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 squares 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.

The data sets to be analysed according to European Pharmacopoeia (1997-2008) Parallel Line Method should be balanced.

10.1.1. Data Preparation

Data is usually given in the form of a table where measurements corresponding to different preparations and dose levels are in separate columns (i.e. treatment groups). Let:

h be the number of preparations (including the standard preparation),

k be the number of treatments and

n be the number of cases in each treatment group. Then, it follows that

d = k / h is the number of dose levels.

The standard is always the first preparation in a column of data.

Consider the following hypothetical 3-dose / 2-preparation example where h = 2, k = 6 and n = 4 and suppose the dose levels are given as 0.125, 0.25 and 0.5.

 

	Preparations
	Standard			Unknown
Cases	Dose 1	Dose 2	Dose 3	Dose 1	Dose 2	Dose 3
1	1.3	2.1	4.1	1.5	2.0	3.9
2	1.7	2.3	4.2	1.1	1.9	4.6
3	1.1	2.7	3.9	0.9	2.1	4.0
4	1.5	2.2	4.3	1.0	2.2	3.7

 

Although this is a well-defined data set for a bioassay, it should be first transformed into a more convenient format for analysis using a statistical package. This is done by stacking all response measurements in a single column. It is also necessary to create a number of categorical data columns (or factors) to keep track of which measurement belongs to which preparation, to which dose group and to which treatment case.

For analysis with UNISTAT, the data for the above example should be entered as follows:

 

Data	Dose	Preparation	Rows	Columns
1.3	.125	Standard	1	1
1.7	.125	Standard	2	1
1.1	.125	Standard	3	1
1.5	.125	Standard	4	1
2.1	.25	Standard	1	2
2.3	.25	Standard	2	2
2.7	.25	Standard	3	2
2.2	.25	Standard	4	2
4.1	.5	Standard	1	3
4.2	.5	Standard	2	3
3.9	.5	Standard	3	3
4.3	.5	Standard	4	3
1.5	.125	Unknown	1	4
1.1	.125	Unknown	2	4
0.9	.125	Unknown	3	4
1.0	.125	Unknown	4	4
2.0	.25	Unknown	1	5
1.9	.25	Unknown	2	5
2.1	.25	Unknown	3	5
2.2	.25	Unknown	4	5
3.9	.5	Unknown	1	6
4.6	.5	Unknown	2	6
4.0	.5	Unknown	3	6
3.7	.5	Unknown	4	6

 

Note that the Dose column contains the actual dose units for all preparations, instead of dose group numbers. This information is needed in Potency and Plot of Treatment Means options. Also, the column Rows is needed for Randomised Block, Latin Squares Design and Crossover Design and Columns is needed for Latin Squares Design and Crossover Design. In Stand-Alone Mode, Rows and Columns variables can be generated automatically by using UNISTAT spreadsheet functions Level(4) and Level(4);B respectively. (see 3.4.2.5. Statistical Functions).

10.1.2. Variable Selection

Once the data is arranged as described above, 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.

The frame Select Data Type (at the top) displays options for the type of Analysis of Variance to be performed. Note that the number of variables to be selected is different for these types of analyses. When the second option Randomised Block Design is selected, four variables will need to be selected.

The third and fourth options Latin Squares Design and Crossover Design require selection of five variables.

After selecting the analysis type, you will need to assign tasks to variables by sending them to the 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.

When all variables are selected, click the [Next] button to proceed to Output Options Dialogue.

10.1.3. Output Options

Output options that have further options under them (i.e. they have further dialogues and windows to display) then an [Opt] button is 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 the default values. If you want to change the default values, you can click on the [Opt] button to display the 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.

[Opt] buttons on this dialogue will allow you to choose from four different types of normality tests, enter assigned potencies for test preparations and edit the plot of treatment means in UNISTAT’s Graphics Editor.

10.1.3.1. Normality Tests for Bioassays

One of the basic assumptions of Parallel Line Method is that for each treatment group (i.e. a unique dose-preparation combination), observations are normally distributed.

Earlier versions of UNISTAT featured a classic Shapiro-Wilk normality test (1965, 1968) as recommended by European Pharmacopoeia (1997-2008). However, this test was shown to be inaccurate and substantially revised by Royston (1995). Also, there are other normality tests which are more powerful than Shapiro-Wilk, such as Cramer-von Mises and Anderson-Darling. Accordingly, as of this version of UNISTAT, we provide the four most commonly used normality tests as part of the Parallel Line Method (see 6.3.3. Normality Tests). A new output options dialogue enables you to display all or any of the four normality tests supported.

If you still wish to use the classic Shapiro-Wilk (1965) and its accompanying overall normality tests as in earlier version of UNISTAT, then you can do so by entering the following line in Documents\Unistat60\Unistat60.ini file under the [Options] section:

   OverallNormality=1

In classic Shapiro-Wilk test, observations are arranged in ascending order for each treatment group and then the following sum is found:

     

The test statistic for each sample is:

     

where S2 is the sum of squared differences from the mean, and a_i i = 1, …, k are the coefficients given by the authors.

