1. What Inferential Statistics Actually Do

Descriptive statistics summarise the sample. Inferential statistics help researchers judge whether an observed pattern is sufficiently compatible with a wider population claim under a stated statistical model.

Inferential methods do not prove that a hypothesis is true. They quantify uncertainty around estimates and help assess whether the observed evidence would be unusual under a specified null model.

The four decisions behind every inferential test

1. What is being estimated? A mean difference, association, proportion difference, odds ratio or another quantity?
2. What is the design? Independent groups, repeated measurements, matched observations or clustered data?
3. What is the outcome type? Continuous, ordinal, nominal, binary or count?
4. What assumptions make the estimate trustworthy? Independence, distributional form, equal variance, linearity or adequate cell counts?

2. Hypothesis Testing Without Misinterpretation

Null and alternative hypotheses

The null hypothesis usually represents no difference or no association. The alternative represents the pattern the researcher is investigating.

H0: Mean trust is equal in trained and untrained employees.
H1: Mean trust differs between trained and untrained employees.

What a p-value means

A p-value is the probability of obtaining data at least as incompatible with the null model as the observed data, assuming the null model and other test assumptions are correct.

What a p-value does not mean

• It is not the probability that the null hypothesis is true.
• It does not measure the size or importance of an effect.
• It does not establish causation.
• It does not rescue a weak design or poor measurement.

Why confidence intervals and effect sizes are essential

A p-value addresses compatibility with a null model. Confidence intervals show the range of estimates reasonably compatible with the data, while effect sizes communicate practical magnitude.

3. Independent-Samples t-Test

Research question it answers

Do two unrelated groups differ in their mean value on a continuous outcome?

Use it when

• There are exactly two independent groups.
• The outcome is continuous or treated as approximately continuous.
• Each participant contributes one independent observation.
• The scientific question concerns a difference in means.

Why this test?

The test directly estimates a mean difference between two independent groups. It is more appropriate than correlation because the predictor is categorical, and more focused than ANOVA when only two groups exist.

Use Welch's t-test when

Group variances are unequal or group sample sizes differ. Welch's version does not require equal variances and is often a safer default than the pooled-variance t-test.

Do not use it when

• The same participants are measured twice.
• There are three or more groups.
• The outcome is binary or nominal.
• Observations are nested within schools, hospitals or organisations.

Trained employees:    M = 4.12, SD = 0.63, n = 86
Untrained employees:  M = 3.58, SD = 0.76, n = 79

Welch's t(151.6) = 4.95, p < .001
Mean difference = 0.54
95% CI [0.32, 0.75]
Cohen's d = 0.77

4. Paired-Samples t-Test

Research question it answers

Did the mean outcome change across two linked measurements?

Use it when

• The same participants are measured at two time points.
• Participants are deliberately matched into pairs.
• The outcome is continuous.
• The analysis concerns the mean of within-pair differences.

Why this test?

The paired t-test removes stable between-person variation by analysing each participant's change score. Treating repeated measurements as independent wastes information and produces incorrect standard errors.

Key assumption

The distribution of the paired differences—not necessarily each raw measurement—should be reasonably compatible with the model, especially in small samples.

Before training: M = 3.21, SD = 0.71
After training:  M = 3.76, SD = 0.66
N = 54 paired observations

Mean change = 0.55
t(53) = 6.18, p < .001
95% CI [0.37, 0.73]
Cohen's dz = 0.84

Alternative

Use the Wilcoxon signed-rank test when the paired outcome is ordinal or the difference scores are severely non-normal in a small sample and a rank-based analysis answers the research question.

5. One-Way ANOVA

Research question it answers

Do mean outcomes differ across three or more independent groups?

Why not perform several t-tests?

Repeated pairwise testing increases the probability of false-positive findings. ANOVA provides an omnibus test of whether at least one group mean differs, followed by controlled post-hoc comparisons.

Use it when

• The outcome is continuous.
• One categorical factor defines three or more independent groups.
• The research question concerns mean differences.
• Observations are independent.

Choose Welch ANOVA when

Variances or group sizes differ materially. Use suitable post-hoc procedures such as Games-Howell rather than methods that assume equal variances.

