
2.25 x Standard S 
0.9879 
0.9835 
Pass 
2.25 x Preparation T 
0.8255 
0.0984 
Pass 
KolmogorovSmirnov Normality Test
Alpha = 0.05
* Lilliefors probability = 0.2 means 0.2 or greater.
Dose x Preparations 
Test Statistic 
* Probability 
Pass/Fail 
1 x Standard S 
0.3050 
0.0851 
Pass 
1 x Preparation T 
0.2922 
0.1177 
Pass 
1.5 x Standard S 
0.2038 
0.2000 
Pass 
1.5 x Preparation T 
0.1639 
0.2000 
Pass 
2.25 x Standard S 
0.1364 
0.2000 
Pass 
2.25 x Preparation T 
0.3115 
0.0703 
Pass 
Cramervon Mises Normality Test
Alpha = 0.05
Dose x Preparations 
Test Statistic 
Probability 
Pass/Fail 
1 x Standard S 
0.0849 
0.1450 
Pass 
1 x Preparation T 
0.0886 
0.1276 
Pass 
1.5 x Standard S 
0.0544 
0.3902 
Pass 
1.5 x Preparation T 
0.0274 
0.8514 
Pass 
2.25 x Standard S 
0.0210 
0.9388 
Pass 
2.25 x Preparation T 
0.1046 
0.0740 
Pass 
AndersonDarling Normality Test
Alpha = 0.05
Dose x Preparations 
Test Statistic 
Probability 
Pass/Fail 
1 x Standard S 
0.4756 
0.1438 
Pass 
1 x Preparation T 
0.5470 
0.0901 
Pass 
1.5 x Standard S 
0.3341 
0.3689 
Pass 
1.5 x Preparation T 
0.1760 
0.8637 
Pass 
2.25 x Standard S 
0.1514 
0.9165 
Pass 
2.25 x Preparation T 
0.5613 
0.0818 
Pass 
10.1.2.1.3. Homogeneity of Variance Tests
Another basic assumption of Parallel Line Method is that variances for different treatment groups are not significantly different from each other.
Earlier versions of UNISTAT featured Bartlett’s chisquare test as recommended by European Pharmacopoeia (19972017), and Hartley’s F test. Here we provide three more homogeneity of variance tests. The computationally demanding Levene’s test is considered to be more powerful than other homogeneity of variance tests.
Test results are displayed as either Pass or Fail, by comparing the calculated pvalue to the alpha value provided in the above dialogue. Smaller pvalues indicate nonhomogeneity.
For a detailed description of these tests see 7.4.2.1. Homogeneity of Variance Test Results.
Example
Homogeneity of Variance Tests
For 6 groups defined by Dose x Preparations.
Alpha = 0.05

Test Statistic 
Probability 
Pass/Fail 
Bartlett’s Chisquare Test 
3.7817 
0.5813 
Pass 
BartlettBox F Test 
0.7637 
0.5760 
Pass 
Cochran’s C (max var / sum var) 
0.3588 
0.2313 
Pass 
Hartley’s F (max var / min var) 
4.6053 
0.0500 
Pass 
Levene’s F Test 
1.5818 
0.1954 
Pass 
Hartley’s F probability = 0.05 means 0.05 or greater.
10.1.2.1.4. Outlier Tests
For more information see section 6.3.4. Outlier Tests.
In addition to the tests provided here outliers can also be detected using standardised residuals. See section 10.0.6. Outlier Detection, Omission and Replacement.
10.1.2.1.5. BoxWhisker, Dot and Bar Plots
This procedure combines boxplot with dot and error bar plots.
Although all three plot types are displayed on the same graph by default, the user can change graph settings to display any combination of the three plot types, as well as changing other characteristics of the graph. To do this click on the [Opt] button situated to the left of this output option and from the Graphics Editor menu select Edit → Width / Notch / Dots.
For more information see section 5.3.1. BoxWhisker, Dot and Bar Plots.
10.1.2.1.6. Normality Plots
By default all treatments are drawn with a line of best fit on a graph with a probit scale Yaxis against the response on a linear Xaxis. If the data lies on a nearstraight line, then it is said to conform to the normal distribution.
All aspects of the graph can be edited an customised by clicking on the [Opt] button situated to the left of this output option. This will display the Graphics Editor. From the menu select Edit → Data Series.
For more information see sections 6.3.3.5. Normality Plots and 5.3.2. Normal Probability Plot.
10.1.2.2. Regression Results
A separate regression line is fitted on each preparation against the natural logarithm of dose. It is also possible to take logs based 10 by entering the following line:
LogBase10=1
C:\ProgramData\Unistat\Unistat10\Unistat10.ini
Example
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Pass/Fail 
Standard S 
248.5800 
113.0060 
1897.7400 
0.9651 
Pass 
Preparation T 
238.5000 
109.5039 
747.8000 
0.9851 
Pass 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Pass/Fail 
Standard S 
247.5150 
111.2549 
2670.7450 
0.9744 
Pass 
Preparation T 
239.5650 




