the Tucker–Lewis index

TLI: Formula, Verified Results, Charts and Interpretation

The Tucker–Lewis Index (Tucker–Lewis index), also called NNFI, compares the target and baseline chi-square-to-degrees-of-freedom ratios and rewards parsimony. Because of that penalty, it can differ from CFI and can occasionally fall outside the zero-to-one interval before software truncation. This guide uses the supplied real-data results, native MathML equations, matching charts, and separate Python, R, SPSS or AMOS, and Excel verification.

Covariance fitEstimator-specificResidual diagnosticsReal data

TLI0.996735

CFI0.997823

Target chi-square30.530

Target degrees of freedom24

Verified result

TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

For Tucker–Lewis index, Tucker–Lewis index = 0.996735 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

Interpretive limit: TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

What TLI measures

The exact estimand and the result this method is allowed to support.

TLI addresses one defined analytical target: The Tucker–Lewis Index (Tucker–Lewis index), also called NNFI, compares the target and baseline chi-square-to-degrees-of-freedom ratios and rewards parsimony. Because of that penalty, it can differ from CFI and can occasionally fall outside the zero-to-one interval before software truncation.

Quantity estimated in this analysis

The tucker–lewis index is reconstructed from the exact variables, matrix, model, panel, or resampling design shown below. The primary output is TLI = 0.996735; CFI = 0.997823 supplies the first supporting check. Tucker–Lewis index = 0.996735 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

For Tucker–Lewis index, the calculation retains full precision until the final display. That matters because the software reports, spreadsheet formulas, chart labels, and narrative must refer to one identical result rather than separately rounded approximations.

Interpretation that is not permitted

Tucker–Lewis index is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled Tucker–Lewis index requires matching robust target and baseline statistics.

For Tucker–Lewis index, this boundary is substantive. A nearby coefficient may share the same data or model, yet it answers a different question. The article therefore names every supporting statistic instead of using broad labels such as “valid,” “good,” or “significant” without the object being evaluated.

Worked conclusion: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

When to use TLI

Research scope, neighboring methods, and excluded claims.

Research question answered

The defensible question is whether the Tucker–Lewis index supports the result stated for the declared dataset and analytical specification. It is answered by reconstruct both chi-square-to-df ratios, followed by retain full precision during subtraction and division. The evidence is bounded by TLI = 0.996735 and its named companion quantities.

For Tucker–Lewis index, changing the case set, expert panel, item block, estimator, factor count, rotation, baseline model, bootstrap design, or criterion definition changes the question. Such a change requires a new result rather than a revision of the wording around the old value.

Nearest methods that answer different questions

CFI: CFI uses noncentrality improvement; Tucker–Lewis index uses relative chi-square-to-df ratios and a stronger complexity penalty.

NFI: NFI uses raw proportional chi-square reduction without the same parsimony adjustment.

These distinctions determine which formula, output table, and chart can legitimately appear in a Tucker–Lewis index post.

Scope limit: TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

Real data used for TLI

Variables, coding, sample or panel size, and the role each input plays.

For Tucker–Lewis index, the model-based analysis uses 649 complete student records and the declared indicator blocks shown in the table. G1, G2, and G3 define Academic Achievement; Medu, Fedu, and reverse-coded TravelAccess define Educational Advantage; goout, Dalc, and Walc define Social-Alcohol Exposure.

For the Tucker–Lewis index, these variables enter a prespecified covariance, composite, or path model. Their order, scaling, factor membership, and missing-data treatment must match the model syntax because Tucker–Lewis index = 0.996735 is conditional on that exact specification.

Variable	Meaning	Mean	SD	Range	Construct
G1	first-period grade	11.3991	2.7453	0–19	Academic Achievement
G2	second-period grade	11.5701	2.9136	0–19	Academic Achievement
G3	final grade	11.9060	3.2307	0–19	Academic Achievement
Medu	mother’s education	2.5146	1.1346	0–4	Educational Advantage
Fedu	father’s education	2.3066	1.0999	0–4	Educational Advantage
TravelAccess	reverse-coded travel accessibility	3.4314	0.7487	1–4	Educational Advantage
goout	frequency of going out	3.1849	1.1758	1–5	Social-Alcohol Exposure
Dalc	workday alcohol use	1.5023	0.9248	1–5	Social-Alcohol Exposure
Walc	weekend alcohol use	2.2804	1.2844	1–5	Social-Alcohol Exposure

Data-to-result trace: Reconstruct both chi-square-to-df ratios is the first data-integrity check, followed by retain full precision during subtraction and division. Both checks are performed before the primary coefficient is interpreted.