If all sample sizes are between 7 and 20 (inclusive), an overall test of normality, which is based on the normal distribution, is also performed according to Shapiro & Wilk (1968).

First, the following ratio is calculated for each sample:

     

and:

     

where a, q and m are the coefficients for k degrees of freedom given by the authors in Shapiro & Wilk (1968).

The test statistic is defined as:

     

with a 1-tail probability from the normal distribution.

10.1.3.2. 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 chi-square test as recommended by European Pharmacopoeia (1997-2008), and Hartley’s F test. As of this version of UNISAT, 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. For a detailed description of these tests see 7.4.2.1. Homogeneity of Variance Test Results.

10.1.3.3. Response Totals and Contrasts

These are the intermediate values calculated directly from raw data and they are used in computing all output statistics. Here we report these values in order to help the user with validating the final results.

First d (number of doses) rows of the table report the sums of all cases in each treatment group. Let:

      , j = 1, … , d, i = 1, … , h

represent the sum of cases for the j^th dose and the i^th preparation in the table.

The next row Total is the sum of these values over dose for each preparation:

       i = 1, … , h

The contrasts are then calculated for each preparation (i = 1, … , h) as follows:

 

No of doses	Linear Contrast (Li)	Quadratic Contrast (Qi)	Cubic Contrast (Ji)
2	S2i – S1I
3	S3i – S1i	S1i – 2S2i + S3i
4	3S4i + S3i – S2i – 3S1i	S1i – S2i – S3i + S3i	3S2i – S1i – S4i – 3S3I

 

10.1.3.4. Validity of Assay

This output option displays an Analysis of Variance (ANOVA) table, which is used to test the Validity of Assay. Also, the residual sum of squares and its degrees of freedom are used in estimating the confidence limits for the Potency (see 10.1.3.7. 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 (1997-2008). For unbalanced designs, the only difference is that the sums are taken up to the maximum number of observation in each treatment group. See 10.2. Slope Ratio Method, section Validity of Assay for a general unbalanced formulation.

Let us first define the three key entries of all ANOVA tables, namely, 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 follows:

 

	Degrees of Freedom	2-dose	3-dose	4-dose
Preparations	h - 1
Linear Regression	1
Non-parallelism	h - 1
Non-linearity	h for 3-dose 2h for 4-dose
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

 

Note that 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, Non-parallelism and Non-linearity) should always add up to Treatments,

2)    Quadratic Regression, Difference of Quadratics and their Residuals should always add up to Non-linearity,

3)    Treatments and their Residuals should always add up to Total.

All four ANOVA methods supported here differ only in their residual terms, where the residual for the latter two designs contain a Row Block term, which is defined as:

     

where:

     

The Latin Squares Design also contains a Column Block term, which is defined as:

     

where:

     

Although the sum of squares and degrees of freedom here are not different from that of Completely Randomised Design, F-statistics and their probability values will be different, as these values are based on the residual sum of squares and degrees of freedom found after removing the effects of Row and Column blocks.

The Crossover Design is different from others in that the column factor usually represents time periods for different treatments and the model includes interaction terms between this variable and others. If the Dose variable has only two values for each preparation then the model is called Twin Crossover Design and it contains interactions between the column factor and Preparations, Linear Regression and Non-Parallelism terms. If there are three dose levels for each preparation, the model is called Triple Crossover Design and it contains additional interactions between the column factor and Quadratic Regression and Quadratic Difference terms.

The Validity of Assay output has different entries for different ANOVA Designs and for different number of dose units used. See 10.2.3. Examples for different types.

10.1.3.5. Regression

A separate regression line is fitted on each preparation against the natural logarithm of dose. The Common Regression is obtained by pooling the difference sum of squares for all preparations and the Total Regression by regressing the response variable on the log of dose variable, without distinguishing between preparations. Note that the slopes of Common Regression and Total Regression are not the same when the design is not balanced (i.e. when it is symmetric or asymmetric). In that case, the slope of Common Regression is used as the common slope in Potency calculations.

The output from this procedure is similar to the first part of output from Heterogeneity of Regression procedure (see 7.4.5. Heterogeneity of Regression).

10.1.3.6. 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.

This output option is equivalent to part of the analysis from Heterogeneity of Regression procedure. Running the same bioassay data set with this procedure, however, may produce slightly different results, since while Heterogeneity of Regression procedure is always based on a 1-way ANOVA model, the current procedure is based on the residual sum of squares and its degrees of freedom as computed for the Validity of Assay output.

European Pharmacopoeia (1997-2008) employs a slightly different algorithm, which is based on linear contrasts as a proxy for the slopes. The two approaches are identical and produce the same probability values. Although we report here the slopes test by default, the linear contrast test output can be displayed instead by entering the following line in Documents\Unistat60\Unistat60.ini file under the [Bioassay] section:

   ParalEuroPharma=1

Note that this Unistat60.ini line also affects the Potency output below.