Residual Variance = 
72.1823 
Degrees of Freedom = 
37 
10.1.2.3. Case (Diagnostic) Statistics
Example
Case (Diagnostic) Statistics

Response 
Dose 
Preparations 
Estimated Response 
Residuals 
Standardised Residuals 
1 
161 
1 
Standard S 
157.7639 
3.2361 
0.5684 
2 
151 
1 
Preparation T 
156.6528 
5.6528 
0.9928 
** 3 
162 
1.5 
Preparation T 
175.4444 
13.4444 
2.3612 
4 
194 
2.25 
Standard S 
195.3472 
1.3472 
0.2366 
5 
176 
1.5 
Standard S 
176.5556 
0.5556 
0.0976 
6 
193 
2.25 
Preparation T 
194.2361 
1.2361 
0.2171 
7 
160 
1 
Preparation T 
156.6528 
3.3472 
0.5879 
8 
192 
2.25 
Preparation T 
194.2361 
2.2361 
0.3927 
9 
195 
2.25 
Standard S 
195.3472 
0.3472 
0.0610 
10 
184 
1.5 
Standard S 
176.5556 
7.4444 
1.3075 
… 
… 
… 
… 
… 
… 
… 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Example
In above example case 3 is reported as an outlier at a two standard deviations threshold. Going back to the spreadsheet, deleting the response value of 162 in row 3 and running the analysis again, we obtain a predicted value of 176.2353 for the omitted outlier.
Case (Diagnostic) Statistics

Response 
Dose 
Preparations 
Estimated Response 
Residuals 
Standardised Residuals 
1 
161 
1 
Standard S 
157.7639 
3.2361 
0.6176 
2 
151 
1 
Preparation T 
157.4436 
6.4436 
1.2298 
* 3 
* 
1.5 
Preparation T 
176.2353 
* 
* 
4 
194 
2.25 
Standard S 
195.3472 
1.3472 
0.2571 
5 
176 
1.5 
Standard S 
176.5556 
0.5556 
0.1060 
6 
193 
2.25 
Preparation T 
195.0270 
2.0270 
0.3869 
7 
160 
1 
Preparation T 
157.4436 
2.5564 
0.4879 
8 
192 
2.25 
Preparation T 
195.0270 
3.0270 
0.5777 
9 
195 
2.25 
Standard S 
195.3472 
0.3472 
0.0663 
10 
184 
1.5 
Standard S 
176.5556 
7.4444 
1.4208 
… 
… 
… 
… 
… 
… 
… 
Cases marked by ‘*’ are predicted.
10.1.2.4. Comparison of Slopes
If an assay with two or more test preparations is found to depart from parallelism significantly, then we ask the question which test preparation’s slope differs from the slope of the standard preparation. A Dunnett’s multiple comparison test is performed to answer this question.
Here we perform a Dunnett Test, Comparisons against a Control Group option, using the slope values from Separate Regression. For details see 7.4.3.8. Dunnett test.
The slopes test described in European Pharmacopoeia (19972017) employs a slightly different algorithm, which is only valid for 2, 3 or 4dose assays. It is based on linear contrasts as a proxy for actual slopes. The two approaches are identical and produce the same probability values. UNISTAT does not have a limitation on the number of distinct dose values.
L_{1} is the linear contrast for the standard preparation,
L_{i} is the linear contrast for the i^{th} test preparation and
s^{2} is the residual mean square value from the ANOVA table (i.e. sum of squares for the overall residual term divided by its degrees of freedom)
The twotailed probability for the test statistic is generated using an algorithm developed by Charles Dunnett (Algorithm AS 251) for α significance level, (h – 1) number of groups. The degrees of freedom is equal to that of the overall residual term of the ANOVA table.
Example
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Probability 
Preparation T – Standard S 
4.3160 
5.9453 
0.7260 
2.0423 
0.4735 
Comparison 
Lower 95% 
Upper 95% 
Pass/Fail 
Preparation T – Standard S 
7.8259 
16.4580 
Pass 
10.1.2.5. Validity of Assay
This output option displays an Analysis of Variance (ANOVA) table, which is used to test the Validity of Assay. The residual sum of squares and its degrees of freedom are used in estimating the confidence limits for the Potency (see section 10.1.2.6. Potency). The three basic tests performed are (i) significance of regression, (ii) parallelism and (iii) linearity. The table may have different entries in its rows depending on the number of doses and / or the ANOVA model employed.
The notation below is for balanced designs as given by European Pharmacopoeia (19972017). For unbalanced designs, the only difference is that the sums are taken up to the maximum number of observations in each treatment group, which can be different. See 10.2. Slope Ratio Method, section 10.2.2.4. Validity of Assay for a general unbalanced formulation.
Let us first define the constant term:
which is the sum total of all cases divided by the total number of treatment groups, the Treatments term:
which is the sum of all squared treatment totals minus the constant term, and the Total term:
which is the sum of all squared cases minus the constant term. The rest of table entries are defined as in the following table. Note that instead of reporting linearity and parallelism, we always report nonlinearity and nonparallelism, so that when an Ftest gives a significant result (i.e. a low Pvalue), it means that a data set is not linear or two or more data sets are not parallel. Where possible, we also report nonlinearity tests for individual preparations, as well as the overall nonlinearity for all preparations.
If you do not wish to display the individual nonlinearity values for preparations enter the following line:
NonLinPrepDisp=0
C:\ProgramData\Unistat\Unistat10\Unistat10.ini
QuadraticDisp=0

