This document discusses testing for homoscedasticity or heteroscedasticity in linearity data using the Cochran test. The Cochran test compares the ratio of the largest variance (C value) to the total variances to the critical C value. If C is less than the critical value, the data is homoscedastic. If C is greater than or equal to the critical value, the data is heteroscedastic. An example is provided to calculate variances, C value, and compare to critical value to determine if three data sets exhibit homoscedasticity or heteroscedasticity.

1. Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)
Chandra Prakash Singh

2. 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)

3. 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)

4. 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)

5. 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 0.48
7 0.727 0.561 0.48 0.431
8 0.68 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 0.57 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 0.23
17 0.434 0.305 0.25 0.219
18 0.418 0.293 0.24 0.209
19 0.403 0.281 0.23 0.2
20 0.389 0.27 0.22 0.192
Critical C values with a 5% significance level.
Decision between homoscedasticity or heteroscedasticity for linearity data
(Cochran test)

6. 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.

7. 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.

8. 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)
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