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Multi-collineartity, Variance Inflation

and Orthogonalization in Regression

The problem of too many variables

Stepwise regression

Collinearity happens to many inexperienced researchers. A common mistake is to put too many regressors into the model. As

what I explained in my example of "fifty ways to improve your grade, " inevitably many of those independent variables will be

too correlated. In addition, when there are too many variables in a regression model i.e. the number of parameters to be

estimated is larger than the number of observations, this model is said to be lack of degree of freedom and thus over-fitting.

The following cases are extreme, but you will get the idea. When there is one subject only, the regression line can be fitted in

any way (left figure). When there are two observations, the regression line is a perfect fit (right figure). When things are

perfect, they are indeed imperfect!

One common approach to select a subset of variables from a complex model is stepwise regression. A stepwise regression is a

procedure to examine the impact of each variable to the model step by step. The variable that cannot contribute much to the

variance explained would be thrown out. There are several versions of stepwise regression such as forward selection,

backward elimination, and stepwise. Many researchers employed these techniques to determine the order of predictors by

its magnitude of influence on the outcome variable (e.g. June, 1997; Leigh, 1996).

However, the above interpretation is valid if and

only if all predictors are independent (But if you

write a dissertation, it doesn't matter. Follow

what your committee advises). Collinear

regressors or regressors with some degree of

correlation would return inaccurate results.

Assume that there is a Y outcome variable and

four regressors X1-X4. In the left panel X1-X4are correlated (non-orthogonal). We cannot tell

which variable contributes the most of the

variance explained individually. If X1 enters the

model first, it seems to contribute the largest

amount of variance explained. X2 seems to be

less influential because its contribution to the

variance explained has been overlapped by the

first variable, and X3 and X4 are even worse.

collinearity: SAS tips by Dr. Alex Yu http://www.creative-wisdom.com/computer/sas/collinear_step

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Indeed, the more correlated the regressors are,

the more their ranked "importance" depends on

the selection order (Bring, 1996). However, we

can interpret the result of step regression as an

indication of the importance of independent

variables if all predictors are orthogonal. In the

right panel we have a "clean" model. The

individual contribution to the variance

explained by each variable to the model is

clearly seen. Thus, we can assert that X1 andX4 are more influential to the dependent

variable than X2 and X3.

Maximum R-square, RMSE, and Mallow's Cp

There are other better ways to perform variable selection such as Maximum R-square, Root Mean Square Error (RMSE), and

Mallow's Cp. Max. R-square is a method of variable selection by examining the best of n-models based upon the largest

variance explanied. The other two are opposite to max. R-square. RMSE is a measure of the lack of fit while Mallow's CP is

the total square errors, as opposed to the best fit by max. R-square. Thus, the higher the R-square is, the better the model is.The lower the RMSQ and Cp are, the better the model is.

For the clarity of illustration, I use only three regressors: X1, X2, X3. The principle illustrated here can be well-applied to the

situation of many regressors. The following output is based on a hypothetical dataset:

Variable R-square RMSE Cp

One-variable models

X3 0.31 2.27 9.40

X2 0.27 2.35 10.90

X1 0.00 2.75 19.41

Two-variable models

X2X3 0.60 1.81 2.70

X1X3 0.33 2.34 11.20

X1X2 0.32 2.35 11.34

Full model

X1X2X3 0.62 1.84 4.00

At first, each regressor enters the model one by one. In all one-variable models, the best variable is X3 according to the max.

R-square criterion (R2=.31). (Now we temporarily ignore RMSE and Cp). Then, all combinations of two-variable models arecomputed. This time the best two predictors are X2 and X3 (R

2=.60). Last, all three variables are used for a full model

(R2=.62). From the one-variable model to the two-variable model, the variance explained gains a substantive improvement

(.60 - .31 = .29). However, from the two-variable to the full model, the gain is trivial (.62 - .60 = .02).


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If you cannot follow the above explanation, this

figure may help you. The x-axis represents the

number of variables while the y-axis represents

the R-square. It clearly indicates a sharp jump

from one to two. But the curve turns into flat

from two to three (see the red arrow).

Now, let's examine RMSE and Cp. Interestingly

enough, in terms of both RMSE and Cp, the full

model is worse than the two-variable model.

The RMSE of the best two-variable is 1.81 but

that of the full model is 1.83 (see the red arrow

in the right panel)! The Cp of the best two is

2.70 whereas that of the full model is 4.00 (see

the red arrow in the following figure)!

Nevertheless, although the approaches of

maximum R-square, Root Mean Square Error,

and Mallow's Cp are different, the conclusion is

the same: One is too few and three are too

many. To perform a variable selection in SAS,

the syntax is "PROC REG; MODEL Y=X1-X3

/SELECTION=MAXR". To plot Max. R-square,

RMSQ, and Cp together, use NCSS (NCSS

Statistical Software, 1999).