TLI assumptions and design requirements

Six conditions checked before the coefficient or decision rule is interpreted.

1. Target and baseline use the same sample

This condition determines whether the input object matches the formula. In the current Tucker–Lewis index analysis, the check is to reconstruct both chi-square-to-df ratios while preserving TLI = 0.996735.

For Tucker–Lewis index, if the condition is not met, the affected matrix, coefficient, cutoff, or path is recomputed from the corrected inputs. The result is not repaired by changing a label or selecting a more favorable software output.

2. Chi-square corrections match

This requirement controls whether the numerical estimate has the interpretation claimed. In the current Tucker–Lewis index analysis, the check is to retain full precision during subtraction and division while preserving CFI = 0.997823.

3. Degrees of freedom are correct

This design condition prevents an attractive coefficient from being attached to the wrong population or model. In the current Tucker–Lewis index analysis, the check is to verify the 24 and 36 degrees of freedom while preserving Target chi-square = 30.530.

4. Models converge

This specification rule keeps the software routes numerically comparable. In the current Tucker–Lewis index analysis, the check is to avoid truncating a theoretical out-of-range value without disclosure while preserving Target degrees of freedom = 24.

5. The baseline model is defined as intended

This diagnostic requirement is checked before a benchmark is applied. In the current Tucker–Lewis index analysis, the check is to compare with CFI to assess complexity effect while preserving Baseline chi-square = 3036.199.

6. Local residuals are inspected

This final condition governs whether the conclusion can survive replication or sensitivity analysis. In the current Tucker–Lewis index analysis, the check is to report residual indices alongside Tucker–Lewis index while preserving Baseline degrees of freedom = 36.

Assumption consequence: TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

TLI hypotheses or decision rule

The statistical question is stated at the correct level for this method.

Statistical question

Tucker–Lewis index is an estimated fit summary rather than a universal null-hypothesis test. The exact-fit chi-square and any close-fit tests retain their own hypotheses.

The value is interpreted against the formula and model context described here; a benchmark does not convert the Tucker–Lewis index into a proof test.

Decision for the worked analysis

The calculation yields TLI = 0.996735. Tucker–Lewis index = 0.996735 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

Tucker–Lewis index = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Language rule: the conclusion names the tested model, construct pair, item set, retained dimensions, or expert panel. It does not convert nonrejection into proof or a benchmark into a universal pass.

TLI formula and worked substitution

Native MathML preserves fractions, roots, summations, matrices, subscripts, and superscripts.

The equation below is the defining mathematical object for TLI. Its symbols are connected to the saved inputs and to Tucker–Lewis index = 0.996735, CFI = 0.997823, Target chi-square = 30.530, Target degrees of freedom = 24.

parsimony-adjusted comparison index equationsNative MathML · no external script

Tucker–Lewis formula

$TLI = \frac{\frac{{χ^{2}}_{b}}{{df}_{b}} - \frac{{χ^{2}}_{m}}{{df}_{m}}}{\frac{{χ^{2}}_{b}}{{df}_{b}} - 1}$

Unlike NFI, parsimony-adjusted comparison index can exceed one or fall below zero before reporting conventions are applied.

Worked substitution

$\begin{array}{rcl} TLI = \frac{\frac{3036.199}{36} - \frac{30.530}{24}}{\frac{3036.199}{36} - 1} \\ TLI & = & 0.996735 \end{array}$

The target model achieves very strong complexity-adjusted improvement over the baseline model.

Symbol and denominator control

For Tucker–Lewis index, the numerator, denominator, matrix order, degrees of freedom, factor count, or panel size shown in the MathML card is retained exactly. A formula from a neighboring method is not substituted even when both produce values on a similar scale.

Full-precision substitution

The spreadsheet and software outputs retain unrounded inputs until the final displayed value. The arithmetic is then reconciled with TLI = 0.996735 and CFI = 0.997823.

Step-by-step TLI calculation

Every stage is tied to a saved value and a method-specific condition.

The worked calculation follows six operations specific to the Tucker–Lewis index. Each operation produces a quantity used by the next step, so a discrepancy is resolved where it originates rather than hidden by rounding.

