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Principal Hessian direction

Li (1992) proposed the method called principal Hessian directions, or pHd. This method examines the matrix $\hat{M}$ given by

\begin{displaymath}
\hat{M} = \frac{1}{n}\sum_{i=1}^n w_i f_i z_iz_i'
\end{displaymath}

where $f_i$ is either equal to $y_i$ (method phdy) or $f_i$ is an ols residual (method phd or phdres), and $w_i$ is the weight for the $i$-th observation (recall again that we assume that $\sum w_i = n$, and if this is not satisfied the program rescales the weights to meet this condition). While all methods produce $\hat{M}$ matrices whose eigenvectors are consistent estimates of vectors in ${\mathcal S}(B)$, the residual methods are more suitable for tests of dimension. See Cook (1998, Chapter 12) for details. Output for phd is again similar to sir, except for the tests. Here is the output for the same setup as before, but for method phdres:
> i2 <- update(i1,method="phdres")
> summary(i2)

Call:
dr(formula = LBM ~ Ht + Wt + log(RCC) + WCC, method = "phdres")

Terms:
LBM ~ Ht + Wt + log(RCC) + WCC

Method:
phd, n = 202.

Eigenvectors:
             Dir1       Dir2      Dir3     Dir4
Ht        0.12764 -0.0003378  0.005550  0.02549
Wt       -0.02163  0.0326138 -0.007342 -0.01343
log(RCC) -0.74348  0.9816463  0.999930 -0.99909
WCC       0.65611 -0.1879008 -0.007408 -0.03157

              Dir1   Dir2   Dir3   Dir4
Eigenvalues 1.4303 1.1750 1.1244 0.3999
R^2(OLS|dr) 0.2781 0.9642 0.9642 1.0000

Asymp. Chi-square tests for dimension:
              Stat df Normal theory Indep. test General theory
0D vs >= 1D 35.015 10     0.0001241    0.005427        0.01811
1D vs >= 2D 20.248  6     0.0025012          NA        0.03200
2D vs >= 3D 10.281  3     0.0163211          NA        0.05530
3D vs >= 4D  1.155  1     0.2825955          NA        0.26625
The column of tests called ``normal theory" were proposed by Li (1992) and require that the predictors are normally distributed. These statistics are asymptotically distributed as Chi-square, with the degrees of freedom shown. When the method is phdres additional tests are provided. Since this method is based on residuals, it gives tests concerning the central subspace for the regression of the residuals on $X$ rather than the response on $X$. The subspace for this residual regression may be, but need not be, smaller than the subspace for the original regression. For example, the column marked ``Indep. test" is essentially a test of $d=0$ versus $d>0$ described by Cook (1998) for the residual regression. Should the significance level for this test be large, we might conclude that the residual regression subspace is of dimension zero. From this we have two possible conclusions: (1) the dimension of the response regression may be 1 if using the residuals removed a linear trend, or (2) the dimension may be 0 if the residuals did not remove a linear trend. Similarly, if the significance level for the independence test is small, then we can conclude that the dimension is at least 1. It could be one if the method is picking up a nonlinear trend in the OLS direction, but it will be 2 if the nonlinearity is in some other direction. The independence test and the final column, also from Cook (1998), use the same test statistic, but different distributions based on different assumptions. Significance levels are obtained by comparing the statistic to the distribution of a random linear combination of Chi-square statistics, each with one df. These statistics do not require normality of the predictors. The way the significance levels in this column are approximated used the method of Bentler and Xie (2000).
next up previous
Next: Quadratic phd Up: Methods available Previous: Sliced average variance estimation
Sandy Weisberg 2002-01-10