Paquete Gameobject

¯. Furthermore the variance of ˆy^∗ from the full model is not less than the variance of ˆy from the subset model. In terms of mean square error

V (ˆy^∗) ≥ M SE(ˆy)

¯^r is positive semidefinite.

4.3 Criteria for Evaluating Subset Regression Mod-els

Two key aspects of the variable selection problem are generating the subset models and deciding if one subset is better than another. In this section we discuss criteria for evaluating and comparing subset regression models.

4.3.1 Coefficient of Multiple Determination

A measure of the adequacy of a regression model that has been widely used is the coefficient of multiple determination, R². Let R²_p denote the coefficient of multiple determination for a subset regression model with p terms, that is, p − 1 regressors and an intercept term β₀. Computationally

R²_p = SS_R(p)

S_yy = 1 − SS_E(p)

S_yy (4.11)

where SS_R(p) and SS_E(p) denote the regression sum of squares and the residual sum of squares, respectively, for a p-term subset model. There are

 K

p − 1



values of R²_p for each value of p, one for each possible subset model of size p. Now R²_p increases as p increases and is a maximum when p = K + 1. Therefore the analyst uses this criterion by adding regressors to the model up to the point where an additional variable is not useful in that it provides only a small increase in R²_p. The general approach is illustrated in Figure 4.1, which represents a hypothetical plot of the maximum value of R²_p for each subset of size p against p. Typically one examines a display such as this and then specifies the number of regressors for the final model as the point at which the ”knee”

in the curve becomes apparent.

Since we cannot find an ”optimum” value of R² for subset regression model, we must look for a ”satisfactory” value. Aitkin [1974] has proposed one solution to this problem by providing a test by which all subset regression models that have an R² not significantly different from the R² for the full model can be identified. let

R²₀ = 1 − (1 − R²_K+1)(1 + d_a,n,K) (4.12)

Figure 4.1: Plot of R²_p against p

where

d_a,n,K = KF_{a,n,n−K−1} n − K − 1

and R²_K+1 is the value of R² for the full model. Aitkin calls any subset of regressor variables producing an R² greater than R²₀ an R²-adequate (α) subset.

Generally it is not straightforward to use R² as an criterion for choosing the number of regressor to include in the model. However, for a fixed number of variables p can be used to compare the

 K

p − 1



subset models so generated. Models having large values of R²_p are preferred.

4.3.2 Adjusted R

To avoid difficulties of interpreting R², some analysts prefer to use adjusted R² statistic, defined for a P -term equation as

R¯²_p = 1 − n − 1 n − p



(1 − R²_p) (4.13)

The ¯R²_p does not necessarily increase as additional regressors are introduced into the model. Infact Edward[1969], Haitovski[1969], and Seber [1977] showed that if s regressors are added to the model, ¯R²_p+s will exceed ¯R²_p iff the partial F -statistic for testing the significance of s additional regressors exceeds 1. Therefore optimum subset model can be chosen with maximum ¯R²_p.

4.3.3 Residual Mean Square

The residual mean square for a subset regression model with p variables, M S_E(p) = SS_E(p)

n − p (4.14)

can be used as a model evaluation criterion. The general behavior of M SE(p) as p increases as in Figure 4.2. Because SS_E(p) always decreases as p increases, M S_E(p)

Figure 4.2: Plot of M S_E(p) against p

initially decreases, then stabilizes, and eventually may increases. The eventual increase in M SE(p) occurs when the reduction in SSE(p) from adding a regressor to the model is not sufficient to compensate for the loss of one degree of freedom in the denominator of (4.14). That is, adding a regressor to a p-term model will cause M S_E(p + 1) to be greater than M SE(p). Advocates of the M SE(p) criterion will plot M SE(p) against and base the choice of p on

1. the minimum M S_E(p),

2. the value of p such that M S_E(p) is approximately equal to M S_E for the full model, or

3. a value of p near the point where the smallest M S_E(p) turns upward.

The subset regression model that minimizes M S_E(p) will also maximize ¯R²_p. To see this, note that

R¯²_p = 1 − n − 1

n − p(1 − R_p²)

= 1 − n − 1 n − p

SS_E(p) S_yy

= 1 − n − 1 S_yy

SSE(p) n − p

= 1 − n − 1

S_yy M SE(p)

Thus the criteria minimum M S_E(p) and maximum ¯R²_p are equivalent.

4.3.4 Mallows’ C

-Statistics

Mallows [1964, 1966, 1973] has proposed a criterion that is related to the mean square error of a fitted value, that is,

E [ˆyi− E(yi)]² = [E(yi) − E(ˆyi)]²+ V (ˆyi) (4.15) where E(y_i) and E(ˆy_i) are the expected responses from the true regression model and p-term subset model, respectively. Thus E(y_i) − E(ˆy_i) is the bias at the i-th data point.

Consequently the two terms on the right-hand side of (4.15) are the squared bias and variance components, respectively, of the mean square error. Let the total squared bias for a p-term equation be

SS_B(p) =

n

X

i=1

[E(y_i) − E(ˆy_i)]²

and define the standardized total total mean square error as

Γ_p = 1 σ²

( _n X

i=1

[E(y_i) − E(ˆy_i)]²+

n

X

i=1

V (ˆy_i) )

= SSB(p) σ² + 1

σ²

n

X

i=1

V (ˆy_i) (4.16)

It can be shown that

n

X

i=1

V (ˆy_i) = pσ²

and that the expected value of the residual sum of squares from a p-term equation is E [SS_E(p)] = SS_B(p) + (n − p)σ²

Substituting for Pn

i=1V (ˆy_i) and SS_B(p) in (4.15) gives Γ_p = 1

σ² E [SS_E(p)] − (n − p)σ²+ pσ²

= E[SSE(p)]

σ² − n + 2p (4.17)

Suppose that ˆσ² is a good estimate of σ². Then replacing E[SS_E(p)] by the observed value SSE(p) produces an estimate of Γp, say

C_p = SS_E(p) ˆ

σ² − n − 2p (4.18)

If the p-term model has negligible bias, then SS_B(p) = 0. Consequently E[SS_E(p)] = (n − p)σ², and

E [C_p|Bias = 0] = (n − p)σ²

σ² − n + 2p = p

When using the C_p criterion, it is helpful to construct a plot of C_p as a function of p for

Figure 4.3: Plot of C_p against p

each regression equation, such as shown in Figure 4.3. Regression equation with little bias will have values of Cp that fall near the line Cp = p (point A in Figure 4.3) while those equations with substantial bias will fall above this line (point B in Figure 4.3).

Generally small values of C_p are desirable. For example, although point C in Figure 4.3 is above the line Cp = p, it is below point A and thus represents a model with lower total error. It may be preferable to accept some bias in the equation to reduce the average prediction error.

To calculate Cp, need an unbiased estimate of σ². Generally, we use the residual mean square for the model. It gives C_p = p = K + 1 for the full model. Using M S_E(K + 1)

from the full model as an estimate of σ² assumes that the full model has negligible bias. If the full model has several regressors that do not contribute significantly to the model (zero regression coefficients), then M SE(K + 1) will often overestimate σ², and consequently the values of C_p will be small. If the C_p statistic is to work properly, a good estimate of σ² must be used.

4.4 Computational Techniques For Variable