Medical Stats
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| Clinical Trial | Planned experiment on human beings which is designed to evaluate the effectiveness of 1 or more forms of treatment
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| Experiment | Series of observations made under conditions controlled or arranged by the investigator
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| Treatment | A pharmaceutical drug or medical treatment (surgery, diet, counselling), or method of organising/delivering health care (education leaflet, phone app)
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| Classification | Systematic arrangement by purpose or phase in drug development
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| Phase I trial | Looks into pharmakinetics of a drug (absorption, metabolism) and toxicity (drug safety). finding maximum tolerated dose.
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| Phase II trial | Initial clinical investigation. Dose finding study. Work out dose and dose schedules and early indication of efficacy
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| Phase III trial | Aim of full scale evaluation, efficacy of a new experimental treatment compared to a standard therapy or placebo, acting as a control
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| Phase IV trial | Effectiveness. Post marketing surveillance for info on long term effects and uncommon side effects.
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| Effect | Difference between what happened to a patient as a result of the treatment vs what would have happened without the treatment
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| Efficacy | The true biological effect of a treatment
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| Effectiveness | Effect of a treatment when widely used in general practice.
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| Efficiency | The time and cost related economics of a treatment.
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| Research Protocol | Formal written document outlining the research plan for a clinical trial
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| Control group's Purpose | Give a yard stick as to what may have happened in absence of experimental treatment. Needed in phase III trials
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| Use of control group in Phase III trials | As investigator/patients expectations and enthusiasm of trial inclusion may affect judgement and outcomes. Some open & uncontrolled trials yeilded spurious results.
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| Gold standard of trial design | Randomised, controlled and double-blinded trial.
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| Randomisation | Treatments assigned according to chance (often equal).
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| Aim of Randomisation | To yield treatment groups that are comparable in terms of extreneous factors.
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| Purpose of Randomisation | Avoid bias due to differences in clinical & demographic characteristics. To support the independence assumption underlying many statistical procedures
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| Simple Randomisation | independent random treatment allocation with fixed probability
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| Pros of simple randomisation | easy to implement
can do analysis via standard statistical methods
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| Cons of simple randomisation | likely to get uneven treatment group numbers esp in small samples
doesnt ensure balance over confounders (e.g. age and disease severity)
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| Block randomisation | Restricted randomisation that works by balancing participant numbers in each group.
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| Block randomisation method | Choose a block at random and assign patients accordingly. Want to keep block sizes short to prevent incomplete blocks (as may be fixed/randomly varying)
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| Block size | A multiple of the number of treatments
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| Cons of block randomisation | Assignment may become known if blocking factor revealed
statistical methods often presume simple randomisation
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| Pros of block randomisation | Has balance. max difference is b/2 for blocks of size b
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| Stratified randomisation | Uses block randomisation in each stratum to aid between group comparability (balance) over important characteristics.
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| Pros of stratified randomisation | Overcomes imbalance with regards to important factors in small samples
Increases efficacy and reduces potential bias
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| Cons of stratified randomisation | no. of strata may limit usefulness
Complicates proceedings
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| Adaptive randomisation - Minimisation | A non random treatment allocation where allocation probabilities adjusted according to patient imbalance
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| Minimisation Purpose | Balance between treatment groups with regard to important prognastic factors.
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| Minimisation Method | Use simple randomisation when groups are balanced. When imbalanced, allocates next patient to treatment so imbalanced minimised via minimisation score.
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| Weighted minimisation | Weighting in favour of treatment which would minimise imbalance
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| Purpose of Blinding/Masking | Avoid bias due to differences in treatment outcome (conscious or unconscious)
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| Single-blind trial | Patient not know which treatment is being received
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| Double-blind trial | Neither patient nor physician knows treatment allocation. GOLD standard.
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| Triple-blind trial | double-blind plus monitoring group & data analyst don't know which group receives experiences & which control.
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| Open trial | All are knowledgable about treatment allocation.
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| Double-dummy | Compares 2 active treatments with different appearances
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| Blind alternative | Incorporate blind assessment or conceal treatment allocation until patient entered into trial
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| Cons of blind trial | More complex as need independent monitoring committee to deal with ethical aspects & trial termination
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| Ethical Issues in clinical trials | can you obtain informed consent? Should you try to continue until investigators convinced 1 better or until entire medical community convinced?
