Fleming Harrington Test: Formula, Verified Results, Python, R, SPSS and Excel
Fleming Harrington Test is presented as a complete, dataset-grounded survival analysis guide. It explains compare survival curves with weights chosen before inspecting the observed separation pattern, the exact formula, assumptions, verified calculations, interpretation, software workflows, matched charts, reports, workbook, internal links, and publication checks. The verified example uses an explicitly prepared teaching endpoint from the uploaded 649-row dataset.
Strong early-weighted separation
With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance.
What does Fleming Harrington Test measure?
a survival-curve contrast targeted to early, middle, or late event times through prespecified weights
Fleming Harrington Test focuses on a survival-curve contrast targeted to early, middle, or late event times through prespecified weights. The estimand must remain separate from related quantities such as ordinary probability, crude event proportion, mean duration, or an unrelated regression coefficient.
Method target
Fleming Harrington Test is selected to compare survival curves with weights chosen before inspecting the observed separation pattern. The method is applied to ordered follow-up times and event indicators, not to a standalone numeric outcome with censoring ignored. The analysis therefore starts from risk sets and event times.
The worked example defines time as absences plus one and the primary event as G3 below 10. For Fleming Harrington Test, these variables are used only to demonstrate a survival-curve contrast targeted to early, middle, or late event times through prespecified weights; they are not presented as naturally observed medical survival times.
What it does not establish
The procedure cannot create causality or a real-world failure process from cross-sectional student records. Its defensible output is the method-specific estimate or test under the stated coding, and selecting ρ and γ after seeing the curves inflates interpretive flexibility; the weighting target should be prespecified.
Selecting ρ and γ after seeing the curves inflates interpretive flexibility; the weighting target should be prespecified.
When should Fleming Harrington Test be used?
Decision logic before software
Time outcome?
Confirm a meaningful duration from a common origin.
Event defined?
State event=1 and censor=0 unambiguously.
Method target?
Match Fleming Harrington Test to the estimand.
Assumptions?
Audit censoring, risk sets, ties, and model form.
Reportable?
Retain numerical evidence and limitations.
Appropriate use
Choose this method when the research question is genuinely about a survival-curve contrast targeted to early, middle, or late event times through prespecified weights and the required assumptions can be defended. It is preferable to a simple mean or binary comparison because it retains event timing and censoring information relevant to uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up.
Fleming Harrington Test is especially useful when its specific estimand is more informative than an ordinary mean comparison or binary event analysis that discards follow-up time.
Inappropriate use
Fleming Harrington Test should be selected from its estimand and weighting or model structure, not from the availability of a command. It is unsuitable for paired observations, inconsistent time origins, or a weight chosen only after inspecting which test is significant. For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22.
Do not publish Fleming Harrington Test output when the matching charts, PDFs, workbook, and dataset describe different definitions or model specifications.
Fleming Harrington Test dataset and variable construction
The exact 649-row teaching structure
The bundled 649-row dataset is used specifically for rho–gamma pooled-survival weights that target a prespecified part of follow-up. The GP/MS event-time table is combined with pooled survival immediately before each event so rho=1 and gamma=0 emphasize earlier separation. The prepared endpoint remains a transparent teaching construction rather than natural clinical, mortality, or equipment-failure follow-up.
| Variable | Role | Coding | Audit note |
|---|---|---|---|
| surv_time | Duration | absences + 1 | Positive values from 1 to 33 |
| surv_event | Primary event | 1 when G3 < 10; 0 otherwise | 100 events and 549 censorings |
| school | Group | GP reference; MS comparison | 423 GP and 226 MS records |
| competing cause | Secondary event | failures > 0 among records without the primary event | 51 competing events |
| predictors | Cox covariates | age, parental education, travel/study time, failures, family relationship, free time, school, gender | Ten-term model |
Fleming Harrington Test assumptions
Conditions required for a defensible result
Independent Groups
Fleming Harrington Test requires independent groups. This condition is evaluated against the prepared duration, event coding, group structure, risk sets, and the method-specific result rather than assumed from the word nonparametric or from successful software execution.
A Common Time Origin
Fleming Harrington Test requires a common time origin. This condition is evaluated against the prepared duration, event coding, group structure, risk sets, and the method-specific result rather than assumed from the word nonparametric or from successful software execution.
