Reports the two standard checks that a correlation matrix is worth factoring at all: the Kaiser-Meyer-Olkin measure of sampling adequacy and Bartlett's test of sphericity. Most published soil quality indices skip both and go straight to the principal components; Theresa et al. (2026) do not, and report KMO 0.81 with Bartlett chi-squared 425.37 on 136 degrees of freedom.
Value
An object of class pca_adequacy, a list with:
- kmo
Overall Kaiser-Meyer-Olkin measure, or
NA- msa
Named vector of per-variable measures, or
NA- kmo_interpretation
Kaiser's label for the overall KMO
- kmo_message
Why KMO could not be computed, when it could not
- bartlett
List of
statistic,df,p_value- n, p
Observations and indicators used
Details
Bartlett's test of sphericity asks whether the correlation matrix is distinguishable from the identity. If it is not – if the indicators are mutually uncorrelated – then there are no components to find and PCA has nothing to reduce. The statistic is $$\chi^2 = -\left(n - 1 - \frac{2p + 5}{6}\right) \ln |R|$$ on \(p(p-1)/2\) degrees of freedom. A small p-value is what you want: it means the matrix is not an identity.
Be aware that this test is close to a formality on real soil data. With a decent sample size almost any set of soil properties rejects sphericity, so passing it is weak evidence. Failing it is strong evidence, and that is the point.
The KMO measure is the more informative of the two. It compares the size of ordinary correlations to the size of partial correlations: $$KMO = \frac{\sum_{i \ne j} r_{ij}^2}{\sum_{i \ne j} r_{ij}^2 + \sum_{i \ne j} a_{ij}^2}$$ where \(a_{ij}\) are the partial correlations. When indicators share common factors, partialling out the others leaves little behind and KMO approaches 1. Kaiser's labels:
| KMO | Verdict |
| below 0.50 | unacceptable |
| 0.50-0.60 | miserable |
| 0.60-0.70 | mediocre |
| 0.70-0.80 | middling |
| 0.80-0.90 | meritorious |
| 0.90 and above | marvellous |
The per-variable measure (msa) is often more useful than the overall
figure: a single indicator with a low MSA is a candidate for removal, and
removing it usually lifts the whole matrix.
A singular correlation matrix. KMO requires inverting the
correlation matrix, which is impossible when indicators are exactly
collinear. This is not an edge case in soil science: particle-size
fractions sum to 100, so Sand, Silt and Clay together
are perfectly collinear by construction, and organic matter and organic
carbon are related by a fixed factor. When the matrix is singular the KMO
fields come back NA with an explanation in $kmo_message,
rather than the function erroring or returning a number computed from a
pseudo-inverse that nobody asked for. Drop one member of each collinear set
and try again.
References
Kaiser, H. F. (1974). An index of factorial simplicity. Bartlett, M. S. (1951). The effect of standardization on a chi-square approximation in factor analysis. Theresa, M. et al. (2026) – an SQI study that reports both.
See also
pca_select_mds, which reports this automatically
Examples
props <- c("pH", "OM", "N", "P", "K", "CEC", "BD")
pca_adequacy(soil_structured[, props])
#> PCA adequacy
#> Observations: 120 Indicators: 7
#>
#> Kaiser-Meyer-Olkin: 0.813 (meritorious)
#> Indicators below 0.60, worth considering for removal:
#> pH 0.353
#>
#> Bartlett's test of sphericity
#> chi-squared = 1012.73, df = 21, p = 5.423e-201
#> The matrix differs from the identity, so there is structure to
#> factor. Note this test rejects on almost any real soil data.
# Particle-size fractions sum to 100, so including all three makes the
# correlation matrix singular and KMO undefined
pca_adequacy(soil_structured[, c("Sand", "Silt", "Clay", "pH", "OM")])
#> PCA adequacy
#> Observations: 120 Indicators: 5
#>
#> Kaiser-Meyer-Olkin: not computable
#> The correlation matrix is singular, so it cannot be inverted and KMO is undefined. This usually means some indicators are exactly collinear -- particle-size fractions summing to 100 is the classic case.
#>
#> Bartlett's test of sphericity
#> Not computable: the correlation matrix is singular.