Degrees of Freedom 
2dose 
3dose 
4dose 
Preparations 
h – 1 



Linear Regression 
1 



Nonparallelism 
h – 1 



Nonlinearity 
h for 3dose 2h for 4dose 



Quadratic Regression 
1 



Difference of Quadratics 
h – 1 



Residual 
h 



Treatments 
k – 1 
M 
M 
M 
Residual 

R 
R 
R 
Total 
nk – 1 
T 
T 
T 
The following relationships should always be true:
1) Degrees of freedom and sum of squares for the first four rows (i.e. Preparations, Linear Regression, Nonparallelism and Nonlinearity) should always add up to Treatments,
2) Quadratic Regression, Difference of Quadratics and their Residuals should always add up to Nonlinearity,
3) Treatments (including plate, row and column effects) and their Residuals should always add up to Total.
Example
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Pass/Fail 
Preparations 
11.111 
1 
11.111 
0.535 
0.4730 
**Fail** 
Linear Regression 
8475.042 
1 
8475.042 
408.108 
0.0000 
Pass 
Nonparallelism 
18.375 
1 
18.375 
0.885 
0.3581 
Pass 
Nonlinearity 
5.472 
2 
2.736 
0.132 
0.8773 
Pass 
Standard S Nonlinearity 
0.028 
1 
0.028 
0.001 
0.9712 
Pass 
Preparation T Nonlinearity 
5.444 
1 
5.444 
0.262 
0.6142 
Pass 
Quadratic Regression 
3.125 
1 
3.125 
0.150 
0.7022 

Quadratic Difference 
2.347 
1 
2.347 
0.113 
0.7402 

Treatments 
8510.000 
5 
1702.000 
81.958 
0.0000 
Pass 
Blocks(Rows) 
412.000 
5 
82.400 
3.968 
0.0116 

Blocks(Columns) 
218.667 
5 
43.733 
2.106 
0.1069 

Residual 
415.333 
20 
20.767 



Total 
9556.000 
35 
273.029 



10.1.2.6. Potency
By default, the program estimates the potency ratio and its confidence limits employing the generalised algorithm given in Finney (1978), which works with unbalanced, symmetric and balanced designs. European Pharmacopoeia (19972017) gives a more restrictive algorithm for balanced assays only. The logarithm of potency ratio is estimated for each test preparation, using the Common Regression slope b.
and:
where:
are the preparation means and A_{i} is the assumed potency of each test preparation. The estimated potency is the antilog of M, Exp(M).
The method of estimating the confidence interval for potency is based on Fieller’s Theorem (see Finney 1978, p. 80). Let us first define the correction factor g as:
The log of confidence limits for the potency ratio of each test preparation is defined as:
where the variance of M_{i} is:
Weights are computed after the estimated potency and its confidence interval are found:
and % Precision is:
According to European Pharmacopoeia (1997) the common slope b is calculated as:
Where:
Z = Log(dose_{i+1}) – Log(dose_{i}), i = 2, … , d
is the log of successive dose ratios. Also define a correction factor:
The log of the corrected potency estimate and its confidence intervals are computed as:
where:
The only difference from the default output here is reporting of C and H constants for validation purposes, where C = 1 / (1 – g).
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.
Example
Potency
Latin Squares Design

Estimated Potency 
Lower 95% 
Upper 95% 
DoF 
% Precision 
Preparation T 
0.9763 
0.9112 
1.0456 
20 
93.33% 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
97.63% 
91.12% 
104.56% 