Establish the analytical object

Action: Reconstruct both chi-square-to-df ratios.

Numerical trace: Tucker–Lewis index = 0.996735; CFI = 0.997823.

For Tucker–Lewis index, condition: target and baseline use the same sample. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Reconstruct the first required quantity

Action: Retain full precision during subtraction and division.

Numerical trace: CFI = 0.997823; Target chi-square = 30.530.

Condition: chi-square corrections match. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Verify the companion quantity

Action: Verify the 24 and 36 degrees of freedom.

Numerical trace: Target chi-square = 30.530; Target degrees of freedom = 24.

Condition: degrees of freedom are correct. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Apply the decision rule

Action: Avoid truncating a theoretical out-of-range value without disclosure.

Numerical trace: Target degrees of freedom = 24; Baseline chi-square = 3036.199.

Condition: models converge. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Inspect local evidence

Action: Compare with CFI to assess complexity effect.

Numerical trace: Baseline chi-square = 3036.199; Baseline degrees of freedom = 36.

Condition: the baseline model is defined as intended. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Reconcile and report

Action: Report residual indices alongside Tucker–Lewis index.

Numerical trace: Baseline degrees of freedom = 36; RMSEA = 0.020492.

For Tucker–Lewis index, condition: local residuals are inspected. This step is repeated after any correction to coding, matrix construction, model syntax, rotation, resampling, or expert-rating denominators.

Final reconciliation: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI results and interpretation

Primary and supporting statistics are kept separate and precisely labeled.

Primary result

0.996735

TLI

Why the result is internally coherent

For Tucker–Lewis index, cFI = 0.997823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

For Tucker–Lewis index, the two quantities are reported together because one is primary and the other supplies context; neither is renamed as the other.

Result item	Exact value	Interpretation restricted to this method
TLI	0.996735	TLI = 0.996735 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.
CFI	0.997823	CFI = 0.997823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.
Target chi-square	30.530	Target chi-square = 30.530 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
Target degrees of freedom	24	Target degrees of freedom = 24 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.
Baseline chi-square	3036.199	Baseline chi-square = 3036.199 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.
Baseline degrees of freedom	36	Baseline degrees of freedom = 36 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.
RMSEA	0.020492	RMSEA = 0.020492 expresses approximate discrepancy per degree of freedom and requires the corresponding confidence interval and estimator correction for complete reporting.
AGFI	0.989823	AGFI = 0.989823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.
GFI	0.994572	GFI = 0.994572 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.
NFI	0.989945	NFI = 0.989945 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.
SRMR	0.035876	SRMR = 0.035876 is the root mean square of standardized residuals; the average must be checked against the largest individual residual cells.
Exact-fit p-value	0.167787	Exact-fit p-value = 0.167787 is a probability under the stated null model and does not quantify practical magnitude.
Sample size	649	Sample size = 649 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.
Observed indicators	9	Observed indicators = 9 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.

Maximum defensible claim: TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

TLI in Python

The Python route calculates or reconstructs the exact named result.

The Python workflow uses semopy, Model to calculate or extract the Tucker–Lewis index from the declared data and analytical specification. It must reproduce Tucker–Lewis index = 0.996735 and retain CFI = 0.997823 as a separate supporting quantity.

The code is read as an executable analysis, not as a printed answer. Its critical verification is to reconstruct both chi-square-to-df ratios; the associated design condition is that target and baseline use the same sample. Tucker–Lewis index is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled Tucker–Lewis index requires matching robust target and baseline statistics.

Python — TLI

import pandas as pd

import numpy as np

df = pd.read_csv("student-por.csv", sep=";") df["TravelAccess"] = 5 - df["traveltime"] vars9 = ["G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc"] X = df[vars9].dropna() from semopy import Model, calc_stats model = Model(""" Achievement =~ G1 + G2 + G3 Education =~ Medu + Fedu + TravelAccess SocialAlcohol =~ goout + Dalc + Walc Achievement ~ Education + SocialAlcohol """) model.fit(df) stats = calc_stats(model) print("TLI") print(stats.T if "tli" == "all" else stats.T.loc[["TLI"]]) print(model.inspect(std_est=True))

Python interpretation: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI in R

The R route declares package, estimator, extraction, rotation, or resampling settings.

