Sampling covariance and SEs of G-study component estimates — vcov.mfrm_multivariate

Quantify how estimated sources of score variation would vary across repeated G-studies with the same assignment design. Supports balanced ANOVA and identifiable MINQUE(0), including incomplete and supported nested sources.

Usage

# S3 method for class 'mfrm_multivariate_gstudy'
vcov(object, score = NULL, weights = NULL, assumption, ...)

Arguments

object: A result from mfrm_g_study().
score: One original score, or exactly two distinct original scores for the sampling covariance of their between-score component covariances. The default selects the first original score.
weights: Optional named weights for one prespecified composite; name every original score. Do not also supply score.
assumption: Required: "normal" asserts independent normal random effects, a common mean for each score and constant component covariances. Assignment and omissions must preserve that model when conditioning on the observed assignment. The assumption is not tested.
...: Reserved; extra arguments are rejected.

Value

A named matrix estimating the sampling covariance of component estimates. Attribute summary is a data frame with Source, Estimate, SamplingVariance, SE, and SEStatus. diagnostics retains the moment rank, singular values, entrywise estimability and matrix PSD check. score, weights, assumption, source_design, source_rows and notes retain interpretation. Save the full result using saveRDS(). MINQUE requires its retained grouping assignments, moment equations and quadratic products; balanced ANOVA uses saved summaries without raw data.

Details

Rows and columns refer to estimated components, such as Person, Task and Residual. This is neither the covariance matrix between individual ratings nor the component matrices between score columns in object. Use attr(result, "summary") to read the original estimates, sampling variance estimates, SEs and availability. A component SE has squared score units; the returned covariance matrix has fourth-power score units. For a between-score covariance, the SE has the product of the two score units. These SEs differ from the D-study's measurement SEM in original score units.

The component estimates are unchanged. This method uses Gaussian product moments to correct plug-in bias in their estimated sampling covariance. It does not fit a likelihood, resample data or construct confidence intervals. In particular, Estimate +/- 1.96 * SE is not a generally justified interval. Use confint.mfrm_multivariate_gstudy() for its separately qualified interval methods. A REML profile's center belongs to a different estimator; the SE returned here describes the original ANOVA/MINQUE estimator.

Calculation

With mean-removing MINQUE quadratic forms $q_s=y^T A_s y$, $E(q_s)=\theta_s$ and $C_{st}=2\mathrm{tr}(A_s V A_t V)$, where $V=\sum_u\theta_u K_u$. Let the linear operator P map $\mathrm{vec}(\theta\theta^T)$ to $\mathrm{vec}(C)$. The normal product-moment identity gives $E\{\mathrm{vec}(qq^T)\}=(I+P)\mathrm{vec}(\theta\theta^T)$. Inverting this moment system gives an unbiased estimator of C for its identifiable entries. This is a consequence of the cited moment identity, not a claim that the reference proposes this package's G-theory procedure. The squared SE, when available, is the corresponding diagonal estimate; the square root is not itself an unbiased SE estimator.

For two scores, write x and y for their variance-component vectors and c for their covariance-component vector. On symmetric matrix coordinates, $E\{\mathrm{sym}(\hat x\hat y^T)\}=\mathrm{sym}(xy^T)+P(cc^T)$ and $E(\hat c\hat c^T)=(I+P/2)(cc^T)+(P/2)\mathrm{sym}(xy^T)$. Solve these coupled equations for the sampling covariance $(P/2)\{\mathrm{sym}(xy^T)+cc^T\}$. Applying only the scalar correction to a covariance component would be incorrect.

Balanced ANOVA reduces to the existing independent-mean-square result A %*% diag(2 * MS^2 / (df + 2)) %*% t(A) for a variance/fixed composite, and the Wishart product-moment correction for a between-score covariance. MINQUE uses the actual observed grouping kernels. Column-block calculations avoid allocating observation-square matrices, but time still increases quadratically with the number of observed ratings. No sampling simulation is performed. Balanced MINQUE uses the same ANOVA limit.

