How to interpret model fit results is probably one of the most frequently asked questions whenever ** Confirmatory Factor Analysis** and

**topics pop up.**

*Structural Equation Modeling (SEM)*Therefore I decided to write a simple guideline that explains where to find and how to interpret all model fit parameters in AMOS, their accepted value range based on a thoughtful literature review on the matter.

In simple terms, model fit in statistics measures the variance between observed and model-implied data using correlation and covariance matrices. Though calculating is a model that fits the data is not too complicated in AMOS, interpreting the results can be a bit challenging at times for students.

The model fit results in AMOS comprises of the following indexes/parameters:

- Chi-Square (CMIN)
- Goodness of Fit Index (GFI)
- Baseline Comparisons in Model Fit
- Parsimony-Adjusted Measures
- Non-Centrality Parameter (NCP)
- Index of Model Fit (FMIN)
- Root Mean Square Error of Approximation (RMSEA)
- Akaike Information Criterion (AIC)
- Expected Cross Validation Index (ECVI)
- Hoelter Index

Next, we will take each of the indexes above, provide a short description for each and add a model fit example for interpretation. Acceptable value ranges will be provided where applied as guidelines for when you write your research paper.

This article concludes with a table containing the most relevant model fit parameters, their ranges, and respective referencing.

**Interpreting CMIN in Model Fit** **Results**

** CMIN** stands for the

**and is used to compare if the observed variables and expected results are statistically significant. In other words, CMIN indicates if the sample data and hypothetical model are an acceptable fit in the analysis.**

*Chi-square value*In Amos, the CMIN result can be found under **View → Text Output → Model Fit → CMIN** and looks like in the following table:

Model | NPAR | CMIN | DF | P | CMIN/DF |

Default model | 4 | 2.119 | 220 | .000 | 2.72 |

Saturated model | 19 | .000 | 0 | ||

Independence model | 24 | 3.765 | 253 | .000 | 5.35 |

**Where:**

*NPAR **= Number of Parameters for each model (default, saturated, and independence). *

*CMIN** = Chi-square value. If significant, the model can be considered unsatisfactory.*

*DF **= Degree of Freedom measures the number of independent values that can diverge without obstructing any limitations in the model.*

*P** = the probability of getting a discrepancy as large as CMIN value if the respective model is correct. *

*CMIN/DF** = discrepancy divided by degree of freedom.*

The value of interest here is the ** CMIN/DF** for the default model and is interpreted as follows:

- If the CMIN/DF value is
**≤****3**it indicates an acceptable fit (Kline, 1998). - If the value is
**≤ 5**it indicates a reasonable fit (Marsh & Hocevar, 1985)

**Interpreting GFI in Model Fit** **Results**

** GFI **stands for

**and is used to calculate the minimum discrepancy function necessary to achieve a perfect fit under maximum likelihood conditions (Jöreskog & Sörbom, 1984; Tanaka & Huba, 1985).**

*Goodness of Fit Index*In Amos, the GFI result can be found under **View → Text Output → Model Fit → RMR, GFI** and looks similar to the table below:

Model | RMR | GFI | AGFI | PGFI |

Default model | .188 | .781 | .649 | .684 |

Saturated model | .000 | 1.000 | ||

Independence model | .194 | .525 | .455 | .507 |

**Where:**

*RMR** = Root Mean Square Residual. The smaller the RMR value the better. An RMR of 0 represents a perfect fit. *

*GFI **= Goodness of Fit Index and takes values of **≤ 1 **where 1 represents a perfect fit.** *

*AGFI **= Adjusted Goodness of Fit Index and indicates the degree of freedom (df) for testing the model. A value of** 1 **indicates a perfect fit. Unlike GFI, AGFI values do not stop at 0.*

*PGFI** = Parsimony Goodness of Fit Index is a modification of GFI (Mulaik et al.,1989) and calculates the degree of freedom for the model.*

The value of interest here is the ** GFI** for the default model and interpreted as follows:

- A value of 1 represents a perfect fit.
- A value
**≥ 0.9**indicates a reasonable fit (Hu & Bentler, 1998). - A value of
**≥ 0.95**is considered an excellent fit (Kline, 2005).

**Interpreting Baseline Comparisons in Model Fit Results**

** Baseline Comparisons** refers to the models automatically fitted by Amos for every analysis, respectively the default, saturated, and independence model.