The R route uses lavaan and the displayed arguments to estimate the Tucker–Lewis index. Package defaults are made explicit because estimator, matrix type, extraction, rotation, baseline, or bootstrap choices can change the result.

R output is reconciled with Tucker–Lewis index = 0.996735 after the analyst retain full precision during subtraction and division. Agreement is expected only when the case set, variable order, and method settings match the Python and workbook calculations.

R — TLI

d <- read.csv2("student-por.csv")

d$TravelAccess <- 5 - d$traveltime

vars9 <- c("G1","G2","G3","Medu","Fedu","TravelAccess","goout","Dalc","Walc")

X <- d[vars9]

library(lavaan)

model <- '

Achievement =~ G1 + G2 + G3

Education =~ Medu + Fedu + TravelAccess

SocialAlcohol =~ goout + Dalc + Walc

Achievement ~ Education + SocialAlcohol

'

fit <- sem(model,data=d,estimator="ML")

fitMeasures(fit,c("tli"))

standardizedSolution(fit)

R interpretation: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI in SPSS or AMOS

The procedure is labeled honestly when base SPSS does not expose the coefficient.

The SPSS or AMOS section shows the procedure that is actually available for the Tucker–Lewis index. When base SPSS does not expose the coefficient, the syntax prepares the correct matrix or model and the coefficient is obtained through AMOS, MATRIX operations, or a validated integration rather than by renaming a different test.

The output must identify Tucker–Lewis index = 0.996735 and the settings needed to reproduce it. The software review specifically verify the 24 and 36 degrees of freedom, while preserving the requirement that degrees of freedom are correct.

SPSS or AMOS — TLI

* TLI is obtained from the prespecified AMOS covariance model.

* Three factors: G1 G2 G3; Medu Fedu TravelAccess; goout Dalc Walc.

* Maximum likelihood, N=649, df=24.

* Request standardized estimates, residual moments, squared multiple correlations, and fit measures.

* Reconcile the exact TLI value with the formula and result ledger in this draft.

SPSS or AMOS interpretation: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI in Excel

The workbook exposes source values, intermediate arithmetic, and the final formula.

The Excel workbook is an arithmetic audit for the Tucker–Lewis index. Named cells retain the inputs, intermediate components, and final formula leading to Tucker–Lewis index = 0.996735; no rounded constant is pasted over a formula cell.

Excel can verify visible calculations and cross-software agreement, but it does not replace estimation, optimization, rotation, or resampling that must occur in statistical software. The workbook therefore focuses on the check to avoid truncating a theoretical out-of-range value without disclosure and documents CFI = 0.997823 independently.

Excel — TLI

Data: 649 rows with documented coding.

Inputs: named cells or ranges required only by TLI.

Calculation: =((ChiSq_Base/DF_Base)-(ChiSq_Model/DF_Model))/((ChiSq_Base/DF_Base)-1)

Audit: compare full-precision Excel output with the Python, R, and SPSS/AMOS values.

Decision: reference the exact result and diagnostics; never paste a rounded value over the formula cell.

Excel interpretation: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI charts and visual diagnostics

Each supplied image is interpreted through its own values and analytical purpose.

Every image below is interpreted as part of the same Tucker–Lewis index analysis. The captions identify what the panel contributes, the exact values visible in the result set, and the condition that would invalidate the reading.

TLI — 01 Tli Primary Metrics — 01 Tli Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for **TLI**. Read **TLI = 0.996735** beside **CFI = 0.997823**; the first quantity is not replaced by the second.
The chart is used to reconstruct both chi-square-to-df ratios. Its interpretation remains valid only when target and baseline use the same sample. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

TLI — 02 Tli Tli Parsimony Components — 02 Tli Tli Parsimony Components
This panel displays the quantities entering the defining equation for **TLI**. Read **CFI = 0.997823** beside **Target chi-square = 30.530**; the first quantity is not replaced by the second.
The chart is used to retain full precision during subtraction and division. Its interpretation remains valid only when chi-square corrections match. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

TLI — 03 Tli Tli Fit Context — 03 Tli Tli Fit Context
This panel provides a visual diagnostic tied to the method’s exact decision rule for **TLI**. Read **Target chi-square = 30.530** beside **Target degrees of freedom = 24**; the first quantity is not replaced by the second.
The chart is used to verify the 24 and 36 degrees of freedom. Its interpretation remains valid only when degrees of freedom are correct. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

