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.
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.
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.
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.
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 |
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Unlike NFI, parsimony-adjusted comparison index can exceed one or fall below zero before reporting conventions are applied.
The target model achieves very strong complexity-adjusted improvement over the baseline model.
Symbol and denominator control
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.
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.
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.
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.
TLI results and interpretation
Primary and supporting statistics are kept separate and precisely labeled.
Primary result
TLI
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.
Why the result is internally coherent
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.
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. |
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.
import pandas as pd
import numpy as npdf = 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))
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.
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)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.
* 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.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.
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.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.
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.
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.
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.
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.
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.
01 Tli Primary Metrics
This panel reconciles the headline estimate with its principal supporting values for TLI. Read Baseline degrees of freedom = 36 beside RMSEA = 0.020492; the first quantity is not replaced by the second.
The chart is used to report residual indices alongside Tucker–Lewis index. Its interpretation remains valid only when local residuals are inspected. 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.
02 Tli Tli Parsimony Components
This panel displays the quantities entering the defining equation for TLI. Read RMSEA = 0.020492 beside AGFI = 0.989823; 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.
03 Tli Tli Fit Context
This panel provides a visual diagnostic tied to the method’s exact decision rule for TLI. Read AGFI = 0.989823 beside GFI = 0.994572; 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.
04 Tli Tli Model Dimensions
This panel provides a visual diagnostic tied to the method’s exact decision rule for TLI. Read GFI = 0.994572 beside NFI = 0.989945; 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.
05 Tli Verified Result Summary
This panel reconciles the headline estimate with its principal supporting values for TLI. Read NFI = 0.989945 beside SRMR = 0.035876; 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 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 |
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.
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.
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.
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.
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.
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 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.
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.
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.
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 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.
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.