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For a random vector with covariance $\Sigma$ and positive coordinate variances, let $D=\mathrm{diag}(\Sigma )$. Its correlation matrix is the covariance of $D^{-1/2}(X-\mathbb{E}X)$, so it is positive semidefinite with unit diagonal. Each entry is the correlation of the corresponding pair of variables. In particular $\mathrm{tr}(C)=p$, explaining why an eigenvalue in correlation-based principal component analysis represents that eigenvalue divided by $p$ of the total variance. The empirical version is the sample correlation matrix.

The calculation is principal component analysis on a correlation matrix, rather than on an unscaled covariance matrix. For each of the eleven rating variables, subtract its column mean and divide by its column standard deviation. If $Z$ is the resulting $43\times 11$ matrix, use the divisor convention of the fitted object to writeHere the scaling standard deviations use divisor $43$, consistently with this convention. Using sample standard deviations with divisor $42$ gives the same correlation matrix and directions, with the corresponding score scale. Every diagonal entry of $R$ is one, so ${\sum }_k{\lambda }_k=\mathrm{tr}R=11$.

A principal component is the linear combination of the standardized ratings along $a_k$. Its principal component score for judge $i$ is $t_{ik}=z_i^T{a}_k$, and its variance is ${\lambda }_k$. The displayed component standard deviations are $\sqrt{{\lambda }_k}$; the explained variance of a principal component is ${\lambda }_k/11$. In particular,The first component explains about $92.122\%$ of the standardized variation, and the first two jointly explain $95.982\%$. Components three, four and five add about $2.321\%$, $0.830\%$ and $0.339\%$. The cumulative line adds these proportions successively; it is not the proportion explained by that component alone. Retaining two components discards about $4.018\%$ of the total standardized squared variation.

The displayed principal component loadings are the unit eigenvector coefficients $a_{jk}$. The first direction has positive, broadly similar weights for all ratings. It measures an overall favorable-rating tendency: a judge high on most ratings has a high first score. The second direction compares integrity and demeanour, with large positive coefficients, against such attributes as decision promptness, case-flow management and physical ability, with negative coefficients. Thus among similarly rated judges, a positive second score indicates relatively stronger integrity/demeanour compared with those operational ratings. This is a description of a data direction, not a causal interpretation. Multiplying a whole eigenvector and its scores by minus one reflects the display but changes nothing substantive.

These principal component loadings are not themselves variable-component correlations. With standardized variables,For example, the integrity correlation with component one is approximately $3.1833\times 0.289\simeq 0.920$. Blank loadings in the printed second column indicate suppressed small entries, not exact zeros. The complete unrounded loading vectors in the fitted object must be used to calculate scores.

A two-dimensional representation plots the 43 pairswith each point labeled by its judge. In the fitted software object these are the first two columns of the score matrix. Nearby points have similar projections of their standardized ratings. The rank-two reconstruction ${\hat{z}}_i=t_{i1}{a}_1+t_{i2}{a}_2$ is the orthogonal projection onto the leading principal plane and minimizes the total squared reconstruction error over all two-dimensional linear subspaces. To see this, for any orthonormal pair of directions the captured variance is the sum of its two Rayleigh quotients, at most ${\lambda }_1+{\lambda }_2$ by diagonalizing $R$; the remaining variance is the total minus that captured sum. Distances in the plot omit the smaller components, so a close projected pair need not be equally close in every original rating. The actual 43 coordinates cannot be recovered from the summary alone; the supplied data matrix or fitted score matrix is needed.

The printed descriptive list omits INTG, although the loading table includes it. INTG denotes integrity and supplies the eleventh variable. The standard data set also contains a contacts variable; the stated eleven-rating analysis uses the ratings, not contacts.

For a population question use a population-averaged logistic model for repeated binary outcomes. Code $z_i=1$ for the new treatment and $0$ for the control, with visit times $t_j=2j$ months for $j=1,2,3$. Let $w_{ij}=(1,z_i,x_i^T,t_j{)}^T$, ${\gamma }_M=({\alpha }_M,{\varphi }_M,{\beta }_M^T,{\delta }_M{)}^T$, andDifferent subjects are independent. Each response has a Bernoulli distribution with this marginal mean, but no independence of a subject's three responses is assumed. The linear time effect and absence of treatment-by-time interaction are substantive mean assumptions; if inappropriate they can be replaced by time indicators and interaction terms.

For complete data, set $m_i=(m_{i1},m_{i2},m_{i3}{)}^T$, $A_i=\mathrm{diag}(m_{ij}(1-m_{ij}))$, and $D_i=\partial m_i/\partial {\gamma }_M^T=A_i{W}_i$, where $W_i$ has rows $w_{ij}^T$. A working exchangeable correlation matrix has unit diagonal and off-diagonal value $\rho \in (-1/2,1)$; set $V_i=A_i^{1/2}C(\rho )A_i^{1/2}$. The generalized estimating equation isWith correct means and independent subjects, a positive-definite working covariance matrix and regularity give consistency even if the working correlation is wrong. Write $H={\sum }_i{D}_i^T{V}_i^{-1}{D}_i$ and $u_i=D_i^T{V}_i^{-1}(Y_i-m_i)$. The fitted subject-level sandwich covariance matrix isevaluated at the estimates. This inference is asymptotic in the number of independent subjects, not the number of visits.