TLI — 04 Tli Tli Model Dimensions — 04 Tli Tli Model Dimensions
This panel provides a visual diagnostic tied to the method’s exact decision rule for **TLI**. Read **Target degrees of freedom = 24** beside **Baseline chi-square = 3036.199**; the first quantity is not replaced by the second.
The chart is used to avoid truncating a theoretical out-of-range value without disclosure. Its interpretation remains valid only when models converge. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

TLI — 05 Tli Verified Result Summary — 05 Tli Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for **TLI**. Read **Baseline chi-square = 3036.199** beside **Baseline degrees of freedom = 36**; the first quantity is not replaced by the second.
The chart is used to compare with CFI to assess complexity effect. Its interpretation remains valid only when the baseline model is defined as intended. A visual pattern that conflicts with the saved table triggers re-estimation or relabeling of the specific chart, not a broad claim that the method has passed.

TLI verification and sensitivity analysis

Six failure modes are checked against the formula, data, output, and charts.

The following diagnostics are not a general checklist. Each one targets a failure mode that can change the calculation or interpretation of TLI.

1. Reconstruct both chi-square-to-df ratios

Begin by reconstruct both chi-square-to-df ratios. For the Tucker–Lewis index, this operation directly connects TLI = 0.996735 with Target chi-square = 30.530. Tucker–Lewis index = 0.996735 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

The governing condition is that target and baseline use the same sample. If it fails, the primary coefficient may be attached to the wrong input object. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with CFI, because CFI uses noncentrality improvement; Tucker–Lewis index uses relative chi-square-to-df ratios and a stronger complexity penalty.

2. Retain full precision during subtraction and division

Next, retain full precision during subtraction and division. For the Tucker–Lewis index, this operation directly connects CFI = 0.997823 with Target degrees of freedom = 24. CFI = 0.997823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

The governing condition is that chi-square corrections match. If it fails, the companion statistic may no longer describe the same model or sample. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with NFI, because NFI uses raw proportional chi-square reduction without the same parsimony adjustment.

3. Verify the 24 and 36 degrees of freedom

The third verification is to verify the 24 and 36 degrees of freedom. For the Tucker–Lewis index, this operation directly connects Target chi-square = 30.530 with Baseline chi-square = 3036.199. Target chi-square = 30.530 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.

The governing condition is that degrees of freedom are correct. If it fails, the decision boundary can move because the required quantity has changed. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with RMSEA, because RMSEA is an absolute approximate-fit measure rather than a baseline comparison.

4. Avoid truncating a theoretical out-of-range value without disclosure

After the core arithmetic is stable, avoid truncating a theoretical out-of-range value without disclosure. For the Tucker–Lewis index, this operation directly connects Target degrees of freedom = 24 with Baseline degrees of freedom = 36. Target degrees of freedom = 24 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.

The governing condition is that models converge. If it fails, software agreement can be artificial if unlike definitions are compared. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with CFI, because CFI uses noncentrality improvement; Tucker–Lewis index uses relative chi-square-to-df ratios and a stronger complexity penalty.

5. Compare with CFI to assess complexity effect

A robustness review must compare with CFI to assess complexity effect. For the Tucker–Lewis index, this operation directly connects Baseline chi-square = 3036.199 with RMSEA = 0.020492. Baseline chi-square = 3036.199 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.

The governing condition is that the baseline model is defined as intended. If it fails, a favorable average can conceal a local failure. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with NFI, because NFI uses raw proportional chi-square reduction without the same parsimony adjustment.

6. Report residual indices alongside TLI

The final reconciliation should report residual indices alongside Tucker–Lewis index. For the Tucker–Lewis index, this operation directly connects Baseline degrees of freedom = 36 with AGFI = 0.989823. Baseline degrees of freedom = 36 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.

The governing condition is that local residuals are inspected. If it fails, the published conclusion can exceed the evidence actually reproduced. The remedy is to correct the relevant coding, matrix, model, rotation, resampling, or panel denominator and rerun the calculation. This check also prevents confusion with RMSEA, because RMSEA is an absolute approximate-fit measure rather than a baseline comparison.

