PNFI in AMOS: How Should Fit and Parsimony Be Balanced?
PNFI rewards models that retain improvement over the baseline model with fewer free parameters. It supports comparisons among models fitted to the same data rather than providing an independent cutoff for declaring a model acceptable.
PNFI equals NFI multiplied by the parsimony ratio. It is usually lower than NFI because it adjusts for degrees of freedom. It is best used to compare competing models with the same observed variables and sample, alongside absolute fit, theoretical plausibility, and other model-selection criteria.
Fit and parsimony must be read together
- Model fit
How well does the model reproduce data?
- Complexity
How many free parameters are needed?
- PNFI
Adds a parsimony perspective
- Theory
Retain a defensible specification
Conceptual diagram. Do not choose a model solely for a higher parsimony index.
Source: Lê Minh · Conceptual illustration, not software output.1. Learning Objectives and Comparison Conditions
After this guide, you should be able to recalculate PNFI from NFI and PRATIO, retrieve the correct AMOS tables, explain why a simpler model may be preferred, and recognize when PNFI comparisons are inappropriate.
Models must be estimated using the same observations, input variables, and baseline model. If sample size, variable lists, or missing-data treatment differ, PNFI no longer provides a clean comparison of parsimony.
- Models use the same data and observations.
- The observed-variable lists are identical.
- The estimator and missing-data treatment are identical.
- The baseline model has the same degrees of freedom.
- Each model has its own theoretical rationale.
- Determine whether models are nested or nonnested before choosing a comparison procedure.
2. What Does PNFI Measure?
The Parsimonious Normed Fit Index (PNFI) adjusts the Normed Fit Index (NFI) by the ratio of the research model’s degrees of freedom to those of the baseline model. IBM Corp. (2022) gives PNFI = NFI x PRATIO, where PRATIO = df_model / df_baseline.
NFI measures improvement of the research model over the baseline, usually the independence model (Bentler and Bonett, 1980). PNFI reduces NFI according to complexity: models with more free parameters have lower df and receive a larger penalty.
PNFI does not measure simplicity alone. A highly constrained model may have high PRATIO but poor fit. The index represents a tradeoff between fit improvement and parsimony.
| Component | Formula or source | Meaning |
|---|---|---|
| NFI | Comparison with the baseline model | Relative improvement in fit |
| PRATIO | df_model / df_baseline | Proportion of degrees of freedom retained |
| PNFI | NFI x PRATIO | Relative fit adjusted for parsimony |
3. Why Consider Parsimony in SEM?
Adding free parameters generally reduces chi-square and improves in-sample fit, even when the new parameters capture noise. A parsimony index attempts to prevent automatic preference for complex models merely because they are more flexible.
Mulaik et al. (1989) argue that model assessment should account for falsifiability and parameter economy. PNFI incorporates this idea into a fit index through degrees of freedom.
Parsimony does not mean selecting the fewest paths at any cost. A model must be sufficiently rich to represent theory and reproduce the data acceptably. A simple model omitting an important structure is not better than a justified complex model.
Prefer a simpler model when the models are theoretically plausible, use the same data, and have defensible fit. PNFI does not justify an underfitting model.
4. Calculating PNFI from AMOS Tables
Suppose NFI = 0,94, the Default model has df = 120, and the Independence model has df = 180. PRATIO = 120 / 180 = 0,667. PNFI = 0,94 x 0,667 = 0,627.
Small differences between hand calculations and AMOS usually result from rounding NFI or PRATIO too early. Retain more digits in intermediate calculations and round PNFI only in the final table.
The formula shows why PNFI is usually below NFI: PRATIO lies between 0 and 1. Do not apply an NFI cutoff directly to PNFI.
| Quantity | Value | Calculation |
|---|---|---|
| NFI | 0,940 | Retrieved from Baseline Comparisons |
| df model | 120 | Default model |
| df baseline | 180 | Independence model |
| PRATIO | 0,667 | 120 / 180 |
| PNFI | 0,627 | 0,940 x 0,667 |
Baseline df refers to the Independence model’s df from the same run. Do not use the saturated model’s df or combine degrees of freedom from different variable sets.
