class: center, middle, inverse, title-slide .title[ # Likelihoods for arbitrary censorship/truncation ] .author[ ### Yue Jiang ] .date[ ### Duke University ] --- ### Estimation for censored data For independently right-censored data, an individual's contribution to the likelihood is `\begin{align*} &\mathrel{\phantom{=}} f(t_i)^{\delta_i}S(t_i)^{1 - \delta_i}\\ &= \lambda(t_i)^{\delta_i}S(t_i)^{\delta_i}S(t_i)^{1 - \delta_i}\\ &= \lambda(t_i)^{\delta_i}S(t_i) \end{align*}` And so the full likelihood, assuming *i.i.d.* failure distributions is `\begin{align*} \prod_{i = 1}^n f(t_i)^{\delta_i}S(t_i)^{1 - \delta_i} = \prod_{i = 1}^n \lambda(t_i)^{\delta_i}S(t_i) \end{align*}` Which may be maximized either parametrically or non-parametrically. --- ### Estimation for censored data What about left-censored data? .question[ - What does it mean for an observation to be left-censored? - What does an *observed* failure time contribute to the likelihood? - What does a *censored* failure time contribute to the likelihood? ] -- <br> .question[ - What is the full contribution for an arbitrary individual to the likelihood for independently censored i.i.d. left-censored data? ] --- ### Estimation for censored data For independently left-censored data, an individual's contribution to the likelihood is `\begin{align*} &\mathrel{\phantom{=}} f(t_i)^{\delta_i}F(t_i)^{1 - \delta_i} \end{align*}` And so the full likelihood, assuming *i.i.d.* failure distributions is `\begin{align*} \prod_{i = 1}^n f(t_i)^{\delta_i}F(t_i)^{1 - \delta_i} \end{align*}` --- ### Estimation for interval censored-data .question[ Using similar arguments to what we've seen before, how about *interval* censored data, where observations are censored in the interval `\((u_{L_i}, u_{U_i}\)`) under the same assumptions? ] --- ### Estimation for censored data `\begin{align*} \prod_{i = 1}^n f(t_i)^{\delta_i}\left(F(t_{U_i}) - F(t_{L_i})\right)^{1 - \delta_i} \end{align*}` --- ### Back to truncation .quesiton[ What is the difference between truncation and censoring? - What might be a real-world example of left-truncated data? - What might be a real-world example of right-truncated data? - What might be a real-world example of *interval*-truncated data? ] --- ### Truncated likelihood contributions (Consider left truncation vs. left censoring). Remember, in left censoring, we know of the existence of someone with a failure at time `\(T < t\)`. However, in left truncated data, we do not observe them at all. .question[ Suppose observations are left truncated at time `\(u\)`. What would be the likelihood contributions of *observed failures* at time `\(t_i\)`? ] --- ### Truncated likelihood contributions We *know* that the failure time `\(T\)` has to be greater than the truncation time `\(u\)` for left truncated data, and observe a failure at time `\(t_i\)`. The individual likelihood contribution is: `\begin{align*} f(t_i | T > u) = \frac{f(t_i)}{S(u)} \end{align*}` --- ### Truncated likelihood contributions Similarly, for right truncation at time `\(w\)`, the likelihood contribution is: `\begin{align*} f(t_i | T < w) = \frac{f(t_i)}{F(w)} \end{align*}` --- ### Truncated likelihood contributions And finally, for interval truncated data between times `\(u\)` and `\(w\)`, the likelihood contribution is: `\begin{align*} f(t_i | u < T < w) = \frac{f(t_i)}{F(w) - F(u)} \end{align*}` --- ### Right censored, left truncated data? .question[ Suppose you have a population of (potentially) right censored data with left truncation. What would the individual likelihood contributions look like? How about the full likelihood to maximize, assuming *i.i.d.* failure times, independent censoring, and common known truncation time? ] --- ### Right censored, left truncated data? `\begin{align*} &\mathrel{\phantom{=}}\prod_{i = 1}^n f(t_i | T > u)^{\delta_i}S(t_i | T > u)^{1 - \delta_i} \\ &= \prod_{i = 1}^n \left(\frac{f(t_i)}{S(u)}\right)^{\delta_i}\left(\frac{S(T_i)}{S(u)}\right)^{1 - \delta_i}\\ &= \prod_{i \in obs} \frac{f(t_i)}{S(u)}\prod_{j \in cens} \frac{S(t_j)}{S(u)} \end{align*}` --- ### Arbitrary combinations .question[ What might the full likelihood look like for arbitrary combinations of censoring and truncation of various types? ] -- We might think of breaking down the population into the following four groups. assuming independence, we could simply multiply the contributions of the following groups: - untruncated and uncensored - untruncated and censored - truncated and uncensored - truncated and censored