For European Pharmacopoeia (1997-2008) linear contrast test we first calculate a test statistic q' for each test preparation:

     

     

where:

L₁ is the linear contrast for the standard preparation,

L_i is the linear contrast for the i^th test preparation and

s² 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 two-tailed probability for the test statistic is generated using an algorithm developed by Charles Dunnett 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.

10.1.3.7. Potency

By default, each test preparation is assigned a Potency of unity, in which case the reported potency is the relative potency ratio. If you want to change this click the [Opt] button situated to the left of the Potency option. Then a further dialogue pops up asking for entry of assigned potency for each test preparation.

By default, the program calculates the potency ratio and its confidence limits employing the generalised algorithm given in Finney (1978), which can work with unbalanced, symmetric and balanced designs. Alternatively, the more restrictive algorithm for balanced assays (see European Pharmacopoeia 1997-2008) can be employed by entering the following line in Documents\Unistat60\Unistat60.ini file under the [Bioassay] section:

   ParalEuroPharma=1

Note that this line also affects the Comparison of Slopes output above. For balanced designs both methods produce exactly the same estimates.

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 assigned 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:

     

where E is the sum of squares for the Linear Regression term and s² is the residual mean squares from the ANOVA table.  is the critical value from the t‑distribution with the same degrees of freedom. The confidence limits computed below are reliable for g < 1. If this condition is not fulfilled, the program will issue a warning message, but still display the confidence limits computed.

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:

     

where E is the sum of squares for the Linear Regression term and 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.

The log of the corrected potency estimate and its confidence intervals are computed as:

     

where:

     

Note that the only difference from the default output here is reporting of C and H constants for validation purposes, where C = 1 / (1 - g).

10.1.3.8. Plot of Treatment Means

This option generates a Plot of Treatment Means against the log of dose. It provides a visual means of inspecting the data, enabling the user to notice immediately whether there is something substantially wrong with the data. In the following example, for instance, the slope of Preparation T is quite different from that of Standard and Preparation U.

Clicking the [Opt] button situated to the left of the Plot of Treatment Means option will place the graph in UNISTAT’s Graphics Editor. Each preparation will be plotted as one data series, with as many points as the number of doses applied. A line of best fit will be drawn for each series, including the standard and all test preparations.

The plot can be further customised and annotated using the tools available under UNISTAT Graphics Window’s Edit menu.

10.1.4. Examples

The following Parallel Line Method examples are based on different Analysis of Variance methods. The data sets were entered into UNISTAT’s spreadsheet and the necessary data manipulations made by using UNISTAT spreadsheet functions (see 10.1.1. Data Preparation). The final data sets were saved in two files; BIOPHARMA6 which contains examples from European Pharmacopoeia (2008, the 6^th edition) and BIOFINNEY containing examples from Finney (1978).

10.1.4.1. Completely Randomised Design with 2 Doses and 3 Preparations

Data is given in Table 5.1.1-I. on p. 582 of European Pharmacopoeia (2008).

Open BIOPHARMA6 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

Completely Randomised Design

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation
0.25 × Standard S	10	332.0000	32.0416
0.25 × Preparation T	10	323.9000	26.9256
0.25 × Preparation U	10	282.2000	29.2339
1 × Standard S	10	248.4000	21.9960
1 × Preparation T	10	244.0000	26.8080
1 × Preparation U	10	250.0000	28.0119

 

Dose×Preparations	Shapiro-Wilk Test	Probability	Kolmogorov-Smirnov Test	* Probability
0.25 × Standard S	0.9565	0.7451	0.1538	0.2000
0.25 × Preparation T	0.9471	0.6348	0.1429	0.2000
0.25 × Preparation U	0.8940	0.1878	0.2235	0.1639
1 × Standard S	0.9302	0.4494	0.2135	0.2000
1 × Preparation T	0.9475	0.6390	0.1446	0.2000
1 × Preparation U	0.9515	0.6864	0.1324	0.2000

 

Dose×Preparations	Cramer-von Mises Test	Probability	Anderson-Darling Test	Probability
0.25 × Standard S	0.0331	0.7759	0.2218	0.7658
0.25 × Preparation T	0.0333	0.7721	0.2311	0.7326
0.25 × Preparation U	0.0895	0.1360	0.5030	0.1549
1 × Standard S	0.0579	0.3692	0.3494	0.3962
1 × Preparation T	0.0341	0.7582	0.2337	0.7232
1 × Preparation U	0.0278	0.8580	0.2055	0.8201

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	1.2810	0.9369
Bartlett-Box F Test	0.2575	0.9362
Cochran's C (max var / sum var)	0.2235	1.0000
Hartley's F (max var / min var)	2.1220	0.0500	p > 0.05
Levene's F Test	0.3738	0.8644

