10.2. Slope Ratio Method
This is a general purpose procedure that can be used to analyse balanced or unbalanced assays with blanks (0dose treatments), plate (row) effects and unlimited numbers of dose levels and test preparations. The algorithm is based on Finney (1978). The Slope Ratio Method specification given in European Pharmacopoeia (19972008), is a restricted special case of this procedure.
10.2.1. Slope Ratio Variable Selection
The data format is as in Parallel Line Method (see 10.1.1. Data Preparation). Measurement data is stacked in a single column, a second column contains the dose level for each measurement and another categorical column indicates which preparation a particular measurement belongs to. An optional row factor can be entered to keep track of the replicates.
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 preparation should always be the standard. It is compulsory to select at least three columns [Data], [Dose] and [Preparation]. The optional [Row Factor] column is usually used to isolate a plate effect (the replicates) and when one is selected, the program assumes that all dose/treatment groups (or cells) have an equal number of replicates.
10.2.2. Slope Ratio Output Options
Let X_{ijk} and Y_{ijk} be the dose and response values for the i^{th} preparation (i = Blank, Standard, Test 1, …, Test n – 1) and the j^{th} dose level of the k^{th} replicate. First, all dose readings are transformed as:
X_{ijk} = Ln(X_{ijk})
where the logarithm is natural (ebased).
10.2.2.1. Normality Tests
As in Parallel Line Method, you can select to display all or any of the four most commonly used normality tests; ShapiroWilk, KolmogorovSmirnov, Cramervon Mises and AndersonDarling (see 10.1.3.1. Normality Tests for Bioassays).
10.2.2.2. Homogeneity of Variance Tests
Five alternative homogeneity of variance tests are performed for unique dosepreparation (treatment) groups as described in section 10.1.3.2. Homogeneity of Variance Tests.
10.2.2.3. Validity of Assay
This output option displays an Analysis of Variance (ANOVA) table, which is used in testing the Validity of Assay. The standard significance tests performed are (i) regression, (ii) intercept and (iii) nonlinearity. The overall nonlinearity test is also broken down to individual tests for each preparation. If blanks (entries with a 0 dose level) exist, there will be an additional term for them. If a [Row Factor] was selected it will appear in the table as a main effect.
Define a cell as a unique combination of dose levels and preparations. For each cell calculate:
Define the overall mean as:
where N is the total number of observations. Also define and as the intercept and slope for each preparation from Separate Regression and as the slope for each preparation from Common Regression (see 10.2.2.4. Regression).
The following definitions are used in calculating the blanks effect:
where Sxx_{i} is as defined in Separate Regression and:
Also define the number of unique dosepreparation combinations excluding blanks as:
The ANOVA table is then constructed as follows.
Due to 
Degrees of Freedom 

Sum of Squares 
Plate 
K – 1 
SSP 

Between Doses 
D – 1 + B 
SSD 

Blanks 
B = 0 or 1 
SSB 

Regression 
n 
SSR 

Intercept 
n – 1 
SSD – SSB – SSR – SSL 

Nonlinearity 
D – 2n 
SSL 

Nonlinearity for Preparation_{i} 
D / n – 2 
SSL_{i} 

Residual 
N – D – B – (K – 1) 
SSE – SSP 
– SSP 
Total 
N – 1 
SST 

10.2.2.4. Regression
Calculate for i = S, T_{1}, …, Tn – 1:
The estimated parameters of the line of best fit for each preparation (i = S, T_{1}, …, Tn – 1) are:
Slope:
Rsquared:
Residual sum of squares:
Standard error of slope:
This information is displayed in the Separate Regression table and used in drawing the best fit lines in Plot of Treatments.
The Common Regression is obtained from a multivariate regression run, after transforming the data into the following form first.