Unavailable or negative results

A moment system can fail to identify some sampling-covariance entries even when point components are identifiable. A contributing balanced stratum with df one is an example for between-score covariance SEs. An SVD checks each target entry's estimability; only unidentified entries become NA. Its rank tolerance is sqrt(.Machine$double.eps) relative to the largest singular value, with relative target reconstruction error below 1e-8.

An unbiased sampling-variance estimate may be negative. It is retained in the returned matrix and SamplingVariance, while its SE is NA and SEStatus explains why. The matrix can also be indefinite with positive diagonal entries. Nothing is clipped or repaired to make it positive semidefinite. A mathematically unbiased variance estimate does not ensure a useful SE in a small sample or correct normal-reference coverage.

References

Magnus, J. R. (1978). The moments of products of quadratic forms in normal variables. Statistica Neerlandica, 32, 201–210. Lemma 6.2, p.209. doi:10.1111/j.1467-9574.1978.tb01399.x .

Rao, C. R. (1971). Estimation of variance and covariance components–MINQUE theory. Journal of Multivariate Analysis, 1, 257–275. doi:10.1016/0047-259X(71)90001-7 .

Examples

data("mfrmr_example_gtheory_crossed", package = "mfrmr")
ratings <- mfrmr_example_gtheory_crossed
# Complete balanced ratings: compute both targets directly.
g <- mfrm_g_study(ratings, c("Content", "Organization"),
                            method = "anova")
sampling <- vcov(g, score = "Content", assumption = "normal")
attr(sampling, "summary") # Component estimates and their sampling SEs.
#>         Source  Estimate SamplingVariance         SE  SEStatus
#> 1       Person 0.9492992      0.076819793 0.27716384 Available
#> 2        Rater 0.8073438      0.238859253 0.48873229 Available
#> 3         Task 1.6162718      0.915385926 0.95675803 Available
#> 4 Person:Rater 1.0182691      0.010283222 0.10140622 Available
#> 5  Person:Task 0.9206047      0.011150699 0.10559687 Available
#> 6   Rater:Task 1.1703825      0.077274940 0.27798370 Available
#> 7     Residual 1.0105340      0.001494044 0.03865286 Available
attr(sampling, "notes")
#> [1] "Sampling covariance of original ANOVA/MINQUE component estimates, conditional on the declared assignment under independent normal random effects. This is not measurement SEM, covariance between individual ratings or the SE of a separate REML estimator."
#> [2] "Gaussian product-moment correction gives unbiased identifiable sampling-covariance entries under the model. Taking square roots is not an unbiasedness guarantee for SEs and does not justify normal-reference confidence intervals."                        
#> [3] "Negative sampling variances are retained, with unavailable SEs. Unidentified entries are NA. An indefinite matrix is not repaired; inspect summary and diagnostics before propagating it to another target."                                                 
#> [4] "Scores and composite weights are prespecified; omission must preserve the model conditional on assignment. No robustness, informative-missingness correction or general confidence-interval coverage is claimed. Save the complete result with saveRDS()."   
# Uncertainty in the covariance between two criteria, by source.
cross_score <- vcov(g, score = c("Content", "Organization"),
                    assumption = "normal")
attr(cross_score, "summary")
#>         Source    Estimate SamplingVariance         SE  SEStatus
#> 1       Person -0.08845568     0.0389471416 0.19735030 Available
#> 2        Rater  0.27503314     0.1301264587 0.36073045 Available
#> 3         Task  0.27652734     0.1332958079 0.36509698 Available
#> 4 Person:Rater -0.12037694     0.0052971546 0.07278155 Available
#> 5  Person:Task -0.11132953     0.0053251089 0.07297334 Available
#> 6   Rater:Task  0.06976932     0.0276174789 0.16618507 Available
#> 7     Residual -0.01083917     0.0007222732 0.02687514 Available

