# Confirmatory Factor Analysis (CFA) in SEM using SPSS Amos

Confirmatory factor analysis is a *Structural Equation Modeling (SEM)* and factor analysis method used to find out if observed variables contribute to latent or unobserved variables. The previous article explained its characteristics. This article demonstrates through a case study how to build a Confirmatory factor analysis** **model in SPSS Amos software.

## Measuring the anxiety level of an individual

The aim of this case study is to estimate the extent to which each of these factors causes anxiety. Through secondary research, it was found that the anxiety theory states that anxiety is caused due to four factors (Antony, 2006):

- Affective
- Cognitive
- Physical
- Behavioural

Each of the above factors further had 3 sub-factors or aspects that are represented in the below figure.

To achieve the aim of the study, 500 individuals were surveyed. The questionnaire consisted of questions regarding all the factors.

## Confirmatory factor analysis model to compute anxiety level

The first step is to create a relationship between the main factors, i.e:

- cognitive,
- affective,
- behavioural, and
- psychological.

This is done using the *SEM* model. Since all the factors are measuring a single variable, i.e. anxiety, covariance needs to be stated between them to draw the linkages between the variables. By doing this, the relationship between the factors is built. The next step is to establish the validity and reliability of the model in order to prove its efficiency.

### Reliability and Validity

The reliability and validity of the model are assessed using four different values i.e. convergent validity, internal consistency, composite reliability, and discriminant validity. The results for the first three measures are shown below.

AVE | CR | Cronbach alpha | |
---|---|---|---|

Affective | 0.59 | 0.71 | 0.70 |

Psychological | 0.57 | 0.73 | 0.72 |

Behavioural | 0.61 | 0.75 | 0.75 |

Cognitive | 0.64 | 0.76 | 0.74 |

*SEM*

**Average Variance Extracted (AVE): **It is the measure for understanding convergent validity i.e. construct’s ability to share items or statements used to depict it. Herein, the value of AVE for all the variables is more than 0.5 i.e. affective – 0.59, psychological – 0.57, behavioural – 0.61, and cognitive is 0.64. Thus, the model has convergent validity.

**Composite Reliability (CR): **It is the method for assessing the contribution or significance of an item by examining the factors loading. Herein, the value of CR is also more than 0.7 for all the constructs i.e. affective – 0.71, psychological – 0.73, behavioural – 0.75, and cognitive – 0.76. Thus, composite reliability is derived for the model.

**Internal Consistency**: It is the reliability method for depicting the factor’s linkage with other factors. Cronbach alpha is the method to measure internal consistency. Herein the value is more than 0.7 for all the variables i.e. affective – 0.70, psychological – 0.72, behavioural – 0.75, and cognitive – 0.74. Thus, there is the presence of internal consistency in the model.

Lastly, discriminant validity is the method for identifying the construct distinction from one another. Herein, the value of construct correlation is compared with the square root of AVE. The below table depicts that as for each of the variables, the correlation value is less than the square root, i.e. 0.80 is more than 0.52, 0.58, and 0.58. Thus, the model has discriminant validity.

Cognitive | Affective | Behavioural | Psychological | |
---|---|---|---|---|

Cognitive | 0.80 | |||

Affective | 0.52 | 0.77 | ||

Behavioural | 0.58 | 0.73 | 0.78 | |

Psychological | 0.58 | 0.63 | 0.75 | 0.76 |

*SEM*model showing anxiety

Hence, with the fulfilment of all reliability and validity conditions, the confirmatory factor analysis model is effective for the assessment of the contribution of the factors in measuring anxiety levels.

### Model fitness of confirmatory factor analysis model in *structural equation modeling (SEM)*

Model fitness refers to the model’s ability to reproduce the existing linkage with other data tested under similar conditions. A well-fitted model ensures consistency and prevents re-working. Thus, it is essential to examine model fitness before assessing the linkage between variables (Kenny, 2020; Shi & Lee, 2019). For this, the model fitness is examined wherein results are shown below.

Name of category | Name of index | Meaning | Index value | Adequate fit |
---|---|---|---|---|

Absolute fit measure | CMIN/Df (normed/relative Chi-Square) | Determine the discrepancy between the fitted and sample covariance matrix by minimizing the sample size impact on the model | 4.94 | Less than 5 |

GFI (Goodness of fit) | The measure defines the replicating capability of the model with the observed covariance matrix | 0.92 | Greater than 0.90 | |

AGFI (adjusted goodness of fit) | Computation of GFI by adjusting against the degree of freedom | 0.88 | Greater than 0.90 | |

RMSEA (root mean square of approximation) | Define model efficiency to fit population covariance matrix with unknown but optimal chosen parameters | 0.09 | Less than 0.10 | |

Incremental fit measure | NFI (normal fit index) | Relative model location of the model between the independence and saturated model | 0.92 | Greater than 0.90 |