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| Ethical issues regarding placebos | Is it ethical if an effective existing treatment is known to exist
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| Declaration of Helsinki | Set of ethical principles regarding human experimentation. For medical community. By WMA. Is a cornerstone document on human research ethics. Not legally binding.
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| Governance arrangement for NHS research ethics committees | Describes what REC's should be like and when their review is needed
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| Example of a complication of clinical trials | Non adherence to protocol - missing follow up visits, not completed full course of treatment
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| Intention to treat set | Analyse as randomised, regardless of adherence. To look at effectiveness.
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| Per Protocol set | Analysis based upon adherers only. To look at efficacy.
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| Cons of per protocol set | Can introduce bias when groups of patients no longer have similar characteristics.
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| When to use intention to treat set | In superiority trials where want to avoid over optimistic estimates
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| Endpoint | Quantative measurement implied or required by the trial objectives. Is determined in each study subject.
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| Hard endpoint | Preferred. Well defined and reliable. No subjectivity. Objectively measured.
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| Soft endpoint | Difficult to define and subjective measures.
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| Student's T-test | Comparison of 2 independent group means. Parametric method.
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| Assumptions of student's T-test | Normality of data and equality of variances across comparison groups.
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| Order of methodology for student's t-test | state research hypothesis (no diff and diff in difference in group means)
estimate D = mean(T) - mean(C)
Compute test statistic under H0. T = (mean(T)-mean(C)-0)/SE(estimate D)
SE(estimate D) = S*sqrt(1/N(T) + 1/N(C))
S depend on underlying variance.
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| S^2 (pooled variance estimate) in student's T-test if common underlying variance | 1/(n-2) * (sum from 1 to N(T) of (each value in T - mean(T))^2 + sum from 1 to N(C) of(each value in C - mean(C))^2
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| S^2 (pooled variance estimate) in student's T-test if S(T) and S(C) are sample standard deviations | ((N(T) -1)*S(T)^2 + (N(C) - 1)*S(C)^2/(N - 2)
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| Confidence interval for student's T-test | T is compared to t-distribution with N-2 degrees of freedom & inference based on if |T| >= t dist (N-2,1 - alpha/2)
D +- t dist (N-2, 1 - alpha/2)
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| Point Estimate | Observed difference in group means D = mean(T) - mean(C)
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| Interpretation if confidence interval for true difference spans zero | there is no statistically significant difference between groups so cannot reject null hypothesis
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| Inference for p value | Calculate alpha for T value. if p < alpha = 0.05. Then is statistically significant
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| Pro of confidence intervals | More informative as can assess plausible range of effect size
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| Welch's test | An approximate test if assume non-constant variance
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| Mann-whitney test | Non parametric alternative to student's T-test
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| 1-sample Paired t-test | Test based on within-pair differences
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| Paired t-test method | Compute mean of within-pair diffs
standard deviation of diffs Sd = sum of squares of differences between diffs and mean diff
standard error of mean of diffs SE(D) = Sd/sqrt(n)
Compute T under H0. T = D/SE(D)
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| Paired t-test confidence interval | Compare T to t-distribution under n-1 degrees of freedom
D +- t(n-1,1-alpha/2)*SE(D)
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| p_1 and p_2 in binary response data | success probabilities for treatment and control groups in binary response data
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| Risk Difference | Difference between risk of an outcome between group 1 and 2. p1 - p2. a/(a+c) - b/(b+d)
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| Relative Risk | Compare risks for 2 groups. p1/p2. (a/(a+
c))/(b/(b+d)). treatment 1 RRx risk of outcome being observed than treatment 2.
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| Disease odds | ratio of success to failure. p_i/(1-p_i)
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| Odds ratio | Measure of association between outcome and exposure. p1/(1-p1) / p2/(1-p2) = ad/bc. treatment 1 ORx more likely to be exposed to outcome than treatment 2.
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| When does odds ratio equal relative risk? | When the occurrence of outcome is less than 10%.
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| Binary data confidence intervals | theta +- z(1-alpha/2)*SE(theta) where theta is risk difference, log of relative risk or log of odds ratio.
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| Standard error for Risk difference BINARY | sqrt(p1(1-p1)/n1 + p2(1-p2)/n2). p1 = a/(a+c) and p2 = b(b+d). n1 = a+c. n2 = b + d
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| Standard error for log of Relative Risk BINARY | sqrt(1/a - 1/(a+c) + 1/b - 1/(b+d)) DONT FORGET TO ANTILOG THE INTERVAL
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| Standard error for log of Odds Ratio BINARY | sqrt(1/a + 1/b + 1/c + 1/d) DONT FORGET TO ANTILOG THE INTERVAL
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| Parallel group design | Different groups of patients are studied in parallel. Estimate of treatment effects is based upon between subject comparisons.