5. Retrieving NFI, PRATIO, and PNFI in AMOS
In AMOS Graphics, open Analysis Properties, enable Fit measures, and run the model. In Text Output, NFI appears under Baseline Comparisons; PRATIO and PNFI appear under Parsimony-Adjusted Measures. The AMOS 29 guide also permits PNFI display on the output path diagram using a text macro.
Always read the Default model row. The Saturated model does not provide a parsimony comparison equivalent to the research model; the Independence model is the baseline used in the formula.
Save chi-square, df, CFI, TLI, RMSEA with its confidence interval, SRMR if available, NFI, PRATIO, PNFI, and AIC and BIC when provided. Place PNFI within this broader comparison profile.
Estimate each competing model
Keep the same data, variables, estimator, and missing-data treatment.
Open Baseline Comparisons
Record the Default model’s NFI and verify the same baseline model.
Open Parsimony-Adjusted Measures
Record PRATIO and PNFI for the Default model.
Cross-check the calculation
Multiply unrounded NFI by PRATIO to detect copying errors.
Construct a multiple-model table
Place fit, df, PNFI, and other criteria in one row per model.
Make a theoretically informed decision
Do not choose a model merely because its PNFI is marginally higher.
6. Hypothetical Numerical Example: Model A or Model B?
The following example is illustrative rather than actual research data. Both models use the same 12 observed variables and 420 cases. The baseline model has df = 180. Model A permits more paths, with NFI = 0,95 and df = 100. Model B removes paths lacking strong justification, with NFI = 0,94 and df = 120.
A’s PNFI is 0,95 x (100 / 180) = 0,528. B’s PNFI is 0,94 x (120 / 180) = 0,627. B has NFI lower by 0,01 but PNFI higher by 0,099 because it retains more degrees of freedom.
PNFI alone cannot justify choosing B. If B has large residuals or omits essential relationships, its parsimony is not worth the cost. If both models have defensible fit and the paths removed from B lack theoretical support, PNFI provides favorable evidence for B.
| Index | Model A | Model B | Interpretation |
|---|---|---|---|
| NFI | 0,950 | 0,940 | A improves fit over the baseline slightly more |
| df | 100 | 120 | B retains more constraints |
| PRATIO | 0,556 | 0,667 | B is more parsimonious according to df |
| PNFI | 0,528 | 0,627 | B has a more favorable fit–parsimony tradeoff |
| Conclusion | More complex | Simpler | Choose only after checking fit and theory |
7. PNFI Has No Universal Cutoff
No single widely accepted threshold establishes adequate PNFI. Its value depends on the variable count, baseline model, degrees of freedom, and relative fit. Rules such as PNFI above 0,50 or 0,60 should therefore be treated as heuristics in some sources rather than decisive standards.
PNFI is more useful for comparing competing models on the same data. Interpret small differences cautiously because there is no simple test distribution applicable to every situation.
Kline (2023) recommends avoiding model selection through a cutoff checklist. Theoretical structure, residuals, parameters, stability, and replicability remain the principal evidence.
Because PNFI is multiplied by PRATIO, it is almost always below NFI. Declaring a model unacceptable because PNFI < 0,90 misunderstands the formula.
8. PNFI, PCFI, AIC, BIC, and Nested-Model Tests
PNFI uses NFI as its basis, whereas PCFI multiplies CFI by the same PRATIO. Both adjust for df while inheriting properties of the original indices. Report both as supplementary evidence if useful, without turning them into separate decision gates.
For nested models, a chi-square difference test directly assesses whether additional constraints reduce fit beyond sampling error. With robust estimators, apply the appropriate correction to the difference test.
AIC and BIC are information criteria. Lower values are preferred within the comparison set, but their penalties differ. Burnham and Anderson (2004) emphasize considering multiple justified models rather than seeking one decisive index.