Traditional method: M = 67.2
Blended method:     M = 72.8
Adaptive method:    M = 76.1

Welch's F(2, 118.4) = 12.67, p < .001
Estimated ω² = .14

Games-Howell:
Adaptive vs Traditional: difference = 8.9, p < .001
Blended vs Traditional:  difference = 5.6, p = .012
Adaptive vs Blended:     difference = 3.3, p = .184

When regression is better

Use regression when adjustment for baseline scores, demographics, interactions or other covariates is required. ANOVA and regression are closely related; regression often provides greater flexibility and transparency.

6. Repeated-Measures and Mixed ANOVA

Repeated-measures ANOVA

Use repeated-measures ANOVA when the same participants are measured at three or more time points or under several conditions.

Mixed ANOVA

Use mixed ANOVA when the design contains both a between-subject factor and a within-subject factor, such as treatment group measured before, immediately after and three months after intervention.

What matters most?

The central question is often the interaction: did groups change differently over time?

Time effect:       F(2, 196) = 22.40, p < .001
Programme effect:  F(1, 98) = 3.18, p = .078
Time × Programme:  F(2, 196) = 6.72, p = .002
Partial η² = .064

When mixed-effects models are preferable

Use linear mixed models when measurement intervals vary, observations are missing, clustering is complex, trajectories differ or the covariance structure needs flexible modelling.

7. Chi-Square Test of Independence

Research question it answers

Are two categorical variables associated?

Use it when

• Both variables are categorical.
• The table contains frequency counts.
• Observations are independent.
• Expected cell counts are sufficiently large.

Why this test?

Chi-square compares observed cell counts with counts expected if the categorical variables were independent.

Alternatives

• Fisher's exact test: sparse two-by-two tables or small expected counts.
• McNemar's test: paired binary outcomes.
• Logistic regression: adjustment for several predictors or estimation of odds ratios.

χ²(1, N = 326) = 9.84, p = .002
Cramer's V = .17
Minimum expected count = 18.7

Incident rate:
Trained staff = 11.2%
Untrained staff = 23.5%

8. Pearson, Spearman and Kendall Correlation

What correlation answers

Correlation estimates the strength and direction of association between two variables.

Why inspect the scatterplot?

A correlation coefficient can conceal non-linearity, clusters and influential outliers. The plot is part of the analysis, not decoration.

Pearson r = .46
95% CI [.36, .55]
p < .001
N = 284

When regression is preferable

Regression is preferable when the researcher needs adjustment for confounders, prediction, interactions, several predictors, or an explicitly directional model.

9. Non-Parametric Alternatives: When and Why

Non-parametric tests are not automatic replacements whenever a normality test is significant. The decision should consider outcome scale, sample size, outliers, distribution shape and the exact estimand.

10. Multiple Testing and Researcher Degrees of Freedom

Testing many outcomes, subgroups or model specifications increases the chance of obtaining apparently significant findings by chance.

Defensible responses

• Specify primary outcomes and hypotheses in advance.
• Limit tests to theory-driven questions.
• Use appropriate multiplicity adjustments where necessary.
• Distinguish confirmatory from exploratory analyses.
• Report all tested outcomes transparently.

11. Thesis Reporting Template

Because the study compared [number] [independent/paired] groups on the continuous outcome [variable], we used [test]. This method was selected because [design-based reason]. Assumptions were assessed using [diagnostics], and [Welch/robust/non-parametric] procedures were used where appropriate. The estimated difference/association was [estimate], 95% CI [lower, upper], p = [value], with an effect size of [value]. This indicates [substantive interpretation], although the [study design/measurement/sampling] does not support [unjustified claim].

12. Final Decision Checklist

• Have you stated whether the question concerns a difference or association?
• Have you identified the outcome's measurement level?
• Have you established whether observations are independent, paired or clustered?
• Have you justified the selected test against realistic alternatives?
• Have you examined assumptions using plots and design knowledge rather than one automatic test?
• Have you reported estimates, confidence intervals and effect sizes?
• Have you kept causal language within the limits of the design?

