Women's empowerment, trust in institutions, food insecurity, job satisfaction, service quality: none is a column in a dataset. Each is inferred from several imperfect questions. Ordinary regression on a summed score treats the sum as the thing and its error as zero, which attenuates every coefficient and hides how well the questions measured anything. SEM makes the measurement explicit and testable.

A diagram is a set of equations. Every arrow is a parameter to estimate, every missing arrow is a restriction (a coefficient fixed at zero), and it is the missing arrows that make the model testable. A model with an arrow between every pair of variables fits any data perfectly and says nothing. The discipline of SEM is deciding which arrows to leave out, before seeing the data, for reasons you can state.

The method is sound and much of its use is not. This course is organised around the decisions, in the order they arise, with the defensible choice at each and the common abuse beside it. A reader who follows it will produce a model that can fail, which is the only kind worth reporting.

Assumed: regression at the level of Econometrics 101, and the survey and scale material in Survey Design 101. Worked numbers are illustrative unless a source is named, with magnitudes chosen to be realistic so the reader learns to read the output. Code is given in lavaan syntax because it is free, readable and the reference implementation; every other package has an equivalent.

Getting this wrong is the commonest specification error in applied SEM and it is invisible in the fit statistics. A formative construct forced into a reflective model shows low loadings and poor 'reliability' that are not defects of measurement but of the model. Jarvis, MacKenzie and Podsakoff (Journal of Consumer Research 2003, 30:199) give the decision rules; Bollen and Lennox (1991) the theory.

Empowerment measured as decision-making, mobility and control over assets raises the question whether there is one empowerment or three. A second-order model puts a general factor above the three, which then load on it; it is identified with three or more first-order factors and says the three are manifestations of one thing. A bifactor model lets every item load on a general factor and on its specific factor at once, and asks how much of the variance is general.

A five-point agreement item is not a continuous measurement. Treating it as one with maximum likelihood works tolerably with five or more categories and roughly symmetric distributions, and fails with skewed items or four or fewer categories: loadings are attenuated and fit is distorted. The ordinal treatment assumes a continuous latent response behind each item, estimates thresholds and polychoric correlations, and fits the model to those by diagonally weighted least squares (WLSMV in Mplus and lavaan).

Convergent validity in the CFA sense is that the items of a construct share enough variance to be measuring one thing. The evidence is the loadings (all substantial, all significant) and the average variance extracted, AVE = mean of the squared standardised loadings, the share of item variance the factor explains on average. Fornell and Larcker (Journal of Marketing Research 1981, 18:39) proposed 0.5 as the floor: the factor explains more of its items than error does.

A scale can have clean loadings, high omega and perfect discriminant validity and still not measure what it claims. Messick (1995) and the Standards for Educational and Psychological Testing (2014) frame validity as the evidence supporting a specific interpretation of scores for a specific use. The CFA is internal-structure evidence. The other kinds come from outside the model.

The cheapest external validity check is to ask whether the latent score differs between groups that theory says it must. A decision-making scale should score higher among women with their own income than without, higher among household heads than daughters-in-law, and should not differ by enumerator. Each is a latent mean comparison (section 04) or a regression of the factor on the grouping variable, and each either supports the interpretation or raises a question the CFA could not.

A South Asian evaluation often fields one instrument in three languages. Pooling the languages in one CFA assumes the items work the same way in each, which is exactly what has to be shown. The sequence above is the standard (Beaton and colleagues, Spine 2000, 25:3186, for health measures; the ITC guidelines for tests), and the last step is the one most often skipped.

Every arrow in the structural model is a hypothesis with a direction, and every absent arrow is the hypothesis that two things are unrelated once the rest of the model is accounted for. Both come from theory, prior evidence, or the logic of the programme, and both are written down before the data are analysed. A model whose arrows were chosen to fit is not a test of anything; it is a description of one sample.

The data supply p(p + 1)/2 unique variances and covariances for p observed variables (plus p means if modelled). The model asks for some number of free parameters. If it asks for more than the data supply, it is under-identified and no unique solution exists. If exactly as many, just-identified: it fits perfectly and tests nothing. If fewer, over-identified, with degrees of freedom equal to the difference, and the fit test has something to test.

Programme membership, age, household size and income are measured once and enter the model as observed variables, with arrows from rectangles. That is correct for variables measured without much error. For a single-item construct that is noisy (a one-question trust measure), the model can still separate error if the analyst fixes the error variance at (1 − reliability) × the observed variance, using a reliability from a prior study.

A covariate that predicts an endogenous latent variable gets an arrow into it, from an observed rectangle. Every endogenous variable the covariate might affect gets one; leaving a covariate out of one equation is a restriction that must be true for the model to be right. Covariates correlate with each other and with the exogenous latents by default (double-headed arrows), and those correlations are estimated, not restricted.