Non-Informative Censoring Within Groups
Fleming Harrington Test requires non-informative censoring within groups. This condition is evaluated against the prepared duration, event coding, group structure, risk sets, and the method-specific result rather than assumed from the word nonparametric or from successful software execution.
Correct Event/Status Coding
Fleming Harrington Test requires correct event/status coding. This condition is evaluated against the prepared duration, event coding, group structure, risk sets, and the method-specific result rather than assumed from the word nonparametric or from successful software execution.
Adequate Risk Sets At Weighted Event Times
For Fleming Harrington Test, assumptions are assessed one by one using counts, curves, residuals, risk sets, or likelihood diagnostics appropriate to the procedure. The weight S(t-) emphasizes earlier event times because pooled survival is largest near the start of follow-up. Successful execution is not counted as evidence that the conditions hold.
Prespecified Weighting Strategy
Fleming Harrington Test requires prespecified weighting strategy. This condition is evaluated against the prepared duration, event coding, group structure, risk sets, and the method-specific result rather than assumed from the word nonparametric or from successful software execution.
Fleming Harrington Test formula and mechanics
Native browser MathML and a plain-language audit trail
Fleming Harrington Test uses this expression to estimate or test a survival-curve contrast targeted to early, middle, or late event times through prespecified weights. Every symbol should be linked to a risk set, event count, survival estimate, covariate, distribution parameter, or weight defined in the surrounding text.
Calculation sequence
- Sort positive durations and verify event/censor coding.
- Construct the exact risk set immediately before each event time.
- Calculate the Fleming Harrington Test contribution defined by the formula.
- Accumulate products, sums, likelihood terms, or weighted contrasts as required.
- Attach uncertainty, diagnostics, and a conclusion that matches the estimand.
Formula interpretation
For Formula interpretation, the Fleming Harrington Test review must connect every symbol to a risk set, event count, likelihood term, or model parameter. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 1 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
The displayed expression uses browser-native MathML and ordinary semantic HTML. Its symbols correspond to the risk sets, event counts, weights, coefficients, or distribution parameters defined in this section; no remote rendering script or equation image is required.
Fleming Harrington Test verified results
Values calculated from the included dataset
| Result item | Verified value |
|---|---|
| Weighted observed component | 62.660 |
| Weighted expected component | 24.524 |
| Variance | 15.208 |
| Standardized z | 9.779 |
| Chi-square | 95.633 |
| p-value | < .001 |
How to interpret Fleming Harrington Test
From statistical output to a restrained conclusion
Primary conclusion
Strong early-weighted separation
For Primary conclusion, the Fleming Harrington Test review must translate the numerical result without overstating causality, equivalence, or natural follow-up. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 2 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
Interpretation order
Fleming Harrington Test in Python
Transparent data preparation and reproducible calculations
This Python section reconstructs a survival-curve contrast targeted to early, middle, or late event times through prespecified weights from explicit arrays and auditable intermediate tables. Uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, so the code below exposes the quantities that determine the final result.
import numpy as np
import pandas as pd
from scipy.stats import chi2rho, gamma = 1.0, 0.0 # prespecify before viewing curves
df = pd.read_csv("dataset.csv")
t = pd.to_numeric(df["absences"]).to_numpy(float) + 1
e = (pd.to_numeric(df["G3"]) < 10).to_numpy(int)
g1 = df["school"].eq("MS").to_numpy()
U = V = 0.0
S_before = 1.0
for tj in np.sort(np.unique(t[e == 1])):
risk = t >= tj
fail = (t == tj) & (e == 1)
n, n1 = risk.sum(), (risk & g1).sum()
dj, d1 = fail.sum(), (fail & g1).sum()
w = (S_before**rho) * ((1.0 - S_before)**gamma)
expected = dj * n1 / n
vj = n1 * (n-n1) * dj * (n-dj) / (n*n*(n-1)) if n > 1 else 0
U += w * (d1 - expected)
V += w*w * vj
S_before *= 1.0 - dj/n
q = U*U/V
print({"rho":rho, "gamma":gamma, "chi2":q, "p":chi2.sf(q,1)})
Python verification checklist
The Python workflow for Fleming Harrington Test begins by printing shapes, status counts, group coding, and intermediate quantities before the final statistic. A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. The saved script implements the fact that the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, which makes the calculation independently auditable.
Only topic-matching URLs from the uploaded register are embedded. When a Python, R, SPSS, or Excel file is absent, the article states that limitation rather than fabricating a filename or borrowing another post’s asset.