1. Why Researchers Use Regression

Regression is used when the researcher wants to estimate how an outcome changes in relation to one or more predictors. Unlike a simple group comparison or bivariate correlation, regression can evaluate several predictors simultaneously, adjust for plausible confounders, represent categorical variables, test interactions and produce predictions.

Questions regression can answer

• How strongly is employee trust associated with AI adoption intention?
• Does training predict adoption after accounting for age and prior experience?
• How much additional variance does organisational culture explain beyond demographic variables?
• Does the effect of transparency differ between experienced and inexperienced users?
• Which factors predict whether a patient returns to hospital?

Regression is preferable to separate tests when

• Several predictors must be considered together.
• Confounding variables need to be adjusted for.
• Predictors include both continuous and categorical variables.
• Interactions or non-linear terms are theoretically important.
• The outcome must be predicted for new observations.

2. Simple Linear Regression

Research question it answers

How much does a continuous outcome change, on average, for a one-unit increase in one predictor?

Use it when

• The outcome is continuous.
• There is one predictor.
• The relationship is reasonably represented by the specified functional form.
• Observations are independent.

Why not use correlation alone?

Correlation is symmetric and describes association. Regression explicitly models an outcome as a function of a predictor and provides an estimated change in the outcome for a one-unit change in the predictor.

Outcome: Employee trust score
Predictor: Perceived transparency score

B = 0.42
SE = 0.05
β = .48
t = 8.40
p < .001
95% CI [0.32, 0.52]
R² = .23

Core assumptions

• Linearity: the mean relationship is appropriately represented by the model.
• Independent observations: one observation does not determine another.
• Homoscedasticity: residual variance is reasonably stable across fitted values.
• Residual behaviour: residuals are suitable for the intended inference.
• No dominant influential observations: conclusions are not driven by a few cases.

3. Multiple Linear Regression

Research question it answers

How is a continuous outcome associated with several predictors considered simultaneously?

Why use it?

Multiple regression estimates the unique association of each predictor with the outcome while holding the other included predictors constant. It also quantifies overall model performance.

Predictor selection must be theoretical

Variables should not be added merely because they are available or statistically significant in preliminary analyses. Predictors should be chosen from theory, prior evidence, design requirements and the purpose of the model.

Three different reasons for including variables

Outcome: Adoption intention
N = 312
R² = .46
Adjusted R² = .45
F(5,306) = 52.11, p < .001

Predictor                  B      SE      β       p
Transparency              .21    .05    .23    <.001
Trust                     .39    .06    .42    <.001
Training received         .18    .07    .11     .011
Prior AI experience       .14    .05    .13     .006
Age                      -.01    .00   -.06     .142

Why adjusted R² matters

R² cannot decrease when predictors are added, even if they contribute little. Adjusted R² penalises unnecessary complexity and is therefore useful when comparing models with different numbers of predictors.

4. Hierarchical Regression

What question does it answer?

Does a theoretically important set of variables explain additional variance after an earlier block of variables has been entered?

When should it be used?

• Predictors have a theoretically justified order.
• Demographics or baseline variables should be entered first.
• The researcher wants to test incremental explanatory value.
• Competing theoretical blocks must be compared.

Why not use automatic stepwise selection?

Stepwise methods allow sample-specific statistical fluctuations to determine the model. They often produce unstable coefficients, inflated performance and weak theoretical interpretation.

Block 1: Age, tenure, education
R² = .08

Block 2: Training and prior AI experience
R² = .21
ΔR² = .13, p < .001

Block 3: Transparency and trust
R² = .47
ΔR² = .26, p < .001

5. Categorical Predictors and Dummy Coding

Regression can include categorical predictors by representing categories through indicator variables.

Reference category

One category is selected as the reference. Each coefficient compares another category with that reference while holding other predictors constant.

Reference category: Local government

University:  B = .24, p = .018
Hospital:    B = -.11, p = .291
Ministry:    B = .37, p = .004

Examiner questions

• Why was this reference category selected?
• Were categories combined, and was that decision justified?
• Are some categories too small for stable estimation?
• Does the coding preserve the substantive meaning of the variable?

6. Interaction Effects and Moderation

What question does an interaction answer?

Does the association between one predictor and the outcome change depending on the level of another variable?

