Assessing Normality Using SPSS and AMOS Before CFA or SEM

1. Learning Objectives and Input Data

After this guide, you should distinguish univariate from multivariate normality, correctly read AMOS’s Tests for normality and outliers tables, trace cases with large Mahalanobis distances, and choose a defensible estimation strategy. The final result is more than pass or fail: it is a diagnostic record with decisions and reasons.

Use the variables actually entering CFA or SEM. Before checking distributions, complete coding, valid-range checks, missing-data treatment, item reversal, and duplicate-record detection. An out-of-range value is a data error rather than evidence about the population distribution.

• The variable list matches the intended CFA or SEM model.
• Missing-value codes are not interpreted as genuine values.
• Reverse-coded items have been processed and checked.
• No values remain outside valid ranges.
• Save an unedited copy of the original data.
• Establish outlier-review criteria before examining results.

2. What Does the Normality Assumption Actually Concern?

In covariance-based SEM with ordinary maximum likelihood, the theoretical assumption concerns the joint distribution of continuous observed variables. Attractive histograms for individual variables therefore do not establish multivariate normality. Conversely, mild skewness in one variable does not automatically invalidate every point estimate.

Nonnormality can distort chi-square statistics, standard errors, and confidence intervals. Consequences depend on the form of skewness, distributional tails, sample size, complexity, and degree of model misspecification. Curran, West, and Finch’s (1996) simulations show that nonnormality and misspecification can jointly impair test-statistic performance.

For ordinal variables with very few categories, the problem extends beyond nonnormality. Continuous-variable models and ordinary ML may be fundamentally inappropriate. Consider a categorical or ordinal estimator in software that directly supports it.

3. Univariate Diagnostics in SPSS

In SPSS, Analyze > Descriptive Statistics > Explore provides histograms, normal Q-Q plots, boxplots, skewness, and kurtosis. Inspect plots before applying thresholds: is there a long tail, bimodality, a ceiling or floor effect, or only mild skewness detectable in a large sample?

Shapiro–Wilk tests exact normality. With large samples, small and practically unimportant deviations can yield p < 0,05. With small samples, the test may lack power. A p-value therefore cannot replace plots, descriptive statistics, and knowledge of the scale.

No pair of skewness and kurtosis cutoffs applies to every design. Kline (2023) notes that some simulation studies treated |skewness| > 2 and |excess kurtosis| > 7 as indications of severe nonnormality, but also emphasizes numerous exceptions and the absence of universal boundaries. Use these benchmarks only as anchors for closer examination rather than automatic rules for estimator selection, transformation, or variable removal.

4. Multivariate Normality and Outliers in AMOS

Mardia (1970) developed measures of multivariate skewness and kurtosis. According to IBM Corp. (2022), AMOS provides multivariate kurtosis and its critical ratio in Assessment of normality. This diagnoses the model’s entire variable set rather than averaging univariate kurtoses.

Under Analysis Properties > Output, select Tests for normality and outliers. After estimation, read the individual-variable rows, Multivariate row, and Observations farthest from the centroid table. The critical ratio increases with deviation and sample size, so do not turn a cutoff into an automatic instruction to change estimators.

Mahalanobis distance measures multivariate distance from the centroid under the covariance structure. A distant case may reflect an entry error, a valid population member, a subgroup, or a missing structure in the data model. Only original records and the sampling design can distinguish these possibilities.

5. Handling Outliers Without Overcleaning

Correct entry errors when supported by questionnaires or original sources. Do not replace a valid value with missing solely because it is distant from the center. For valid cases, options include retaining them, using a less sensitive estimator, modeling subgroups, or reporting sensitivity analyses.

Deleting outliers after observing that their removal improves fit is an outcome-dependent decision. This may reduce representativeness and produce overly optimistic reports. If cases are excluded because they fall outside the target population or violate recruitment criteria, apply those criteria consistently.

Winsorization or log transformation can alter a scale’s meaning. Use them only when the transformation suits the variable, is interpretable, and is explicitly documented. Do not transform individual Likert items merely to produce bell-shaped histograms.

6. From Diagnostics to Estimator Selection

Ordinary ML may remain useful with modest deviations, sufficient samples, and manageable model complexity, but justify the decision through the sensitivity of standard errors and fit. Robust corrections adjust test statistics or standard errors for nonnormality; they neither change the data nor repair misspecification (Savalei, 2014). Kline (2023) likewise emphasizes assessing assumptions and fit through an evidence profile rather than one cutoff.

AMOS supports bootstrapping and the Bollen–Stine bootstrap. Bollen and Stine (1992) use resampling under the null model to assess overall fit when distributional assumptions are violated. Bootstrap confidence intervals are more suitable for many indirect effects, but bootstrapping does not make data normal.