Fleming Harrington Test in R
Independent survival-analysis validation
R provides an independent implementation of the same a survival-curve contrast targeted to early, middle, or late event times through prespecified weights. The script states status coding, factor references, and the function or manual calculation needed for this method instead of relying on defaults.
library(survival)
df <- read.csv("dataset.csv", stringsAsFactors=FALSE)
df$time <- as.numeric(df$absences) + 1
df$event <- ifelse(as.numeric(df$G3) < 10, 1, 0)
df$group <- ifelse(df$school == "MS", 1, 0)weighted_two_sample <- function(time, event, group, kind="logrank", rho=0, gamma=0) {
event_times <- sort(unique(time[event == 1]))
U <- 0; V <- 0; Sminus <- 1
for (tt in event_times) {
at_risk <- time >= tt
is_event <- time == tt & event == 1
n1 <- sum(at_risk & group == 1); n0 <- sum(at_risk & group == 0)
d1 <- sum(is_event & group == 1); d0 <- sum(is_event & group == 0)
n <- n1 + n0; d <- d1 + d0
if (kind == "breslow") w <- n
else if (kind == "tarone") w <- sqrt(n)
else if (kind == "fh") w <- Sminus^rho * (1-Sminus)^gamma
else w <- 1
expected1 <- d * n1 / n
U <- U + w * (d1 - expected1)
if (n > 1) V <- V + w^2 * n1*n0*d*(n-d)/(n^2*(n-1))
Sminus <- Sminus * (1 - d/n)
}
c(U=U, variance=V, z=U/sqrt(V), chisq=U^2/V,
p=pchisq(U^2/V, df=1, lower.tail=FALSE))
}
weighted_two_sample(df$time, df$event, df$group, kind="fh", rho=1, gamma=0)
R validation checklist
For Fleming Harrington Test, the R section is written to reproduce the same estimand and endpoint as the Python and Excel calculations. The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. Reference levels and all nondefault options are displayed so the direction cannot change silently.
Fleming Harrington Test in SPSS
Syntax-first setup and output audit
The SPSS workflow separates native procedures from extensions and preserves the event value in saved syntax. It is reviewed against the same dataset counts and interpretation used by the other software sections.
COMPUTE surv_time = absences + 1.
COMPUTE surv_event = (G3 < 10).
VALUE LABELS surv_event 0 'Censored' 1 'Event'.
EXECUTE.
KM surv_time BY school
/STATUS=surv_event(1)
/PRINT TABLE MEAN
/PLOT SURVIVAL HAZARD
/TEST LOGRANK BRESLOW TARONE.
COXREG surv_time WITH age Medu Fedu studytime failures famrel
/STATUS=surv_event(1)
/METHOD=ENTER age Medu Fedu studytime failures famrel
/PRINT=CI(95) GOODFIT SUMMARY.Fleming Harrington Test in Excel
A visible calculation and reconciliation workbook
The Excel workbook exposes the arithmetic behind w_j = Ŝ(t_j−)^ρ[1 − Ŝ(t_j−)]^γ and reconciles selected rows with the programmatic output. It is an auditable calculation, not a black-box result.
Excel step 1
Create one row per unique event time.
Excel step 2
Calculate pooled and group-specific risk sets.
Excel step 3
Calculate observed and expected group events.
Excel step 4
Apply the method-specific weight before summing U and V.
Excel step 5
Use =CHISQ.DIST.RT(U^2/V,1) for the p-value.
Excel controls
For Excel controls, the Fleming Harrington Test review must make the spreadsheet an auditable calculation rather than a decorative download. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 3 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
Fleming Harrington Test charts and chart-specific interpretation
First chart full-width; remaining charts arranged in pairs
Media placement follows the verified workbook: the first chart spans the content width, later charts form responsive pairs, and every downloadable file remains tied to this post’s method and dataset definition.
Python chart 1 — Fleming Harrington Test
Python chart 1: shows the prepared 1–33 duration distribution, event/censor pattern, and where the weighted two-sample test obtains most of its information. For this topic, the display should be read with the event definition and the fact that the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up.
Python chart 2 — Fleming Harrington Test
Python chart 2: summarizes the principal Fleming Harrington Test output and the numerical components behind the reported conclusion. With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance.
Python chart 3 — Fleming Harrington Test
Python chart 3: examines the diagnostic path most relevant to the assumptions of this weighted two-sample test. The review priority is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; visible structure is a warning rather than decoration.
