Hi LLLi,

In your reply, you mentioned "Formulas: *θ*’_{L},=[1+ln(*θ*_{L})/ln(*r* )] and *θ*’_{U}=[1+ln(*θ*_{U})/ln(*r*)]. From [0.8, 1.25] and *r* 10 we get [0.9031, 1.0969] and from [0.5, 2] and *r* 10 [0.6990, 1.3010]".

While in Smith's paper, it is said that "the CI of Rdnm completely outside (0.8, 1.25), indicating a disproportionate increase". I am confused about the difference between the reference interval. Which reference interval we should use to evalute whether DP or not?

You are quoting the footnote *b* below Table 2. I guess this is just a typo. Smith used a slightly different terminology (compared to Chow/Liu and Hummel *et al*.). He starts from 0.80 and 1.25 (Θ_{L} and Θ_{U}; page 1279, first paragraph). The transformed acceptance range (he calls it *“the critical region”* and later *“the reference interval”*) is derived in Eq. (4). That’s the same one I used above. Now look at page 1282, left column, second paragraph, which reads:

The corresponding 90% CI (0.679, 0.844) fell outside the reference interval (0.903, 1.097) defined by Eq. (4) for r = 10 and Θ_{U} = 1/Θ_{L} = 1.25, indicating a disproportionate change in C_{max} across the dose range studied.

In other words from the original range [0.80, 1.25] he gets the transformed one [0.903, 1.097] and *this* is what you should use (of course depending on the actual *r* in the study). *q.e.d.*

Furthermore, the paper also said that the Rdnm value of 1 would denote ideal dose-proportionality.

Correct. Let *β* ≡ 1 and *r* ⊂ ℝ. Then *R*_{dnm} = *r*^{ β–1} = *r*^{ 0} = 1. Less mathematical: If the slope is *exactly* 1 then for *any* possible ratio of dose levels *R*_{dnm} will be *exactly* 1.

Rdnm = PK/corresponding dose?

No. *r* = the ratio of the highest/lowest dose and *R*_{dnm} = *r*^{ β–1}. You do not dose-normalize in this model.

For power model of Rdnm, what is Y and what is x?

In my project I used the *linearized* power model, which is

ln(*Y*_{j}) = *α* + *β* · ln(*x*_{j}),

where *Y* is he respective PK metric (AUC, C_{max}, …) and *x* the dose; both at level *j*. Most people prefer the linearized model over the original one – which is

*Y*_{j} = *α* · *x*_{j}^{ β}

because the latter requires nonlinear fitting. If you have a Phoenix/NLME license go ahead with the “pure” model. Anyhow, I would not recommend that because in a regulatory setting the former is more easy to assess than the latter.

However, there is a situation which demands nonlinear fitting: A power model with an *intercept, i.e.*,

*Y*_{j} = *α* + *λ* · *x*_{j}^{ β}.

You would need this model for dosing an endogenous compound and measurable basal levels.