 

Response Totals and Contrasts

Dose	Standard S	Preparation T	Preparation U	Total
0.25	3320.0000	3239.0000	2822.0000
1	2484.0000	2440.0000	2500.0000
Total	5804.0000	5679.0000	5322.0000	16805.0000
Linear Contrast	-836.0000	-799.0000	-322.0000	-1957.0000

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	4706800.417	1	4706800.417
Preparations	6256.633	2	3128.317	4.086	0.0223
Linear Regression	63830.817	1	63830.817	83.377	0.0000
Non-parallelism	8218.233	2	4109.117	5.367	0.0075
Treatments	78305.683	5	15661.137
Residual	41340.900	54	765.572
Total	119646.583	59	2027.908

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
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	R-squared
Standard S	257.5833	-47.0559	49559.1333	0.5629
Preparation T	251.3333
Preparation U	233.4833

 

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	**
Preparation T – Standard S	0.9678	-26.0022	31.3401

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	1.0000	1.1420	0.7836	1.6869
Preparation U	1.0000	1.6689	1.1481	2.5550

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0348	19.7070	68.6179
Preparation U	0.0377	8.1229	68.7960

 

G =	0.0482
C =	1.0507

 

 

Looking at the plot of treatment means we can see that Preparation U line is not parallel to Standard S and Preparation T lines. This can also be picked up from the non-parallelism 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 Add-In Mode, you can simply select the block A1:C41 and repeat the analysis. In Stand-Alone 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 41-60 Omitted

Selected by C4 Select

Completely Randomised Design

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation
0.25 × Standard S	10	332.0000	32.0416
0.25 × Preparation T	10	323.9000	26.9256
1 × Standard S	10	248.4000	21.9960
1 × Preparation T	10	244.0000	26.8080

 

Dose×Preparations	Shapiro-Wilk Test	Probability	Kolmogorov-Smirnov Test	* Probability
0.25 × Standard S	0.9565	0.7451	0.1538	0.2000
0.25 × Preparation T	0.9471	0.6348	0.1429	0.2000
1 × Standard S	0.9302	0.4494	0.2135	0.2000
1 × Preparation T	0.9475	0.6390	0.1446	0.2000

 

Dose×Preparations	Cramer-von Mises Test	Probability	Anderson-Darling Test	Probability
0.25 × Standard S	0.0331	0.7759	0.2218	0.7658
0.25 × Preparation T	0.0333	0.7721	0.2311	0.7326
1 × Standard S	0.0579	0.3692	0.3494	0.3962
1 × Preparation T	0.0341	0.7582	0.2337	0.7232

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	1.1985	0.7534
Bartlett-Box F Test	0.4029	0.7509
Cochran's C (max var / sum var)	0.3475	0.6641
Hartley's F (max var / min var)	2.1220	0.0500	p > 0.05
Levene's F Test	0.4381	0.7271

 

Response Totals and Contrasts

Dose	Standard S	Preparation T	Total
0.25	3320.0000	3239.0000
1	2484.0000	2440.0000
Total	5804.0000	5679.0000	11483.0000
Linear Contrast	-836.0000	-799.0000	-1635.0000

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	3296482.225	1	3296482.225
Preparations	390.625	1	390.625	0.529	0.4718
Linear Regression	66830.625	1	66830.625	90.491	0.0000
Non-parallelism	34.225	1	34.225	0.046	0.8308
Treatments	67255.475	3	22418.492
Residual	26587.300	36	738.536
Total	93842.775	39	2406.225

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
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	R-squared
Standard S	249.3250	-58.9702	26621.5250	0.7151
Preparation T	243.0750

 

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

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	1.0000	1.1118	0.8250	1.5136

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0214	34.6983	74.2012

 

G =	0.0455
C =	1.0476

 

 

Note that the estimated potency was calculated with the default assigned potency value of 1 for Preparation U.

In Stand-Alone 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 value.

10.1.4.2. Completely Randomised Design with 5 Doses and 4 Preparations

Data is given in Table 5.1.4-I. on p. 585 of European Pharmacopoeia (2008).

Open BIOPHARMA6 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the first option Completely Randomised Design and then select columns C15, C16 and L17 respectively as [Data], [Dose] and [Preparation]. 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 Anderson-Darling Test and Report summary statistics boxes. Then click [Back] and [Finish] to display the following output:

