UTS

UTS Topics (Weeks 1-7)

• Week 1: Survival time, censored/truncated data, characteristics (✅ 2026-04-07 09:26)
• Week 2: $S(t)$, $h(t)$, $H(t)$, mrl$(t)$—definitions, relationships, proofs (✅ 2026-04-07 09:26)
• Week 3: Exponential, Weibull, Gamma distributions & properties (✅ 2026-04-07 09:26)
• Week 4: Right/Left/Interval censoring, truncation types (✅ 2026-04-07 09:26)
• Week 5: Likelihood construction for censored/truncated data (✅ 2026-04-07 09:26)
• Week 6: Kaplan-Meier, Nelson-Aalen, Greenwood’s formula, CI (✅ 2026-04-07 09:26)

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Topic Frequency Analysis (UTS 2020-2024)

Counts

Based on analysis of UTS 2020, 2021, 2022, and Responsi Prep 2024:

Basic Quantities & Relationships

Topic	2020	2021	2022	2024	Total
Hazard function $h(t)$ from $S(t)$	✅	✅	✅	✅	4/4
Median calculation	✅	✅	✅	✅	4/4
PDF $f(t)$ from hazard	✅	✅		✅	3/4
Mean/Expected value $E[T]$		✅			1/4
Mean Residual Life mrl$(t)$			✅	✅	2/4
Cumulative hazard $H(t)$			✅	✅	2/4
Relationships ($S=e^{-H}$, $h=f/S$)	✅		✅	✅	3/4

Parametric Models

Topic	2020	2021	2022	2024	Total
Weibull distribution			✅	✅	2/4
Exponential distribution		✅			1/4

Censoring & Truncation

Topic	2020	2021	2022	2024	Total
Right censoring (Type I) identification	✅	✅		✅	3/4
Left truncation identification	✅	✅			2/4
Likelihood construction	✅	✅		✅	3/4
Survival time definition (origin/scale/event)		✅			1/4

Non-Parametric Estimation

Topic	2020	2021	2022	2024	Total
Kaplan-Meier calculation		✅		✅	2/4
Nelson-Aalen estimation		✅		✅	2/4
Survival curve interpretation	✅			✅	2/4
Confidence interval for hazard		✅			1/4

High-Frequency Topics (≥3 appearances)

Must Master

1. Hazard function calculation — Given $S(t)$ or other quantities, find $h(t)$
2. Median survival time — Find $t$ where $S(t)=0.5$
3. Likelihood construction — For right-censored and left-truncated data
4. Censoring type identification — Type I, Type II, left truncation

Reference to Material Notes

Topic Category	Reference Note
Hazard, Survival, Relationships	def-hazard-function_202603281500, def-survival-function_202603281500
Median	def-median-life_202603281500
Mean Residual Life	def-mean-residual-life_202603281500
Weibull	def-weibull-distribution-survival_202603281500
Exponential	def-exponential-distribution-survival_202603281500
Likelihood	def-likelihood-censored-data_202603281500
Kaplan-Meier	def-kaplan-meier-estimator_202603281500
Nelson-Aalen	def-nelson-aalen-estimator_202603281500

UTS 2020 Specific Questions

Q1: Given $S(t)=\frac{{\theta }^4}{({\theta }^2+t^2{)}^2}$, find:

• (a) Hazard function $h(t)$
• (b) Median of $T$

Q2: Survival curve interpretation for cancer patients

• Which cancer type is more dangerous?
• Approximate survival at 5 months
• Hazard at specific weeks

Q3: Censoring scheme + likelihood (Uniform distribution)

• 5 light bulbs: 3 fail at 5, 9, 13 hours; 2 censored at $4+p$ hours

Q4: Left truncation + likelihood

• Mice study with delayed entry at different ages

UTS 2021 Specific Questions

Q1: Given $h(t)=\frac{2t}{1+t^2}$, find:

• (a) Median
• (b) PDF $f(t)$
• (c) Mean $E[T]$

Q2: Transformer study (1990-2020)

• Define $T$, origin, scale, event
• Identify censoring/truncation for 4 cases
• Likelihood for exponential model

Q3: Kaplan-Meier calculation

• Two drug groups with censored data
• Compare effectiveness

Q4: Nelson-Aalen confidence interval

• Given CI for $H(t_j)$ and $H(t_{j+1})$, find deaths at $t_{j+1}$

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Mathematical Techniques in UTS

Integration Techniques

Technique	Example from UTS	GeneralForm	Cheatsheet Link
Power rule	${\int }_0^c\frac{4t}{27c}dt=\frac{2c^2}{27c}$ (normalization)		Power Rule
Logarithmic integral	$\int \frac{2t}{1+t^2}dt=\ln (1+t^2)$	$\int \frac{f^{\prime }}{f}=\ln f$	General Power of Function
Inverse logarithmic	$\int \frac{1}{80-t}dt=-\ln |80-t|$	$\int \frac{f^{\prime }}{f}=\ln |f|$	Log Integrals
Arctan integral	${\int }_0^{\infty }\frac{1}{1+t^2}dt=\frac{\pi }{2}$	$\frac{{\int }_1}{a^2+x^2}=\frac{1}{a}\arctan \frac{x}{a}$	Inverse Trig Integrals
Linear function	${\int }_t^a\frac{a-u}{a}du$ (MRL calculation)		Basic Rules
Integration by parts	${\int }_t^{\infty }(u-t)f(u)du$ (MRL proof)	$\int udv=uv-\int vdu$	Integration by Parts
Improper integrals	$E[T]={\int }_0^{\infty }S(t)dt$		Definite Integrals

Derivative Techniques

Technique	Example from UTS		Cheatsheet Link
Chain rule	$\frac{d}{dt}({\theta }^2+t^2{)}^{-2}=-4t({\theta }^2+t^2{)}^{-3}$		Chain Rule
Quotient rule	$h(t)=\frac{f(t)}{S(t)}$		Quotient Rule
Logarithmic derivative	$h(t)=-\frac{d}{dt}\ln S(t)$		Log Derivatives
Power rule	$\frac{d}{dt}{t}^n=n{t}^{n-1}$		Power Rule
Log-likelihood derivative	$\frac{d}{d\lambda }\ln L=\frac{r}{\lambda }-\sum t_i$ (MLE)		Stats Applications

Algebraic Techniques

Technique	Example from UTS
Exponential equations	$e^{-\lambda m^{\alpha }}=0.5\Rightarrow m={\left(\frac{\ln 2}{\lambda }\right)}^{1/\alpha }$
Logarithmic equations	$-\ln S(t)=\ln (1+t^2)\Rightarrow S(t)=\frac{1}{1+t^2}$
Systems of equations	Solve ${\hat{S}}_{KM}(20)=0.953564$ for unknown censoring count
Linear CI equations	Solve CI bounds for $\hat{H}$ to find variance and deaths

Key Patterns to Memorize

$\int \frac{f^{\prime }(t)}{f(t)}dt=\ln |f(t)|+C\text{(appears in }H(t)\text{ calculations)}$

$\int \frac{1}{a-t}dt=-\ln |a-t|+C\text{(for }h(t)=\frac{1}{a-t}\text{ type hazard)}$

${\int }_0^{\infty }\frac{1}{1+t^2}dt=\frac{\pi }{2}\text{(arctan integral for mean)}$

$\frac{d}{dt}\ln S(t)=\frac{S^{\prime }(t)}{S(t)}=-h(t)\text{(hazard from survival)}$

$\text{MLE for exponential: }\hat{\lambda }=\frac{r}{\sum t_i}\text{(}r=\text{number of events)}$