Percent CI 
Lower 95% 
Upper 95% 
Pass/Fail 
Preparation T 
100.00% 
93.33% 
107.09% 
Pass 
PotencyPctLower: 93.3289382317094 GT 75 Pass
AND
PotencyPctUpper: 107.0925498258 LT 125 Pass
G = 
0.0107 
C = 
1.0108 
10.1.2.7. DoseResponse Plot
By default, this option generates a plot of treatment means against the log of dose, with standard error bars for each mean. Each preparation will be plotted as one data series, with as many points as the number of doses applied. A line of best fit and its confidence limits will be drawn for each series. This provides a visual means of inspecting the data, enabling the user to notice immediately whether there is something substantially wrong with the data.
Alternatively, individual response values can be plotted against the log of dose, all with or without trend lines and confidence intervals.
Clicking the [Opt] button situated to the left of DoseResponse Plot output option will place the graph in UNISTAT’s Graphics Editor. Here, all aspects of the graph can be edited and customised.
Selecting Edit → DoseResponse Plot… (or doubleclicking on the graph area) will pop a dialogue where what is displayed on the plot can be controlled.
If you uncheck the Treatment means with error bars box, individual response values will be plotted against the log of dose.
10.1.3. Parallel Line Examples
The following Parallel Line Method examples are based on different Analysis of Variance methods. Data sets were entered into UNISTAT’s spreadsheet and necessary data manipulations made by using UNISTAT spreadsheet functions (see 10.0.1. Data Preparation and 10.0.2. Doses, Dilutions and Potency). Final data sets were saved in two files; BIOPHARMA9 which contains examples from European Pharmacopoeia (2017, the 9^{th} Edition) and BIOFINNEY containing examples from Finney (1978).
10.1.3.1. Completely Randomised Design with 2 Doses and 3 Preparations
Data is given in Table 5.1.1I. on p. 4364 of European Pharmacopoeia (9^{th} Edition).
Open BIOPHARMA9 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the first option Completely Randomised Design and then select columns C1, C2 and L3 respectively as [Data], [Dose] and [Preparation]. Click [Next] to proceed to Output Options Dialogue. Click [All] to perform all tests in one go and click [Finish]. The following output is obtained:
Parallel Line Method
Valid Number of Cases: 60, 0 Omitted
Completely Randomised Design
Summary Statistics
Data variable: Response
Subsample selected by: Dose x Preparations

Valid Cases 
Mean 
Variance 
Standard Error 
0.25 x Standard S 
10 
332.0000 
1026.6667 
10.1325 
0.25 x Preparation T 
10 
323.9000 
724.9889 
8.5146 
0.25 x Preparation U 
10 
282.2000 
854.6222 
9.2446 
1 x Standard S 
10 
248.4000 
483.8222 
6.9557 
1 x Preparation T 
10 
244.0000 
718.6667 
8.4774 
1 x Preparation U 
10 
250.0000 
784.6667 
8.8581 
AndersonDarling Normality Test
Alpha = 0.05
Dose x Preparations 
Test Statistic 
Probability 
Pass/Fail 
0.25 x Standard S 
0.2218 
0.7658 
Pass 
0.25 x Preparation T 
0.2311 
0.7326 
Pass 
0.25 x Preparation U 
0.5030 
0.1549 
Pass 
1 x Standard S 
0.3494 
0.3962 
Pass 
1 x Preparation T 
0.2337 
0.7232 
Pass 
1 x Preparation U 
0.2055 
0.8201 
Pass 
Homogeneity of Variance Tests
For 6 groups defined by Dose x Preparations.
Alpha = 0.05

Test Statistic 
Probability 
Pass/Fail 
Levene’s F Test 
0.3738 
0.8644 
Pass 
Dixon’s Outlier Test
Alpha = 0.05
Onetailed tests
Dose x Preparations 
Dixon’s Q 
Table Q 
Pass/Fail 
0.25 x Standard S Q(Min) 
0.1351 
0.4779 
Pass 
Q(Max) 
0.2889 
0.4779 
Pass 
0.25 x Preparation T Q(Min) 
0.0714 
0.4779 
Pass 
Q(Max) 
0.1333 
0.4779 
Pass 
0.25 x Preparation U Q(Min) 
0.1299 
0.4779 
Pass 
Q(Max) 
0.0429 
0.4779 
Pass 
1 x Standard S Q(Min) 
0.1667 
0.4779 
Pass 
Q(Max) 
0.3333 
0.4779 
Pass 
1 x Preparation T Q(Min) 
0.0857 
0.4779 
Pass 
Q(Max) 
0.1351 
0.4779 
Pass 
1 x Preparation U Q(Min) 
0.0429 
0.4779 
Pass 
Q(Max) 
0.1410 
0.4779 
Pass 
N = 10, Q(Min)=(X(2)X(1))/(X(N1)X(1)), Q(Max)=(X(N)X(N1))/(X(N)X(2))
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
248.4000 
60.3047 
13594.4000 
0.7199 
Preparation T 
244.0000 
57.6357 
12992.9000 
0.7107 
Preparation U 
250.0000 
23.2274 
14753.6000 
0.2600 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
257.5833 
47.0559 
49559.1333 
0.5629 
Preparation T 
251.3333 