In Amos, the Baseline Comparisons results can be found under **View → Text Output → Model Fit → Baseline Comparisons** and look similar to the table below:

Model | NFIDelta1 | RFI rho1 | IFIDelta2 | TLIrho2 | CFI |

Default model | .957 | .890 | .966 | .900 | .965 |

Saturated model | 1.000 | 1.000 | 1.000 | ||

Independence model | .000 | .000 | .000 | .000 | .000 |

**Where:**

*NFI **= Normed Fit Index also referred to as Delta 1 (Bollen, 1898b), and consists of values scaling between (terribly fitting) independence model and (perfectly fitting) saturated model. A value of 1 shows a perfect fit while models valued **< 0.9** can be usually improved substantially (Bentler & Bonett, 1980).*

*RFI **= Relative Fit Index and derived from NFI where values closed to 1 indicate a very good fit while 1 indicates a perfect fit.*

*IFI **= Incremental Fit Index where values closed to 1 indicates a very good fit while 1 indicates a perfect fit.*

*TLI **= Tucker-Lewis coefficient also known as Bentler-Bonett non-normed fit index (NNFI) ranges from (but not limited to) 0 to 1 where a value closer to 1 represents a very good fit while 1 represents a perfect fit.*

*CFI **= Comparative Fit Index has value truncated between 0 and 1 where values closed to 1 show a very good fit while 1 represents the perfect fit (**Hu & Bentler, 1999**)*

The value of interest here is CFI** **for the default model. A CFI value of

**≥ 0.95**is considered an excellent fit for the model (West et al., 2012).

**Interpreting Parsimony-Adjusted Measures in Model Fit Results**

** Parsimony-Adjusted Measures** refers to relative fit indices that are adjusted for the majority of indices discussed so far.

Think of *adjustments* as penalties for less parsimonious models. In other words, the more complex the model the lower the fit index as generally a simpler explanation of a phenomenon is favored over a complex one.

In Amos, the Parsimony-Adjusted Measures results can be found under **View → Text Output → Model Fit → Parsimony-Adjusted Measures** and look something like this:

Model | PRATIO | PNFI | PCFI |

Default model | .970 | .984 | .991 |

Saturated model | .000 | .000 | .000 |

Independence model | 1.000 | .000 | .000 |

**Where:**

*PRATIO **= Parsimony Ratio that calculates the number of constraints in a model and is used to compute the PNFI and PCFI indices.*

*PNFI **= Parsimony Normed Fixed Index expressing the result of parsimony adjustment (James, Mulaik & Brett, 1982) to the Normed Fixed Index (NFI).*

*PCFI **= Parsimony Comparative Fix Index expressing the result of parsimony adjustment applied to the Comparative Fit Index (CFI).*

**Interpreting NCP in Model Fit Results**

** NCP** stands for

**expressing the degree to which a null hypothesis is false.**

*Non-Centrality Parameter*In Amos, the NCP results can be found under **View → Text Output → Model Fit → NCP** and look similar to this:

Model | NCP | LO 90 | HI 90 |

Default model | 2.201 | 2.871 | 7.889 |

Saturated model | .000 | .000 | .000 |

Independence model | 1.765 | 3.860 | 5.986 |

**Where:**

**NCP** = Non-Centrality Parameter value with boundaries expressed by LO (NcpLo) and Hi (NcpHi) respectively the lower and higher boundaries of 90% confidence interval for the NCP.

**LO 90** = Lower boundary (NcpLo method) of a 90% confidence interval for the NCP.

**HI 90** = Upper boundary (NcpHi method) of a 90% confidence interval for the NCP.

From the example table above,** **the population NCP for the default model is between 2.87 and 7.88 with a confidence level of approximately 90 percent.

**Interpreting FMIN in Model Fit Results**

** FMIN** stands for

**and is reported when CMIN does not have a positive result usually caused by a larger sample size.**

*Index of Model Fit*In Amos, the FMIN results can be found under **View → Text Output → Model Fit → FMIN** and are displayed as in the following example:

Model | FMIN | F0 | LO 90 | HI 90 |

Default model | 1.590 | 1.019 | 1.371 | 1.683 |

Saturated model | .000 | .000 | .000 | .000 |

Independence model | 1.181 | 1.524 | 1.542 | 1.522 |

**Where:**

*FMIN **= Index of Model Fit with boundaries expressed by LO and Hi respectively the lower and higher boundaries of 90% confidence interval for the FMIN. A value closer to 0 represents a better model fit for the observed data with 0 being the perfect fit. *

*F0 **= Confidence interval*

*LO 90 **= Lower boundary of a 90% confidence interval of the FMIN.*

*HI 90 **= Higher boundary of a 90% confidence interval of the FMIN.*

**Interpreting RMSEA in Model Fit Results**

RMSEA stands for Root Mean Square Error of Approximation and measures the difference between the observed covariance matrix per degree of freedom and the predicted covariance matrix (Chen, 2007).

In Amos, the RMSEA results can be found under **View → Text Output → Model Fit → RMSEA** and are expressed as in the following table:

Model | RMSEA | LO 90 | HI 90 | PCLOSE |

Default model | .073 | .074 | .077 | .000 |

Independence model | .035 | .035 | .038 | .000 |

**Where:**

**RMSEA** = Root Mean Square Error of Approximation where values higher than 0.1 are considered poor, values between 0.08 and 0.1 are considered borderline, values ranging from 0.05 to 0.08 are considered acceptable, and values *≤ 0.05 **are considered excellent (MacCallum et al, 1996).*

*LO 90 **= Lower boundary (RmseaLo) of a 90% confidence interval of the RMSEA.*

*HI 90 **= Higher boundary (RmseaHi) of a 90% confidence interval of the RMSEA.*

*PCLOSE ** = P-value of the null hypothesis*

The value of interest here is represented by ** RMSEA **in the default model field where values

**indicate a better model fit (**

*≤ 0.05**MacCallum et al, 1996*).

