The Cochran and Cox Approximation Pair T test Unequal Variance

[mertcan]

Hi everyone I hope you are well. Maybe as you know according to Behners-Fisher problem (unequal variance case of samples) there are some kind of approximations. I have recently covered the Satterthweiths Approximations and comprehended the logic of it. But I got stuck with the Cochran-Cox approximation which is the following: Approximation of the probability level of the approximate t statistic is the value of p such that $$t^{'} = \frac {\bar x_1-\bar x_2} {\sqrt { s_1^2/n1 + s_2^2/n2}} =(s_1^2/n1*t_1+s_2^2/n2*t_2)/(s_1^2/n1+s_2^2/n2)$$
where t1 and t2 are the critical values of the t distribution corresponding to a significance level of p and sample sizes of n1 and n2, respectively besides s_1 and s_2 are corresponds to sample variances. The number of degrees of freedom is undefined when n1=!n2. Could you help me derive it?

 

[mmwave]

Hello,

Thank you for your post. The Cochran-Cox approximation is a method used to approximate the probability level of the t statistic in cases where the samples have unequal variances. It is derived from the Behrens-Fisher problem, which deals with the comparison of means from two populations with unequal variances.

To derive the Cochran-Cox approximation, we start with the t statistic formula:

$$t = \frac {\bar x_1 - \bar x_2} {\sqrt {\frac {s_1^2} {n_1} + \frac {s_2^2} {n_2}}}$$

We can then rewrite this formula as:

$$t = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt {s_1^2 + s_2^2}}$$

Next, we substitute the sample variances with their corresponding critical values from the t distribution:

$$t = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt {s_1^2 + s_2^2}} = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt {\frac {(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2} {n_1 + n_2 - 2}}}$$

We can then simplify this formula to:

$$t = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt {(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}} = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt {\frac {n_1 s_1^2 + n_2 s_2^2} {n_1 + n_2 - 2}}}$$

Finally, we can use the Satterthwaite approximation to further simplify this formula:

$$t = \frac {(\bar x_1 - \bar x_2) \sqrt {n_1 + n_2}} {\sqrt