Parallel Line Method

Completely Randomised Design

Normality Tests

Smaller probabilities indicate non-normality.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation	Anderson-Darling Test	Probability
0.0625 × Standard S	3	-3.0745	0.0884	0.2663	0.3634
0.125 × Standard S	3	-2.3963	0.0960	0.2303	0.4841
0.25 × Standard S	3	-1.8351	0.0377	0.1976	0.5929
0.5 × Standard S	3	-1.1664	0.1318	0.3365	0.2031
1 × Standard S	3	-0.6352	0.0293	0.1941	0.6090
0.0625 × Preparation T	3	-2.3435	0.0181	0.4878	0.0565
0.125 × Preparation T	3	-1.7891	0.0628	0.1896	0.6303
0.25 × Preparation T	3	-1.0725	0.0417	0.2231	0.5077
0.5 × Preparation T	3	-0.5503	0.1416	0.1896	0.6304
1 × Preparation T	3	0.1691	0.1422	0.2501	0.4141
0.0625 × Preparation U	3	-2.5719	0.1036	0.3560	0.1719
0.125 × Preparation U	3	-2.0017	0.0710	0.2307	0.4827
0.25 × Preparation U	3	-1.3045	0.0181	0.3835	0.1353
0.5 × Preparation U	3	-0.6183	0.0912	0.2070	0.5544
1 × Preparation U	3	-0.0480	0.0940	0.1910	0.6236
0.0625 × Preparation V	3	-2.4852	0.0275	0.4878	0.0565
0.125 × Preparation V	3	-1.8745	0.1040	0.4518	0.0768
0.25 × Preparation V	3	-1.1606	0.0206	0.3403	0.1967
0.5 × Preparation V	3	-0.5539	0.0713	0.4738	0.0637
1 × Preparation V	3	0.0468	0.0165	0.4238	0.0975

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	25.6778	0.1394
Bartlett-Box F Test	1.3733	0.1319
Cochran's C (max var / sum var)	0.1514	0.8834
Hartley's F (max var / min var)	74.4176
Levene's F Test	2.1145	0.0230

 

Response Totals and Contrasts

Dose	Standard S	Preparation T	Preparation U	Preparation V	Total
0.0625	-9.2236
0.125	-7.1888
0.25	-5.5054
0.5	-3.4992
1	-1.9055
0.0625		-7.0305
0.125		-5.3672
0.25		-3.2176
0.5		-1.6508
1		0.5072
0.0625			-7.7158
0.125			-6.0051
0.25			-3.9135
0.5			-1.8550
1			-0.1439
0.0625				-7.4555
0.125				-5.6234
0.25				-3.4819
0.5				-1.6618
1				0.1404
Total	-27.3225	-16.7590	-19.6333	-18.0822	-81.7970

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	111.513	1	111.513
Preparations	4.475	3	1.492	223.395	0.0000
Linear Regression	47.584	1	47.584	7125.912	0.0000
Non-parallelism	0.019	3	0.006	0.933	0.4339
Non-linearity	0.074	12	0.006	0.926	0.5307
Treatments	52.152	19	2.745
Residual	0.267	40	0.007
Total	52.419	59	0.888

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
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	R-squared
Standard S	-0.5621	0.9085	0.3600	0.9925
Preparation T	0.1422
Preparation U	-0.0495
Preparation V	0.0539

 

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
Preparation V - Standard S	0.4262	-0.0345	0.1141
Preparation T - Standard S	0.8033	-0.0519	0.0967

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	1.0000	2.1710	2.0272	2.3270
Preparation U	1.0000	1.7581	1.6435	1.8820
Preparation V	1.0000	1.9701	1.8406	2.1103

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0012	181.8564	93.3790
Preparation U	0.0011	287.1655	93.4785
Preparation V	0.0011	224.6942	93.4289

 

G =	0.0006
C =	1.0006

 

 

Note that all samples have an assigned potency of 20 μg protein/ml. Next click on the [Last Procedure Dialogue] button on the Output Medium Toolbar. This will display the Output Options Dialogue again. Click the [Opt] button situated to the left of the Potency option, enter 20 for each test preparation and click [Finish].

Parallel Line Method

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	20.0000	43.4197	40.5448	46.5397
Preparation U	20.0000	35.1628	32.8697	37.6403
Preparation V	20.0000	39.4018	36.8126	42.2058

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0012	0.4546	93.3790
Preparation U	0.0011	0.7179	93.4785
Preparation V	0.0011	0.5617	93.4289

 

G =	0.0006
C =	1.0006

 

10.1.4.3. Randomised Block Design with 4 Doses and 2 Preparations

Data is given in Table 5.1.3.-I on p. 585 of European Pharmacopoeia (2008).

Open BIOPHARMA6 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the second option Randomised Block Design and the select columns C10, C11, L12 and C13 respectively as [Data], [Dose], [Preparation] and [Row Factor]. Click [Next] to proceed to the Output Options Dialogue.