Preparation U 
233.4833 



Residual Variance = 
884.9845 
Degrees of Freedom = 
56 
Case (Diagnostic) Statistics

Response 
Dose 
Preparations 
Estimated Response 
Residuals 
Standardised Residuals 
1 
300 
0.25 
Standard S 
322.8167 
22.8167 
0.7670 
2 
310 
0.25 
Standard S 
322.8167 
12.8167 
0.4308 
3 
330 
0.25 
Standard S 
322.8167 
7.1833 
0.2415 
4 
290 
0.25 
Standard S 
322.8167 
32.8167 
1.1031 
5 
364 
0.25 
Standard S 
322.8167 
41.1833 
1.3844 
6 
328 
0.25 
Standard S 
322.8167 
5.1833 
0.1742 
** 7 
390 
0.25 
Standard S 
322.8167 
67.1833 
2.2584 
8 
360 
0.25 
Standard S 
322.8167 
37.1833 
1.2499 
9 
342 
0.25 
Standard S 
322.8167 
19.1833 
0.6448 
10 
306 
0.25 
Standard S 
322.8167 
16.8167 
0.5653 
11 
289 
1 
Standard S 
257.5833 
31.4167 
1.0561 
12 
221 
1 
Standard S 
257.5833 
36.5833 
1.2297 
13 
267 
1 
Standard S 
257.5833 
9.4167 
0.3165 
… 
… 
… 
… 
… 
… 
… 
Cases marked by ‘**’ are outliers at 2 x Standard Deviation.
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Preparation U – Standard S 
37.0773 
12.6231 
2.9372 
2.2713 
Preparation T – Standard S 
2.6690 
12.6231 
0.2114 
2.2713 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Preparation U – Standard S 
0.0093 
8.4061 
65.7484 
**Fail** 
Preparation T – Standard S 
0.9678 
26.0022 
31.3401 
Pass 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
6256.633 
2 
3128.317 
4.086 
0.0223 
Linear Regression 
63830.817 
1 
63830.817 
83.377 
0.0000 
Nonparallelism 
8218.233 
2 
4109.117 
5.367 
0.0075 
Nonlinearity 
0.000 
0 



Treatments 
78305.683 
5 
15661.137 
20.457 
0.0000 
Residual 
41340.900 
54 
765.572 


Total 
119646.583 
59 
2027.908 


Potency

Estimated Potency 
Lower 95% 
Upper 95% 
DoF 
% Precision 
Preparation T 
1.1420 
0.7836 
1.6869 
54 
68.62% 
Preparation U 
1.6689 
1.1481 
2.5550 
54 
68.80% 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
114.20% 
78.36% 
168.69% 
Preparation U 
166.89% 
114.81% 
255.50% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
68.62% 
147.71% 
Preparation U 
100.00% 
68.80% 
153.10% 
G = 
0.0482 
C = 
1.0507 
Looking at the plot of treatment means we can see that Preparation U line is not nearly parallel to Standard S and Preparation T lines. This can also be picked up from the nonparallelism test in Validity of Assay (0.0075), which is significant at 5% level. The Comparison of Slopes test also reports a significant difference between Preparation U and Standard S slopes.
This assay can still be useful by omitting Preparation U and performing the analysis for Standard S and Preparation U. In Excel AddIn Mode, you can simply select the block A1:C41 and repeat the analysis. In StandAlone Mode, you can define a Select Row column to omit these rows from the analysis, without actually deleting them from the spreadsheet. To do this, click somewhere on column 4, and select Data → Select Row option from UNISTAT’s spreadsheet menus. The colour of C4 will change. This indicates that all rows with a 0 entry in this column will be omitted from the subsequent analyses.
When the analysis is repeated without Preparation U, the following results are obtained:
Parallel Line Method
Rows 4160 Omitted
Selected by C4 Select
Valid Number of Cases: 40, 20 Omitted
Completely Randomised Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
248.4000 
60.3047 
13594.4000 
0.7199 
Preparation T 
244.0000 
57.6357 
12992.9000 
0.7107 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
249.3250 
58.9702 
26621.5250 
0.7151 
Preparation T 
243.0750 



Residual Variance = 
719.5007 
Degrees of Freedom = 
37 
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Preparation T – Standard S 
2.6690 
12.3983 
0.2153 
2.0281 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Preparation T – Standard S 
0.8308 
22.4759 
27.8138 
Pass 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
390.625 
1 
390.625 
0.529 
0.4718 
Linear Regression 
66830.625 
1 
66830.625 
90.491 
0.0000 
Nonparallelism 
34.225 
1 
34.225 
0.046 
0.8308 
Nonlinearity 
0.000 
0 



Treatments 
67255.475 
3 
22418.492 
30.355 
0.0000 
Residual 
26587.300 
36 
738.536 


