FrontPage StatisticalRegression

## 1. Statistical (stepwise) regression ¶

```REGRESSION
/DEPENDENT api00
/METHOD=ENTER ell acs_k3 avg_ed meals .
```
 Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. Correlations B Std. Error Beta Zero-order Partial Part 1 (Constant) 709.639 56.240 12.618 .000 ell -.843 .196 -.147 -4.307 .000 -.766 -.217 -.091 acs_k3 3.388 2.333 .032 1.452 .147 .185 .075 .031 avg_ed 29.072 6.924 .156 4.199 .000 .793 .212 .089 meals -2.937 .195 -.655 -15.081 .000 -.902 -.615 -.319

From the left column, zero-order (r2 ), partial (pr2 ), and semi-partial or part (sr2 ).
We know that the meal's contribution alone is the biggest in the explanation of Y variance (-.319). Then, ell, avg_ed, acs_k3.
Therefore, the beta (standardized coefficient) value of meals is the largest (We can compare the beta values across the IV since they were standardized).

### 1.1. forward selection ¶

Forward selection
Forward selection begins with no predictors in the regression equation. The predictor variable that has the highest correlation with the criterion variable is entered into the equation first. The rest variables are entered into the equation depending on the contribution of each predictor.
가장 큰 상관관계를 갖는 meals 변인이 먼저 투입되어 regression이 진행된다. 이 때의 zero-order r squared 값은 -.902 에 달한다. 다른 변인이 고려되지 (투입되지) 않은 상태이므로, partial, part 값들도 zero-order와 동일한 값을 갖는다.
다음 단계에서 상관관계가 다음으로 높은 변인이 투입되어 regression이 진행된다 (partial correlation값에 의해서 판단).

```REGRESSION
/DEPENDENT api00
/METHOD=FORWARD ell acs_k3 avg_ed meals .
```
 Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. Correlations B Std. Error Beta Zero-order Partial Part 1 (Constant) 892.894 6.830 130.731 .000 meals -4.046 .100 -.902 -40.531 .000 -.902 -.902 -.902 2 (Constant) 887.826 6.718 132.163 .000 meals -3.469 .154 -.773 -22.565 .000 -.902 -.758 -.488 ell -.951 .197 -.166 -4.840 .000 -.766 -.242 -.105 3 (Constant) 781.172 27.182 28.738 .000 meals -3.004 .189 -.670 -15.855 .000 -.902 -.633 -.336 ell -.819 .195 -.143 -4.191 .000 -.766 -.212 -.089 avg_ed 27.828 6.881 .149 4.044 .000 .793 .204 .086

 Excluded Variablesd Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolerance 1 ell -.166a -4.840 .000 -.242 .398 acs_k3 .009a .383 .702 .020 .962 avg_ed .175a 4.711 .000 .236 .339 2 acs_k3 .020b .920 .358 .047 .951 avg_ed .149b 4.044 .000 .204 .329 3 acs_k3 .032c 1.452 .147 .075 .936

### 1.2. Backward selection ¶

Backward selection
Backward elimination begins with all predictor variables in the regression equation and sequentially removes them. Two removal criteria are available.

```REGRESSION
/DEPENDENT api00
/METHOD=BACKWARD ell acs_k3 avg_ed meals .
```
All entered
Then, one that is not significantly contributes is removed. . . .
 Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. Correlations B Std. Error Beta Zero-order Partial Part 1 (Constant) 709.639 56.240 12.618 .000 ell -.843 .196 -.147 -4.307 .000 -.766 -.217 -.091 acs_k3 3.388 2.333 .032 1.452 .147 .185 .075 .031 avg_ed 29.072 6.924 .156 4.199 .000 .793 .212 .089 meals -2.937 .195 -.655 -15.081 .000 -.902 -.615 -.319 2 (Constant) 781.172 27.182 28.738 .000 ell -.819 .195 -.143 -4.191 .000 -.766 -.212 -.089 avg_ed 27.828 6.881 .149 4.044 .000 .793 .204 .086 meals -3.004 .189 -.670 -15.855 .000 -.902 -.633 -.336

 Excluded Variablesb Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolerance 2 acs_k3 .032a 1.452 .147 .075 .936

### 1.3. Stepwise selection ¶

Stepwise selection
Stepwise selection is a combination of forward and backward procedures.
Step 1
The first predictor variable is selected in the same way as in forward selection. If the probability associated with the test of significance is less than or equal to the default .05, the predictor variable with the largest correlation with the criterion variable enters the equation first.
Step 2
The second variable is selected based on the highest partial correlation. If it can pass the entry requirement (PIN=.05), it also enters the equation.
Step 3
From this point, stepwise selection differs from forward selection: the variables already in the equation are examined for removal according to the removal criterion (POUT=.10) as in backward elimination.
Step 4
Variables not in the equation are examined for entry. Variable selection ends when no more variables meet entry and removal criteria.

```REGRESSION
/DEPENDENT api00
/METHOD=STEPWISE ell acs_k3 avg_ed meals
```
 Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. Correlations B Std. Error Beta Zero-order Partial Part 1 (Constant) 892.894 6.830 130.731 .000 meals -4.046 .100 -.902 -40.531 .000 -.902 -.902 -.902 2 (Constant) 887.826 6.718 132.163 .000 meals -3.469 .154 -.773 -22.565 .000 -.902 -.758 -.488 ell -.951 .197 -.166 -4.840 .000 -.766 -.242 -.105 3 (Constant) 781.172 27.182 28.738 .000 meals -3.004 .189 -.670 -15.855 .000 -.902 -.633 -.336 ell -.819 .195 -.143 -4.191 .000 -.766 -.212 -.089 avg_ed 27.828 6.881 .149 4.044 .000 .793 .204 .086

 Excluded Variablesd Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolerance 1 ell -.166a -4.840 .000 -.242 .398 acs_k3 .009a .383 .702 .020 .962 avg_ed .175a 4.711 .000 .236 .339 2 acs_k3 .020b .920 .358 .047 .951 avg_ed .149b 4.044 .000 .204 .329 3 acs_k3 .032c 1.452 .147 .075 .936   