Use an interaction when

• The theory predicts conditional effects.
• A treatment may work differently across groups.
• The slope of one predictor may vary by context.
• Average effects could conceal meaningful heterogeneity.

Why main effects become conditional

When an interaction is present, a main-effect coefficient represents the effect when the interacting variable equals its reference value or zero. It should not be interpreted as a universal average effect without care.

Transparency:                 B = .18, p = .012
AI experience:               B = .09, p = .083
Transparency × Experience:   B = .13, p = .004

Simple slopes:
Low experience:   B = .17, p = .028
High experience:  B = .43, p < .001

Reporting requirement

Interactions should be interpreted with predicted values, marginal effects or simple slopes—not only by reporting the product-term p-value.

7. Binary Logistic Regression

Research question it answers

Which variables predict the probability of a binary outcome?

Use it when

• The outcome has two categories.
• Several predictors must be considered simultaneously.
• Adjusted odds ratios or predicted probabilities are required.
• Linear regression would produce unsuitable probability predictions.

Why not linear regression?

Linear regression assumes a continuous outcome and can generate predicted probabilities below zero or above one. Logistic regression models log-odds and keeps predicted probabilities within the valid range.

Outcome: Adopted system (1 = yes)

Predictor              OR      95% CI        p
Training received     2.31    1.42–3.76    <.001
Transparency          1.68    1.29–2.18    <.001
Prior experience      1.44    1.10–1.89     .008
Age                   0.98    0.95–1.01     .176

AUC = .79
Brier score = .16

Odds ratios are not risk ratios

When an outcome is common, an odds ratio can appear substantially larger than the corresponding risk ratio. Predicted probabilities often communicate results more clearly to non-technical readers.

Model evaluation

• Discrimination: can the model separate cases with and without the outcome?
• Calibration: do predicted probabilities agree with observed frequencies?
• Overall accuracy: does performance exceed simple baselines?
• Clinical or practical value: would the model improve decisions?

8. Poisson and Negative-Binomial Regression

Research question they answer

How do predictors relate to a count outcome such as number of incidents, publications, hospital visits or software failures?

Why not linear regression?

Count outcomes are non-negative, often skewed and may have variance that changes with the mean.

Negative-binomial regression

Training received:
IRR = 0.66
95% CI [0.52, 0.84]
p = .001

9. Explanation Versus Prediction

An explanatory model seeks interpretable estimates related to theory. A predictive model seeks accurate performance on new data. The two goals overlap but are not identical.

Why train/test separation matters

Evaluating predictive performance on the same data used to build the model produces optimistic estimates. A test set, cross-validation or bootstrap validation is required.

Common predictive metrics

• RMSE or MAE for continuous outcomes.
• AUC for discrimination in binary outcomes.
• Brier score for probabilistic accuracy.
• Sensitivity and specificity at stated thresholds.
• Calibration plots and calibration slope.

10. Multicollinearity and Unstable Coefficients

Multicollinearity occurs when predictors contain overlapping information. It can increase standard errors, destabilise coefficients and make individual predictor interpretation difficult.

How to respond

• Inspect correlations and variance inflation factors.
• Consider whether variables measure the same construct.
• Avoid deleting theoretically essential confounders solely to reduce VIF.
• Combine indicators only when conceptually justified.
• Use regularisation for prediction-focused models where appropriate.

11. Outliers, Leverage and Influence

An observation can be unusual in the outcome, unusual in its predictor values, or influential on the fitted model. These are different problems.

Cases should not be deleted simply because they make results less significant. Researchers should verify data accuracy, investigate substantive explanations, conduct sensitivity analyses and document decisions transparently.

12. Common Model-Building Errors

• Adding every available variable without theoretical justification.
• Using stepwise selection as the principal scientific strategy.
• Controlling for mediators or colliders without recognising the consequence.
• Interpreting standardised coefficients as causal importance.
• Reporting R² without assessing residuals or validation.
• Testing numerous interactions without prior rationale.
• Removing non-significant variables until the preferred conclusion appears.
• Describing odds ratios as changes in probability.