Parallel Line Method

Randomised Block Design

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation
1 × Standard S	5	246.6000	6.7305
1 × Preparation T	5	237.4000	6.4653
1.5 × Standard S	5	203.0000	6.1644
1.5 × Preparation T	5	195.4000	7.5033
2.25 × Standard S	5	162.4000	17.2714
2.25 × Preparation T	5	150.4000	5.5946
3.375 × Standard S	5	107.4000	5.7706
3.375 × Preparation T	5	104.4000	7.2319

 

Dose×Preparations	Shapiro-Wilk Test	Probability	Kolmogorov-Smirnov Test	* Probability
1 × Standard S	0.7977	0.0777	0.3237	0.0942
1 × Preparation T	0.9171	0.5116	0.1982	0.2000
1.5 × Standard S	0.7607	0.0373	0.3418	0.0568
1.5 × Preparation T	0.9649	0.8416	0.2156	0.2000
2.25 × Standard S	0.9904	0.9809	0.1729	0.2000
2.25 × Preparation T	0.8523	0.2018	0.2660	0.2000
3.375 × Standard S	0.8977	0.3974	0.2724	0.2000
3.375 × Preparation T	0.9718	0.8866	0.2125	0.2000

 

Dose×Preparations	Cramer-von Mises Test	Probability	Anderson-Darling Test	Probability
1 × Standard S	0.1018	0.0770	0.5636	0.0681
1 × Preparation T	0.0413	0.5834	0.2658	0.5150
1.5 × Standard S	0.1095	0.0592	0.6235	0.0447
1.5 × Preparation T	0.0381	0.6470	0.2243	0.6507
2.25 × Standard S	0.0238	0.8949	0.1660	0.8710
2.25 × Preparation T	0.0666	0.2535	0.4027	0.2095
3.375 × Standard S	0.0559	0.3616	0.3395	0.3235
3.375 × Preparation T	0.0355	0.6990	0.2191	0.6722

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	9.7854	0.2011
Bartlett-Box F Test	1.4146	0.1953
Cochran's C (max var / sum var)	0.5000	0.0039
Hartley's F (max var / min var)	9.5304	0.0500	p > 0.05
Levene's F Test	1.3017	0.2813

 

Response Totals and Contrasts

Dose	Standard S	Preparation T	Total
1	1233.0000	1187.0000
1.5	1015.0000	977.0000
2.25	812.0000	752.0000
3.375	537.0000	522.0000
Total	3597.0000	3438.0000	7035.0000
Linear Contrast	-2291.0000	-2220.0000	-4511.0000
Quadratic Contrast	-57.0000	-20.0000	-77.0000
Cubic Contrast	-87.0000	10.0000	-77.0000

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	1237280.625	1	1237280.625
Preparations	632.025	1	632.025	11.722	0.0019
Linear Regression	101745.605	1	101745.605	1887.111	0.0000
Non-parallelism	25.205	1	25.205	0.467	0.4998
Non-linearity	259.140	4	64.785	1.202	0.3321
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
Blocks(Rows)	876.750	4	219.188	4.065	0.0101
Residual	1509.650	28	53.916
Total	105048.375	39	2693.548

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
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	R-squared
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

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	1.0000	1.0741	1.0291	1.1214

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0004	1971.4511	95.8129

 

G =	0.0022
C =	1.0022

 

 

The assigned potency for the test preparation is 20,000 IU/vial and we also need to apply a correction factor of 0.89512 because dilutions were not exactly equipotent on the basis of the assigned potency. Next click on the [Last Procedure Dialogue] button on the Output Medium Toolbar. This will display the Output Options Dialogue again. Click the [Opt] button situated to the left of the Potency option, enter 20000 * 0.89512 and click [Finish]. You can also enter the complete expression as:

      20000 * (670*16.7/25)/(20000*1/40)

as UNISTAT numeric input boxes now accept formulas.

Parallel Line Method

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	17902.4000	19228.4755	18423.3508	20075.1776

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0004	0.0000	95.8129

 

G =	0.0022
C =	1.0022

 

10.1.4.4. Latin Squares Design with 3 Doses and 2 Preparations

Data is given in Table 5.1.2.-II on p. 584 of European Pharmacopoeia (2008).

Note that 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 BIOPHARMA6 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the third option Latin Squares Design and then select columns C5, C6, L7, C8 and C9 respectively as [Data], [Dose], [Preparation], [Row Factor] and [Column Factor]. Click [Next] to proceed to Output Options Dialogue.

Parallel Line Method

Latin Squares Design

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation
1 × Standard S	6	158.6667	6.5929
1 × Preparation T	6	156.1667	4.7081
1.5 × Standard S	6	176.5000	5.3572
1.5 × Preparation T	6	174.6667	8.6641
2.25 × Standard S	6	194.5000	4.8477
2.25 × Preparation T	6	195.5000	4.0373

 

Dose×Preparations	Shapiro-Wilk Test	Probability	Kolmogorov-Smirnov Test	* Probability
1 × Standard S	0.8581	0.1827	0.3050	0.0851
1 × Preparation T	0.8118	0.0749	0.2922	0.1177
1.5 × Standard S	0.8965	0.3536	0.2038	0.2000
1.5 × Preparation T	0.9757	0.9284	0.1639	0.2000
2.25 × Standard S	0.9879	0.9835	0.1364	0.2000
2.25 × Preparation T	0.8255	0.0984	0.3115	0.0703