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| What inference test to use for parallel group design? | 2 independent samples t-test
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| Paired design | Patient recieves both treatments. Then estimate of treatment effect based on within subject comparison
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| What inference test to use to paired design | 1-sample t-test on within subject differences
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| Crossover Design | Patient recieves sequence of treatments. Order determined by randomisation. Treatment effect based on within-subject comparisons
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| Treatment Periods | times treatments are administered
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| Pros of cross-over design | Patients act as own control as uses within-subject comparisons - eliminates between patient variation.
Smaller sample size required for same no of observations
Same degree of precision in estimation with fewer observations
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| Cons of cross-over design | Inconvinience to patients as multiple treatments mean longer time under observation
Patients may withdraw
Treatment effect not constant over time (p-by-t interaction)
Carry over effect
Analysis more complex - may be systematic diffs between periods
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| Carry over effect | Persistence of treatment applied in one period in a subsequence period of treatment
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| How to deal with carry over effect | Wash out period
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| Wash out Period | Period in trial during which effect of treatment given previously believed to disappear.
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| When are cross-over trials useful? | Chronic diseases that are relatively stable
Single dose trials of biological equivalence rather than long term trials
Drugs with rapid and reversible effects
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| AB/BA design | A 2 treatment x 2 period cross over trial.
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| Simple analysis for a cross over design with no period effect | Use paired t-test
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| Assumptions underlying use of paired t-test | Normally distributed differences
Unbiased - that E(d) is the true treatment effect
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| Factors that may cause differences to not be distributed at random about true treatment effect | period effect
period by treatment interaction
carry over effect
patient by treatment interaction
patient by period interaction
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| When is a treatment effect estimate unbiased | If expectation of estimate is the estimate.
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| Why do you need adequate sample sizes? | ethics
budget constraints
time constraints
trial should be large enough to give reliable answer to research question
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| P value | Probability, p, of obtaining test result at least as extreme as that observed assuming H0 is true
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| Size of test | Given by alpha, usually alpha = 0.05
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| When p value less than alpha? | Reject H0 and conclude data is inconsistent with null.
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| Type I error rate | alpha, probability of rejecting H0 given H0 is true (saying theres a diff when theres not)
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| Type II error rate | beta, probability of failing to reject H0 when H1 true (saying no diff when there is a diff)
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| Power of test | 1 - beta, probability of rejecting H0 when H1 is true (correctly saying is a diff)
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| What does power of a test depend on? | the statistical test being used
size of test (alpha)
variability of observations (sigma^2)
H1 (size of diff 'triangle')
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| What does symbol triangle* represent? | The clinically relevant difference in a gauss test per say
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| One & two sample one sided gauss test | Used to determine sample size
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| Epidemiology | Study of distributions and determinants of disease in human populations
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| What are descriptive studies used for? | Generating research hypotheses and resource allocation
Gaining info regarding disease frequency in populations
Info on distributions - age, geography, time, etc
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| Disease Determinants aka risk factors | Factors precipitating disease (biological or enviro or social)
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| Target population | Population about which we wish to draw inferences for
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| Study population | Population from which data is collected
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| Generalisability | Whether can use study population results to draw accurate conclusions about target pop
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| Factors affecting study sample choice | generalisability of results trading off with availability and cost
opportunistic sample is readily identifiable and likely to be cooperative
Preference is a random sample of target population
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| Routinely collected data | A source for data in epidemiological studies. E.g. hospital data bases, birth/death registers, census data)
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| Sources for data in epidemiological studies | routinely collected data
data purposely collected by investigators via surveys and follow up
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| Purpose of routinely collected data | vital to monitor public health and health planning
can use to investigate possible associations between routinely available attributes and rate of indecence from a particular disease
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| Con of routinely collected data | May be of limited quality - subject to regional variation and often dont contain required individual level info.