13. Thesis Reporting Template

We used [model] because the outcome [variable] was [continuous/binary/count] and the research question required [adjusted association/prediction/incremental explanation]. Predictors were selected based on [theory/prior evidence/design], rather than automated significance testing. Diagnostics assessed [linearity, residual variance, influence, collinearity, calibration or overdispersion]. The model explained/predicted [performance measure]. Holding other included variables constant, [predictor] was associated with [estimate and confidence interval]. Because the study was [cross-sectional/observational], the coefficient is interpreted as an association rather than a causal effect.

14. Regression Decision Checklist

• Is the model compatible with the outcome type?
• Is each predictor included for a stated theoretical or methodological reason?
• Have categorical variables been coded and interpreted correctly?
• Have non-linear patterns and interactions been considered where theory requires them?
• Have collinearity, residuals and influential observations been examined?
• Are odds ratios, rate ratios and regression coefficients interpreted accurately?
• Has predictive performance been validated on unseen data?
• Are causal claims consistent with the research design?

1. Why Latent-Variable Analysis Is Needed

Many concepts used in social science, management, education, psychology and health research cannot be observed directly. Trust, organisational commitment, anxiety, service quality, perceived usefulness and research self-efficacy are examples of latent constructs. Researchers infer them from several observed indicators such as questionnaire items, ratings or behavioural measures.

A total or average score assumes that all selected items represent the intended construct adequately. Factor analysis investigates whether that assumption is supported by the pattern of relationships among the items.

2. Principal Components Analysis Versus Exploratory Factor Analysis

PCA and EFA are often treated as interchangeable, but they answer different questions.

If the aim is to discover the latent dimensions underlying questionnaire responses, EFA is usually conceptually preferable. If the aim is merely to compress variables while retaining as much total information as possible, PCA may be appropriate.

3. Exploratory Factor Analysis: The Decisions That Matter

When EFA is appropriate

• The dimensionality of a new or substantially adapted scale is uncertain.
• Items may represent several related constructs.
• The instrument is being used in a population or cultural setting where its structure has not been established.
• Theory provides broad expectations but not a sufficiently precise measurement model.

Step 1: Establish factorability

Researchers should examine the correlation matrix, item distributions and evidence that indicators share enough variance to support factor extraction. The Kaiser–Meyer–Olkin statistic and Bartlett's test can support this judgement, but they should not replace inspection of the data.

Step 2: Choose an extraction method

Common-factor methods such as principal-axis factoring or maximum likelihood are generally preferred when the purpose is latent construct discovery. Maximum likelihood supports model comparisons and confidence intervals when its distributional assumptions are reasonably satisfied.

Step 3: Decide how many factors to retain

The eigenvalue-greater-than-one rule is easy to apply but can overextract or underextract factors. Stronger decisions combine:

• Parallel analysis.
• The scree plot.
• Model fit or residual evidence where available.
• Interpretability and theoretical coherence.
• Whether each factor has enough strong and conceptually meaningful indicators.

Step 4: Select an appropriate rotation

Oblique rotation is often more realistic because psychological and organisational constructs rarely operate in complete isolation. If factors prove almost uncorrelated, that finding can be reported; forcing orthogonality prematurely can distort the structure.

Step 5: Interpret loadings and cross-loadings

A factor loading indicates how strongly an item relates to a factor. No universal cut-off determines whether an item is acceptable. The decision depends on sample size, construct breadth, item quality and the pattern of competing loadings.

Item                          Factor 1   Factor 2
TR1: Management is transparent   .78       .09
TR2: Decisions are explained     .74       .14
TR3: Information is accessible   .69       .18
AD1: I intend to use the system  .11       .82
AD2: I would recommend adoption  .17       .76
AD3: I expect continued use      .25       .71

Extraction: Principal-axis factoring
Rotation: Direct oblimin
Factor correlation: .36

Items should not be deleted mechanically

Deleting an item solely because its loading falls just below a conventional threshold can damage content validity. Item removal should consider wording, theoretical coverage, redundancy, cross-loading, communalities and the consequences for the meaning of the construct.

4. Confirmatory Factor Analysis

CFA evaluates a measurement structure specified in advance. The researcher identifies which items should measure which latent constructs and then examines how well that model reproduces the observed covariance matrix.