Severe nonnormality, few-category ordinal data, complex missingness, or requirements for robust standard errors and scaled chi-square may require an estimator in other software. Yuan and Bentler (2000) present likelihood adjustments for nonnormal and missing data; choices must follow the data type and inferential objective.

7. Hypothetical Numerical Example: From Diagnostics to Decisions

The following example is illustrative rather than actual research data. A CFA model has 18 Likert items with 5 categories and N = 412. Univariate skewness ranges from -1,18 to 0,74; excess kurtosis ranges from -0,62 to 2,35. Q-Q plots show right tails for three items; AMOS reports a multivariate kurtosis critical ratio of 12,84.

Three observations with Mahalanobis p2 < .001 are investigated. One has a value of 55 on an item ranging from 1 to 5 and is corrected to 5 after checking the questionnaire. The remaining two are valid, very-high-intensity users and are retained.

The team uses ML as the primary model, reports 2.000 bootstrap samples for confidence intervals, and obtains Bollen–Stine p = .074 as supplementary fit evidence. Removing the two valid cases in a sensitivity analysis changes loadings by at most .018 and leaves structural-path conclusions unchanged. Retention therefore has clearer evidential support than mechanical deletion.

8. Bollen–Stine, Bootstrap Standard Errors, and Robust ML Are Not Interchangeable

The Bollen–Stine bootstrap generates a reference distribution for the model-fit statistic under the null hypothesis of model correctness. Bollen–Stine p therefore addresses global fit rather than automatically correcting every parameter’s standard error or confidence interval (Bollen and Stine, 1992; Kline, 2023).

Bootstrapping standard errors and confidence intervals is a separate procedure. It resamples to describe the empirical distribution of parameter estimates. If a study requires inference for indirect effects or coefficients with skewed distributions, request bootstrapping for those specific parameters rather than merely enabling Bollen–Stine.

Robust ML retains the basic ML estimates but adjusts standard errors and test statistics for nonnormality. Savalei (2014) shows that robust corrections have multiple forms and assumptions, so reports must name the specific estimator. AMOS does not provide every commonly used robust variant; cross-checking in other software is sometimes necessary.

9. Few-Category Likert Data: A Measurement Issue Beyond Normality

Likert items with four or five categories are sometimes treated as approximately continuous when distributions are not extreme and samples are sufficiently large. With two or three categories, severe skewness, or many sparse frequency cells, however, Pearson correlations and ordinary ML may poorly represent relationships between continuous latent variables.

Kline (2023) recommends choosing methods according to measurement level and data conditions. One option is ordinal CFA using polychoric correlations and an estimator such as WLSMV in supporting software. Flora and Curran (2004) show that robust WLS outperforms full WLS in many ordinal-data simulation conditions, particularly with smaller samples; this supports matching estimators to measurement levels rather than applying default ML to every Likert item.

If AMOS with ML is retained, describe it as an approximation, report individual-item distributions, and conduct sensitivity analyses rather than claiming that ML is always robust. AMOS 29 does not offer a categorical estimator equivalent to WLSMV, so ordinal comparisons require supporting software and the same model specification.

10. Designing Sensitivity Analyses Before Drawing Conclusions

Sensitivity analysis asks whether scientific conclusions change under another reasonable estimation choice. It provides a practical assessment of nonnormality’s consequences rather than debating whether data are perfectly normal (Yuan and Bentler, 2000; Savalei, 2014).

Do not create dozens of models. Prespecify one primary analysis and one or two comparisons targeting the actual risk: bootstrap parameter confidence intervals, robust ML in other software, ordinal CFA for few-category items, or analyses with and without cases under review.

11. Common Interpretation Errors

12. APA 7 Reporting Template and Checklist

The following template uses hypothetical values. Replace bracketed values with actual results; state software versions, bootstrap count, confidence-interval type, and reasons for handling each case.

• Check ranges, missing data, and reversed items before normality.
• Distinguish univariate and multivariate normality.
• Inspect histograms and Q-Q plots rather than only p-values.
• State the software’s kurtosis convention.
• Read the Multivariate row in AMOS.
• Trace every large-Mahalanobis-distance case to the original data.
• Do not delete valid cases merely to improve fit.
• Explain the choice of ML, bootstrapping, or a robust estimator.
• Describe the bootstrap count and confidence-interval type.
• Use Bollen–Stine for its intended fit-assessment purpose.
• Conduct sensitivity analysis when decisions may affect results.
• Cross-check every in-text citation against the reference list.