Dependent 
Independent Variables 


Variable 
Standard 
Test 1 
Test n 
Blank 
Y_{0jk} 
0 
0 
0 
Replicates 
… 
… 
… 
… 
Standard 
Y_{Sjk} 
X_{Sjk} 
0 
0 
Replicates 
… 
… 
… 
… 
Test 1 
Y_{1jk} 
0 
X_{1jk} 
0 
Replicates 
… 
… 
… 
… 
Test n 
Y_{njk} 
0 
0 
X_{njk} 
Replicates 
… 
… 
… 
… 
The estimated parameters are displayed in Common Regression table.
10.2.2.5. Potency
By default, each test preparation is assigned a potency of unity. If you want to change this click the [Opt] button situated to the left of the Potency option. In this case, a further dialogue pops up asking for entry of assigned potency for each test preparation.
For each test preparation, the potency ratio is calculated as follows:
For confidence intervals of M first define Vss, Vii, Vsi, i = T_{1}, …, T_{n – 1} as the values corresponding to elements of (X’X)^{1} matrix from the Common Regression run. First define:
Then the confidence interval for potency ratio of each test preparation is calculated using Fieller’s Theorem (see Finney 1978, p. 156):
where the variance of M_{i} is:
and the approximate variance of M_{i} is (when g is negligible):
M_{i} is the relative potency and M_{iL} and M_{iU} are the confidence limits for the relative potency. The estimated potency and its confidence interval are obtained by multiplying these relative values by the assigned potency supplied by the user for each test preparation separately.
Weights are computed after the estimated potency and its confidence interval are found:
and % Precision is:
10.2.2.6. Plot of Treatments
This option generates a Plot of Treatments against dose levels. Standard and each test preparation are plotted in separate series and a line of best fit is drawn for each one of them. The coefficients of lines are as in Separate Regression output.
Clicking the [Opt] button situated to the left of the Plot of Treatments option will place the graph in UNISTAT’s Graphics Editor. The plot can be further customised and annotated using the tools available under the UNISTAT Graphics Window’s Edit menu.
The same plot is drawn here using the XY Plots procedure, this time with confidence intervals for regression lines included.
10.2.3. Slope Ratio Examples
Example 1
Data is given in Table 5.2.1I on p. 588 of European Pharmacopoeia (2008). The data is rearranged as described in section 10.1.1. Data Preparation and saved in columns 2426 of BIOPHARMA6.
Although the data set contains blanks (0 dose treatments), they need to be removed from the analysis. In Excel AddIn Mode, you can simply select the block X10:Z57. In StandAlone Mode, you can define C26 as 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 26, and select Data → Select Row option from UNISTAT’s spreadsheet menus. The colour of C26 will change. This indicates that all rows with a 0 entry in this column will be omitted from subsequent analyses.
Select Bioassay → Slope Ratio Method. In StandAlone Mode select columns C23, C24 and L25 respectively as [Data], [Dose] and [Preparation] from the Variable Selection Dialogue. In Excel AddIn Mode, you will need to select the three highlighted columns in the same order. Click [Next] to proceed to Output Options Dialogue. If you do not want to display all normality tests click on the [Opt] button situated to the left of Normality Tests option. Click [None] and then check the ShapiroWilk Test and Report summary statistics boxes. Then click [Back] and [Finish].
In StandAlone Mode, do not forget to reset column 4 after you finish this example, 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 C26 will change back to its original value.
Slope Ratio Method
Rows 18 Omitted
Selected by C26 Select
Normality Tests
Smaller probabilities indicate nonnormality.
* Lilliefors probability = 0.2 means 0.2 or greater.
DosexPreparations 
Valid Cases 
Mean 
Standard Deviation 
ShapiroWilk Test 
Probability 
1 x Standard S 
8 
0.1351 
0.0025 
0.8969 
0.2707 
2 x Standard S 
8 
0.2176 
0.0021 
0.8816 
0.1952 
3 x Standard S 
8 
0.2996 
0.0027 
0.8269 
0.0551 
1 x Preparation T 
8 
0.1200 
0.0011 
0.8599 
0.1199 
2 x Preparation T 
8 
0.1898 
0.0012 
0.8042 
0.0318 
3 x Preparation T 
8 
0.2554 
0.0018 
0.9255 
0.4763 
Homogeneity of Variance Tests

Test Statistic 
Probability 

Bartlett’s Chisquare Test 
8.5820 
0.1269 

BartlettBox F Test 
1.7315 
0.1239 

Cochran’s C (max var / sum var) 
0.3079 
0.3345 

Hartley’s F (max var / min var) 
6.2344 
0.0500 
p > 0.05 
Levene’s F Test 
2.2830 
0.0635 

Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Constant 
1.976 
1 
1.976 


Regression 
0.192 
2 
0.096 
24849.565 
0.0000 
Intercept 
0.000 
1 
0.000 
0.001 
0.9780 
Nonlinearity 
0.000 
2 
0.000 
2.984 
0.0614 
Standard S Nonlinearity 
0.000 
1 
0.000 
0.086 
0.7702 
Preparation T Nonlinearity 
0.000 
1 
0.000 
5.882 
0.0197 
Treatments 
0.192 
5 
0.038 


Residual 
0.000 
42 
0.000 


Total 
0.192 
47 
0.004 


Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
0.0530 
0.0822 
0.0001 
0.9989 
Preparation T 
0.0530 
0.0677 
0.0001 
0.9992 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
0.0530 
0.0822 
0.0002 
0.9990 
Preparation T 

0.0677 


Potency
Test Preparation 
Assigned Potency 
Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
1.0000 
0.8231 
0.8171 
0.8292 
Test Preparation 
Variance 
Weight 
% Precision 
Preparation T 
0.0000 
110860.3771 
99.2644 
G = 
0.0001 
C = 
1.0001 
Example 2
Data is given in Table 5.2.2I on p. 589 of European Pharmacopoeia (2008).
Open BIOPHARMA6 and select Bioassay → Slope Ratio Method. The blank preparation is already omitted from this data set. From the Variable Selection Dialogue select columns C27, C28 and L29 Preparations respectively as [Data], [Dose] and [Preparation]. Click [Next] to proceed to Output Options Dialogue. Click on the [Opt] button situated to the left of Normality Tests option, click [None] and then check the Cramervon Mises Test and Report summary statistics boxes and click [Back]. Click the [Opt] button situated to the left of the Potency option. Enter the assigned potency value 15 for both preparations, click [Back] and [Finish].
Slope Ratio Method
Normality Tests
Smaller probabilities indicate nonnormality.
DosexPreparations 
Valid Cases 
Mean 
Standard Deviation 
Cramervon Mises Test 
Probability 
1 x Standard S 
2 
18.0000 
0.0000 
* 
* 
2 x Standard S 
2 
23.6500 
1.2021 
0.0419 
0.4774 
3 x Standard S 
2 
30.4000 
0.0000 
* 
* 
4 x Standard S 
2 
36.1500 
0.6364 
0.0419 
0.4774 
1 x Preparation T 
2 
15.9500 
1.2021 
0.0419 
0.4774 
2 x Preparation T 
2 
23.6500 
0.7778 
0.0419 
0.4774 
3 x Preparation T 
2 
28.1500 
1.0607 
0.0419 
0.4774 
4 x Preparation T 
2 
36.1000 
2.4042 
0.0419 
0.4774 
1 x Preparation U 
2 
15.5500 
0.2121 
0.0419 
0.4774 
2 x Preparation U 
2 
19.4000 
1.1314 
0.0419 
0.4774 
3 x Preparation U 
2 
23.6500 
0.7778 
0.0419 
0.4774 
4 x Preparation U 
2 
27.2000 
0.2828 
0.0419 
0.4774 
Homogeneity of Variance Tests

Test Statistic 
Probability 
Bartlett’s Chisquare Test 
5.2396 
0.8129 
BartlettBox F Test 
0.5751 
0.8146 
Cochran’s C (max var / sum var) 
0.4510 
0.2364 
Hartley’s F (max var / min var) 
128.4444 

Levene’s F Test 


Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Constant 
14785.770 
1 
14785.770 


Regression 
1087.665 
3 
362.555 
339.498 
0.0000 
Intercept 
3.474 
2 
1.737 
1.626 
0.2371 
Nonlinearity 
5.065 
6 
0.844 
0.791 
0.5943 
Standard S Nonlinearity 
0.446 
2 
0.223 
0.209 
0.8144 
Preparation T Nonlinearity 
4.453 
2 
2.227 
2.085 
0.1670 
Preparation U Nonlinearity 
0.166 
2 
0.083 
0.078 
0.9257 
Treatments 
1096.205 
11 
99.655 


Residual 
12.815 
12 
1.068 


Total 
1109.020 
23 
48.218 


Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
11.7500 
6.1200 
2.2960 
0.9939 
Preparation T 
9.7250 
6.4950 
13.4085 
0.9692 
Preparation U 
11.6500 
3.9200 
2.1760 
0.9860 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard S 
11.0417 
6.3561 
21.3544 
0.9807 
Preparation T 