Total 
93842.775 
39 
2406.225 


Potency

Estimated Potency 
Lower 95% 
Upper 95% 
DoF 
% Precision 
Preparation T 
1.1118 
0.8250 
1.5136 
36 
74.20% 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
111.18% 
82.50% 
151.36% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
74.20% 
136.14% 
G = 
0.0455 
C = 
1.0476 
In StandAlone Mode, do not forget to reset column 4, otherwise the Select Row function will be effective in subsequent procedures you run. To do this, click somewhere on column 4, and select Data → Select Row option again, or select Formula → Quick Formula from the menu and enter data. The colour of C4 will change back to its original state.
10.1.3.2. Completely Randomised Design with 5 Doses and 4 Preparations
Data is given in Table 5.1.4I. on p. 4367 of European Pharmacopoeia (9^{th} Edition).
Open BIOPHARMA9 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the first option Completely Randomised Design and then select columns C17, C18, L19 and L20 respectively as [Data], [Dose], [Preparation] and [Dilution]. Click [Next] to proceed to Output Options Dialogue, select only the following output options and click [Finish]:
Parallel Line Method
Valid Number of Cases: 60, 0 Omitted
Completely Randomised Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
0.5998 
0.8813 
0.0904 
0.9920 
Preparation T 
0.1355 
0.9037 
0.1208 
0.9898 
Preparation U 
0.0226 
0.9278 
0.0843 
0.9933 
Preparation V 
0.0714 
0.9211 
0.0459 
0.9963 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
0.5621 
0.9085 
0.3600 
0.9925 
Preparation T 
0.1422 



Preparation U 
0.0495 



Preparation V 
0.0539 



Residual Variance = 
0.0065 
Degrees of Freedom = 
55 
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Preparation U – Standard S 
0.0466 
0.0304 
1.5296 
2.4415 
Preparation V – Standard S 
0.0398 
0.0304 
1.3074 
2.4415 
Preparation T – Standard S 
0.0224 
0.0304 
0.7365 
2.4415 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Preparation U – Standard S 
0.3036 
0.0278 
0.1209 
Pass 
Preparation V – Standard S 
0.4262 
0.0345 
0.1141 
Pass 
Preparation T – Standard S 
0.8033 
0.0519 
0.0967 
Pass 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
4.475 
3 
1.492 
223.395 
0.0000 
Linear Regression 
47.584 
1 
47.584 
7125.912 
0.0000 
Nonparallelism 
0.019 
3 
0.006 
0.933 
0.4339 
Nonlinearity 
0.074 
12 
0.006 
0.926 
0.5307 
Standard S Nonlinearity 
0.017 
3 
0.006 
0.850 
0.4747 
Preparation T Nonlinearity 
0.028 
3 
0.009 
1.410 
0.2538 
Preparation U Nonlinearity 
0.018 
3 
0.006 
0.886 
0.4564 
Preparation V Nonlinearity 
0.011 
3 
0.004 
0.559 
0.6455 
Treatments 
52.152 
19 
2.745 
411.053 
0.0000 
Residual 
0.267 
40 
0.007 


Total 
52.419 
59 
0.888 


Potency
Assumed potency of Preparation T: 20 Āµg protein/ml
Assumed potency of Preparation U: 20 Āµg protein/ml
Assumed potency of Preparation V: 20 Āµg protein/ml

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
43.4197 
40.5448 
46.5397 
Preparation U 
35.1628 
32.8697 
37.6403 
Preparation V 
39.4018 
36.8126 
42.2058 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
217.10% 
202.72% 
232.70% 
Preparation U 
175.81% 
164.35% 
188.20% 
Preparation V 
197.01% 
184.06% 
211.03% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
93.38% 
107.19% 
Preparation U 
100.00% 
93.48% 
107.05% 
Preparation V 
100.00% 
93.43% 
107.12% 
G = 
0.0006 
C = 
1.0006 
10.1.3.3. Randomised Block Design with 4 Doses and 2 Preparations
Data is given in Table 5.1.3.I on p. 4366 of European Pharmacopoeia (9^{th} Edition).
Open BIOPHARMA9 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the second option Randomised Block Design and the select columns C11, C12, L13, C14 and L15 respectively as [Data], [Dose], [Preparation], [Row Factor] and [Dilution]. Click [Next] to proceed to the Output Options Dialogue.
Parallel Line Method
Valid Number of Cases: 40, 0 Omitted
Randomised Block Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
248.5800 
113.0060 
1897.7400 
0.9651 
Preparation T 
238.5000 
109.5039 
747.8000 
0.9851 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
247.5150 
111.2549 
2670.7450 
0.9744 
Preparation T 
239.5650 



Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Preparation T – Standard S 
3.5022 
5.1221 
0.6837 
2.0484 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Preparation T – Standard S 
0.4998 
6.9901 
13.9944 
Pass 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
632.025 
1 
632.025 
11.722 
0.0019 
Linear Regression 
101745.605 
1 
101745.605 
1887.111 
0.0000 
Nonparallelism 
25.205 
1 
25.205 
0.467 
0.4998 
Nonlinearity 
259.140 
4 
64.785 
1.202 
0.3321 
Standard S Nonlinearity 
238.140 
2 
119.070 
2.208 
0.1287 
Preparation T Nonlinearity 
21.000 
2 
10.500 
0.195 
0.8241 
Quadratic Regression 
148.225 
1 
148.225 
2.749 
0.1085 
Quadratic Difference 
34.225 
1 
34.225 
0.635 
0.4323 
Residual 
76.690 
2 
38.345 


Treatments 
102661.975 
7 
14665.996 
272.015 
0.0000 
Blocks(Rows) 
876.750 
4 
219.188 
4.065 
0.0101 
Residual 
1509.650 
28 
53.916 


Total 
105048.375 
39 
2693.548 


Potency
Randomised Block Design
Assigned potency of Standard S: 670 IU/mg
Predilution of Standard S: 16.7 mg / 25 ml; 1 ml / 40 ml
Assumed potency of Preparation T: 20000 IU/vial
Predilution of Preparation T: 1 vial / 40 ml; 1 ml / 40 ml

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
19228.4755 
18423.3508 
20075.1776 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
107.41% 
102.91% 
112.14% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
95.81% 
104.40% 
G = 
0.0022 
C = 
1.0022 
10.1.3.4. Latin Square Design with 3 Doses and 2 Preparations
Data is given in Table 5.1.2.II on p. 4367 of European Pharmacopoeia (9^{th} Edition).
The entry and transformation of this data set is more complicated than the two previous examples. In order to assign the correct dose levels and preparation groups, information given in Table 5.1.2.I is essential. Ensure that the way the factor columns are created is understood well.
Open BIOPHARMA9 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the third option Latin Square Design and then select columns C5, C6, L7, C8, C9 and L10 respectively as [Data], [Dose], [Preparation], [Row Factor], [Col Factor] and [Dilution]. Click [Next] to proceed to Output Options Dialogue.
Parallel Line Method
Valid Number of Cases: 36, 0 Omitted
Latin Squares Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
158.6389 
44.1879 
478.3611 
0.8895 
Preparation T 
155.7778 
48.5040 
573.1111 
0.8901 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
157.7639 
46.3460 
1069.8472 
0.8879 
Preparation T 
156.6528 



Residual Variance = 
32.4196 
Degrees of Freedom = 
33 
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Preparation T – Standard S 
4.3160 
4.5883 
0.9407 
2.0860 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Preparation T – Standard S 
0.3581 
5.2551 
13.8871 
Pass 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Preparations 
11.111 
1 
11.111 
0.535 
0.4730 
Linear Regression 
8475.042 
1 
8475.042 
408.108 
0.0000 
Nonparallelism 
18.375 
1 
18.375 
0.885 
0.3581 
Nonlinearity 
5.472 
2 
2.736 
0.132 
0.8773 
Standard S Nonlinearity 
0.028 
1 
0.028 
0.001 
0.9712 
Preparation T Nonlinearity 
5.444 
1 
5.444 
0.262 
0.6142 
Quadratic Regression 
3.125 
1 
3.125 
0.150 
0.7022 
Quadratic Difference 
2.347 
1 
2.347 
0.113 
0.7402 
Treatments 
8510.000 
5 
1702.000 
81.958 
0.0000 
Blocks(Rows) 
412.000 
5 
82.400 
3.968 
0.0116 
Blocks(Columns) 
218.667 
5 
43.733 
2.106 
0.1069 
Residual 
415.333 
20 
20.767 


Total 
9556.000 
35 
273.029 


Potency
Latin Squares Design
Assigned potency of Standard S: 4855 IU/mg
Predilution of Standard S: 25.2 mg / 24.5 ml
Assumed potency of Preparation T: 5600 IU/mg
Predilution of Preparation T: 21.4 mg / 23.95 ml

Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
5456.3665 
5092.3689 
5843.3620 

Relative Potency 
Lower 95% 
Upper 95% 
Preparation T 
97.63% 
91.12% 
104.56% 

Percent CI 
Lower 95% 
Upper 95% 
Preparation T 
100.00% 
93.33% 
107.09% 
G = 
0.0107 
C = 
1.0108 
10.1.3.5. Twin Crossover Design
Data is given in Table 5.1.5II. on p. 4368 of European Pharmacopoeia (9^{th} Edition).
Open BIOPHARMA9 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the fourth option Crossover Design and select columns C21, C22, L23, C24, C25 and L26 respectively as [Data], [Dose], [Preparation], [Row Factor] and [Col Factor]. Click [Next] to proceed to Output Options Dialogue and then click [Finish]. The following output is obtained:
Parallel Line Method
Valid Number of Cases: 64, 0 Omitted
Crossover Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
101.1875 
47.6991 
19294.1875 
0.3119 
Test 
91.5625 
20.1977 
23815.8750 
0.0618 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
96.4219 
33.9484 
44563.5781 
0.1658 
Test 
96.3281 