**Interpreting AIC in Model Fit Results**

** AIC** stands for

**(Akaike, 1987) and is used to measure the quality of the statistical model for the data sample used. The AIC is a score represented by a single number and used to determine model is the best fit for the data set.**

*Akaike Information Criterion*In Amos, the AIC results can be seen under **View → Text Output → Model Fit → AIC** as shown in the example below:

Model | AIC | BCC | BIC | CAIC |

Default model | 4.201 | 1.647 | 7.728 | 6.728 |

Saturated model | 2.000 | 8.698 | 7.811 | 7.101 |

Independence model | 0.765 | 3.823 | 6.749 | 4.749 |

**Where:**

*AIC** = Akaike Information Criterion score useful only when compared with other AIC scores of the same data set. The lower the AIC value the better. *

*BCC = **Browne-Cudeck Criterion specifically used to analyze the moment structures and impose a higher penalty for less parsimonious models.*

*BIC =** Bayes Information Criterion applies a higher penalty to complex models in contrast with AIC, BCC, CAIC and therefore has a greater propensity of choosing **parsimonious models**.*

*CAIC =** Consistent Akaike Information Criterion (Atilgan & Bozdogan, 1987) is reported only when the means and intercepts are not explicit in the case of a single group. CAIC applies a penalty for complex models higher than AIC and BCC but less severe than BIC.*

**Interpreting ECVI in Model Fit Results**

** ECVI** stands for

**(Browne & Cudeck, 1993) measures the predicting future of a model using simple transformation of chi-square similar to AIC (excepting the constant scale factor).**

*Expected Cross Validation Index*In Amos, the ECVI results can be found under **View → Text Output → Model Fit → ECVI** and look like in the example below:

Model | ECVI | LO 90 | HI 90 | MECVI |

Default model | 2.881 | 1.233 | 3.545 | 2.900 |

Saturated model | .434 | .434 | .434 | .529 |

Independence model | 2.301 | 1.319 | 3.299 | 2.309 |

**Where:**

*ECVI** = **Expected Cross Validation Index **where a smaller value represents a better model fit.*

*LO 90** = lower limit of a 90% confidence interval for the population ECVI.*

*HI 90** = upper limit of a 90% confidence interval for the population ECVI.*

*MECVI **= excepting a scale factor used in computation, the MECVI is similar to Browne-Cudeck Criterion (BCC).*

**Interpreting HOELTER Index in Model Fit Results**

The Hoelter index is used to measure if the chi-square is significant or not.

In Amos, the Hoelter Index results can be found under **View → Text Output → Model Fit → HOELTER** with indices expressed as follows:

Model | HOELTER.05 | HOELTER.01 |

Default model | 228 | 201 |

Independence model | 241 | 208 |

**Where:**

**HOELTER .05 **= measures if the sample size can be accepted at the 0.05 level for the default model. To paraphrase, if your sample size is higher than the value specified for the default model at 0.05 level, the default model should be rejected.

**HOELTER .01 **= calculates if the sample size for the default model can be accepted at the 0.01 level. Respectively, if the sample size is higher than the number specified for the default model at 0.01 level, you may reject the default model.

**Model Fit Cheat Sheet**

This model fit cheat sheet summarizes some of the most important parameters and their accepted values accordingly to the literature.

Acronym | Explication | Accepted fit | Reference |

Likelihood Ratio | P-value | ≥ 0.05 | Joreskog & Surbom (1996); |

Relative X2 | (X2/df) | ≤ 2 = acceptable fit | Tabachnick & Fidell (2007); |

CMIN/DF | Chi-square divided by Degree of Freedom | ≤ 3 = acceptable fit≤ 5 = reasonable fit | Kline (1998); Marsh & Hocevar (1985); |

GFI | Goodness of Fit Index | 1 = perfect fit≥ 0.95 = excellent fit≥ 0.9 = acceptable fit | Kline (2005); Hu & Bentler (1998); |

AGFI | Adjusted Goodness of Fit Index | ≥ 0.90 = acceptable fit | Tabachnick & Fidell (2007); |

CFI | Comparative Fit Index | 1 = perfect fit≥ 0.95 = excellent fit≥ .90 = acceptable fit | West et al. (2012); Fan et al. (1999); |

RMSEA | Root Mean Square Error of Approximation | ≤ 0.05 = reasonable fit | MacCallum et al (1996); |

RMR | Root Mean Squared Residual | ≤ 0.05 = acceptable fit≤ 0.07 = acceptable fit | Diamantopoulos & Siguaw (2000); Steiger (2007); |

SRMR | Standardized Root Mean Squared Residual | ≤ 0.05 = acceptable fit | Diamantopoulos & Siguaw (2000); |

CN | Critical N | ≥ 200 = acceptable fit | Joreskog & Sorbom (1996); |

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