Python chart 4 — Fleming Harrington Test
Python chart 4: places uncertainty, residuals, weighted contributions, or fitted discrepancies on a distributional scale. It supports the model or test audit but does not replace the natural-scale result or its confidence interval.
Python chart 5 — Fleming Harrington Test
Python chart 5: collects the key verified metrics used in the article, including sample information and the method-specific estimate. Every displayed value must reconcile with dataset.csv and the downloadable Python output.
R chart 1 — Fleming Harrington Test
R chart 1 independently reproduces the prepared duration, event, and censoring structure for Fleming Harrington Test. Read it with the declared event definition before comparing groups or fitted quantities.
R chart 2 — Fleming Harrington Test
R chart 2 presents the benchmark output using R conventions. Its values should agree with the Python calculation after reference levels, tie handling, weighting, and status coding are aligned.
R chart 3 — Fleming Harrington Test
R chart 3 focuses on the diagnostic evidence for Fleming Harrington Test. Visible departures or sparse-tail behavior should trigger a sensitivity analysis rather than a cosmetic interpretation.
R chart 4 — Fleming Harrington Test
R chart 4 displays uncertainty or residual structure on the scale used by the R workflow. It supports the numerical audit but does not replace the natural-scale estimate and its limitation.
R chart 5 — Fleming Harrington Test
R chart 5 consolidates the principal metrics used in the R output. Every annotation must reconcile with dataset.csv, the printed result, and the matched downloadable file.
Fleming Harrington Test diagnostics and sensitivity analysis
Evidence required beyond the primary number
Data diagnostics
Before interpreting the primary result, verify the 649-row count, 100 events, 549 censorings, 1–33 duration range, GP/MS composition, tied times, and missing values. The method-specific review then asks analysts to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices.
Method diagnostics
For Fleming Harrington Test, diagnostic evidence is tied to the formula and result table. The unsigned chi-square needs curve direction and signed score contributions for substantive interpretation. The article then uses the instruction to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, preserving qualifications where the result is fragile.
Sensitivity diagnostics
Compare Fleming Harrington Test with log-rank equal weights, Breslow early risk-set weights, Tarone–Ware square-root weights, Fleming–Harrington prespecified early/late weights. Explain whether the substantive conclusion changes and why.
Full Fleming Harrington Test publication audit
Method-specific checkpoints for content, data, formulas, results, and assets
1. Research estimand
At research estimand, the article must move from terminology to evidence. State the exact population quantity and contrast before examining results. Its defining computation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and the audit should show where the required quantities appear in the CSV or derived table.
For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22. This evidence is read alongside the checkpoint rather than used as a substitute for it. The recommended diagnostic is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, and the final interpretation should remember that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
2. Time origin
This checkpoint asks whether time origin has been translated into executable analysis. Document what time zero represents and reject records measured from a different baseline. The method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; therefore a generic survival-analysis explanation is not enough for this post.
The weight S(t-) emphasizes earlier event times because pooled survival is largest near the start of follow-up. The post links this evidence to the formula and saved output rather than repeating generic advice. A defensible review will show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, and it will state clearly that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
3. Event and status coding
Event and status coding defines the checkpoint for this article. Describe every status value in words and verify its frequency before fitting. Because the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, the reviewer must connect the source rows to that mechanism rather than infer correctness from the software label.
A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. This is the concrete evidence used for the checkpoint. The sensitivity plan is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; the narrative must not forget that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
4. Censoring definition
Censoring definition defines the checkpoint for this article. Explain why a censored observation contributes to earlier risk sets and not later events. Because the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, the reviewer must connect the source rows to that mechanism rather than infer correctness from the software label.
The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. Reproducing that figure from the bundled CSV is required before publication. The diagnostic sequence should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, while the substantive statement recognizes that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
5. Duration scale
A strong account of duration scale names the decision and shows its consequence. Audit the numerical time scale and any recoding used to obtain it. Since the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, hidden defaults at this point would propagate into every later value.
The unsigned chi-square needs curve direction and signed score contributions for substantive interpretation. This is the concrete evidence used for the checkpoint. The sensitivity plan is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; the narrative must not forget that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
6. Risk-set or likelihood construction
This checkpoint asks whether risk-set or likelihood construction has been translated into executable analysis. Trace the core estimating equation to observable rows and event times. The method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; therefore a generic survival-analysis explanation is not enough for this post.