Residual Variance = 
730.5505 
Degrees of Freedom = 
61 
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Test – Standard 
27.5014 
8.4520 
3.2538 
2.0484 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Test – Standard 
0.0030 
10.1882 
44.8146 
**Fail** 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Nonparallelism 
1453.5156 
1 
1453.5156 
1.0638 
0.3112 
DaysxPreparations 
31.6406 
1 
31.6406 
0.0232 
0.8801 
DaysxLinear Regression 
50.7656 
1 
50.7656 
0.0372 
0.8485 
Error Between 
38258.8125 
28 
1366.3862 


Blocks(Rows) 
39794.7344 
31 
1283.7011 


Preparations 
0.1406 
1 
0.1406 
0.0010 
0.9747 
Linear Regression 
8859.5156 
1 
8859.5156 
64.5324 
0.0000 
Days 
478.5156 
1 
478.5156 
3.4855 
0.0724 
DaysxNonparallelism 
446.2656 
1 
446.2656 
3.2506 
0.0822 
Error Within 
3844.0625 
28 
137.2879 


Total 
53423.2344 
63 
847.9878 


Potency
Crossover Design
Assumed potency of Test: 40 IU/ml

Estimated Potency 
Lower 95% 
Upper 95% 
Test 
40.1106 
33.4162 
48.1646 

Relative Potency 
Lower 95% 
Upper 95% 
Test 
100.28% 
83.54% 
120.41% 

Percent CI 
Lower 95% 
Upper 95% 
Test 
100.00% 
83.31% 
120.08% 
G = 
0.0650 
C = 
1.0695 
Although the plot of treatment means and the Comparison of Slopes test seem to indicate deviation from parallelism, the nonparallelism test in Validity of Assay (0.3112) is not significant at 5% level.
10.1.3.6. Triple Crossover Design
Table 10.3.1. on p. 205 from Finney, D. J. (1978) is an example with three dose levels and two preparations.
Open BIOFINNEY and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the fourth option Crossover Design and select columns C15, C16, S17, C18 and C19 respectively as [Data], [Dose], [Preparation], [Row Factor] and [Col Factor]. Click [Next] to proceed to Output Options Dialogue and click [Finish].
Parallel Line Method
Valid Number of Cases: 60, 0 Omitted
Crossover Design
Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
3.4547 
0.2272 
8.1200 
0.0576 
Test 
4.1272 
0.5713 
5.2830 
0.3725 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
3.4547 
0.3993 
13.9718 
0.1798 
Test 
3.9695 



Residual Variance = 
0.2451 
Degrees of Freedom = 
57 
Comparison of Slopes
Comparison 
Difference 
Standard Error 
q Stat 
Table q 
Test – Standard 
0.3441 
0.1209 
2.8455 
2.0639 
Comparison 
Probability 
Lower 95% 
Upper 95% 
Result 
Test – Standard 
0.0089 
0.5937 
0.0945 
**Fail** 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Nonparallelism 
0.5688 
1 
0.5688 
1.6032 
0.2176 
Quadratic Regression 
0.8687 
1 
0.8687 
2.4484 
0.1307 
DaysxPreparations 
0.3481 
1 
0.3481 
0.9810 
0.3318 
DaysxLinear Regression 
0.0031 
1 
0.0031 
0.0086 
0.9267 
DaysxQuadratic Difference 
1.9026 
1 
1.9026 
5.3624 
0.0294 
Error Between 
8.5153 
24 
0.3548 


Blocks(Rows) 
12.2066 
29 
0.4209 


Preparations 
0.3330 
1 
0.3330 
4.7403 
0.0395 
Linear Regression 
3.0636 
1 
3.0636 
43.6087 
0.0000 
Days 
0.0150 
1 
0.0150 
0.2141 
0.6477 
Quadratic Difference 
0.0261 
1 
0.0261 
0.3716 
0.5478 
DaysxNonparallelism 
0.0164 
1 
0.0164 
0.2335 
0.6333 
DaysxQuadratic Regression 
0.0216 
1 
0.0216 
0.3075 
0.5844 
Error Within 
1.6861 
24 
0.0703 


Total 
17.3685 
59 
0.2944 


Potency

Estimated Potency 
Lower 95% 
Upper 95% 
Test 
0.2754 
0.1783 
0.3923 
G = 
0.0977 
C = 
1.1083 