Use CFA when

• A validated scale is being tested in a new sample.
• Theory predicts a specific number of constructs and item assignments.
• The researcher needs to compare competing measurement models.
• Latent constructs will later be used in SEM.

EFA and CFA should not be performed as if they were the same exercise

EFA searches for a plausible structure; CFA tests a specified structure. Exploring and confirming the same model in the same sample can capitalise on chance. A stronger design uses separate development and validation samples, a random split where sample size permits, or independent replication.

Identification

A model must contain enough information to estimate its parameters uniquely. Researchers normally set the scale of each factor by fixing one loading or fixing the factor variance. Constructs with too few indicators can be weakly identified and difficult to defend.

Correlated residuals

Modification indices may suggest correlating item errors. Such changes should be made only when there is a substantive reason, such as nearly identical wording, a shared method effect or item-specific content. Adding residual correlations merely to improve fit can conceal a poorly specified scale.

5. Reliability, Convergent Validity and Discriminant Validity

A measurement model should be evaluated as a coherent whole rather than through one statistic.

Cronbach's alpha alone cannot establish unidimensionality or validity. A very high alpha may reflect redundant items, while a moderate alpha may occur in a broad but theoretically defensible construct.

Construct              CR      Omega    AVE     HTMT maximum
Transparency          .89      .88      .67         .71
Trust                  .91      .90      .72         .78
Adoption intention     .87      .86      .63         .78

6. Structural Equation Modelling

SEM combines a measurement model with a structural model. The measurement component links latent constructs to indicators. The structural component estimates relationships among those constructs.

SEM is useful when

• Key variables are latent and measured with several indicators.
• Several dependent relationships must be estimated simultaneously.
• Indirect effects are central to the theory.
• Measurement error should be represented explicitly.
• Competing theoretical models need to be compared.

SEM is not automatically superior to regression. It requires adequate measurement, sufficient information, a defensible model and a sample capable of supporting the intended complexity.

7. Understanding Model Fit Without Worshipping Cut-offs

Model fit asks how closely the model-implied covariance structure resembles the observed structure. No single index is sufficient, and conventional cut-offs are guidelines rather than universal laws.

Local fit is as important as global fit

A model can achieve acceptable global indices while containing weak loadings, implausible factor correlations, large residuals or unsupported paths. Researchers should inspect parameter estimates, residuals, standard errors, convergence warnings and theoretically important areas of misfit.

χ²(51) = 86.40, p = .001
CFI = .962
TLI = .951
RMSEA = .047, 90% CI [.029, .064]
SRMR = .039
Standardised loadings = .62 to .86

CB-SEM and PLS-SEM

Covariance-based SEM is generally used for theory testing, model comparison and estimation of a covariance structure. Partial least squares SEM is component-based and often emphasises prediction and explained variance. The choice should be driven by the research objective and measurement assumptions—not by the belief that one method requires no assumptions or automatically solves small-sample problems.

8. Mediation: Testing an Indirect Mechanism

Mediation asks whether the association between an antecedent and an outcome operates partly through an intervening variable.

• Path a: antecedent to mediator.
• Path b: mediator to outcome, conditional on the antecedent.
• Indirect effect: the product a × b.
• Direct effect: the remaining antecedent–outcome relationship after including the mediator.
• Total effect: direct plus indirect effect.

The indirect effect should normally be evaluated with a bootstrap confidence interval or another method that respects its non-normal sampling distribution. Requiring a significant total effect before testing mediation is not logically necessary.

Transparency → Trust:          a = .52, p < .001
Trust → Adoption intention:   b = .41, p < .001
Direct effect:               c′ = .18, p = .012
Indirect effect:            ab = .21
Bootstrap 95% CI [.14, .30]

9. Moderation: When an Effect Changes Across Conditions

Moderation asks whether the strength or direction of a relationship differs according to another variable. In regression, this is represented by an interaction. In SEM, moderation may involve observed variables, latent interactions or multigroup comparisons.

Example question

Does the relationship between training quality and adoption intention become stronger when managerial support is high?

