Reading matrix results

How to read the standardised results of the matrix operators this instance offers. Each operator lists its output fields: what the value means, the labelled bands of a published convention where one applies, what its sign says, and the caveats to keep in mind.

Matrix operators

MAT_COLLINEARITY

Whether candidate predictors overlap so much that a regression on them would give unstable coefficients, and which ones are tangled. See its catalog entry.

scalars.max_vif

The largest variance inflation factor among the predictors (each one’s is also listed): how many times larger that predictor’s coefficient variance is than it would be if it were unrelated to the other predictors. 1 means no overlap; tolerance, its reciprocal, is the share of the predictor’s spread the others do not explain.

Caveats:

• No convention bands it. Cut-offs of 5 or 10 are rules of thumb only: O’Brien (2007) shows they are arbitrary, and a large sample can make a high value harmless while a small one can make a modest value matter.
• It depends only on the predictors, not on any outcome; it says how unstable coefficients would be, not whether a predictor matters.
• Under params.repair it is read off the nearest consistent correlation table, not the observed one.

scalars.condition_number

The largest of Belsley’s condition indices: how close the predictors (with the intercept, unless params.center is true) come to an exact straight-line dependency. 1 means none; large values mean some combination of predictors nearly cancels out.

Caveats:

• No convention bands it. Belsley, Kuh and Welsch’s guideline (an index above about 30 where two or more variables put a large share of their variance on that dimension) is a rule of thumb; read the variance-decomposition proportions to see which variables are involved.
• With the intercept included (the default) a predictor whose values sit far from zero relative to their spread raises it even when the predictors are unrelated; params.center true removes that part.

primary.values

The predictors’ correlation table that the variance inflation factors and the centered diagnostics are read from: each off-diagonal cell is the correlation of its row and column predictors, from -1 to +1.

Sign:

• +: the two fields tend to rise together
• -: one field tends to fall as the other rises

Caveats:

• A predictor can be heavily inflated with no single large pairwise correlation, when it is close to a combination of several others; read the variance inflation factors, not just this table.

MAT_CORRELATION

How closely every pair of numeric fields moves together, in a line or in rank order, as one square table of values from -1 to 1. See its catalog entry.

primary.values

Each off-diagonal cell is the correlation of its row and column members under params.method, from -1 to +1. pearson (default): r, how closely the two follow a straight line together; spearman: rho, r on the members’ ranks, how steadily one rises or falls with the other; kendall: tau-b, the share of agreeing minus disagreeing row pairs, tie-adjusted. 0 means no such link. The diagonal is 1.

Bands (convention: Cohen (1988); a labelled convention, not a rule):

Sign:

• +: the two fields tend to rise together
• -: one field tends to fall as the other rises

Caveats:

• Correlation is not causation: a third factor may drive both fields.
• A null cell means a member had no spread (constant, or too few rows), so its correlation is undefined, not 0.
• A Pearson r near zero rules out only a straight-line link; a curved relationship can still be strong.
• The bands are Cohen’s for r; Kendall’s tau-b runs smaller than r or rho for the same strength (about two thirds of rho), so read a kendall cell against lower cut-offs.
• The matrix reports no p-values; run TEST_PEARSON_R, TEST_SPEARMAN_R or TEST_KENDALL_TAU on a pair to test it.

scalars.determinant

The determinant of the correlation table: 1 when no member is linearly related to the others, falling toward 0 as members become linear combinations of one another.

Caveats:

• Null when the table is not positive definite, including when any cell is null.
• No published convention bands it; a value near 0 flags near-redundant members.

MAT_FACTOR

Which hidden qualities a set of related measures reflects, and how strongly each measure is tied to each, before building scale scores. See its catalog entry.

primary.values

Each cell is a loading: how strongly the row’s measure is tied to the column’s factor. Without a rotation, or with varimax, it is the correlation between the two; under promax or oblimin it is the measure’s own tie to that factor with the other factors held fixed. A factor is named by the measures with the largest loadings.

Sign:

• +: the measure rises with the factor
• -: the measure falls as the factor rises; a factor’s overall sign is a convention (each column is made to sum positive)

Caveats:

• Unrotated loadings put most measures on the first factor; a rotation gives a cleaner pattern without changing how well the factors fit (the communalities).
• Under an oblique rotation the factors correlate (auxiliary.factor_correlation), so loadings on correlated factors overlap.
• A uniqueness at or below 0, or held at the maximum-likelihood floor (a Heywood case, flagged with a warning), means the fit is improper; try fewer factors or drop a near-duplicate measure.

scalars.kmo

The Kaiser-Meyer-Olkin measure of sampling adequacy: how much of the measures’ correlation is shared across the whole set rather than tied to single pairs, from 0 to 1. Higher means a factor analysis is more worthwhile.