The result sits between a scientific weighting choice and a multiple-testing problem when several weights are explored. Any discrepancy across Python, R, SPSS, or Excel must be traced to definitions or defaults. In addition, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; the reader should be told that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
7. Ties and discretization
Ties and discretization can invalidate an otherwise polished article. Declare how simultaneous event times are aggregated or approximated. The reason is specific to this procedure: it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. The final wording should state any unresolved limitation rather than hide it behind a p-value.
For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22. Reproducing that figure from the bundled CSV is required before publication. The diagnostic sequence should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, while the substantive statement recognizes that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
8. Reference coding
Reference coding defines the checkpoint for this article. Establish reference coding before assigning better or worse direction. Because the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, the reviewer must connect the source rows to that mechanism rather than infer correctness from the software label.
The weight S(t-) emphasizes earlier event times because pooled survival is largest near the start of follow-up. Reproducing that figure from the bundled CSV is required before publication. The diagnostic sequence should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, while the substantive statement recognizes that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
9. Missing-data handling
At missing-data handling, the article must move from terminology to evidence. Reconcile every omitted row and confirm that exclusions do not change status coding. Its defining computation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and the audit should show where the required quantities appear in the CSV or derived table.
A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. The article should retain this value in a saved table and connect it to its matching chart. Reviewers should also show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, because the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
10. Dependence and clustering
Dependence and clustering defines the checkpoint for this article. Assess whether repeated, matched, or nested records require robust or multilevel treatment. Because the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, the reviewer must connect the source rows to that mechanism rather than infer correctness from the software label.
The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. The article should retain this value in a saved table and connect it to its matching chart. Reviewers should also show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, because the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
11. Information and event adequacy
Information and event adequacy receives an explicit pass, warning, or fail assessment. Print the status mapping and reconcile each event total with the CSV. This is necessary because the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and a different construction would answer a different survival question.
The unsigned chi-square needs curve direction and signed score contributions for substantive interpretation. Reproducing that figure from the bundled CSV is required before publication. The diagnostic sequence should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, while the substantive statement recognizes that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
12. Tail support
Tail support is reviewed separately from statistical significance. Separate stable follow-up from the thin tail before generalizing results. For this weighted two-sample test, the core operation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; the prose, formula, table, and chart must all describe that same operation.
The result sits between a scientific weighting choice and a multiple-testing problem when several weights are explored. It provides an audit anchor, not an automatic scientific conclusion. The method-specific safeguard is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. Interpret the displayed effect under the constraint that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
13. Uncertainty interval
Uncertainty interval receives an explicit pass, warning, or fail assessment. Report sampling uncertainty on the natural scale and reproduce its calculation. This is necessary because the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and a different construction would answer a different survival question.
For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22. The numerical record supports a method-specific audit, but it does not remove design limitations. To test robustness, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; when stating direction, note that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
14. Null hypothesis and p-value
Null hypothesis and p-value is reviewed separately from statistical significance. Explain what the p-value conditions on and what it cannot establish. For this weighted two-sample test, the core operation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; the prose, formula, table, and chart must all describe that same operation.
The weight S(t-) emphasizes earlier event times because pooled survival is largest near the start of follow-up. It provides an audit anchor, not an automatic scientific conclusion. The method-specific safeguard is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. Interpret the displayed effect under the constraint that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
15. Effect magnitude
Use effect magnitude to challenge the draft rather than merely document it. Translate the numerical output into the method’s own effect scale. The relevant technical fact is that the estimator uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, which determines what must be checked in the stored output.
A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. The numerical record supports a method-specific audit, but it does not remove design limitations. To test robustness, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; when stating direction, note that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
16. Software defaults
Software defaults receives an explicit pass, warning, or fail assessment. Record package versions, defaults, factor coding, convergence, and tie settings. This is necessary because the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and a different construction would answer a different survival question.
The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. The post links this evidence to the formula and saved output rather than repeating generic advice. A defensible review will show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, and it will state clearly that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
17. Cross-software reconciliation
Cross-software reconciliation receives an explicit pass, warning, or fail assessment. Reconcile output differences by checking definitions before blaming numerical software. This is necessary because the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and a different construction would answer a different survival question.
For 17. Cross-software reconciliation, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 4 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
18. Chart-to-table audit
Chart-to-table audit defines the checkpoint for this article. Reject any image or download whose filename, values, or method label belongs to another post. Because the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, the reviewer must connect the source rows to that mechanism rather than infer correctness from the software label.