PNFI does not replace AIC, BIC, or chi-square difference testing. Each addresses a different question about fit, penalties, and comparative evidence.
| Tool | Question | Favorable direction | Condition |
|---|---|---|---|
| PNFI or PCFI | What is fit after adjustment for df? | Higher within the comparison set | Same data and baseline |
| Chi-square difference | Do additional constraints significantly reduce fit? | Depends on H0 and alpha | Nested models |
| AIC | Relative expected information loss? | Lower | Same data and suitable likelihood |
| BIC | Fit–penalty balance with a stronger complexity penalty? | Lower | Same data and suitable likelihood |
| Fit indices | Does each model have defensible fit? | Based on the evidence profile | Do not use a single cutoff |
9. Common PNFI Interpretation Errors
| Weak practice | Problem | Remedy |
|---|---|---|
| PNFI > 0,50, so the model is acceptable | Turns a heuristic into an absolute cutoff | Use PNFI to support comparison |
| PNFI is lower than NFI, so the model is poor | Ignores the PRATIO penalty | Explain the formula before comparing |
| Comparing PNFI across different variable sets | Baseline models and df are not equivalent | Compare only the same data and variables |
| Choosing the highest PNFI regardless of fit | May favor underfitting | Check RMSEA, CFI, TLI, SRMR, and residuals |
| Adding paths to increase NFI | df decreases and PNFI may decline | Add only justified parameters |
| Reporting one PNFI without comparison | Its model-selection role is unclear | State a descriptive purpose or provide a competing-model table |
10. APA 7 Reporting Template and Checklist
The following template uses hypothetical A and B example values. State that models use the same data and explain the theoretical rationale for every difference in constraints.
Example report
Two competing models were estimated using the same N = 420 and 12 observed variables. Model A had NFI = .950, PRATIO = .556, and PNFI = .528; Model B had NFI = .940, PRATIO = .667, and PNFI = .627. Although Model B’s NFI was lower by .010, its PNFI was higher by .099 because it retained more degrees of freedom. Considering RMSEA = [ ], CFI = [ ], TLI = [ ], SRMR = [ ], and [AIC/BIC or chi-square difference], Model [ ] was preferred because [theoretical and empirical rationale]. PNFI was used as comparative evidence rather than an independent cutoff.
- Models use the same sample and observed variables.
- The estimator and missing-data treatment are identical.
- Record NFI, model df, and baseline df.
- Recalculate PRATIO and PNFI for cross-checking.
- Read the correct Default model row.
- Do not apply NFI cutoffs to PNFI.
- Check absolute fit and residuals.
- Identify models as nested or nonnested.
- Use AIC, BIC, or chi-square difference tests where appropriate.
- Support the final decision with theory rather than PNFI ranking alone.
Conclusion
PNFI highlights the cost of adding parameters: fit relative to the baseline may improve while parsimony declines. It is most useful when models are compared on the same data and each has theoretical justification.
There is no magical PNFI value validating a model. A good report presents NFI, PRATIO, PNFI, and competing criteria, then explains why the selected model better balances fit, parsimony, and meaning.
References
- Bentler, P. M., & Bonett, D. G. (1980). Significance tests and goodness of fit in the analysis of covariance structures. Psychological Bulletin, 88(3), 588-606.https://doi.org/10.1037/0033-2909.88.3.588
- Mulaik, S. A., James, L. R., Van Alstine, J., Bennett, N., Lind, S., & Stilwell, C. D. (1989). Evaluation of goodness-of-fit indices for structural equation models. Psychological Bulletin, 105(3), 430-445.https://doi.org/10.1037/0033-2909.105.3.430
- Burnham, K. P., & Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods & Research, 33(2), 261-304.https://doi.org/10.1177/0049124104268644
- Kline, R. B. (2023). Principles and practice of structural equation modeling (5th ed.). Guilford Press.https://www.guilford.com/books/Principles-and-Practice-of-Structural-Equation-Modeling/Rex-Kline/9781462551910
- IBM Corp. (2022). IBM SPSS Amos 29 user's guide.https://www.ibm.com/docs/en/SSLVMB_29.0.0/pdf/IBM_SPSS_Amos_User_Guide.pdf
Academic note: The example values illustrate the formula. PNFI depends on the comparison structure and baseline model; do not use one study’s value as a mandatory cutoff for another.