Bands (convention: Kaiser (1974); a labelled convention, not a rule):

Caveats:

• Null when the correlation table is singular (a measure is an exact combination of others).

scalars.bartlett_p

Follows the shared reading: see Reading a p-value.

scalars.bartlett_chisq

Bartlett’s test statistic: how far the correlation table is from one in which no measure is related to any other. Large values mean the measures are related enough to factor.

Caveats:

• With many rows it is almost always large; read KMO for how worthwhile the analysis is.

scalars.rmsr

The root-mean-square residual: the typical size of the correlations the factors leave unexplained, on the correlation scale. Smaller means the factors reproduce the correlations more closely.

Caveats:

• No convention bands it here; it always shrinks as factors are added, so weigh it against how readable the extra factor is.

scalars.empirical_p

Follows the shared reading: see Reading a p-value.

scalars.empirical_chisq

The empirical chi-square: the sum of the squared leftover correlations times the inference size. It measures the residual size, on df degrees of freedom, and is not a likelihood test of the model.

Caveats:

• It rests on the inference size: the rows (listwise), the summed weight or the effective sample size under a weight, or the smallest pair under pairwise deletion.
• Null p when df is 0 or below: the model then has no spare correlations to test against.

scalars.likelihood_p

Follows the shared reading: see Reading a p-value.

scalars.likelihood_chisq

Maximum likelihood only: the likelihood-ratio chi-square, with Bartlett’s correction, of the k-factor model against a free correlation table, on df degrees of freedom. A small p says the factors leave correlation unexplained.

Caveats:

• With many rows it flags even trivial misfit; read tli and rmsea beside it.
• It assumes the measures are multivariate normal and rests on the inference size (the rows, the summed weight or the effective sample size).

scalars.tli

Maximum likelihood only: the Tucker-Lewis index, how much better the factor model fits per degree of freedom than a model of unrelated measures. Near 1 or above means a close fit; it is not capped at 1.

Caveats:

• No convention bands it here; cut-offs such as .95 come from confirmatory models under specific conditions.
• Null when df is 0 or below.

scalars.rmsea

Maximum likelihood only: the root-mean-square error of approximation, the misfit per degree of freedom after allowing for sampling, as a point estimate. 0 means no misfit beyond chance; smaller is better.

Caveats:

• No convention bands it here; cut-offs such as .05 or .08 are rules of thumb; read rmsea_lower and rmsea_upper for its uncertainty.
• Null when df is 0 or below; with few rows or few df it is unstable.

scalars.rmsea_lower

Maximum likelihood only: the lower bound of the RMSEA confidence interval at conf_level (90% by default, as psych reports it). 0 when even a perfectly fitting model would often give a chi-square this large.

Caveats:

• It inverts the noncentral chi-square and assumes multivariate normal measures; with few rows the interval is wide.
• Null when df is 0 or below, or when no bound exists on the searched range.

scalars.rmsea_upper

Maximum likelihood only: the upper bound of the RMSEA confidence interval at conf_level (90% by default). A small upper bound says the misfit per degree of freedom is plausibly small.

Caveats:

• The bounds divide by the inference size minus one where the point estimate divides by the inference size itself (psych’s rule), so with very few rows the point can sit outside the interval; such a bound is null.
• Null when df is 0 or below.

scalars.bic

Maximum likelihood only: the Bayesian information criterion of the fit, the likelihood chi-square minus df times the log of the inference size. Compare it across factor counts on the same measures: the lower value is preferred.

Caveats:

• Only differences between fits of the same measures and rows mean anything; the value alone does not.
• Null when df is 0 or below.

MAT_PARTIAL_CORRELATION

How closely each pair of numeric fields moves together once other fields are held fixed, as one square table of values from -1 to 1. See its catalog entry.

primary.values

Each off-diagonal cell is the correlation of its row and column members once the held-fixed fields are taken out of both (every other member, or the params.control fields), from -1 to +1. 0 means no straight-line link is left after holding them fixed. The diagonal is 1.