#	Verification operation	Condition protected	Saved quantity traced
1	reconstruct both chi-square-to-df ratios	target and baseline use the same sample	TLI = 0.996735
2	retain full precision during subtraction and division	chi-square corrections match	CFI = 0.997823
3	verify the 24 and 36 degrees of freedom	degrees of freedom are correct	Target chi-square = 30.530
4	avoid truncating a theoretical out-of-range value without disclosure	models converge	Target degrees of freedom = 24
5	compare with CFI to assess complexity effect	the baseline model is defined as intended	Baseline chi-square = 3036.199
6	report residual indices alongside TLI	local residuals are inspected	Baseline degrees of freedom = 36

Diagnostic conclusion: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Failure boundary: TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

TLI compared with related methods

Differences in estimand, formula, and conclusion determine the correct choice.

Method choice depends on the estimand, model, and data structure. These three comparisons explain why the post uses the Tucker–Lewis index formula and output rather than a nearby procedure.

CFI

CFI uses noncentrality improvement; Tucker–Lewis index uses relative chi-square-to-df ratios and a stronger complexity penalty.

In the current analysis, CFI = 0.997823 remains evidence for the Tucker–Lewis index; it is not relabeled as a CFI result. CFI = 0.997823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

NFI

NFI uses raw proportional chi-square reduction without the same parsimony adjustment.

In the current analysis, Target chi-square = 30.530 remains evidence for the Tucker–Lewis index; it is not relabeled as a NFI result. Target chi-square = 30.530 is read with its degrees of freedom, estimator, sample size, and p-value; it is not a stand-alone effect size.

RMSEA

RMSEA is an absolute approximate-fit measure rather than a baseline comparison.

In the current analysis, Target degrees of freedom = 24 remains evidence for the Tucker–Lewis index; it is not relabeled as a RMSEA result. Target degrees of freedom = 24 is retained as a distinct supporting quantity for the Tucker–Lewis index; it is not substituted for the primary result.

Selection rule: The Tucker–Lewis Index (TLI), also called NNFI, compares the target and baseline chi-square-to-degrees-of-freedom ratios and rewards parsimony. Because of that penalty, it can differ from CFI and can occasionally fall outside the zero-to-one interval before software truncation.

How to report TLI

A complete result paragraph includes the value, analytical object, settings, and limitation.

Results paragraph

TLI was evaluated using the declared data, specification, and software settings. The primary result was TLI = 0.996735; CFI = 0.997823 and Target chi-square = 30.530 supplied supporting context. TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

The report then states the limitation explicitly: Tucker–Lewis index is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled Tucker–Lewis index requires matching robust target and baseline statistics.

Settings that must accompany the result

target and baseline use the same sample; chi-square corrections match; degrees of freedom are correct; models converge.

For Tucker–Lewis index, these details identify the exact version of the analysis and make cross-software reconciliation possible.

Verification actions retained in the record

reconstruct both chi-square-to-df ratios; retain full precision during subtraction and division; verify the 24 and 36 degrees of freedom; avoid truncating a theoretical out-of-range value without disclosure.

The final wording is revised only after those operations reproduce the saved values.

Reporting standard: name the statistic, value, analytical object, sample or panel size, method settings, and limitation in the same result paragraph.

16A

TLI decision scenarios

For TLI, worked conflicts show how the conclusion changes when an input, assumption, or supporting statistic fails.

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Consider a review in which TLI = 0.996735 is reproduced but CFI = 0.997823 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to reconstruct both chi-square-to-df ratios and verify that target and baseline use the same sample.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with CFI only for method selection: CFI uses noncentrality improvement; TLI uses relative chi-square-to-df ratios and a stronger complexity penalty. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Input-definition sensitivity: Retain full precision during subtraction and division

Consider a review in which Target chi-square = 30.530 is reproduced but Target degrees of freedom = 24 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain full precision during subtraction and division and verify that chi-square corrections match.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with NFI only for method selection: NFI uses raw proportional chi-square reduction without the same parsimony adjustment. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

Consider a review in which Baseline chi-square = 3036.199 is reproduced but Baseline degrees of freedom = 36 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the 24 and 36 degrees of freedom and verify that degrees of freedom are correct.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with RMSEA only for method selection: RMSEA is an absolute approximate-fit measure rather than a baseline comparison. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Local-chart conflict: Avoid truncating a theoretical out-of-range value without disclosure

Consider a review in which RMSEA = 0.020492 is reproduced but AGFI = 0.989823 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid truncating a theoretical out-of-range value without disclosure and verify that models converge.

