The work on bootstrap confidence intervals aimed to form confidence intervals for just one application — contributions (or percentage contributions) of individual variables to a quadratic form. The new methods extended the range of pivotal quantities that could be used as pivots and the methods performed markedly bet- ter than alternatives. Clearly the performance of the methods should be explored in other applications. In particular the bootstrap methods could be used to con- struct interval estimates for the contributions of individual regressor to a multiple regression.
In the work on evaluating the contributions of individual regressors to a mul- tiple regression, the primary new feature was to use the cross-products of the response and predictors in forming orthogonal components. A subsidiary idea was to use regression coefficient as weights when forming the components. This idea was used to derive NM3 from NM1 and it could be applied to modify any of the RW and NM2 measures. Also the weighting scheme could be generalized, i.e., could use the weight (| bβj|)α to Xj. Setting α = 0 gives no weight, and importance
of weighting would increase with the increase of α. The generalized weighting scheme could also be applied to NM3. The use of different weighting scheme with
various measures is a topic that needs further work.
In the work on collinearity identification, the use of simulated data enabled methods to be examined in conditions where the structure underlying the data was known. This work suggested that the transformation matrices of the cos-max and cos-square transformations can provide insight into the collinearity structure of a dataset, even when the dataset has multiple, overlapping collinearities. This work was limited and more work with simulated data needs to be done.
The methods used in this thesis are generally designed for the case where the number of observations is greater than the number of variables. Due to the development of data collection technology, in recent years data sets often have a comparatively small number of observations and a large number of variables. Data sets with a large number of variables compared to the number of observations are called high-dimensional data. Examples of such data sets include microarray data, Netflix movie rating data. Further research needs to modify the transformations developed in this thesis so that they are applicable to high-dimensional data.
More generally, the work in this thesis illustrates that transformations to or- thogonality have varied applications. There are likely to be numerous other appli- cations in which the transformations would prove useful, so research is needed to find some of these applications. Also, the cos-max and cos-square transformations have different properties. For example, one has the rotation invariance property and the other has the duplicate invariance property. Further work is also needed that compares the two transformations critically.
Appendix A
We illustrate the method of taking a sample from all possible p! orderings by using the data from Vandaele(1976). The dataset has 14 measurements and were originally collected from the F BI’s Uniform Crime Report and other government sources to identify the variables that are responsible for crime rates in 1960 based on the data form 47 states of the USA.
Among the 14 variables we have used 13 variables for our study (we have not used the indicator variable). Crime rate is the response variable and the other 12 variables are considered as regressors. We have calculated the relative weights (RW) measure ofJohnson(2000), the general dominance (GD) measure of Budescu (1993)/LMG measure ofLindeman et al. (1980) for these 12 regressors. The regressors have 12! possible orderings of variables, from which we have taken a random sample of 500 orderings. We have calculated the sequential R2 values of each variable for each ordering. Finally, the average of the sequential R2 values
for each variable from these 500 orderings were calculated to approximate the true value of LMG/GD.
Table 1 gives the values of the RW, GD/LMG measures obtained from the U.S. crime data. The last two rows of this table are the average and standard deviation, respectively, of 1000 approximate LMG contributions. The average of
Table 1: LMG analysis for US crime data X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 RW 0.035 0.060 0.183 0.170 0.019 0.035 0.046 0.018 0.020 0.040 0.082 0.059 GD 0.041 0.056 0.189 0.168 0.015 0.033 0.035 0.020 0.016 0.034 0.076 0.083 Mean 0.041 0.056 0.189 0.168 0.015 0.033 0.035 0.020 0.016 0.034 0.076 0.083 SD 0.001 0.002 0.008 0.008 0.001 0.002 0.002 0.002 0.001 0.002 0.005 0.002
RW: Relative importance byJohnson(2000) GD: General dominance
Mean and SD are the mean and standard deviation of sample contributions R2= 0.767
the approximate LMG contributions are close to the true GD/LMG contributions with small standard deviations. This illustrates that GD/LMG contributions can be approximated by taking a sample of the orderings of regressors and this will reduce the number of model estimations and consequently reduce the time required for computation.
Approximating GD by taking a sample of subset models from the dominance analysis formulation is not feasible. General dominance is the average of condi- tional dominance at all levels from 0 to p − 1 so, rather than taking a simple random subset of models from all possible models, we need to take samples sep- arately from each level. Level 0 has only one row, so there is no need to take a sample. Also level p − 1 has p rows, but each column has only one element. So again if we take a sample from that level, conditional dominance from that level will be missing for some variables and will affect the general dominance. For level p − 2, we have to take a large number of samples to get a conditional dominance value for all columns. We also need to take a large number of rows for other higher levels. Also, since each element of a row is the difference between the R2 values
of two models, each row has some connection with the previous level. As a result, to approximate the true output from the sample we cannot reduce the number of
models that must be estimated and consequently computation time is not reduced much. If we take a small sample of rows the sample results overestimate the true output.