 

Dose×Preparations	Cramer-von Mises Test	Probability	Anderson-Darling Test	Probability
1 × Standard S	0.0849	0.1450	0.4756	0.1438
1 × Preparation T	0.0886	0.1276	0.5470	0.0901
1.5 × Standard S	0.0544	0.3902	0.3341	0.3689
1.5 × Preparation T	0.0274	0.8514	0.1760	0.8637
2.25 × Standard S	0.0210	0.9388	0.1514	0.9165
2.25 × Preparation T	0.1046	0.0740	0.5613	0.0818

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	3.7817	0.5813
Bartlett-Box F Test	0.7637	0.5760
Cochran's C (max var / sum var)	0.3588	0.2313
Hartley's F (max var / min var)	4.6053	0.0500	p > 0.05
Levene's F Test	1.5818	0.1954

 

Response Totals and Contrasts

Dose	Standard S	Preparation T	Total
1	952.0000	937.0000
1.5	1059.0000	1048.0000
2.25	1167.0000	1173.0000
Total	3178.0000	3158.0000	6336.0000
Linear Contrast	215.0000	236.0000	451.0000
Quadratic Contrast	1.0000	14.0000	15.0000

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	1115136.000	1	1115136.000
Preparations	11.111	1	11.111	0.535	0.4730
Linear Regression	8475.042	1	8475.042	408.108	0.0000
Non-parallelism	18.375	1	18.375	0.885	0.3581
Non-linearity	5.472	2	2.736	0.132	0.8773
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
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

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
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	R-squared
Standard S	157.7639	46.3460	1069.8472	0.8879
Preparation T	156.6528

 

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

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	1.0000	0.9763	0.9112	1.0456

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0011	963.9008	93.3289

 

G =	0.0107
C =	1.0108

 

 

The assigned potency for the test preparation is 5600 IU/mg and we also need to apply a correction factor of 0.99799 because dilutions were not exactly equipotent on the basis of the assigned potency. Next click on the [Last Procedure Dialogue] button on the Output Medium Toolbar. This will display the Output Options Dialogue again. Click the [Opt] button situated to the left of the Potency option, enter 5600 * 0.99799 and click [Finish]. You can also enter the complete expression as:

      5600 * (4855*25.2/24.5)/(5600*21.4/23.95)

as UNISTAT numeric input boxes now accept formulas.

Parallel Line Method

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Preparation T	5588.7440	5456.3512	5092.3546	5843.3456

 

Test Preparation	Variance	Weight	% Precision
Preparation T	0.0011	0.0000	93.3289

 

G =	0.0107
C =	1.0108

 

10.1.4.5. Twin Crossover Design

Data is given in Table 5.1.5-II. on p. 586 of European Pharmacopoeia (2008).

Open BIOPHARMA6 and select Bioassay → Parallel Line Method. From the Variable Selection Dialogue select the fourth option Crossover Design and select columns C18, C19, L20, C21 and C22 respectively as [Data], [Dose], [Preparation], [Row Factor] and [Column Factor]. Click [Next] to proceed to Output Options Dialogue. Click the [Opt] button situated to the left of the Potency option. Enter 40 as the assigned potency for the unknown. Click [Back] to get back to output options, click [All] to perform all tests in one go and then click [Finish]. The following output is obtained:

Parallel Line Method

Crossover Design

Normality Tests

Smaller probabilities indicate non-normality.

* Lilliefors probability = 0.2 means 0.2 or greater.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation
1 × Standard	16	101.1875	30.5160
1 × Test	16	91.5625	26.4448
2 × Standard	16	68.1250	18.8428
2 × Test	16	77.5625	29.8060

 

Dose×Preparations	Shapiro-Wilk Test	Probability	Kolmogorov-Smirnov Test	* Probability
1 × Standard	0.9507	0.5009	0.1380	0.2000
1 × Test	0.9229	0.1876	0.1453	0.2000
2 × Standard	0.9112	0.1216	0.1901	0.1220
2 × Test	0.9278	0.2253	0.1260	0.2000

 

Dose×Preparations	Cramer-von Mises Test	Probability	Anderson-Darling Test	Probability
1 × Standard	0.0470	0.5319	0.2966	0.5482
1 × Test	0.0542	0.4284	0.4079	0.3070
2 × Standard	0.0806	0.1890	0.4974	0.1809
2 × Test	0.0636	0.3197	0.4091	0.3049

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	3.7999	0.2839
Bartlett-Box F Test	1.2736	0.2815
Cochran's C (max var / sum var)	0.3240	0.6856
Hartley's F (max var / min var)	2.6228
Levene's F Test	2.1683	0.1011

 

 