Stepwise regression based on AICc

Although the result of stepwise regression depends on the order of entering predictors, JMP (SAS Institute, 2010) allows the

user to select or deselect variables in any order. The process is so interactive that the analyst can easily determine whether

certain variables should be kept or dropped. In addition to Mallows' CP, JMP shows Akaike's information criterion correction

(AICc) to indicate the balance between fitness and simplicity of the model.

The original Akaike's information criterion (AIC) without correction, developed by Hirotsugu Akaike (1973), is in alignment to

Ockhams razor: Given all things being equal, the simplest model tends to be the best one; and simplicity is a function of the

number of adjustable parameters. Thus, a smaller AIC suggests a "better" model. Specifically, AIC is a fitness index for trading

off the complexity of a model against how well the model fits the data. The general form of AIC is: AIC = 2k 2lnL where k is

the number of parameters and L is the likelihood function of the estimated parameters. Increasing the number of freeparameters to be estimated improves the model fitness, however, the model might be unnecessarily complex. To reach a

balance between fitness and parsimony, AIC not only rewards goodness of fit, but also includes a penalty that is an increasing

function of the number of estimated parameters. This penalty discourages over-fitting and complexity. Hence, the best

model is the one with the lowest AIC value. Since AIC attempts to find the model that best explains the data with a minimum

of free parameters, it is considered an approach favoring simplicity.

AICc is a further step beyond AIC in the sense that AICs imposes a greater penalty for additional parameters. The formula of

AICs is:

AICc = AIC + (2K(K+1)/(n-k-1))

where n = sample size and k = the number of parameters to be estimated.


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Burnham and Anderson (2002) recommend replacing AIC with AICc, especially when the sample size is small and the number

of parameters is large. Actually, AICc converges to AIC as the sample size is getting larger and larger. Hence, AICc should be

used regardless of sample size and the number of parameters.

Bayesian information criterion (BIC) is similar to AIC, but its penalty is heavier than that of AIC. However, some authors

believe that AIC and AICc are superior to BIC for a number of reasons. First, AIC and AICc is based on the principle of

information gain. Second, the Bayesian approach requires a prior input but usually it is debatable. Third, AIC is asymptotically

optimal in model selection in terms of the least squared mean error, but BIC is not asymptotically optimal (Burnham &

Anderson, 2004; Yang, 2005).

JMP provides the users with the options of AICc and BIC for model refinement. To start running stepwise regression withAICc or BIC, use Fit models and then choose Stepwise from Personality. These short movie clips show the first and the

second steps of constructing an optimal regression model with AICc (Special thanks to Michelle Miller for her help in

recording the movie clips).

Partial least squares regression

There are other ways to reduce the number of variables such as factor analysis, principal component analysis and partial least

squares. The philosophy behind these methods is very different from variable selection methods. In the former group of


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procedures "redundant" variables are not excluded. Rather they are retained and combined to form latent factors. It is believed

that a construct should be an "open concept" that is triangulated by multiple indicators instead of a single measure (Salvucci,

Walter, Conley, Fink, & Saba, 1997). In this sense, redundancy enhances reliability and yields a better model.

However, factor analysis and principal component analysis do not have the distinction between dependent and independent

variables and thus may not be applicable to research with the purpose of regression analysis. One way to reduce the number of

variables in the context of regression is to employ the partial least squares (PLS) procedure. PLS is a method for constructing

predictive models when the variables are too many and high collinear (Tobias, 1999). Besides collinearity, PLS is also robust

against other data structural problems such as skew distributions and omission of regressors (Cassel, Westlund, & Hackl,

1999). It is important to note that in PLS the emphasis is on prediction rather than explaining the underlying relationships

between the variables.

Like principal component analysis, the basic idea of PLS is to extract several latent factors and responses from a large number

of observed variables. Therefore, the acronym PLS is also taken to mean projection to latent structure. The slide show

below illustrates the idea of factor extraction. Please press the next button to start the slide show (this Macromedia flash

slideshow is made by Gregory Van Eekhout):

The following is an example of the SAS code for PLS: PROC PLS; MODEL; y1-y5 = x1-x100; Note that unlike an ordinary

least squares regression, PLS can accept multiple dependent variables. The output shows the percent variation accounted for

each extracted latent variable:

Number oflatent

variables

Model effectsCurrent

Model effectsTotal

DVCurrent

DVTotal

1 39.35 39.35 28.70 28.70

2 29.94 69.29 25.58 54.28

3 7.93 77.22 21.86 76.14

4 6.40 83.62 6.45 82.59

5 2.07 85.69 16.96 99.54

Update: 2012

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