# Multi-collineartity, Variance Inflation and Orthogonalization in Regression

## Polynomial regression

 Not all regression models are linear. In some situations the relationship among variables may be non-linear. A classic example is stress-performance relationship. Initially pressure could lead to better efficiency. But if the stress is too intense, performance will decrease due to physical or mental break down. Another classic example is the relationship between performance and ability. Contrary to popular belief, increasing ability in a discipline or a specific task does not lead to a linear increase in performance. Many teachers are frustrated with the phenomenon that many low achievers do not show improvement in test scores despite tremendous efforts contributed by both teachers and students. It is because low-ability learners do not have the required skills to perform even the basic function. Once they master the basic skills, their performance gain would be proportional to their ability gain. The curve hits an inflection point and turn virtually flat again when their ability are matured. For example, the score difference in a writing test between a master and a Ph.D. may be minimal. The technical term for this S-shaped curve is ogive.
In curvilinear cases, polynomial regressions, which involve quadratic, cubic, or quartic terms, should be implemented. The equations of polynomial regressions are listed in the following:
 Quadratic: Y = A + B1X + B2X2 Cubic: Y = A + B1X + B2X2 + B3X3 Quartic: Y = A + B1X + B2X2 + B3X3 + B4X4
Which term should be used depends on the number of "turns" (inflection points) on the non-linear curve. In case 1 there is only one turn on the curve and a quadratic term should be used. In case 2 there are two inflection points and thus a cubic term should be applied.

## Gram-Schmidt method

Can you smell the smoke of multi-collinearity? Are the quadratic, cubic, quartic and the original variables highly correlated? Yes, of course. The first three are derived from raising power of the original variable. To avoid the problem of multi-collinearity, again you should "orthogonalize" the vectors. Again, centered-score regression can be used for partial orthogonalization (Neter, Wasserman, & Kutner, 1990). Nonetheless, the Gram-Schmidt method, which is a full orthogonalization approach, is considered a better approach. The explanation of Gram-Schmidt method is beyond the scope of this tutorial. Please consult the book by Saville and Wood (1991) for detail. Nevertheless, the concept of orthogonalization remains the same here.

One of the easiest way to perform Gram-Schmidt orthogonalization is using Mathemetica. It takes only two lines of command syntax to transform the vectors (see the following panel). The first step is to load the Orthogonalization function. It is important to note that the symbols before and after the phrase "Orthogonalization" are an accent mark(`) (the key is located at the upper left corner of the keyboard), not quotation mark ('). The second step is to output the original vectors to new orthogonal vectors.

## PROC ORTHOREG

Another way to orthogonalize the vectors in the regression is to employ PROC ORTHOREG in SAS. This procedure is specifically developed for ill-conditioned data and polynomial model. The orthogonalization method here is Gentleman-Givens transformations. The following example is a labor statistics dataset in the SAS manual. Price level, GNP, unemployment rate, size of armed forces, population, and year are used to predict employment rate. The raw variables are strongly correlated and it is believed that the regression model is a quadratic model. Because collinearity became a threat to the stability of the model, PROC ORTHOREG instead of PROC REG or PROC GLM is used in the estimation.

```	proc orthoreg;  model Employment =
Prices   Prices*Prices
GNP      GNP*GNP
Jobless  Jobless*Jobless
Military Military*Military
PopSize  PopSize*PopSize
Year     Year*Year;
```