How many knots for a spline fit of baseline hazard? I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) section 5.1 Smooth Baseline Hazard. On p. 108 they suggest two strategies for expanding time varying parameters in basis functions. The first one is to use a small number of basis functions, e.g. $m=4$ or $m=5$. Now recall equation 5.3 for polynomial splines of degree $d$ from p. 106 (it might contain a typo):
$$
\gamma_0(t) = \gamma_{00} + \gamma_{01}t + \dots + \gamma_{0d}t^d + \sum_{i=1}^{m_s} \gamma_{k+1}(t-\tau_i)^d_+ \tag{5.3}
$$
where $(t-\tau_i)^d_+$ are truncated power functions. If we combine the advice with the equation, we get that with a cubic spline ($d=3$) we would be splitting the timeline into... a single interval: $m=m_s+d+1$ and so with $m=5$ and $d=3$ we get $m_s=1$. Does that make sense?


The problem is (once again for this book) overloading or poor editing of symbols, in this case $m$.
The number of knots for splitting the timeline is given by $m_s$. That's what you choose. In the truncated power series basis, you then need $m=m_s+d+1$ total basis functions (including a constant function for the intercept), $d$ more than the $m_s +1$ implied in the text just above the equation.
There is nothing I see in this part of the Tutz and Schmid presentation about splines that is specific to discrete-time data. I find Frank Harrell's explanation in Section 2.4.4 of his course notes or book to be easier to follow, with useful links to further reading.
Those presentations also clarify that a cubic spline restricted to be linear beyond the outermost knots, a natural spline, only needs (in this terminology) $m_s-1$ coefficients (plus the intercept). So you often don't need to use up many degrees of freedom to get smooth fitting without penalization.
With respect to the statement "Choosing a small number of basis functions, say 4 or 5, such that numerically stable estimates of the coefficients exist," I assume that Tutz and Schmid were thinking about the usual implementation via restricted splines. Restricted splines involve fewer basis functions than the full truncated power basis. For example, Frank Harrell recommends 3 to 5 knots in the references above (3 to 5 basis functions if you include the constant for the intercept as Tutz and Schmid do).
Tutz and Schmid don't pay much further attention to unpenalized regression splines in the rest of that Chapter. They seem to prefer penalized splines, which they cover in detail in contexts of modeling both time and continuous covariates. For penalized splines the number of initial knots doesn't matter too much, as the penalization smooths things out.