# Incomplete assignments also work, but their moment calculation is longer.
# These saved results use the same ratings with rows 1, 8 and 23 omitted.
# Omitted ratings are absent observations, never zero scores.
incomplete <- readRDS(system.file("examples", "gstudy-sampling.rds",
                                  package = "mfrmr", mustWork = TRUE))
incomplete$study$design
#> $person
#> [1] "Person"
#> 
#> $rater
#> [1] "Rater"
#> 
#> $task
#> [1] "Task"
#> 
#> $scores
#> [1] "Content"      "Organization"
#> 
#> $facets
#>   Rater    Task 
#> "Rater"  "Task" 
#> 
#> $count_columns
#>    Rater     Task 
#> "Raters"  "Tasks" 
#> 
#> $counts
#> Person  Rater   Task 
#>     40      8      6 
#> 
#> $levels
#> $levels$Person
#>  [1] "1"  "2"  "3"  "4"  "5"  "6"  "7"  "8"  "9"  "10" "11" "12" "13" "14" "15"
#> [16] "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" "26" "27" "28" "29" "30"
#> [31] "31" "32" "33" "34" "35" "36" "37" "38" "39" "40"
#> 
#> $levels$Rater
#> [1] "1" "2" "3" "4" "5" "6" "7" "8"
#> 
#> $levels$Task
#> [1] "1" "2" "3" "4" "5" "6"
#> 
#> 
#> $rows
#> [1] 1917
#> 
#> $nesting
#> NULL
#> 
#> $child_counts
#> NULL
#> 
#> $balanced
#> [1] FALSE
#> 
#> $complete
#> [1] FALSE
#> 
#> $potential_cells
#> [1] 1920
#> 
#> $observed_fraction
#> [1] 0.9984375
#> 
#> $method
#> [1] "Multivariate MINQUE(0)"
#> 
#> $score_convention
#> [1] "Means over common random raters and tasks"
#> 
#> $calculation_version
#> [1] 2
#> 
attr(incomplete$variance, "summary")
#>         Source  Estimate SamplingVariance         SE  SEStatus
#> 1       Person 0.9461647      0.076504531 0.27659452 Available
#> 2        Rater 0.8072029      0.239431991 0.48931788 Available
#> 3         Task 1.6208397      0.921356363 0.95987310 Available
#> 4 Person:Rater 1.0126988      0.010217691 0.10108260 Available
#> 5  Person:Task 0.9266872      0.011308107 0.10633958 Available
#> 6   Rater:Task 1.1786624      0.078418884 0.28003372 Available
#> 7     Residual 1.0088781      0.001580505 0.03975556 Available
attr(incomplete$covariance, "summary")
#>         Source    Estimate SamplingVariance         SE  SEStatus
#> 1       Person -0.08472677     0.0389429831 0.19733977 Available
#> 2        Rater  0.27916523     0.1303902391 0.36109589 Available
#> 3         Task  0.27819934     0.1336031299 0.36551762 Available
#> 4 Person:Rater -0.11681953     0.0052861207 0.07270571 Available
#> 5  Person:Task -0.11212412     0.0053908081 0.07342212 Available
#> 6   Rater:Task  0.06643709     0.0278149937 0.16677828 Available
#> 7     Residual -0.01494854     0.0007523036 0.02742815 Available
attr(incomplete$covariance, "notes")
#> [1] "Sampling covariance of original ANOVA/MINQUE component estimates, conditional on the declared assignment under independent normal random effects. This is not measurement SEM, covariance between individual ratings or the SE of a separate REML estimator."
#> [2] "Gaussian product-moment correction gives unbiased identifiable sampling-covariance entries under the model. Taking square roots is not an unbiasedness guarantee for SEs and does not justify normal-reference confidence intervals."                        
#> [3] "Negative sampling variances are retained, with unavailable SEs. Unidentified entries are NA. An indefinite matrix is not repaired; inspect summary and diagnostics before propagating it to another target."                                                 
#> [4] "Scores and composite weights are prespecified; omission must preserve the model conditional on assignment. No robustness, informative-missingness correction or general confidence-interval coverage is claimed. Save the complete result with saveRDS()."   
# To repeat the full incomplete-data calculation:
# source(system.file("examples", "gstudy-sampling.R", package = "mfrmr"))
# incomplete <- gstudy_sampling_example()