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| Limits of routinely collected data | Coverage - difficult to define morbidity so incomplete coverage
Accuracy - diagnosis of cause of death and illness can be wrong
Availability - confidential safe guards may limit data availability
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| Reasons for incomplete coverage of routinely collected data | Inconsistent reporting of infectious diseases by practitioners and cancer registers may miss non hospital cases
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| Disease Iceberg | moving down it, cases increase but severity decreases
died, hospital, GP, self report, population screening
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| Incidence (I) of disease | number of new cases of disease within a specified time period
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| Prevalence (P) of disease | number of existing cases of disease at particular point of time. Probabiity of disease prior to seeing test result. (a+c)/N
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| Calculation for incidence of disease | number of develop disease in specified time period / sum of length of time in which each person in population is at risk for
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| Calculation for prevalence of disease | number with disease at time t / number in population at risk at time t
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| Person-time of observation | Total observation time. Often express as person-years. Used for incidence rates.
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| Purpose of prevalence measures | better for descriptive studies than analytical, suggests possible causes
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| Purpose of incidence measures | Better to study as can establish sequence of events. Not susceptible to bias by survival
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| Cumulative Incidence Risk | Alternative measure of disease occurance. Calculate: number of people who get disease in specified period / number of people free of disease at beginning of period. Is the risk of developing a disease.
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| Crude mortality rate | number of deaths in a specified time period divided by average population at risk in period, multiplied by length of study period
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| How to assess utility of a diagnostic test | Apply test to a number of individuals where true disease status known (based on GOLD standard)
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| Sensitivity | Errors in testing. Proportion of truly diseased persons in tested population who are identified as diagnosed by screening test. P(T|D) (prob of pos test given diseased). a/(a+c)
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| Specificity | Proportion of truly non diseased who are identified as so by screening test. P(bar(T)|bar(D)). d/(b+d)
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| Positive Predictive Value PPV | Proportion truly diseased of those testing positive. Probability of diseased after a positive test. P(D|T). a/(a+b)
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| Negative Predictive Value NPV | Proportion not diseased of those testing negative. P(bar(D)|bar(T)). Probability of not diseased after a negative test. c/(c+d)
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| High cut off value | More specific test
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| Low cut off value | More sensitive test
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| Intermediate cut off value | Smaller overall errors. Need to balance high sensitivity and low specificity.
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| Receiver Operating Characteristic curve | Plot of sensitivity against (1-specificity) for various cut off points. Cut off choice is top-left-most plot on curve.
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| PPV using sensitivity, prevalence and specificity | (sens x prev)/((sens x prev) + (1 - spec)(1-prev))
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| NPV using sensitivity, prevalence, specificity | (spec x (1-prev))/((spec x (1- prev))+((1-sens) x prev))
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| Likelihood Ratio | Used to make therapeutic decisions regarding tests. P(T|D)/P(T|bar(D)) = sens/(1-spec). Test informative if value high. Is the ratio of posterior disease odds after pos test to prior disease odds, so tells how much odds of disease increase after pos test.
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| Weight | Relative importance to give to sensitivity
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| When to give high weight to sensitivity? | If disease high threatening and treatment has few side effects. (dont want false negatives).
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| When to give high weight to specificity? | If disease not serious and treatment has many side effects. (dont want false positives)
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| How to maximise weight with respect to study | Choose w = number of positive samples/total number of samples. Want to maximise M = w x sens + (1-w) x spec. 1-w is weight of spec.
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| How to maximise weight with respect to population criterion | w = number with disease in population / total population size. Which is disease prevalence.
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| Screening test | Test for particular disease given to population at risk who are asymptomatic
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| Criteria for disease screening | Natural history of disease should be well understood
should be an important health issue
test should be acceptable to population
there should exist a suitable screening test
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| Why must be careful in disease screening | As truly diseased population likely to be small so many disease free will test positive
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| Observational studies | Investigator role is passive so exposures not manipulated
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| Intervention studies | Investigator role is active. Groups exposed to intervention of interest (eg treatment). Are experimental studies (clinical trials)
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| Why are intervention studies the gold standard in terms of causality | can handle confounding with randomisation and exposure known to preceed disease.
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| ethical Issue with intervention studies | rarely acceptable to force exposures on people
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| Cross-sectional studies | surveys also feature but only to give info for snapshot in time - are less useful. Can only use to measure prevalence.
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| Most common observational studies | cohort and case studies.
