Linear-Model Deletion Diagnostics for Gene Expression (or similar) Data Structures

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Description

This is an extension of standard linear-model diagnostics for use with gene-expression datasets, in which the same model was run simultaneously on each row of a response matrix.

Usage

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 dfbetasPerGene(lmobj)

 CooksDPerGene(lmobj)

 dffitsPerGene(lmobj)

 Leverage(lmobj)

Arguments

lmobj

An object produced by lmPerGene.

Details

Deletion diagnostics gauge the influence of each observation upon model fit, by calculating values after removal of the observation and comparing to the complete-data version.

DFFITS_i measures the distance on the response scale, between fitted values with and without observation y\_i, at point i. The distance is normalized by the regression standard error and the point's leverage (see below).

Cook's D_i is the square of the distance, in parameter space, between parameter estimates witn and without observation y\_i, normalized and rescaled by standard errors and by a factor depending upon leverage.

DFBETAS_{i,j} breaks the square root of Cook's D into its Euclidean components for each parameter j - but uses a somewhat different scaling function from Cook's D.

The leverage is the diagonal of the "hat matrix" X'(X'X)^{-1}X'. This measure provides the relative weight of observation y\_i in the fitted value y-hat\_i. Typically observations with extreme X values (or belonging to smaller groups if model variables are categorical) will have high leverage.

All these functions exist for standard regression, see influence.measures.

The functions described here are extensions for the case in which the response is a matrix, and the same linear model is run on each row separately.

For more details, see the references below.

All functions are implemented in matrix form, which means they run quite fast.

Value

dfbetasPerGene A G x n x p array, where G, n are the number of rows and columns in the input's expression matrix, respectively, and p the number of parameters in the linear model (including intercept)

CooksDPerGene A G x n matrix.

dffitsPerGene A G x n matrix.

Leverage A vector of length n, corresponding to the diagonal of the "hat matrix".

Note

The commonly-cited reference alert thresholds for diagnostic measures such as Cook's $D$ and DFBETAS, found in older references, appear to be out of date. See LaMotte (1999) and Jensen (2001) for a more recent discussion. Our suggested practice is to inspect any samples or values that are visibly separate from the pack.

Author(s)

Robert Gentleman, Assaf Oron

References

Belsley, D. A., Kuh, E. and Welsch, R. E. (1980) Regression Diagnostics. New York: Wiley.

Cook, R. D. and Weisberg, S. (1982) Residuals and Influence in Regression. London: Chapman and Hall.

Williams, D. A. (1987) Generalized linear model diagnostics using the deviance and single case deletions. Applied Statistics *36*, 181-191.

Fox, J. (1997) Applied Regression, Linear Models, and Related Methods. Sage.

LaMotte, L. R. (1999) Collapsibility hypotheses and diagnostic bounds in regression analysis. Metrika 50, 109-119.

Jensen, D.R. (2001) Properties of selected subset diagnostics in regression. Statistics and Probability Letters 51, 377-388.

See Also

influence.measures for the analogous simple regression diagnostic functions

Examples

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data(sample.ExpressionSet)
layout(1)
lm1 = lmPerGene( sample.ExpressionSet,~score+type)
CD = CooksDPerGene(lm1)
### How does the distribution of mean Cook's distances across samples look?

boxplot(log2(CD) ~ col(CD),names=colnames(CD),ylab="Log Cook's
Distance",xlab="Sample")
### There are a few gross individual-observation outliers (which is why we plot on the log
### scale), but otherwise no single sample pops out as problematic. Here's
### one commonly-used alert level for problems:
lines(c(-5,30),rep(log2(2/sqrt(26)),2),col=2)


DFB = dfbetasPerGene(lm1)

### Looking for simultaneous two-effect outliers - 500 genes times 26
### samples makes 13000 data points on this plot

plot(DFB[,,2],DFB[,,3],main="DFBETAS for Score and Type (all genes)",xlab="Score Effect
Offset (normalized units)",ylab="Type Effect Offset (normalized units)",pch='+',cex=.5)
lines(c(-100,100),rep(0,2),col=2)
lines(rep(0,2),c(-100,100),col=2)

DFF = dffitsPerGene(lm1)
summary(apply(DFF,2,mean))

Lev = Leverage(lm1)
table(Lev)
### should have only two unique values because this is a dichotomous one-factor model