Alternative-method challenge: Compare with CFI to assess complexity effect

Consider a review in which GFI = 0.994572 is reproduced but NFI = 0.989945 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare with CFI to assess complexity effect and verify that the baseline model is defined as intended.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with NFI only for method selection: NFI uses raw proportional chi-square reduction without the same parsimony adjustment. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Replication and reporting decision: Report residual indices alongside TLI

Consider a review in which SRMR = 0.035876 is reproduced but Exact-fit p-value = 0.167787 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to report residual indices alongside TLI and verify that local residuals are inspected.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with RMSEA only for method selection: RMSEA is an absolute approximate-fit measure rather than a baseline comparison. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Consider a review in which Sample size = 649 is reproduced but Observed indicators = 9 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to reconstruct both chi-square-to-df ratios and verify that target and baseline use the same sample.

Input-definition sensitivity: Retain full precision during subtraction and division

Consider a review in which TLI = 0.996735 is reproduced but CFI = 0.997823 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain full precision during subtraction and division and verify that chi-square corrections match.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with NFI only for method selection: NFI uses raw proportional chi-square reduction without the same parsimony adjustment. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

Consider a review in which Target chi-square = 30.530 is reproduced but Target degrees of freedom = 24 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the 24 and 36 degrees of freedom and verify that degrees of freedom are correct.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with RMSEA only for method selection: RMSEA is an absolute approximate-fit measure rather than a baseline comparison. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Local-chart conflict: Avoid truncating a theoretical out-of-range value without disclosure

Consider a review in which Baseline chi-square = 3036.199 is reproduced but Baseline degrees of freedom = 36 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to avoid truncating a theoretical out-of-range value without disclosure and verify that models converge.

Alternative-method challenge: Compare with CFI to assess complexity effect

Consider a review in which RMSEA = 0.020492 is reproduced but AGFI = 0.989823 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to compare with CFI to assess complexity effect and verify that the baseline model is defined as intended.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with NFI only for method selection: NFI uses raw proportional chi-square reduction without the same parsimony adjustment. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Replication and reporting decision: Report residual indices alongside TLI

Consider a review in which GFI = 0.994572 is reproduced but NFI = 0.989945 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to report residual indices alongside TLI and verify that local residuals are inspected.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with RMSEA only for method selection: RMSEA is an absolute approximate-fit measure rather than a baseline comparison. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Consider a review in which SRMR = 0.035876 is reproduced but Exact-fit p-value = 0.167787 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to reconstruct both chi-square-to-df ratios and verify that target and baseline use the same sample.

Input-definition sensitivity: Retain full precision during subtraction and division

Consider a review in which Sample size = 649 is reproduced but Observed indicators = 9 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to retain full precision during subtraction and division and verify that chi-square corrections match.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with NFI only for method selection: NFI uses raw proportional chi-square reduction without the same parsimony adjustment. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

Consider a review in which TLI = 0.996735 is reproduced but CFI = 0.997823 is not. For the Tucker–Lewis index, the disagreement cannot be settled by averaging the two outputs because they describe different components of the analysis. The first action is to verify the 24 and 36 degrees of freedom and verify that degrees of freedom are correct.

If the discrepancy persists, the analyst identifies whether the cause is coding, matrix construction, model identification, estimator, rotation, baseline definition, resampling, or panel denominator. The result is compared with RMSEA only for method selection: RMSEA is an absolute approximate-fit measure rather than a baseline comparison. The published conclusion remains TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI downloads and reproducibility files

All linked files belong to the same analysis and remain on onlineinternetcafe.com.

PYTLI Python reportReproduces TLI = 0.996735 and documents how to reconstruct both chi-square-to-df ratios.Open file →RTLI R reportReproduces TLI = 0.996735 and documents how to retain full precision during subtraction and division.Open file →SPSSTLI SPSS/AMOS outputReproduces TLI = 0.996735 and documents how to verify the 24 and 36 degrees of freedom.Open file →XLSXTLI Excel workbookReproduces TLI = 0.996735 and documents how to avoid truncating a theoretical out-of-range value without disclosure.Open file →

The four files belong to one TLI analysis. Their primary values, variable order, method settings, and chart labels must agree; a mismatch is resolved in the source calculation before the WordPress draft is published.

TLI frequently asked questions

Answers use the worked result and the exact method boundary.

What does TLI measure?

