DISTRIBUTION FUNCTIONS, MEAN SQUARED ERRORS, AND CONFIDENCE LIMITS FOR RIDGE REGRESSION ESTIMATORS

A large number of ridge regression estimators have been proposed and used with little knowledge of their true distributions. Because of this lack of knowledge, these estimators cannot be used to test hypotheses or to form confidence intervals.^ This paper presents a basic technique for deriving the exact distribution functions for a class of generalized ridge estimators. The technique is applied to five prominent generalized ridge estimators. Graphs of the resulting distribution functions are presented. The actual behavior of these estimators is found to be considerably different than the behavior which is generally assumed for ridge estimators.^ This paper also uses the derived distributions to examine the mean squared error properties of the estimators. A technique for developing confidence intervals based on the generalized ridge estimators is also presented. ^

THE IMPROVEMENT REGION IN RIDGE REGRESSION

One of the difficulties in the practical application of ridge regression is that, for a given data set, it is unknown whether a selected ridge estimator has smaller squared error than the least squares estimator. The concept of the improvement region is defined, and a technique is developed which obtains approximate confidence intervals for the value of ridge k which produces the maximum reduction in mean squared error. Two simulation experiments were conducted to investigate how accurate these approximate confidence intervals might be. ^

Isolation and characterization of cDNAs encoding novel homeoproteins from chondrocytes

Bone morphogenesis is a complex biological process. The multistep process of chondrogenesis is the most important aspect of endochondral bone formation. To study the mechanisms which control this multistep pathway of chondrogenesis during embryonic development, I started by isolating cDNAs encoding novel transcriptional factors from chondrocytes. Several such cDNAs encoding putative homeoproteins were identified from a rat chondrosarcoma cDNA preparation. I have been concentrating on characterizing two of these cDNAs. The deduced amino acid sequence of the first homeoprotein, Cart-1, contains a prd-type homeodomain. Northern hybridization and RNase protection analysis revealed that Cart-1 RNAs were present at high levels in a well differentiated rat chondrosarcoma tumor and in a cell line derived from this tumor. Cart-1 transcripts were also detected in primary chondrocytes, but not in numerous other cell types except very low levels in testis. In situ hybridization of rat embryos at different stages of development revealed relatively high levels of Cart-1 RNAs in prechondrocytic mesenchymal cells and in early chondrocytes of cartilage primordia. It is speculated that Cart-1 might play an important role in chondrogenesis. The second putative homeoprotein, rDlx, contains a Distal-less-like homeodomain. rDlx RNAs were also present at high levels in the rat chondrosarcoma tumor and in the cell line derived from this tumor. In situ hybridization of rat embryos revealed high levels of rDlx transcripts in the developing cartilages and perichondria of mature cartilages. rDlx transcripts were also detected in a number of nonchondrogenic tissues such as forebrain, otic vesicles, olfactory epithelia, apical ectodermal ridge (AER) of limb buds, the presumptive Auerbach ganglia of gastrointestinal tract. The unique expression pattern of rDlx suggests that it might play important roles in chondrogenesis and other aspects of embryogenesis. ^