svydiags (version 0.6)

svycollinear: Condition indexes and variance decompositions in linear models fitted with complex survey data

Description

Compute condition indexes and variance decompositions for diagnosing collinearity in fixed effects, linear regression models fitted with data collected from one- and two-stage complex survey designs.

Usage

svycollinear(mod, intcpt=TRUE, w, Vcov, sc=TRUE, svyglm.obj, rnd=3, fuzz=0.3)

Value

\(p \times (p+1)\) data frame, \(\mathbf{\Pi}\). The first column gives the condition indexes of \(\widetilde{\mathbf{X}}\). Values of 10 or more are usually considered to potentially signal collinearity of two or more columns of \(\widetilde{\mathbf{X}}\). The remaining columns give the proportions (within columns) of variance of each estimated regression coefficient associated with a singular value decomposition into p terms. Columns \(2, \ldots, p+1\) will each approximately sum to 1. Note that some `proportions' can be negative due to the nature of the variance decomposition. If two proportions in a given row of \(\mathbf{\Pi}\) are relatively large and its associated condition index in that row in the first column of \(\mathbf{\Pi}\) is also large, then near dependencies between the covariates associated with those elements are influencing the regression coefficient estimates.

Arguments

mod

Either (i) an \(n \times p\) matrix of real-valued covariates used in fitting a linear regression; \(n\) = number of observations, \(p\) = number of covariates in model, excluding the intercept; the matrix mod should not contain columns for the strata and cluster identifiers (unless those variables are part of the model). No missing values are allowed. The matrix can be created with model.matrix; see the example below. The model list component of an svyglm object cannot be used as it does not have factor variables coded as zero-ones.

Or, (ii) a model object produced by svyglm in the survey package.

intcpt

TRUE if the model contains an intercept; FALSE if not.

w

\(n\)-vector of survey weights used in fitting the model. No missing values are allowed.

Vcov

Variance-covariance matrix of the estimated slopes in the regression model; component cov.unscaled in an svyglm object. This matrix can also be extracted with vcov(m) where m is an svyglm object.

sc

TRUE if the columns of the weighted model matrix \(\tilde{\mathbf{X}}\) (defined in Details) should be scaled for computing condition indexes; FALSE if not. If TRUE, each column of \(\tilde{\mathbf{X}}\) is divided by its Euclidean norm, \(\sqrt{\tilde{\mathbf{x}}^T \tilde{\mathbf{x}}}\).

svyglm.obj

Is mod a svyglm.obj object? TRUE or FALSE. If mod is a matrix, set svyglm.obj to FALSE.

rnd

Round the output to rnd decimal places.

fuzz

Replace any variance decomposition proportions that are less than fuzz by '.' in the output.

Author

Richard Valliant

Details

svycollinear computes condition indexes and variance decomposition proportions to use for diagnosing collinearity in a linear model fitted from complex survey data as discussed in Liao and Valliant (2012). All measures are based on \(\widetilde{\mathbf{X}} = \mathbf{W}^{1/2}\mathbf{X}\) where \(\mathbf{W}\) is the diagonal matrix of survey weights and X is the \(n \times p\) matrix of covariates. In a full-rank model with p covariates, there are p condition indexes, defined as the ratio of the maximum eigenvalue of \(\widetilde{\mathbf{X}}\) to its minimum eigenvalue. Before computing condition indexes, as recommended by Belsley (1991), the columns are normalized by their individual Euclidean norms, \(\sqrt{\tilde{\mathbf{x}}^T\tilde{\mathbf{x}}}\), so that each column has unit length. The columns are not centered around their means because that can obscure near-dependencies between the intercept and other covariates (Belsley 1984).

Variance decompositions are for the variance of each estimated regression coefficient and are based on a singular value decomposition of the variance formula. Proportions of the model variance, \(Var_M(\hat{\mathbf{\beta}}_k)\), associated with each column of \(\widetilde{\mathbf{X}}\) are displayed in an output matrix described below.

References

Belsley, D.A., Kuh, E. and Welsch, R.E. (1980). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley-Interscience.

Belsley, D.A. (1984). Demeaning conditioning diagnostics through centering. The American Statistician, 38(2), 73-77.

Belsley, D.A. (1991). Conditioning Diagnostics, Collinearity, and Weak Data in Regression. New York: John Wiley & Sons, Inc.

Liao, D, and Valliant, R. (2012). Condition indexes and variance decompositions for diagnosing collinearity in linear model analysis of survey data. Survey Methodology, 38, 189-202.

Lumley, T. (2010). Complex Surveys. New York: John Wiley & Sons.

Lumley, T. (2023). survey: analysis of complex survey samples. R package version 4.2.

See Also

svyvif

Examples

require(survey)
    # example from svyglm help page
data(api)
dstrat <- svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)
m1 <- svyglm(api00~ell+meals+mobility, design=dstrat)
    # send model object from svyglm
CI.out <- svycollinear(mod = m1, w=apistrat$pw, Vcov=vcov(m1), sc=TRUE, svyglm.obj=TRUE,
        rnd=3, fuzz= 0.3)

    # use model.matrix to create matrix of covariates in model
data(nhanes2007)
newPSU <- paste(nhanes2007$SDMVSTRA, nhanes2007$SDMVPSU, sep=".")
nhanes.dsgn <- svydesign(ids = ~newPSU,
                         strata = NULL,
                         weights = ~WTDRD1, data=nhanes2007)
m1 <- svyglm(BMXWT ~ RIDAGEYR + as.factor(RIDRETH1) + DR1TKCAL +
    DR1TTFAT + DR1TMFAT + DR1TSFAT + DR1TPFAT, design=nhanes.dsgn)
X <- model.matrix(~ RIDAGEYR + as.factor(RIDRETH1) + DR1TKCAL + DR1TTFAT + DR1TMFAT
            + DR1TSFAT + DR1TPFAT,
        data = data.frame(nhanes2007))
CI.out <- svycollinear(mod = X, w=nhanes2007$WTDRD1, Vcov=vcov(m1), sc=TRUE, svyglm.obj=FALSE,
        rnd=2, fuzz=0.3)