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| Cohort study | Most useful observational study in disease causes. Tracks 2 or more groups forward from exposure to outcome. classify according to exposures. exposure vs non exposure rows and outcome vs non outcome columns
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| Pros of cohort study | Strongest evidence for causal r.ship as exposure known to preceed outcome
Can measure risk, incidence rate & survival time
can examine many events after 1 exposure
maintain temporal sequence
best way to ascertain incidence & natural disease history
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| Rationale of cohort studies | Follow cohorts through time - record disease occurrance and compare exposure groups with respect to disease outcome- then interpret
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| Cons of cohort studies | large & costly
selection bias
loss to follow up
confounding variables
exposure can change in study period
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| Case control study | Case & control groups defined & selected according to disease status. Look back in time to ascertain each person's exposure status
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| Pros of case control study | efficient in terms of time and money (if exposure frequency not too low)
can study rare diseases
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| Cons of case control study | choosing control group can lead to selection bias
obtaining exposure history so may have recall bias
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| Risk difference for cohort studies (binary exposure) | p1 - p0. a/(a+b) - c/(c+d)
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| Relative risk for cohort studies (binary exposure) | p1/p0. (a/(a+b)/c/(c+d)).
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| Disease odds in cohort studies. binary exposure | p/1-p. Ratio of success to failure
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| Odds ratio in cohort studies. binary exposure. | Compares exposed and not exposed groups. p1/(1-p1) / p0/(1-p0) = ad/bc
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| In case control studies - estimate disease odds ratio not disease risk... | p1/(1-p1) / p0/(1-p0)
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| Disease odds ratio | similar to odds ratio if disease is rare.
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| Exposure odds for cases (D) | P(E|D)/P(bar(E)|D)
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| Exposure odds for controls (bar(D)) | P(E|bar(D))/P(bar(E)|bar(D))
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| Exposure odds ratio | Ratio of 2 exposure odds. Equal to disease odds ratio. ad/bc
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| How to increase precision of estimate of disease odds ratio? | More controls.
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| A confounder | 3rd variable associated with exposure of interest. Independently associated with risk of disease. partially or fully explains relationship between E and D.
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| Consequences of confounders | Creates a false relationship between E and D
Masks true relationship between E and D
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| Design based approaches to control confounding | Randomisation (balance over potential confounders)
Restriction (Limit participation to individuals similar in relation to confounders)
Matching (select controls to be similar to cases in terms of confounders)
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| Analysis based approaches to control confounding | Stratification
Standardisation (control confounding using external pop to adjust for age, gender etc)
Multivariate analysis/regression models (include confounding variable in model)
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| Stratification (in analysis based approach to control confounding) | Examine exposure-disease associations within strata of confounding variable. Estimate pooled estimate of association measure, adjusting for confounding effect.
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| Mantel-haenszel estimate | Estimate of association between outcome and exposure after adjusting for confounding. Estimates a common odds ratio so use log in confidence interval
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| Standard error for mantel-haenszel | very long. sum(a+d/n x ad/n)/2xsum(ad/n)^2 + same for bc + sum((a+d/n x ad/n)+(same for bc))/2sum(ad/n)sum(bc/n)
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| If odds ratio confidence interval spans 1 for mantel-haenzel | association between exposure and outcome not significant at 5% level
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| If reduction in odds ratio from joint odds ratio to individual in mantel-haenzel is small | The degree of confounding is small
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| Unstratified matching in case control studies | Choose controls randomly
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| Stratified matching in case control studies | Match controls to cases according to confounding variable. May be group or individual matching.
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| Rationale of matched case control study | Eliminate confounders by design
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| Pros of matching in case control study | control confounders by elimination
gain in efficiency
minimises selection bias
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| Cons of matching in case control study | More complex study design
cant study effect of matching variables on outcome of interest
overmatching (matching variable linked to exposure not disease)
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| Maximum likelihood estimate of exposure odds ratio in matched case-control study | b/c. standard error is log of estimate = sqrt(1/b + 1/c)
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| Standardisation | process aimed at removing confounding by choosing a standard population with known distribution of confounders
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| Direct standardisation | disease rates in population of interest are applied to standard population counts
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| Indirect standardisation | disease rates in standard population applies to population of interest
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| Direct standardised event rate | expected event rate in standard population, if age-specific event rates in study population prevailed
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| How to adjust event rate for age | category specific event rates for each population being compared will be applied to a single standard population
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| Direct age standardised event rate per 1000 | 1000/total size of standard pop x sum of ((observed number of events in ith age group of study population/size of ith age group of study population)x size of ith age group of standard population)
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| Person years | average population size x study length
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| When to use indirect standardisation of event rates | when age specific rates aren't available for population of interest
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| Methods of indirect standardisation of event rates | Produce standardised event ratio (SER) or standardised mortality rate (SMR)
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| SMR | O/E. where O is total number of observed events in all age groups of study population and E is sum of (observed number of events in ith age gorup of standard population / size of ith age group of standard population) x size of ith age group of study pop
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| Requirement for direct standardised rates | need age specific rates for all population studies
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| Requirement for indirect standardised rates | need total number of cases observed
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| Standardised incidence / mortality rate | ratio of 2 indirectly standardised rates
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| Pro of indirect standardised rates | More stable in case of small numbers of events
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| Con of indirect standardised rates | Need age specific rates for standard population
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| Survival analysis | data analysis in the form of time from some well defined time origin to occurrence of some event or endpoint.
