# 㞚性数掉分析|STAT 770/BIOS 805/BIOSTAT695/PSY 525/625/SOCI612/STA 4504/STA 517Analysis of Categorical Data代写

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$$\operatorname{Pr}\left(T_{i}>t \mid T_{i} \geq t\right)=1-p_{i t} .$$
The discrete-time survivor function can be expressed as the product of the conditional probabilities of having “survived” all previous time points or time intervals, as
$$S_{i t}=\operatorname{Pr}\left(T_{i} \geq t\right)=\prod_{s=1}^{t-1}\left(1-p_{i s}\right) .$$
The unconditional event probability (or probability of experiencing the event at time $t$ ) is the discrete-time analog of the continuous-time probability distribution function and may be written as
\begin{aligned} \operatorname{Pr}\left(T_{i}=t\right) &=\operatorname{Pr}\left(T_{i}=t \mid T_{i} \geq t\right) \operatorname{Pr}\left(T_{i} \geq t\right) \ &=p_{i t} S_{i t} \ &=p_{i t} \prod_{s=1}^{t-1}\left(1-p_{i s}\right) \end{aligned}

## STAT 770/BIOS 805/BIOSTAT695/PSY 525/625/SOCI612/STA 4504/STA 517 COURSE NOTES ：

We assume that, conditional on $v$, the hazard rate is a product of an underlying hazard $\lambda(t)$ and the multiplicative frailty,
$$\lambda(t \mid v)=\lambda(t) v,$$
and that frailty (v) follows a gamma distribution,
$$g(v)=\frac{\alpha^{\alpha} v^{\alpha-1}}{\Gamma(\alpha)} \exp (-\alpha v) \quad \text { where } \alpha>0,$$
with mean $\mathrm{E}(v)=1$ and $\operatorname{var}(v)=1 / \alpha=\phi$.

# 㞚性数掉分析 | Analysis of Categorical Data代写 STAT 341代考

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$$f(u)=\exp (u) \exp {-\exp (u)}$$
where $\mathrm{E}(U)=0.5704$ and $\operatorname{var}(U)=\pi^{2} / 6$. The cumulative distribution function has a convenient closed form
$$F(u)=1-\exp {-\exp (u)} .$$

For individual-level data, we may again use the random utility function or latent variable framework
$$y_{i}^{*}=\mathbf{x}{i}^{\prime} \beta+\varepsilon{i},$$

## STAT 341COURSE NOTES ：

Similarly, subscript $+j$ stands for the column marginal total:
$$f_{+j}=\sum_{i=1}^{I} f_{i j} \quad \text { and } \quad F_{+j}=\sum_{i=1}^{I} F_{i j},$$
and subscript $++$ represents the grand total:
$$f_{++}=\sum_{j=1}^{J} \sum_{i=1}^{I} f_{i j} \quad \text { and } \quad F_{++}=\sum_{j=1}^{J} \sum_{i=1}^{I} F_{i j} .$$