For almost any structural model there exist others, with different arrows, that imply exactly the same covariance matrix and therefore fit identically. Reverse a path between two exogenous-free variables, replace a path with a correlation, swap a mediator and an outcome: the χ² does not move. MacCallum and colleagues (Psychological Bulletin 1993, 114:185) found that published models routinely had dozens of equivalent alternatives and that authors never mentioned them.

The default SEM fits covariances only; means are ignored and every intercept is free. A mean structure adds the item intercepts and the latent means to the model and is needed for three purposes: comparing latent means across groups (does the treatment group have higher empowerment, on the latent scale?), measurement invariance beyond the metric level (section 08), and growth models where the trajectory's intercept and slope are the latent variables.

Attitude and behaviour items in field surveys are skewed: most respondents agree, or most report never. Under ordinary ML, skew and kurtosis inflate the χ² (the model is rejected too often), deflate the standard errors (paths look more significant than they are), and distort the fit indices in the direction of rejection. The parameter estimates themselves are mostly fine. The fix is a robust estimator and it costs nothing.

Listwise deletion (drop any case with any missing item) throws away information, biases estimates unless the data are missing completely at random, and can halve a sample with twenty items and 5% missingness each. Full-information maximum likelihood uses every case's available data under the weaker missing-at-random assumption and is the default in Mplus and one option away in lavaan (missing = "fiml"). Enders (Applied Missing Data Analysis, Guilford, 2010) is the reference.

Resample the cases with replacement, refit the model, record every parameter; repeat 1,000 to 5,000 times. The spread of the estimates is a standard error that assumes nothing about normality, and the percentiles are a confidence interval that can be asymmetric, which matters for products of coefficients (indirect effects) whose sampling distribution is skewed. Section 07 relies on it.

The model implies a covariance matrix; the data supply one; the χ² statistic, (n − 1) times the fit function, tests whether the difference is zero. A significant χ² means the model does not reproduce the data exactly. With a few hundred cases almost every model is rejected, because no model is exactly right and the test has power to see it; with fifty cases almost none is, because the test cannot see anything. The test is correct and its verdict is rarely the question.

For every fixed parameter, the software reports the expected drop in χ² if it were freed. Free the largest, refit, repeat, and any model will reach any fit threshold. MacCallum, Roznowski and Necowitz (Psychological Bulletin 1992, 111:490) showed that such specification searches capitalise on chance: the modifications rarely replicate in a new sample, and the final model is a description of the sample's noise. This is the single most common way SEM papers in applied journals are made to 'fit'.

Two models are nested if one is the other with some parameters fixed. The difference in their χ² values, on the difference in their degrees of freedom, tests whether the restriction holds. This is the principled way to ask most SEM questions: is a path zero (fix it and compare)? Are two factors one (fix their correlation to 1 and compare)? Does the model hold across groups (section 08)? Is the second-order factor adequate (compare with the correlated-factors model)?

Anderson and Gerbing (Psychological Bulletin 1988, 103:411) argued for fitting the measurement model as a CFA (all factors correlated) before imposing any structural paths, so that misfit in the measurement is found and fixed where it belongs and the structural model is tested against a measurement model that is known to hold. The structural model is nested in the CFA (it replaces free correlations with paths and zeros), so the χ² difference between them is the test of the structural restrictions alone.

A moderator changes the size or sign of an effect: membership raises decision-making more where women's baseline mobility is low, or less in households with a resident mother-in-law. Statistically, an interaction term. With observed variables it is the product X × W, mean-centred, as in regression. With a latent moderator or latent predictor the product of latents is needed, and that is where SEM has to work harder.

The indirect effect may itself vary with a moderator: savings mediate membership's effect on decision-making where a bank is far away and not where one is near. Conditional process models (Preacher, Rucker and Hayes, Multivariate Behavioral Research 2007, 42:185; Hayes 2022) estimate the indirect effect at values of the moderator and test whether it changes, using the index of moderated mediation (Hayes, 2015): the slope of the indirect effect on the moderator, with a bootstrap interval.

With the same constructs measured at two or more waves, the cross-lagged panel model regresses each construct at wave 2 on both constructs at wave 1, and the cross-lagged paths (trust at 1 to participation at 2, and the reverse) are read as evidence of direction. Hamaker, Kuiper and Grasman (Psychological Methods 2015, 20:102) showed the standard model confounds within-person change with stable between-person differences, and can find cross-lagged 'effects' that are entirely trait differences. The random-intercept cross-lagged panel model separates the two and needs three or more waves.