For the actual incomplete data, define $R_{ij}=1$ if visit $j$ is observed, with $R_{i0}=1$ and monotone dropout. Let ${\mathcal{H}}_{i,j-1}$ contain treatment, baseline covariates and responses observed before visit $j$. Under sequential missing at random, specifyThe assumption is that the retention decision, conditional on this history, does not additionally depend on unseen current or future outcomes. Fit the $q_{ij}$, for example by visit-specific logistic regressions. A simple valid choice uses working independence and the inverse-observation-weighted estimating equations for longitudinal dropout:Terms with $R_{ij}=0$ are zero and require no missing response. Given the full response vector, sequential missing at random makes $\mathbb{E}(R_{ij}/{\rho }_{ij}\mid Y_i,z_i,x_i)=1$. Consequently the weighted score has the same expectation as the complete-data score, proving its mean-zero property. Correct retention probabilities, positivity and a correctly specified marginal mean are required. Use cluster-robust uncertainty, accounting for fitted retention-model coefficients, or resample entire subjects and refit both models. Under missing completely at random, the simpler unweighted observed-response GEE can suffice; ordinary unweighted GEE is not valid under arbitrary history-dependent missing at random.

The treatment coefficient gives a population conditional-on-baseline odds ratio $e^{{\varphi }_M}$ at each visit. To communicate an absolute population treatment benefit, standardize predicted probabilities over the full baseline population, rather than only completers:A causal reading additionally uses the randomized controlled trial assignment, consistency of the treatment definition, no interference and the dropout assumptions; the statistical model alone does not supply those conditions.

Factor analysis models a large set of correlated measurements using a smaller collection of shared latent factors and variable-specific variation. A classical orthogonal factor model isTake $q<p$, centered errors, and positive specific variances for the inverse-covariance and likelihood formulas below. The factor loadings form the $p\times q$ matrix $L$, and the model covariance isThus for $i\ne k$, $\mathrm{Cov}(X_i,X_k)={\sum }_j{L}_{ij}{L}_{kj}$; the common latent factors explain off-diagonal correlation. The communality of variable $i$ is $h_i^2={\sum }_j{L}_{ij}^2$, and $\mathrm{Var}(X_i)=h_i^2+{\psi }_i$. For standardized variables this is $1=h_i^2+{\psi }_i$. The specific variance represents measurement noise or genuinely unshared variation; it is not another common latent factor.

This differs from principal component analysis. A principal component maximizes projected total variance and is a determined linear score after choosing its direction; a factor model attempts to explain shared covariance while reserving diagonal specific variance. Low-dimensional PCA truncation can help initialize a factor fit, but it does not justify setting all specific variances to zero. Latent factors are unobserved, not directly observed scores. When the latent factors and the errors are jointly Gaussian, their conditional mean iswhich supplies one factor-score estimate, with remaining uncertainty. This follows from the block conditional normal formula in Question 1; it does not make the latent factor uniquely recoverable from one observation.

For the Gaussian likelihood, use the empirical covariance $S_0=n^{-1}{\sum }_i(x_i-\bar{x})(x_i-\bar{x}{)}^T$. Fit the factor model by minimizing $\log |L{L}^T+\Psi |+\mathrm{tr}(S_0(L{L}^T+\Psi {)}^{-1})$ subject to admissible specific variances. Other approaches estimate communalities and extract latent factors from the reduced correlation matrix. Choose $q$ using substantive plausibility, a scree plot, fitted residual correlations and model-fit assessment. Under local identifiability, a likelihood goodness-of-fit count for the covariance model iswhere the subtracted rotational dimension must be removed from the loading parameter count. This count alone does not establish identifiability, and boundary fits with zero specific variance need special care.

Factor rotation addresses interpretability and nonuniqueness. For every orthogonal matrix $Q$, replace $(L,F)$ by $(LQ,Q^{T}F)$; then $(LQ)(LQ{)}^T=L{L}^T$, so the fit is unchanged. The absolute orientation of unrestrained latent factors is therefore not identified by the covariance. Varimax rotation chooses orthogonal axes favoring a simple pattern of large and small squared loadings. Oblique rotation can permit correlated latent factors, provided their covariance is transformed consistently. Sign changes and permutations are further harmless relabelings; an interpretation must state its chosen convention.

The sketch uses four standardized hypothetical measurements sharing two orthogonal latent factors. Rotating the factor coordinates changes the loading descriptions while preserving every implied covariance and communality; the scree plot belongs to that same constructed covariance matrix. Factor analysis separates shared variation from specific variance, and its latent factors require an identifying or interpretive orientation.