The result sits between a scientific weighting choice and a multiple-testing problem when several weights are explored. The numerical record supports a method-specific audit, but it does not remove design limitations. To test robustness, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; when stating direction, note that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
19. Sensitivity specification
Use sensitivity specification to challenge the draft rather than merely document it. Repeat the analysis under a defensible neighboring specification and explain the comparison. The relevant technical fact is that the estimator uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, which determines what must be checked in the stored output.
For 19. Sensitivity specification, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 5 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
20. Scientific limitation
This checkpoint asks whether scientific limitation has been translated into executable analysis. Keep inference inside the observed design, coding, and follow-up window. The method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; therefore a generic survival-analysis explanation is not enough for this post.
For 20. Scientific limitation, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 6 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
21. Generalizability boundary
The reviewer should pause at generalizability boundary and reproduce the relevant step. Separate computational correctness from scientific validity and causal interpretation. In this analysis the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; that mechanism sets the boundary for correct interpretation.
A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. The post links this evidence to the formula and saved output rather than repeating generic advice. A defensible review will show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, and it will state clearly that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
22. Reproducible record
Treat reproducible record as an analytical decision. Preserve the CSV, transformation rules, code, output, metadata, and matched URLs. Here the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; a reproducible audit therefore records the relevant inputs, intermediate quantities, and settings before accepting the displayed result.
The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. Any discrepancy across Python, R, SPSS, or Excel must be traced to definitions or defaults. In addition, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; the reader should be told that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
23. Publication language
At publication language, the article must move from terminology to evidence. Make the published record independently reproducible and free of unsupported wording. Its defining computation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and the audit should show where the required quantities appear in the CSV or derived table.
The unsigned chi-square needs curve direction and signed score contributions for substantive interpretation. Any discrepancy across Python, R, SPSS, or Excel must be traced to definitions or defaults. In addition, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; the reader should be told that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
24. SEO and asset consistency
SEO and asset consistency can invalidate an otherwise polished article. Reject any image or download whose filename, values, or method label belongs to another post. The reason is specific to this procedure: it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. The final wording should state any unresolved limitation rather than hide it behind a p-value.
The result sits between a scientific weighting choice and a multiple-testing problem when several weights are explored. That result becomes publishable only after its risk-set, likelihood, or coding trail is reconciled. A useful next check is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. Directional language must remain consistent with the rule that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
25. Event-time weight function
At event-time weight function, the article must move from terminology to evidence. Define the decision operationally and show how it was checked. Its defining computation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and the audit should show where the required quantities appear in the CSV or derived table.
For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22. The article should retain this value in a saved table and connect it to its matching chart. Reviewers should also show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, because the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
26. Crossing survival curves
Use crossing survival curves to challenge the draft rather than merely document it. Connect this checkpoint to a saved calculation rather than a generic claim. The relevant technical fact is that the estimator uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, which determines what must be checked in the stored output.
For 26. Crossing survival curves, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 7 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
27. Observed and expected events
Observed and expected events can invalidate an otherwise polished article. Describe every status value in words and verify its frequency before fitting. The reason is specific to this procedure: it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. The final wording should state any unresolved limitation rather than hide it behind a p-value.
A general Fleming–Harrington family also permits gamma above zero to emphasize later differences. The result is meaningful only within the constructed endpoint and observed follow-up. The audit should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, then frame direction according to the principle that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
28. Weighted variance
The reviewer should pause at weighted variance and reproduce the relevant step. Report sampling uncertainty on the natural scale and reproduce its calculation. In this analysis the procedure uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; that mechanism sets the boundary for correct interpretation.
The selected rho and gamma values must be chosen before inspecting which weighting gives the smallest p-value. The numerical record supports a method-specific audit, but it does not remove design limitations. To test robustness, show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices; when stating direction, note that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
29. Alternative weighting families
Alternative weighting families is reviewed separately from statistical significance. Choose a plausible alternative model or weight before judging robustness. For this weighted two-sample test, the core operation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; the prose, formula, table, and chart must all describe that same operation.
The unsigned chi-square needs curve direction and signed score contributions for substantive interpretation. It provides an audit anchor, not an automatic scientific conclusion. The method-specific safeguard is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. Interpret the displayed effect under the constraint that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
30. Direction of separation
Direction of separation is reviewed separately from statistical significance. Verify that category ordering agrees across tables, coefficients, and prose. For this weighted two-sample test, the core operation uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up; the prose, formula, table, and chart must all describe that same operation.