6.0561 


Preparation U 

4.1228 


Potency
Test Preparation 
Assigned Potency 
Estimated Potency 
Lower 95% 
Upper 95% 
Preparation T 
15.0000 
14.2920 
13.3681 
15.2711 
Preparation U 
15.0000 
9.7295 
8.8542 
10.6088 
Test Preparation 
Variance 
Weight 
% Precision 
Preparation T 
0.0008 
5.2437 
93.5355 
Preparation U 
0.0007 
6.1678 
91.0034 
G = 
0.0056 
C = 
1.0056 
Example 3
Table 7.10.2. on p. 161 from Finney, D. J. (1978) is an example with blanks, four replicates and two preparations.
Open BIOFINNEY and select Bioassay → Slope Ratio Method. From the Variable Selection Dialogue select columns C12 Data, C13 Dose and S14 Preparations respectively as [Data], [Dose] and [Preparation]. Click [Next] to proceed to Output Options Dialogue. Click [All] to select all output options and then click [Finish]. The potency ratio and its confidence limits are calculated with the default assigned potency of 1. The following output is obtained.
Slope Ratio Method
Normality Tests
Smaller probabilities indicate nonnormality.
* Lilliefors probability = 0.2 means 0.2 or greater.
DosexPreparations 
Valid Cases 
Mean 
Standard Deviation 
0 x Blank 
4 
41.7500 
3.3040 
0.5 x Standard 
4 
100.0000 
3.5590 
1 x Standard 
4 
161.5000 
4.9329 
0.5 x Test 
4 
85.0000 
4.7610 
1 x Test 
4 
122.2500 
1.2583 
DosexPreparations 
ShapiroWilk Test 
Probability 
KolmogorovSmirnov Test 
* Probability 
0 x Blank 
0.9157 
0.5130 
0.2521 
0.2000 
0.5 x Standard 
0.8947 
0.4051 
0.2500 
0.2000 
1 x Standard 
0.9646 
0.8081 
0.1939 
0.2000 
0.5 x Test 
0.9110 
0.4877 
0.2357 
0.2000 
1 x Test 
0.8949 
0.4064 
0.3287 
0.1554 
DosexPreparations 
Cramervon Mises Test 
Probability 
AndersonDarling Test 
Probability 
0 x Blank 
0.0443 
0.5124 
0.2706 
0.4502 
0.5 x Standard 
0.0518 
0.3989 
0.3151 
0.3280 
1 x Standard 
0.0304 
0.7840 
0.1973 
0.7044 
0.5 x Test 
0.0463 
0.4814 
0.2783 
0.4263 
1 x Test 
0.0676 
0.2335 
0.3610 
0.2343 
Homogeneity of Variance Tests

Test Statistic 
Probability 
Bartlett’s Chisquare Test 
4.3575 
0.3598 
BartlettBox F Test 
1.1126 
0.3498 
Cochran’s C (max var / sum var) 
0.3372 
0.8137 
Hartley’s F (max var / min var) 
15.3684 

Levene’s F Test 
3.3168 
0.0390 
Validity of Assay
Due To 
Sum of Squares 
DoF 
Mean Square 
FStat 
Prob 
Constant 
208488.200 
1 
208488.200 


Regression 
31456.914 
2 
15728.457 
1089.731 
0.0000 
Blanks 
2.161 
1 
2.161 
0.150 
0.7043 
Intercept 
34.225 
1 
34.225 
2.371 
0.1444 
Nonlinearity 
0.000 
0 



Standard Nonlinearity 
0.000 
0 



Test Nonlinearity 
0.000 
0 



Treatments 
31493.300 
4 
7873.325 


Residual 
216.500 
15 
14.433 


Total 
31709.800 
19 
1668.937 


Separate Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
38.5000 
123.0000 
111.0000 
0.9855 
Test 
47.7500 
74.5000 
72.7500 
0.9745 
Common Regression

Intercept 
Slope 
Residual SS 
Rsquared 
Standard 
42.1429 
118.6286 
252.8857 
0.9920 
Test 

81.2286 


Potency
Test Preparation 
Assigned Potency 
Estimated Potency 
Lower 95% 
Upper 95% 
Test 
1.0000 
0.6847 
0.6464 
0.7236 
Test Preparation 
Variance 
Weight 
% Precision 
Test 
0.0003 
3046.6783 
94.3986 
G = 
0.0021 
C = 
1.0021 