10.2. Slope Ratio Method
This is a general purpose procedure that can be used to analyse balanced or unbalanced assays with blanks (0-dose 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 (1997-2008), is a restricted special case of this procedure.
10.2.1. 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 dose-preparation 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. Output Options
Let Xijk and Yijk be the dose and response values for the ith preparation (i = Blank, Standard, Test 1, …, Test n - 1) and the jth dose level of the kth replicate. First, all dose readings are transformed as:
Xijk = Ln(Xijk)
where the logarithm is natural (e-based).
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; Shapiro-Wilk, Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling (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 dose-preparation (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) non-linearity. The overall non-linearity 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 Sxxi is as defined in Separate Regression and:
Also define the number of unique dose-preparation 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 |
|
|
Non-linearity |
D – 2n |
SSL |
|
|
Non-linearity for Preparationi |
D / n – 2 |
SSLi |
|
|
Residual |
N – D – B – (K – 1) |
SSE – SSP |
- SSP |
|
Total |
N - 1 |
SST |
|
10.2.2.4. Regression
Calculate for i = S, T1, …, Tn - 1:
The estimated parameters of the line of best fit for each preparation (i = S, T1, …, Tn - 1) are:
Slope: 
R-squared: 
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 |
Y0jk |
0 |
0 |
0 |
|
Replicates |
… |
… |
… |
… |
|
Standard |
YSjk |
XSjk |
0 |
0 |
|
Replicates |
… |
… |
… |
… |
|
Test 1 |
Y1jk |
0 |
X1jk |
0 |
|
Replicates |
… |
… |
… |
… |
|
Test n |
Ynjk |
0 |
0 |
Xnjk |
|
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 = T1, …, Tn - 1 as the values corresponding to elements of (X'X)-1 matrix from the Common Regression run. First define:
where s2 is the residual mean squares and 
is the critical value from the
t-distribution with degrees of freedom of the overall residual term from the
ANOVA table. Note that if divided by s2, Vss, Vii, Vsi give the
variance / covariance matrix of the Common Regression
coefficients.
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 Mi is:
and the approximate variance of Mi is (when g is negligible):
Note that Mi is the relative potency and MiL and MiU 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 X-Y Plots procedure, this time with confidence intervals for
10.2.3. Examples
Example 1
Data is given in Table 5.2.1-I on p. 588 of European Pharmacopoeia (2008). The data is rearranged as described in section 10.1.1. Data Preparation and saved in columns 24-26 of BIOPHARMA6.
Although the data set contains blanks (0 dose treatments), they need to be removed from the analysis. In Excel Add-In Mode, you can simply select the block X10:Z57. In Stand-Alone 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 Stand-Alone Mode select columns C23, C24 and L25 respectively as [Data], [Dose] and [Preparation] from the Variable Selection Dialogue. In Excel Add-In 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 Shapiro-Wilk Test and Report summary statistics boxes. Then click [Back] and [Finish].
In Stand-Alone 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 1-8 Omitted
Selected by C26 Select
Normality Tests
Smaller probabilities indicate non-normality.
* Lilliefors probability = 0.2 means 0.2 or greater.
|
Dose×Preparations |
Valid Cases |
Mean |
Standard Deviation |
Shapiro-Wilk Test |
Probability |
|
1 × Standard S |
8 |
0.1351 |
0.0025 |
0.8969 |
0.2707 |
|
2 × Standard S |
8 |
0.2176 |
0.0021 |
0.8816 |
0.1952 |
|
3 × Standard S |
8 |
0.2996 |
0.0027 |
0.8269 |
0.0551 |
|
1 × Preparation T |
8 |
0.1200 |
0.0011 |
0.8599 |
0.1199 |
|
2 × Preparation T |
8 |
0.1898 |
0.0012 |
0.8042 |
0.0318 |
|
3 × Preparation T |
8 |
0.2554 |
0.0018 |
0.9255 |
0.4763 |
Homogeneity of Variance Tests
|
|
Test Statistic |
Probability |
|
|
Bartlett's Chi-square Test |
8.5820 |
0.1269 |
|
|
Bartlett-Box 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 |
F-Stat |
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 |
|
Non-linearity |
0.000 |
2 |
0.000 |
2.984 |
0.0614 |
|
Standard S Non-linearity |
0.000 |
1 |
0.000 |
0.086 |
0.7702 |
|
Preparation T Non-linearity |
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 |
R-squared |
|
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 |
R-squared |
|
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 5.2.2-I on p. 589 of European Pharmacopoeia (2008).