Bands (convention: Cohen (1988); a labelled convention, not a rule):

Sign:

• +: the two fields tend to rise together
• -: one field tends to fall as the other rises

Caveats:

• Correlation is not causation: a third factor may drive both fields.
• A partial r can be much smaller than, or even opposite in sign to, the plain r when the held-fixed fields drive both members; holding fixed a field that is itself an outcome of the pair can also create a link that is not there.
• A null cell means the input had an undefined correlation (a member with no spread, or too few rows), so no cell can be computed.
• Under params.repair the cells come from the nearest consistent correlation table, not the observed one; the PULSE_MATRIX_NOT_PSD warning says how far it moved.
• The matrix reports no p-values.

MAT_PCA

How many underlying dimensions a set of related measures covers and which measures belong to each, so a few summaries can stand in for many. See its catalog entry.

primary.values

Each cell is a loading: how strongly the row’s measure is tied to the column’s component, the correlation between the two on a correlation input. A component is named by the measures with the largest loadings; components are listed from the one that summarises the most spread down.

Sign:

• +: the measure rises with the component
• -: the measure falls as the component rises; a component’s overall sign is a convention (its largest loading is made positive, or under a rotation its loadings sum positive)

Caveats:

• Unrotated, a measure loading on several components at once is common and does not mean it measures several things; params.rotation varimax redistributes the kept variance into a cleaner pattern without changing the communalities.
• Components summarise shared spread; they do not show that an underlying cause exists.
• On a covariance input the loadings are in the measures’ own units, so they are not comparable across measures.

scalars.kmo

The Kaiser-Meyer-Olkin measure of sampling adequacy: how much of the measures’ correlation is shared across the whole set rather than tied to single pairs, from 0 to 1. Higher means a component summary is more worthwhile.

Bands (convention: Kaiser (1974); a labelled convention, not a rule):

Caveats:

• Null when the correlation table is singular (a measure is an exact combination of others).
• The per-measure version of this reading is also reported; a measure with a low value fits the set poorly.

scalars.bartlett_p

Follows the shared reading: see Reading a p-value.

scalars.bartlett_chisq

Bartlett’s test statistic: how far the correlation table is from one in which no measure is related to any other. Large values mean the measures are related enough to summarise.

Caveats:

• With many rows it is almost always large; read KMO for how worthwhile a summary is.
• It rests on the inference size: the rows (listwise), the summed weight or the effective sample size under a weight, or the smallest pair under pairwise deletion.

MAT_RELIABILITY

How consistently a set of rating items measures one thing, before you add them up into a score, with a check of how each item fits the rest. See its catalog entry.

scalars.alpha

Cronbach’s alpha: how consistently the items measure one shared quality, from the share of the summed score’s spread that the items share. 1 means the items move in lockstep; it can fall below 0 when items pull against each other, which usually means a reverse-worded item was not reversed.

Bands (convention: George & Mallery (2003); a labelled convention, not a rule):

Caveats:

• Alpha rises with the number of items, so a long battery can reach a high alpha with weakly related items.
• A high alpha does not show the items measure only one thing.
• Alpha assumes every item is equally tied to the shared quality; when the ties differ it understates reliability, which omega allows for.

scalars.alpha_standardized

Alpha computed on the items’ correlations instead of their raw spreads: the alpha the battery would have if every item were first put on the same scale.

Bands (convention: George & Mallery (2003); a labelled convention, not a rule):

Caveats:

• It differs from alpha when the items’ spreads differ; report the one that matches how the score is built (raw sums or standardised items).

scalars.omega

McDonald’s omega: the share of the summed score’s spread due to the one shared quality, from a one-factor fit that lets each item’s tie to it differ. Read on the same 0 to 1 scale as alpha; it is usually at least as high.

Caveats:

• No published convention bands omega; it is not read against the alpha bands.
• Null with a warning when the battery has 2 items, when the fit puts an item’s leftover spread at or below zero (a Heywood case), or when a pairwise table is inconsistent and params.repair is not set. When the fit does not converge, omega is kept as the last iterate’s value and a warning says so; treat it as provisional.
• The fit assumes one shared quality; when the items reflect several, omega from one factor misstates reliability.

scalars.mean_inter_item_r

The average correlation among the items, the strength of the typical pair; standardized alpha is built from it and the item count.

Caveats:

• An average hides spread: one item unrelated to the rest pulls it down; check the table and item_total_r.
• Like any correlation it shows the items move together, not that one causes another.