Linearity, cochran test

1. Linearity
The Regulatory Requirements
Chandra Prakash Singh
Research Scientist

2. VALIDATION OF ANALYTICAL PROCEDURES: TEXT AND
METHODOLOGY Q2(R1)
LINEARITY -The linearity of an analytical procedure is its ability (within a given range) to
obtain test results which are directly proportional to the concentration (amount) of analyte in
the sample.
Linearity should be evaluated by visual inspection of a plot of signals as a function of analyte
concentration or content.
If there is a linear relationship, test results should be evaluated by appropriate statistical
methods, for example, by calculation of a regression line by the method of least squares.
In some cases, to obtain linearity between assays and sample concentrations, the test data may
need to be subjected to a mathematical transformation prior to the regression analysis.
The correlation coefficient, y-intercept, slope of the regression line and residual sum of
squares should be submitted. A plot of the data should be included.
In addition, an analysis of the deviation of the actual data points from the regression line may
also be helpful for evaluating linearity.

3. Linearity Data
Visual evaluation
Is it visually linear
Mathematical treatment
of data for adaptation of
the model
Variables evaluation (Hypothesis Test)
NO
YES
Is It Homoscedastic? Homoscedastic (OLS)Heteroscedastic (WLS)
YESNO
Anova and correlation (OLS)
Anova and correlation
(WLS)
r value and F test
Residues analysis
Were Requirements Met? Linearity proven
Linearity not proven
(the method may not
be accepted)
NO YES
Brazilian Health Surveillance Agency

4. 1. At least five concentrations in the range of 80 to 120%.
2. Prepare 3 curves separately since the stock solution preparation.
3. Send dispersion graph.
4. Prove the homoscedasticity of the curve with Cochran test.
5. Calculate and report the slope and y-intercept.
6. Apply F test to evaluate the significance of the slope.
7. Evaluate the correlation coefficient r2> 0.990 and determination coefficient
r>0.980 and send the graphic.
8. Send residual analysis (residual plot).
9. Send chromatograms.
10. Significance for all statistical analysis shall be 5%.
Linearity – Regulatory Requirements

5. To test homoscedasticity or heteroscedasticity for linearity data, we must consider the
variance of the y results for each x value.
If the variance of y is constant than situation called homoscedasticity.
If changes of y variance are observed than situation called heteroscedasticity.
The variance of y values for each level of concentration is calculated as follows:
Where: j represents the j-th result for the replicates set i;
m represents the number of measures (replicates) for the analytical response y in each point.
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)

6. In order to evaluate whether the data is homoscedastic or not, it is recommended to use
the Cochran test. The hypotheses are:
The null hypothesis (H0) is that the variances are all equal to each other and the
alternative hypothesis (H1) is that at least one of the variances is different from the
other ones.
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)

7. The statistics to be tested is:
Where the C value is the ratio between the greater variance observed for y datasets and
the sum of y variances observed for all levels of concentration.
The C value calculated should be compared to the critical value at a 5% significance
level. The conclusion of the test will be:
If C < Ccritical, the null hypothesis is accepted (homoscedastic data).
If C ≥ Ccritical, the null hypothesis is rejected (heteroscedastic data).
Critical C values at a 5% significance level are described in below table.
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)

8. Number of points
No. of measures (replicates) for variable y
2 3 4 5
5 0.841 0.684 0.598 0.544
6 0.781 0.616 0.532 0.480
7 0.727 0.561 0.480 0.431
8 0.680 0.516 0.438 0.391
9 0.638 0.478 0.403 0.358
10 0.602 0.445 0.373 0.331
11 0.570 0.417 0.348 0.308
12 0.541 0.392 0.326 0.288
13 0.515 0.371 0.307 0.271
14 0.492 0.352 0.291 0.255
15 0.471 0.335 0.276 0.242
16 0.452 0.319 0.262 0.230
17 0.434 0.305 0.250 0.219
18 0.418 0.293 0.240 0.209
19 0.403 0.281 0.230 0.200
20 0.389 0.270 0.220 0.192
Critical C values with a 5% significance level.
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)

9. Set -1 Set -2 Set -3
Level
Conc.
(mcg/mL)
Response
(Area)
Level
Conc.
(mcg/mL)
Response
(Area)
Level
Conc.
(mcg/mL)
Response
(Area)
L-1 10.1 101 L-1 9.9 100 L-1 10.0 100
L-2 20.2 200 L-2 19.8 198 L-2 20.0 201
L-3 30.3 303 L-3 29.7 297 L-3 30.0 300
L-4 40.4 404 L-4 39.6 400 L-4 40.0 400
L-5 50.5 505 L-5 49.5 495 L-5 50.0 501
L-6 60.6 603 L-6 59.4 594 L-6 60.0 600
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)
Example:
Decision between homoscedasticity or heteroscedasticity for three set of linearity data.

10. Step-1 - Calculate the variance at each level.
Step-2 - Calculate the sum of all variance in y.
Step-3 – Calculate greater variance in y.
Step-4 – Use the Cochran test using the null hypothesis (H0) and alternative hypothesis (H1).
Step-5 – Calculate the C value by Cochran test .
Step-6 - The C value calculated should be compared to the critical value at a 5% significance
level.
Step -7 - The conclusion of the test will be:
If C < Ccritical, the null hypothesis is accepted (homoscedastic data).
If C ≥ Ccritical, the null hypothesis is rejected (heteroscedastic data).
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)
Where the C value is the ratio between
the greater variance observed for y
datasets and the sum of y variances
observed for all levels of concentration.

11. Level
Set-1 Set-2 Set-3
Variance
Response (Area) Response (Area) Response (Area)
L-1 101 100 100 0.3
L-2 200 198 201 2.3
L-3 303 297 300 9.0
L-4 404 400 400 5.3
L-5 505 495 501 25.3
L-6 603 594 600 21.0
Observed C < C critical, Hence the null
hypothesis is accepted and linearity data is
homoscedastic data.
Sum of VAR 63.3
Max of VAR 25.3
C 0.4
C critical 0.616
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)
Question
Thanks