Interpretation requires conditional effects

A significant interaction term is not the final substantive conclusion. Researchers should estimate and display the effect of the predictor at meaningful values of the moderator, or use regions of significance where appropriate.

Training quality:                 B = 0.24, p < .001
Managerial support:              B = 0.19, p = .004
Training × managerial support:   B = 0.11, p = .009

Conditional effect of training:
Low support (-1 SD):              B = 0.13, 95% CI [0.03, 0.23]
Mean support:                     B = 0.24, 95% CI [0.16, 0.32]
High support (+1 SD):             B = 0.35, 95% CI [0.24, 0.46]

Multigroup analysis and measurement invariance

Before comparing latent means or structural paths across groups, researchers should establish that constructs are measured sufficiently similarly. A typical invariance sequence considers:

• Configural invariance: the broad factor pattern is comparable.
• Metric invariance: factor loadings are sufficiently comparable.
• Scalar invariance: item intercepts or thresholds are sufficiently comparable.
• Residual invariance: item residual variances are comparable when required by the research claim.

Without adequate invariance, an apparent group difference may reflect measurement differences rather than a genuine difference in the construct or relationship.

10. Sample Size and Estimation

Rules such as “ten cases per item” or “200 participants is always enough” are too crude. Required sample size depends on model complexity, factor loadings, indicator reliability, missingness, distribution, estimator and the size of effects being estimated.

Strong planning uses simulation or power analysis tailored to the proposed model. Researchers should also select estimators that match the indicators:

• Maximum likelihood or robust maximum likelihood for suitable continuous indicators.
• Ordinal estimators for Likert-type indicators treated as ordered categories where appropriate.
• Bootstrapping or robust standard errors when distributional conditions require them.

11. Common Errors in Factor Analysis and SEM

• Calling PCA a latent-factor analysis without justification.
• Retaining factors only because their eigenvalues exceed one.
• Using Varimax automatically even when constructs are expected to correlate.
• Deleting items repeatedly until fit indices cross preferred thresholds.
• Reporting Cronbach's alpha as proof of validity.
• Using modification indices without theoretical justification.
• Testing a heavily modified CFA model on the same data and calling it confirmed.
• Reporting only CFI, RMSEA or R² while ignoring weak parameters and local misfit.
• Assuming SEM establishes causality.
• Claiming mediation from cross-sectional correlations without acknowledging temporal ambiguity.
• Interpreting a moderation coefficient without presenting conditional effects.
• Comparing groups without assessing measurement invariance.

12. Examiner-Ready Reporting Templates

EFA template

Exploratory factor analysis was used because the dimensional structure of the adapted instrument had not been established in the target population. Common-factor extraction was selected to investigate latent constructs, and an oblique rotation was used because the dimensions were theoretically expected to correlate. The number of factors was determined using parallel analysis, scree-plot evidence, interpretability and item-loading patterns rather than the eigenvalue rule alone.

CFA template

A confirmatory factor model was specified before estimation, with each indicator loading only on its theorised construct. Model adequacy was evaluated using multiple global fit indices, standardised loadings, residuals, factor correlations and theoretically meaningful local diagnostics. Model modifications were made only where they were substantively defensible and are reported transparently.

SEM and mediation template

The structural model tested whether [antecedent] was related to [outcome] directly and indirectly through [mediator]. The indirect effect was evaluated using a bootstrap confidence interval. The measurement model was assessed before structural paths were interpreted. Because the design was [cross-sectional/observational], the paths are interpreted as theory-consistent associations rather than definitive causal effects.

13. Final Decision Checklist

• Is the goal data reduction, latent structure exploration or confirmation of a specified model?
• Does the extraction method match that goal?
• Was factor retention based on several defensible sources of evidence?
• Does the rotation reflect plausible relationships among constructs?
• Were item decisions based on theory and content coverage as well as statistics?
• Was the measurement model evaluated before structural relationships?
• Were reliability and validity assessed with more than Cronbach's alpha?
• Were global fit, local fit and parameter plausibility considered together?
• Were mediation and moderation effects interpreted directly with confidence intervals?
• Were group comparisons preceded by an appropriate invariance assessment?
• Are causal claims consistent with the design, timing and assumptions?