Open BIOPHARMA6 and select Bioassay → Slope Ratio Method. Note that 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 Cramer-von 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 non-normality.
|
Dose×Preparations |
Valid Cases |
Mean |
Standard Deviation |
Cramer-von Mises Test |
Probability |
|
1 × Standard S |
2 |
18.0000 |
0.0000 |
* |
* |
|
2 × Standard S |
2 |
23.6500 |
1.2021 |
0.0419 |
0.4774 |
|
3 × Standard S |
2 |
30.4000 |
0.0000 |
* |
* |
|
4 × Standard S |
2 |
36.1500 |
0.6364 |
0.0419 |
0.4774 |
|
1 × Preparation T |
2 |
15.9500 |
1.2021 |
0.0419 |
0.4774 |
|
2 × Preparation T |
2 |
23.6500 |
0.7778 |
0.0419 |
0.4774 |
|
3 × Preparation T |
2 |
28.1500 |
1.0607 |
0.0419 |
0.4774 |
|
4 × Preparation T |
2 |
36.1000 |
2.4042 |
0.0419 |
0.4774 |
|
1 × Preparation U |
2 |
15.5500 |
0.2121 |
0.0419 |
0.4774 |
|
2 × Preparation U |
2 |
19.4000 |
1.1314 |
0.0419 |
0.4774 |
|
3 × Preparation U |
2 |
23.6500 |
0.7778 |
0.0419 |
0.4774 |
|
4 × Preparation U |
2 |
27.2000 |
0.2828 |
0.0419 |
0.4774 |
Homogeneity of Variance Tests
|
|
Test Statistic |
Probability |
|
Bartlett's Chi-square Test |
5.2396 |
0.8129 |
|
Bartlett-Box 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 |
F-Stat |
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 |
|
Non-linearity |
5.065 |
6 |
0.844 |
0.791 |
0.5943 |
|
Standard S Non-linearity |
0.446 |
2 |
0.223 |
0.209 |
0.8144 |
|
Preparation T Non-linearity |
4.453 |
2 |
2.227 |
2.085 |
0.1670 |
|
Preparation U Non-linearity |
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 |
R-squared |
|
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 |
R-squared |
|
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 non-normality.
* Lilliefors probability = 0.2 means 0.2 or greater.
|
Dose×Preparations |
Valid Cases |
Mean |
Standard Deviation |
|
0 × Blank |
4 |
41.7500 |
3.3040 |
|
0.5 × Standard |
4 |
100.0000 |
3.5590 |
|
1 × Standard |
4 |
161.5000 |
4.9329 |
|
0.5 × Test |
4 |
85.0000 |
4.7610 |
|
1 × Test |
4 |
122.2500 |
1.2583 |
|
Dose×Preparations |
Shapiro-Wilk Test |
Probability |
Kolmogorov-Smirnov Test |
* Probability |
|
0 × Blank |
0.9157 |
0.5130 |
0.2521 |
0.2000 |
|
0.5 × Standard |
0.8947 |
0.4051 |
0.2500 |
0.2000 |
|
1 × Standard |
0.9646 |
0.8081 |
0.1939 |
0.2000 |
|
0.5 × Test |
0.9110 |
0.4877 |
0.2357 |
0.2000 |
|
1 × Test |
0.8949 |
0.4064 |
0.3287 |
0.1554 |
|
Dose×Preparations |
Cramer-von Mises Test |
Probability |
Anderson-Darling Test |
Probability |
|
0 × Blank |
0.0443 |
0.5124 |
0.2706 |
0.4502 |
|
0.5 × Standard |
0.0518 |
0.3989 |
0.3151 |
0.3280 |
|
1 × Standard |
0.0304 |
0.7840 |
0.1973 |
0.7044 |
|
0.5 × Test |
0.0463 |
0.4814 |
0.2783 |
0.4263 |
|
1 × Test |
0.0676 |
0.2335 |
0.3610 |
0.2343 |
Homogeneity of Variance Tests
|
|
Test Statistic |
Probability |
|
Bartlett's Chi-square Test |
4.3575 |
0.3598 |
|
Bartlett-Box 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 |
F-Stat |
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 |
|
Non-linearity |
0.000 |
0 |
|
|
|
|
Standard Non-linearity |
0.000 |
0 |
|
|
|
|
Test Non-linearity |
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 |
R-squared |
|
Standard |
38.5000 |
123.0000 |
111.0000 |
0.9855 |
|
Test |
47.7500 |
74.5000 |
72.7500 |
0.9745 |
Common Regression
|
|
Intercept |
Slope |
Residual SS |
R-squared |
|
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 |
