By definition a variable with degrees of freedom is the sum of squares of independent, 2   standard normal variables N(0,1). Remember also that the unbiased estimate of 2 is the sample variance defined by From which we obtain = (n-1) s2. Since this is the sum of squares of normal variables n-1 {u2, which are all independent with means 0 and variance 2, what we have un} …, demonstrated is that, for a random sample of size from a normal population, n , independently of FINALLY let’s return to the for the random normal sample {y1, logLikelihood y2, yn} …, discussed earlier. The last form we looked at on page 4 was Rather than look at the of this set of variables, we look instead at the logLikelihood of the transformed set of variables {u1, for which the Jacobian was logLikelihood u2, un} …, seen to be 1 (and remember = and ): u1 n Given that is normally distributed with mean  and variance 2, we now can separate u1 n out the two terms. Thus, the of the transformed set of variables {u1, logLikelihood u2, …, un} The first is the likelihood for from which can be maximized to provide the ML/REML u1 estimate of The second is the likelihood for the set of variables {u2, which are . u3, …, un} all independent of and this provides the REML estimate of 2. u1,
By definition a 2 variable with  degrees of freedom is the sum of squares of  independent, standard normal variables N(0,1). Remember also that the unbiased estimate of 2 is the sample variance defined by From which we obtain = (n-1) s2. Since this is the sum of squares of n-1 normal variables {u2, …, un} which are all independent with means 0 and variance 2, what we have demonstrated is that, for a random sample of size n from a normal population, , independently of FINALLY let’s return to the logLikelihood for the random normal sample {y1, y2, …, yn} discussed earlier. The last form we looked at on page 4 was Rather than look at the logLikelihood of this set of variables, we look instead at the logLikelihood of the transformed set of variables {u1, u2, …, un} for which the Jacobian was seen to be 1 (and remember u1 = n and ): Given that u1 is normally distributed with mean n  and variance 2, we now can separate out the two terms. Thus, the logLikelihood of the transformed set of variables {u1, u2, …, un} The first is the likelihood for u1 from which can be maximized to provide the ML/REML estimate of . The second is the likelihood for the set of variables {u2, u3, …, un} which are all independent of u1, and this provides the REML estimate of 2.
An%20introduction%20to%20REML.docx