Response Totals and Contrasts

Dose	Standard	Test	Total
Days: 1
1	765.0000	719.0000
2	557.0000	579.0000
Total	1322.0000	1298.0000	2620.0000
Days: 2
1	854.0000	746.0000
2	533.0000	662.0000
Total	1387.0000	1408.0000	2795.0000
Preparations
Total	2709.0000	2706.0000	5415.0000
Linear Contrast
Days: 1	-208.0000	-140.0000	-348.0000
Days: 2	-321.0000	-84.0000	-405.0000
Total	-529.0000	-224.0000	-753.0000

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	458159.7656	1	458159.7656
Non-parallelism	1453.5156	1	1453.5156	1.0638	0.3112
Days×Preparations	31.6406	1	31.6406	0.0232	0.8801
Days×Linear 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
Days×Non-parallelism	446.2656	1	446.2656	3.2506	0.0822
Error Within	3844.0625	28	137.2879
Total	53423.2344	63	847.9878

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
Standard	101.1875	-47.6991	19294.1875	0.3119
Test	91.5625	-20.1977	23815.8750	0.0618

 

Common Regression

	Intercept	Slope	Residual SS	R-squared
Standard	96.4219	-33.9484	44563.5781	0.1658
Test	96.3281

 

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	**

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Test	40.0000	40.1106	33.4162	48.1646

 

Test Preparation	Variance	Weight	% Precision
Test	0.0074	0.0772	83.3102

 

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 non-parallelism test in Validity of Assay (0.3112) is not significant at 5% level.

10.1.4.6. 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 [Column Factor]. Click [Next] to proceed to Output Options Dialogue. Click on the [Opt] button situated to the left of Normality Tests option and check the Shapiro-Wilk Test and Report summary statistics boxes. Then click [Back] and [Finish].

Parallel Line Method

Crossover Design

Normality Tests

Smaller probabilities indicate non-normality.

 

Dose×Preparations	Valid Cases	Mean	Standard Deviation	Shapiro-Wilk Test	Probability
0.5 × Standard	10	3.7120	0.1946	0.9485	0.6508
1.25 × Test	10	4.0700	0.3391	0.9285	0.4330
1 × Standard	10	3.2550	0.7950	0.9026	0.2341
2.5 × Test	10	3.4630	0.5483	0.9046	0.2459
2 × Standard	10	3.3970	0.4072	0.9038	0.2408
5 × Test	10	3.2780	0.3720	0.9058	0.2532

 

Homogeneity of Variance Tests

	Test Statistic	Probability
Bartlett's Chi-square Test	18.0814	0.0028
Bartlett-Box F Test	3.6473	0.0027
Cochran's C (max var / sum var)	0.4548	0.0035
Hartley's F (max var / min var)	16.6848	0.0100	p < 0.01
Levene's F Test	6.6315	0.0001

 

Response Totals and Contrasts

Dose	Standard	Test	Total
Days: 1
1	18.7200	19.8100
2	14.6900	19.0100
3	17.0300	16.1400
Total	50.4400	54.9600	105.4000
Days: 2
1	18.4000	20.8900
2	17.8600	15.6200
3	16.9400	16.6400
Total	53.2000	53.1500	106.3500
Preparations
Total	103.6400	108.1100	211.7500
Linear Contrast
Days: 1	-1.6900	-3.6700	-5.3600
Days: 2	-1.4600	-4.2500	-5.7100
Total	-3.1500	-7.9200	-11.0700
Quadratic Contrast
Days: 1	6.3700	-2.0700	4.3000
Days: 2	-0.3800	6.2900	5.9100
Total	5.9900	4.2200	10.2100

 

Validity of Assay

Due To	Sum of Squares	DoF	Mean Square	F-Stat	Prob
Constant	747.3010	1	747.3010
Non-parallelism	0.5688	1	0.5688	1.6032	0.2176
Quadratic Regression	0.8687	1	0.8687	2.4484	0.1307
Days×Preparations	0.3481	1	0.3481	0.9810	0.3318
Days×Linear Regression	0.0031	1	0.0031	0.0086	0.9267
Days×Quadratic 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
Days×Non-parallelism	0.0164	1	0.0164	0.2335	0.6333
Days×Quadratic Regression	0.0216	1	0.0216	0.3075	0.5844
Error Within	1.6861	24	0.0703
Total	17.3685	59	0.2944

 

Separate Regression

	Intercept	Slope	Residual SS	R-squared
Standard	3.4547	-0.2272	8.1200	0.0576
Test	4.1272	-0.5713	5.2830	0.3725

 

Common Regression

	Intercept	Slope	Residual SS	R-squared
Standard	3.4547	-0.3993	13.9718	0.1798
Test	3.9695

 

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	**

 

Potency

Test Preparation	Assigned Potency	Estimated Potency	Lower 95%	Upper 95%
Test	1.0000	0.2754	0.1783	0.3923

 

Test Preparation	Variance	Weight	% Precision
Test	0.0326	372.0902	64.7516

 

G =	0.0977
C =	1.1083