The Tucker–Lewis Index (TLI), also called NNFI, compares the target and baseline chi-square-to-degrees-of-freedom ratios and rewards parsimony. Because of that penalty, it can differ from CFI and can occasionally fall outside the zero-to-one interval before software truncation.

What is the main result in this TLI analysis?

TLI = 0.996735. TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

What does the result not prove?

TLI is not a residual index, not explained variance, and not numerically identical to CFI. Robust or scaled TLI requires matching robust target and baseline statistics.

Which supporting value should be reported with the primary result?

CFI = 0.997823 is the first companion quantity. CFI = 0.997823 belongs to the declared covariance model and estimator; its baseline, complexity adjustment, or residual weighting must match the displayed formula.

Which assumption is most likely to change the interpretation?

For TLI, the first requirement is that target and baseline use the same sample. The result is recomputed if that condition is not satisfied.

What is the most important numerical verification?

The analyst must reconstruct both chi-square-to-df ratios. That operation traces TLI = 0.996735 to the formula and saved inputs.

Why can software packages disagree on TLI?

Disagreement can arise because chi-square corrections match or because the packages implement different estimators, matrices, baselines, rotations, standardizations, bootstrap rules, or coefficient definitions. Matching labels alone is not enough.

How is TLI different from CFI?

CFI uses noncentrality improvement; TLI uses relative chi-square-to-df ratios and a stronger complexity penalty.

How should a chart be interpreted?

Each chart is tied to a named output such as Target chi-square = 30.530. It supports a local calculation or diagnostic and does not replace the full numerical result.

How should TLI be reported?

Report TLI = 0.996735, the required supporting quantities, sample or panel size, exact method settings, and this qualified conclusion: TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

TLI = 0.996735 indicates excellent incremental fit after accounting for target-model complexity. The value remains conditional on a correctly specified independence baseline and matching estimator corrections.

What TLI measures

Quantity estimated in this analysis

Interpretation that is not permitted

When to use TLI

Research question answered

Nearest methods that answer different questions

Real data used for TLI

TLI assumptions and design requirements

1. Target and baseline use the same sample

2. Chi-square corrections match

3. Degrees of freedom are correct

4. Models converge

5. The baseline model is defined as intended

6. Local residuals are inspected

TLI hypotheses or decision rule

Statistical question

Decision for the worked analysis

TLI formula and worked substitution

Symbol and denominator control

Full-precision substitution

Step-by-step TLI calculation

Establish the analytical object

Reconstruct the first required quantity

Verify the companion quantity

Apply the decision rule

Inspect local evidence

Reconcile and report

TLI results and interpretation

Primary result

Why the result is internally coherent

TLI in Python

TLI in R

TLI in SPSS or AMOS

TLI in Excel

TLI charts and visual diagnostics

01 Tli Primary Metrics

02 Tli Tli Parsimony Components

03 Tli Tli Fit Context

04 Tli Tli Model Dimensions

05 Tli Verified Result Summary

01 Tli Primary Metrics

02 Tli Tli Parsimony Components

03 Tli Tli Fit Context

04 Tli Tli Model Dimensions

05 Tli Verified Result Summary

TLI verification and sensitivity analysis

1. Reconstruct both chi-square-to-df ratios

2. Retain full precision during subtraction and division

3. Verify the 24 and 36 degrees of freedom

4. Avoid truncating a theoretical out-of-range value without disclosure

5. Compare with CFI to assess complexity effect

6. Report residual indices alongside TLI

TLI compared with related methods

CFI

NFI

RMSEA

How to report TLI

Results paragraph

Settings that must accompany the result

Verification actions retained in the record

TLI decision scenarios

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Input-definition sensitivity: Retain full precision during subtraction and division

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

Local-chart conflict: Avoid truncating a theoretical out-of-range value without disclosure

Alternative-method challenge: Compare with CFI to assess complexity effect

Replication and reporting decision: Report residual indices alongside TLI

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Input-definition sensitivity: Retain full precision during subtraction and division

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

Local-chart conflict: Avoid truncating a theoretical out-of-range value without disclosure

Alternative-method challenge: Compare with CFI to assess complexity effect

Replication and reporting decision: Report residual indices alongside TLI

Boundary-case interpretation: Reconstruct both chi-square-to-df ratios

Input-definition sensitivity: Retain full precision during subtraction and division

Software-definition reconciliation: Verify the 24 and 36 degrees of freedom

TLI downloads and reproducibility files

TLI frequently asked questions

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