The result sits between a scientific weighting choice and a multiple-testing problem when several weights are explored. Reproducing that figure from the bundled CSV is required before publication. The diagnostic sequence should show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices, while the substantive statement recognizes that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
31. Multiple weight searches
Multiple weight searches receives an explicit pass, warning, or fail assessment. Define the decision operationally and show how it was checked. This is necessary because the method uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, and a different construction would answer a different survival question.
For the prespecified rho=1 and gamma=0 weighting, the verified chi-square is 95.6330 with p about 1.38×10^-22. It provides an audit anchor, not an automatic scientific conclusion. The method-specific safeguard is to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. Interpret the displayed effect under the constraint that the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs.
32. Proportional-hazards context
Use proportional-hazards context to challenge the draft rather than merely document it. Connect this checkpoint to a saved calculation rather than a generic claim. The relevant technical fact is that the estimator uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up, which determines what must be checked in the stored output.
For 32. Proportional-hazards context, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 8 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
33. Early-versus-late evidence
Early-versus-late evidence can invalidate an otherwise polished article. Document the evidence and the consequence of a warning or failure. The reason is specific to this procedure: it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. The final wording should state any unresolved limitation rather than hide it behind a p-value.
For 33. Early-versus-late evidence, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 9 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
34. Practical weighting target
Practical weighting target can invalidate an otherwise polished article. Define the decision operationally and show how it was checked. The reason is specific to this procedure: it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. The final wording should state any unresolved limitation rather than hide it behind a p-value.
For 34. Practical weighting target, the Fleming Harrington Test review must record a method-specific publication checkpoint and the evidence required to pass it. This method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; therefore the editor should display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices. The bundled example supplies the following numerical anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. Checkpoint 10 passes only when the formula, code output, table, chart caption, and interpretation describe the same event definition and censoring rule.
Final Fleming Harrington Test release decision
This draft is released only when its exact formula, event definition, software settings, numerical result, chart captions, download files, and contextual links agree. The central computational mechanism is that it uses pooled survival raised to rho and one minus pooled survival raised to gamma to target prespecified portions of follow-up. That statement differentiates the article from the other twenty survival posts and prevents a shared template from substituting for method-specific explanation.
The final robustness record directs the editor to show the weight curve, weighted observed-minus-expected contributions, crossing curves, and alternative rho-gamma choices. The directional interpretation remains: the test statistic is unsigned; curves and weighted contributions identify where and in which direction separation occurs. Because the example is built from absences and G3 in a student-performance dataset, publication must keep the teaching-purpose limitation visible and must not recast the endpoint as clinical survival, mortality, equipment failure, or causal evidence.
Fleming Harrington Test compared with related methods
Choose the method by estimand, not menu proximity
| Related method | Comparison question |
|---|---|
| log-rank equal weights | Log-rank equal weights uses equal event-time weights and is the conventional overall curve comparison under proportional-hazards sensitivity. Compare it with the present method by checking the estimand, censor handling, weight or distribution, uncertainty, and practical interpretation; retain Fleming Harrington Test only when a survival-curve contrast targeted to early, middle, or late event times through prespecified weights is the actual target. |
| Breslow early risk-set weights | Breslow early risk-set weights gives greatest influence to early times where pooled risk sets are largest. Compare it with the present method by checking the estimand, censor handling, weight or distribution, uncertainty, and practical interpretation; retain Fleming Harrington Test only when a survival-curve contrast targeted to early, middle, or late event times through prespecified weights is the actual target. |
| Tarone–Ware square-root weights | Tarone–ware square-root weights provides intermediate early emphasis by using the square root of the pooled risk set. Compare it with the present method by checking the estimand, censor handling, weight or distribution, uncertainty, and practical interpretation; retain Fleming Harrington Test only when a survival-curve contrast targeted to early, middle, or late event times through prespecified weights is the actual target. |
| Fleming–Harrington prespecified early/late weights | Fleming–harrington prespecified early/late weights uses rho and gamma to target a planned part of follow-up. Compare it with the present method by checking the estimand, censor handling, weight or distribution, uncertainty, and practical interpretation; retain Fleming Harrington Test only when a survival-curve contrast targeted to early, middle, or late event times through prespecified weights is the actual target. |
How to report Fleming Harrington Test
A complete, restrained result statement
Reporting template
“A Fleming Harrington Test analysis used 649 records from dataset(100).csv. Duration was defined as absences plus one, and the event indicator equaled one when G3 was below 10; 100 events and 549 right-censored observations were available. With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. The analysis documented event coding, reference groups, risk sets, ties, assumptions, software settings, diagnostics, matching files, and the educational nature of the prepared survival endpoint.”