Comparing women in Bihar and Tamil Nadu on a decision-making scale, or the same women before and after a programme, or respondents who answered in Hindi and in Bangla, assumes that the items relate to the construct in the same way in each group. If a Bangla item loads more weakly, or has a different intercept (Bangla speakers answer 'yes' more often at the same true level), the group difference in scores mixes real difference with measurement difference and cannot be separated. Invariance testing checks the assumption, level by level.

When scalar invariance fails, the aim is to find the few items responsible, free their intercepts or thresholds, and check that the rest hold. Modification indices in the constrained model point to the items; free them one at a time, largest first, retesting after each, and stop when ΔCFI against the metric model is within tolerance. Byrne, Shavelson and Muthén's rule is that at least two invariant items per factor (the marker plus one) are needed for the latent mean comparison to be identified and meaningful; more is better.

Longitudinal invariance is tested the same way, with waves as 'groups' in a single model where the same respondents appear at each wave and each item's error is allowed to correlate with itself across waves (the same person's idiosyncratic response to item d2 persists). Without it, a change in the latent mean between baseline and endline could be a change in how the items are understood, which a programme that talks to women about decision-making might well produce.

Once metric invariance holds, structural paths can be compared across groups: is the effect of savings on decision-making the same for women with and without their own income? Fit the structural model in both groups at once, then constrain the path equal and test the constraint. This is moderation by a categorical variable (section 07), done with full measurement models and a proper test.

Partial least squares path modelling (Wold, 1982; Lohmöller, 1989) builds each construct as a weighted composite of its indicators, with weights chosen iteratively to maximise the explained variance of the endogenous composites, then estimates the structural paths by ordinary regression among the composites. There is no common factor, no separation of measurement error, and no overall fit function to test. What there is: a prediction-oriented model that runs on small samples, handles formative constructs natively, and converges when CB-SEM will not.

Ordinary PLS path coefficients between reflective constructs are biased toward zero, because the composites carry measurement error, and loadings are biased upward. Consistent PLS (Dijkstra and Henseler, MIS Quarterly 2015, 39:297) corrects both using ρ_A, so that PLSc estimates for a common-factor model are consistent, and for a well-specified model come close to CB-SEM's. It is one click in SmartPLS and one argument in SEMinR and cSEM.

When the constructs are reflective and the sample allows it, fitting the model both ways is cheap and informative. If the CB-SEM fits and the PLSc paths agree with it, the result does not depend on the estimator. If CB-SEM rejects the model while PLS reports strong paths, the paths are estimated on a structure the data contradict, and the PLS result is the weaker one. Papers that do this are rare; referees notice them.

The arrows in a path diagram are causal hypotheses. Estimating them assumes the causal structure; it does not test it. Bollen and Pearl ('Eight myths about causality and structural equation models', in Morgan, ed., Handbook of Causal Analysis for Social Research, Springer, 2013) list the misreadings on both sides: that SEM proves causation (it cannot), and that SEM is merely correlational (it is a language for stating causal assumptions and deriving what they imply). The coefficients are causal effects if, and only if, the assumptions hold.

Imai, Keele and Tingley (Psychological Methods 2010, 15:309) formalised what mediation needs: sequential ignorability, meaning that the treatment is as good as random given covariates, and that the mediator is as good as random given treatment and covariates. The first is delivered by randomisation; the second is delivered by nothing, because the mediator was not assigned. Their sensitivity analysis asks how strong an unmeasured confounder of the mediator-outcome path would have to be (the correlation ρ between the two error terms) to reduce the indirect effect to zero.

Every construct in the model measured from one respondent, in one sitting, on the same response format, shares variance that has nothing to do with the constructs: the respondent's mood, acquiescence, social desirability, the desire to be consistent, the enumerator's manner. Podsakoff, MacKenzie, Lee and Podsakoff (Journal of Applied Psychology 2003, 88:879) catalogued the sources and the remedies; their 2012 review (Annual Review of Psychology 63:539) updated them. The bias inflates correlations among self-reported constructs and therefore every path between them.

Women who join a self-help group differ from those who do not in ways the model does not measure, and those ways affect decision-making. Membership is endogenous, and its path in an observational SEM is biased by whatever drives both. This is the same problem as in any regression, and SEM offers the same remedies: an instrument (a variable that affects membership and affects the outcome only through it), a design (randomisation, a discontinuity), or an honest limitation.

A directed acyclic graph is a path diagram with the unmeasured variables drawn in and no estimation implied. Drawing one for the study, including the things you cannot measure (household bargaining power, the husband's attitudes, the enumerator's effect), shows which paths are identified by adjustment, which need a design, and which are hopeless. Pearl's back-door criterion gives the rule: a path X → Y is identified if every back-door path from X to Y is blocked by measured covariates.

Academic Writing 101 in this series covers the prose. The SEM-specific rule is that the measurement results come before the structural results and are read as a condition on them: a reader who does not trust the constructs has no reason to read the paths.