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## F(a, K — 1, N — K — 1) is the ath upper quantile of a central F K — 1 and N — K —

If the preliminary tests for the equality of slopes and the equality of intercepts are not rejected at the 0.25 level of significance for the null hypothesis of batch similarity of the degradation lines among batches, all batches are considered from the same population of production batches with a common degradation pattern. As a result, model (11.3.1) reduces to

= a + f3X,, + i = 1, . . . ,K, j= 1, . . ,n„ (11.4.1) where a and p are the common intercept and slope for model (11.4.1) and the same normality assumption is posed for

The procedure for estimating an expiration dating period of single batch described in Sec. 11.2 can be applied directly to the pooled stability data from all batches. The least-squares estimates of p, a, and o-2 are then given by

SS(T) b

S(T)' a, = Y — bX.,

Syy(T) — bS (T) S` (11.4.2) N — 2

where S(T), Sw(T), SS(T), X , and N are as defined in (11.3.14). The leastsquares estimate of the mean degradation line at time point t = x is given as

y(x) = a, + bcx (11.4.3) with its least-squares estimate of the variance

firy(x)i s _ X)2-I (11.4.4) LN

Therefore, the 95% lower confidence limit for the mean degradation line is given as

Lc(x) = a, + b,x — N — 2)SE(x), where

(11.4.5)

SE(x) = "V' [y(x)],

and t(0.05, N — 2) is the 5% upper quantile of a central t distribution with N — 2 degrees of freedom. Hence the overall expiration dating period can be estimated as the small root xL(c) of the following quadratic equation:

[11 — (a, + bcx)]2 = t2(0.05, N — [ 1 + (11.4.6) N Sx,(T)]• The conditions for the existence of the root xL(c) for the quadratic equation above are the same as those given in (11.2.10) and (11.2.11) for a single batch. However, the standard error of slope and intercept estimated from the pooled stability data should be used to evaluate the two conditions.