Include
Avoid calling hazard a probability, treating censoring as missingness, or converting a nonsignificant result into proof of equality. The final sentence should answer the stated estimand and no broader question.
Avoid
A complete report states the prepared time origin, event and censor codes, sample and event counts, group or predictor reference, exact method, formula, estimate or statistic, uncertainty, and the relevant diagnostics. It then gives this result: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance.
Fleming Harrington Test downloads
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Fleming Harrington Test frequently asked questions
Method-specific answers for draft review
What does Fleming Harrington Test measure?
Fleming Harrington Test is used for the estimand defined in this article. It uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up. The interpretation remains conditional on the stated time origin, event code, censoring rule, group or predictor coding, and any distributional or proportionality assumptions.
When should Fleming Harrington Test be used?
Use Fleming Harrington Test when the research objective requires rho–gamma pooled-survival weights that target a prespecified part of follow-up and the assumptions listed in the article are defensible. The method is inappropriate when a different event type, time emphasis, adjustment strategy, or hazard shape is the scientific target.
What data are used in this Fleming Harrington Test example?
The GP/MS event-time table is combined with pooled survival immediately before each event so rho=1 and gamma=0 emphasize earlier separation. All values come from the uploaded 649-row file and the disclosed absences-plus-one/G3 event construction.
What is the main Fleming Harrington Test result?
The result is summarized by this verified anchor: With ρ = 1 and γ = 0, the Fleming–Harrington statistic was χ² = 95.633, p < .001, emphasizing earlier differences while retaining a formal weighted log-rank variance. It should be read together with the method-specific assumptions, uncertainty, and the teaching-endpoint limitation rather than as a stand-alone causal conclusion.
How does censoring affect Fleming Harrington Test?
Censored records contribute to risk sets or likelihood survival terms until their observed duration. Their handling matters because the method uses prespecified rho and gamma weights to emphasize selected portions of pooled follow-up; treating censoring as an event or deleting censored rows would change the estimate and usually bias the analysis.
How are ties handled in Fleming Harrington Test?
The prepared durations are integer-valued, so tied times are common. The article states the exact pooled-event rule, weight, or Efron/Breslow approximation used for Fleming Harrington Test, and software results should be reconciled only after those defaults match.
Can Fleming Harrington Test be completed in Python?
Yes. The Python section reconstructs the data fields and exposes the intermediate quantities required for Fleming Harrington Test. It prints the benchmark result and supports the diagnostic task to display the weight curve, weighted contributions, crossing curves, and alternative prespecified rho–gamma choices.
Can Fleming Harrington Test be completed in R?
Yes. The R section uses a method-appropriate survival or competing-risk routine, declares factor references and tie or weighting settings, and provides an independent check of the benchmark result for Fleming Harrington Test.
Can Fleming Harrington Test be completed in SPSS?
SPSS is used only where a native procedure matches Fleming Harrington Test. When no exact native command exists, the post describes SPSS as a data-management, charting, or integration route and does not rename a different test or model.
How does Excel support Fleming Harrington Test?
Excel supports Fleming Harrington Test by displaying pooled survival, rho–gamma weight, weighted score, variance, chi-square, and p-value in visible cells. The matching workbook must reproduce selected Python and R benchmark values and retain the exact event, censoring, group, tie, and interval definitions.
What is the largest reporting mistake for Fleming Harrington Test?
The largest Fleming Harrington Test reporting error is choosing rho and gamma after inspecting the curves and then reporting the resulting p-value as prespecified. The article also keeps the teaching-endpoint limitation visible so the worked result is not presented as causal or naturally observed survival evidence.
Which internal guides support Fleming Harrington Test?
Start with Log Rank Test because it provides the nearest check on rho–gamma pooled-survival weights that target a prespecified part of follow-up. Use Breslow Test, Tarone Ware Test, Kaplan Meier Survival Curve to compare weighting, probability scale, model assumptions, or software implementation; each link has a specific methodological role rather than serving as generic navigation.