Approximating the Distribution of the Sample R 2 in Best Subset Regressions

Diehr, George; Hoflin, Donald R.

doi:10.2307/1267954

Cited by 12 publications

(29 citation statements)

References 7 publications

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“…These problems have been the subject of researches by Wilkinson and Dallal (1981) and Westfall et al (1998) for problem 1 and by Diehr and Hoflin (1974), Rencher and Pun (1980), Copas andLong (1991), andFreedman et al (1992) for problem 2. These problems have been the subject of researches by Wilkinson and Dallal (1981) and Westfall et al (1998) for problem 1 and by Diehr and Hoflin (1974), Rencher and Pun (1980), Copas andLong (1991), andFreedman et al (1992) for problem 2.…”

Section: Forward Selection: a Huge Type I Errormentioning

confidence: 99%

“…This problem was amplified by the introduction of spatial analyses using principal coordinates of neighbor matrices (PCNM; Legendre 2002, Borcard et al 2004) and Moran's eigenvector maps (MEM; Dray et al 2006), which, by construction, generate a large number of variables modeling the spatial relationships among the sampling sites. However, forward selection is known to overestimate the amount of explained variance, which is measured by the coefficient of multiple determination (R 2 ; Diehr andHoflin 1974, Rencher andPun 1980). An automatic selection procedure is used in many cases to objectively select a subset of explanatory variables.…”

Section: Introductionmentioning

confidence: 99%

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Forward Selection of Explanatory Variables

Blanchet¹,

Legendre²,

Borcard³

2008

Ecology

1,821

1,273

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This paper proposes a new way of using forward selection of explanatory variables in regression or canonical redundancy analysis. The classical forward selection method presents two problems: a highly inflated Type I error and an overestimation of the amount of explained variance. Correcting these problems will greatly improve the performance of this very useful method in ecological modeling. To prevent the first problem, we propose a two-step procedure. First, a global test using all explanatory variables is carried out. If, and only if, the global test is significant, one can proceed with forward selection. To prevent overestimation of the explained variance, the forward selection has to be carried out with two stopping criteria: (1) the usual alpha significance level and (2) the adjusted coefficient of multiple determination (Ra(2)) calculated using all explanatory variables. When forward selection identifies a variable that brings one or the other criterion over the fixed threshold, that variable is rejected, and the procedure is stopped. This improved method is validated by simulations involving univariate and multivariate response data. An ecological example is presented using data from the Bryce Canyon National Park, Utah, U.S.A.

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Section: Forward Selection: a Huge Type I Errormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Forward Selection of Explanatory Variables

Blanchet¹,

Legendre²,

Borcard³

2008

Ecology

1,821

1,273

View full text Add to dashboard Cite

show abstract

“…The third involves the suitability of forward selection to applied problems, particularly since optimal algorithms (all possible subsets) and stepwise procedures are widely used. Diehr and Hoflin (1974) and Rencher and Pun (1980) provide approximation formulas, based on their simultations, for the null distribution of the sample R2 in all possible subsets and stepwise regression, respectively. We have not done so for our simulation even though the distributions are similar to those discussed in the earlier papers.…”

Section: And Gerald E Dallalmentioning

confidence: 99%

“…Because of its complexity the family of R2 distributions for various subset selection methods has been examined by Monte Carlo simulation in a number of studies (Diehr and Hoflin 1974;Wilkinson 1979;Rencher and Pun 1980). These studies examined the conditional distribution of R2 given the number of candidate predictors, the number of predictors selected, and a particular selection rule.…”

Section: Introductionmentioning

confidence: 99%

Tests of Significance in Forward Selection Regression With anF-to-Enter Stopping Rule

Wilkinson

Dallal

1981

Technometrics

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A Monte Carlo simulation is used to estimate the upper percentage points of the null distribution of the sample squared multiple correlation coefftcient (R') when the number of predictors selected is determined by a stopping rule. In the study, the sample size n and the number of candidate predictors m satisfy 2 2 m 2 20 and 10 I n -m -1 I 200, while the F threshold ranges from two to four. Tables of the upper five percent and upper one percent sample R* values are presented and an example is given to illustrate the use of the tables.

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“…Hockings (1976) proposed that biased estimators may offer improvements over the usual least-squares estimator when selection has occurred. Finally, Diehr and Hoflin ( 1974) obtained approximate percentile points to the unknown distribution of the sample squared multiple correlation in best subset regression.…”

mentioning

confidence: 99%

Estimation of the Squared Cross-Validity Coefficient in the Context of Best Subset Regression

Kennedy

1988

Applied Psychological Measurement

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A monte carlo study was conducted to examine the performance of several strategies for estimating the squared cross-validity coefficient of a sample regression equation in the context of best subset regression. Data were simulated for populations and experimental designs likely to be encountered in practice. The results indicated that a formula presented by Stein (1960) could be expected to yield estimates as good as or better than cross-validation, or several other formula estimators, for the populations considered. Further, the results suggest that sample size may play a much greater role in validity estimation in subset selection than is true in situations where selection has not occurred.

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Approximating the Distribution of the Sample R 2 in Best Subset Regressions

Cited by 12 publications

References 7 publications

Forward Selection of Explanatory Variables

Forward Selection of Explanatory Variables

Tests of Significance in Forward Selection Regression With anF-to-Enter Stopping Rule

Estimation of the Squared Cross-Validity Coefficient in the Context of Best Subset Regression

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