LM in R: understanding levels, factors, & coefficients
- N차원의 linear 모델은 N개의 factor에 의해서만 가능하다.
- level이 3개 이상이 되면 regression space의 수가 늘어날 뿐 차원이 증가하지는 않는다.
- lm은 anova와 달리 lm의 모든 coefficient들에 대한 one-sample t-test를 통해서 p를 제공한다.
- 그래서 anova보다 훨씬 풍부하고 정교한 정보를 주고 post-hoc조차 필요없게 된다.
sample data
| grade | Male | Female | all |
|---|---|---|---|
| 1st | 73.6 | 73.5 | 73.55 |
| 2nd | 74.2 | 72.5 | 73.35 |
| 3rd | 83.2 | 83.3 | 83.25 |
1 way ANOVA as single regression
lm(formula = Score ~ Grade, data = data)
| Coefficients | ||
|---|---|---|
| (Intercept) | Grade2nd | Grade3rd |
| 73.55 | -0.2 | 9.7 |
single regression
y = 73.55 - 0.2 x (1st:0, 2nd:1)
y = 73.55 + 9.7 x (1st:0, 2nd:1)
| Coefficients | Estimate | Std. Error | t value | Pr(>|t|) |
|---|---|---|---|---|
| (Intercept) | 73.5500 | 0.7214 | 101.957 | < 2e-16 *** |
| Grade2nd | -0.2000 | 1.0202 | -0.196 | 0.845 |
| Grade3rd | 9.7000 | 1.0202 | 9.508 | 2.31e-13 *** |
| Analysis of Variance Table | Df | Sum | Sq Mean | Sq F value | Pr(>F) |
|---|---|---|---|---|---|
| Grade | 2 | 1280.93 | 640.47 | 61.537 | 5.783e-15 *** |
| Residuals | 57 | 593.25 | 10.41 |
2 way ANOVA as multiple regression
lm(formula = Score ~ Grade*Gender, data = data)
| Coefficients | |||||
|---|---|---|---|---|---|
| (Intercept) | Grade2nd | Grade3rd | GenderM | Grade2nd:GenderM | Grade3rd:GenderM |
| 73.5 | 0.7 | 9.7 | 0.1 | -1.8 | 0 |
multiple regression
y = 73.5 + 0.7 x1 (1st:0, 2nd:1) + 0.1 x2 (F:0, M:1) - 1.8 x1 x2
y = 73.5 + 9.7 x1 (1st:0, 3rd:1) + 0.1 x2 (F:0, M:1) - 0 x1 x2
| Coefficients | Estimate | Std. Error | t value | Pr(>|t|) |
|---|---|---|---|---|
| (Intercept) | 7.350e+01 | 1.035e+00 | 71.000 | < 2e-16 *** |
| GenderM | 1.000e-01 | 1.464e+00 | 0.068 | 0.946 |
| Grade2nd | 7.000e-01 | 1.464e+00 | 0.478 | 0.634 |
| Grade3rd | 9.700e+00 | 1.464e+00 | 6.626 | 1.67e-08 *** |
| GenderM:Grade2nd | -1.800e+00 | 2.070e+00 | -0.869 | 0.388 |
| GenderM:Grade3rd | 4.353e-14 | 2.070e+00 | 0.000 | 1.000 |
| Analysis of Variance Table | Df | Sum Sq | Mean Sq | F value | Pr(>F) |
|---|---|---|---|---|---|
| Gender | 1 | 3.75 | 3.75 | 0.3499 | 0.5566 |
| Grade | 2 | 1280.93 | 640.47 | 59.7636 | 2.05e-14 *** |
| Gender:Grade | 2 | 10.80 | 5.40 | 0.5039 | 0.6070 |
| Residuals | 54 | 578.70 | 10.72 |
Hosung Nam
