Decision between homoscedasticity or heteroscedasticity for linearity data (Cochran test)

Decision between homoscedasticity or heteroscedasticity for linearity data (Cochran test)

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 (H ) is that the variances are all equal to each other and the alternative hypothesis (H 1 ) 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 < C critical , the null hypothesis is accepted (homoscedastic data).   If C ≥ C critical , 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 ) Thanks