La exposición está oscuro pero los ejemplos y la discusión en la página. 101 hacer las intenciones claras.
Recordemos que el objetivo (para la situación con un solo continuo de la covariable $x$) es generalizar la regresión logística de la caja
$$\text{logit}(y) = \beta_0 + \beta_1 x$$
to a relatively simple nonlinear expression of the form
$$\text{logit}(y) = \beta_0 + \beta_1 F_1(x) + \cdots + \beta_J F_J(x).$$
"Fractional polynomials" [sic] are expressions of the form
$$F(x) = x^p (\log(x))^q$$
for suitably chosen powers $p$ and $p$, with $p$ a natural number and $p$ a real number close to $1$. It is intended that if a high power $q$ of the logarithm is included, then all lower powers $p-1, p-2, \ldots, 1, 0$ will also be included. To be practicable and interpretable, H&L suggest restricting the values of $p$ to the set $P$ = $\{-2, -1, -1/2, 0, 1/2, 1, 2, 3\}$ ($0$ corresponds to $\el registro$, as usual) and $q$ to the set $\{0,1\}$.
When we limit the fractional polynomial to just two terms ($J=2$), the only possibilities according to these rules are of the form
$$F_1(x) = x^{p_1}, F_2(x) = x^{p_2}$$
for $p_1 \ne p_2$ or
$$F_1(x) = x^p, F_2(x) = x^p\log(x).$$
(The case $p=0$ corresponds to using $F_1(x) = \log(x)$ and $F_2(x) = (\log(x))^2$.)
These possibilities can be uniquely determined by a non-decreasing sequence of $J=2$ elements of $P$. The sequence $(p_1,p_2)$ with $p_2 \gt p_1$ specifies the first kind of fractional polynomial and the sequence $(p_1,p_2) = (p,p)$ specifies the second kind. Because $P$ has eight elements, this gives $\binom{8+1}{2} = 36$ possibilities for $J=2$. For instance, your case of $(-1,-1)$ specifies the model
$$\text{logit}(y) = \beta_0 + \beta_1 \frac{1}{x} + \beta_2 \frac{\log(x)}{x}.$$
(H&L go on to recount an approximate procedure in which partial likelihood ratio tests are used to fit the best model with $J=1$ (there are just eight of these) and then the best model with $J=2$ is fit. Each contributes approximately $2J$ grados de libertad en la prueba de chi-cuadrado.)
Por supuesto, para estar realmente seguro de lo R está haciendo, usted debe buscar en el código fuente, o el ajuste del modelo y de la trama de las predicciones con los datos, o ambos.