CFI (comparative fit index) | NFI revised form wherein Discrepancy between the hypothesized model and data is computed by considering the sample size | 0.93 | Greater than 0.90 | |

TLI (Tucker Lewis index) | Modified NFI model enabling model examination with smaller sample size | 0.91 | Greater than 0.90 | |

IFI (Incremental fit index) | Adjusted NFI model for sample size and degree of freedom | 0.93 | Greater than 0.90 | |

Parsimonious fit measure | PGFI (parsimony goodness of fit index) | Modified GFI model wherein loss of a degree of freedom is considered | 0.57 | Greater than 0.50 |

PCFI (parsimony comparative fit index) | Modified CFI model wherein loss of a degree of freedom is considered | 0.68 | Greater than 0.50 | |

PNFI (parsimony normed fit index) | Modified NFI model wherein loss of a degree of freedom is considered | 0.67 | Greater than 0.50 |

*SEM*

The above table revealed that for absolute fitness all the indices values are approximately fulfilling the required criteria i.e. CMIN/Df is 4.94 < 5, GFI is 0.92 > 0.9, RMSEA is 0.09 < 0.10, and even AGFI is 0.88 ≈ 0.9 (Hooper et al., 2008). Further, for incremental fitness too, NFI is 0.92 > 0.9, CFI is 0.93 > 0.9, TLI is 0.91 > 0.9 and IFI is 0.93 > 0.9 (Hooper et al., 2008).. Even for parsimonious fitness, the indices value is such that PGFI is 0.57 > 0.5, PCFI is 0.68 > 0.5 and PNFI is 0.67 > 0.5 (Hooper et al., 2008). Hence, the model as fulfil all the requirement, thus is suitable for building linakge between factors and determining contribution of variables in measuring anxiety level.

### Linkage examination

In order to identify the factors contributing to anxiety level measurement, all the sub-factors were assessed separately. The results are shown in the table below.

Variables | Construct | Estimate | S.E. | C.R. | P |
---|---|---|---|---|---|

Reduced QOL | Affective | 1.00 | |||

Anger | Affective | 1.34 | 0.11 | 12.69 | 0.00 |

Sadness | Affective | 1.25 | 0.10 | 12.53 | 0.00 |

Sweatiness | Psychological | 1.00 | |||

Dry Mouth | Psychological | 0.79 | 0.07 | 11.45 | 0.00 |

Heart rate | Psychological | 1.38 | 0.08 | 17.80 | 0.00 |

Isolation | Behavioural | 1.00 | |||

Compulsions | Behavioural | 1.27 | 0.09 | 15.00 | 0.00 |

Avoidance | Behavioural | 1.29 | 0.09 | 14.66 | 0.00 |

Obsessions | Cognitive | 1.00 | |||

Poor Concentration | Cognitive | 1.14 | 0.09 | 12.34 | 0.00 |

Fearlessness | Cognitive | 1.13 | 0.08 | 14.12 | 0.00 |

*SEM*

Firstly, the ‘p-value’ is relevant in order to assess whether there is a significant relationship between the sub-factors and anxiety or not. This ‘p-value’ must be less than 0.05 for the relationship to exist (Kock, 2016). In this case, all the sub-factors or aspects have a ‘p-value’ of 0.00, therefore there is a significant relationship.

Next, the ‘Estimate’ value of the variables is relevant. In the case of many sub-factors such as reduced QOL, Anger, and Sadness it is high. This shows high factor loading. Similarly for other constructs too, the factor loading is above 0.5. Thus, this shows that affective, cognitive, behavioural, and psychological factors have an important and positive contribution in measuring the anxiety level of an individual.

Confirmatory factor analysis helps to determine the efficiency of the construct. It is a key step and analysis in an *SEM* model. Since the model is proven to be effective, each of the selected factors has a positive contribution in measuring the main construct i.e. affective, cognitive, behavioural, and psychological factors together compute individual anxiety levels.

#### References

- Antony, M. M. (2006). Assessment of Anxiety and the Anxiety Disorders: An Overview.
*Practitioner’s Guide to Empirically Based Measures of Anxiety*, 9–17. https://doi.org/10.1007/0-306-47628-2_2 - Hooper, D., Coughlan, J., & Mullen, M. R. (2008). Structural Equation Modeling : Guidelines for Determining Model Fit.
*The Electronic Journal of Business Research Methods*,*6*(1), 53–60. - Kenny, D. A. (2020).
*Measuring Model Fit*. http://www.davidakenny.net/cm/fit.htm - Kock, N. (2016). Hypothesis testing with confidence intervals and P values in PLS-SEM.
*International Journal of E-Collaboration*,*12*(3), 1–6. - Shi, D., & Lee, T. (2019). Understanding the Model Size Effect on SEM Fit Indices.
*Educational and Psychological Measurement*,*79*(2), 310–334. https://doi.org/10.1177/0013164418783530

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