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| Examples of time origins for survival analysis | time of trial entry or diagnosis time
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| Time-to-event data | can be time to death or time to failure of prosthetic etc
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| Positive event example in time-to-event data | hospital discharge
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| Adverse event example in time-to-event data | Death
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| Neutral event example in time-to-event data | Cessation of breast feeding
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| Lifetime random variable | time-to-event rate is random variable T
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| Why can't apply time-to-event data to standard methods of analysis | Event times are positive continuous, typically skewed, and subject to censoring
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| When does censoring occur? | if endpoint not observed
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| Right censoring | event time greater than last follow up time
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| Left censoring | Event time before time of last follow up but is unknown
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| Interval censoring | event time falls in a specified interval
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| Reasons for right censoring | period of observation end prior to event occurrence
loss to follow up
completing event precludes further follow up - eg death
event may not be inevitable (eg pregnancy)
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| Why one treatment may have more censoring at the end than other | results in increased survival
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| Why does period of observation time vary between patients? | patients accrued sequentially over time and followed up to fixed date so period of observation varies between patients
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| Assumption of survival analysis when acru patients over time | Prognosis doesn't depend on entry time to study
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| Aims of survival analysis | model survival times for a group & make predictions
assess effects of covariates on survival
compare survival distribution for 2 or more groups
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| The functions of interest when summarising survival data | Survivor function & hazard function
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| Survival time of individual i | observation of a non negative r.v. T
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| Distribution function of T | F(t)
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| Probability density function of T | f(t)
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| Link between Distribution and pdf of T | F(t) = integral (from 0 to t) of f(s) ds = P(T <= t). f(t) = d/dt F(t)
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| Survivor function | Probability an individual survives to time t. S(t) = P(T>t) = 1 - F(t) = integral (from t to infinity) f(s) ds. Monotone decreasing function S(0) = 1
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| Link between pdf and survivor function | f(t) = -d/dt S(t)
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| Hazard function | specify instantaneous rate of failure at T=t given survival to time t. h(t) = f(s)/S(t). RATE not probability
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| How to approximate probability an individual who survived to time t will experience event in interval (t,t+delta*t) | h(t)*delta*t
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| Cumulative/integrated hazard function | H(t) = integral (from 0 to t) of h(u) du. Probability of failure at time t given survival until time t
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| Decreasing hazard function | Clinically less likely, such as in risk after organ transplantEs
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| Emperical survivor function | estimate of survivor function. hat(S)(t). Step function with steps at observed failure times.
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| Kaplan-meier estimate | Estimates survivor function. first step in presence of censored data. Is the product limit estimator
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| event times and number in sample at risk at each event time and number who fail at this time notation | t_j and n_j and d_j
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| Estimated probability of survival through interval j in kaplan meier survivor function | p_j = (n_j - d_j)/n_j) = 1 - d_j/n_j
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| Assumptions for kaplan-meier estimate | independence of event times
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| Kaplan meier estimator | S(t) = multiplication of all p_j's for k=r+1 subintervals
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| Kaplan-meier when no censoring | emperical survivor estimator
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| Greenwood's formula | for estimated variance. S(t)^2 x sum of (d_j/(n_j(n_j - d_j)))
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| Make a smoother plot for survivor function estimate | More data and more distinct failure times
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| End point of the kaplan-meier estimator curve | the estimated survival % at t years
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| Why can't use t-test to compare group means in survival distribution? | Because of censoring
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| Log-rank test | formal comparison of 2 groups survival distribution. Can compare more than 2 groups.
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|
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| Informal comparison of 2 groups survival distribution | Use Kaplan-meier curve
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| Cox proportional hazards model | commonly used to flexibly model covariate effects on the hazard function.
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|
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