10장: 단순회귀분석(Simple Linear Regression)

[안재형]

##### 10장: 단순회귀분석(Simple Linear Regression)

###

> cars = read.csv("cars.csv")
> out = lm(dist~speed, data=cars)
> summary(out)

Call:
lm(formula = dist ~ speed, data = cars)

Residuals:
    Min      1Q  Median      3Q     Max
-29.069  -9.525  -2.272   9.215  43.201

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) -17.5791     6.7584  -2.601   0.0123 * 
speed         3.9324     0.4155   9.464 1.49e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.38 on 48 degrees of freedom
Multiple R-squared: 0.6511,     Adjusted R-squared: 0.6438
F-statistic: 89.57 on 1 and 48 DF,  p-value: 1.490e-12

> with(cars, cor(speed,dist))^2

> with(cars, cor.test(speed,dist))

# Scatter plot with trend

> plot(dist ~ speed, data=cars, col="blue")

> abline(out, col="red")

### No intercept model

> lm(dist~speed-1, data=cars) 

Call:
lm(formula = dist ~ speed - 1, data = cars)

Coefficients:
speed 
2.909 

# no intercept model with sqrt-root transformation: sqrt(dist) = 0.39675*speed

> out2 = lm(sqrt(dist) ~ speed - 1, data=cars)

> summary(out2)

Call:
lm(formula = sqrt(dist) ~ speed - 1, data = cars)

Residuals:
     Min       1Q   Median       3Q      Max
-2.27813 -0.69718  0.02079  0.79650  3.38978

Coefficients:
      Estimate Std. Error t value Pr(>|t|)   
speed  0.39675    0.01015   39.09   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.167 on 49 degrees of freedom
Multiple R-squared: 0.9689,     Adjusted R-squared: 0.9683
F-statistic:  1528 on 1 and 49 DF,  p-value: < 2.2e-16

> plot(sqrt(dist)~speed, data=cars)
> abline(out2)  

# 회귀진단 

par(mfrow=c(2,2))
plot(out2)

> qqnorm(resid(out2))

> qqline(resid(out2))

> shapiro.test(resid(out2))

### one_comp.csv

> one.comp=read.csv("one_comp.csv")

> plot(conc~time, data=one.comp)

> plot(conc~time, data=one.comp, log="y")

> summary(lm(log(conc)~time, data=one.comp))

Call:
lm(formula = log(conc) ~ time, data = one.comp)

Residuals:
     Min       1Q   Median       3Q      Max
-0.10308 -0.05859  0.00874  0.04190  0.09032

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  3.72031    0.03809   97.68 3.10e-12 ***
time        -0.63771    0.01485  -42.95 9.68e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.06901 on 7 degrees of freedom
Multiple R-squared: 0.9962,     Adjusted R-squared: 0.9957
F-statistic:  1845 on 1 and 7 DF,  p-value: 9.676e-10

# 잘못된 모형

> out = lm(conc ~ time, data=one.comp)

> par(mfrow=c(2,2))

> plot(out)

# 결과물 추출

> out = lm(dist ~ speed, data=cars)
> names(out)
 [1] "coefficients"  "residuals"     "effects"       "rank"          "fitted.values"
 [6] "assign"        "qr"            "df.residual"   "xlevels"       "call"        
[11] "terms"         "model"       
> out$rank
[1] 2
> out$df.residual
[1] 48

> B = coef(out)
> B
(Intercept)       speed
 -17.579095    3.932409

> fitted(out)
        1         2         3         4         5         6         7         8
-1.849460 -1.849460  9.947766  9.947766 13.880175 17.812584 21.744993 21.744993
        9        10        11        12        13        14        15        16
21.744993 25.677401 25.677401 29.609810 29.609810 29.609810 29.609810 33.542219
       17        18        19        20        21        22        23        24
33.542219 33.542219 33.542219 37.474628 37.474628 37.474628 37.474628 41.407036
       25        26        27        28        29        30        31        32
41.407036 41.407036 45.339445 45.339445 49.271854 49.271854 49.271854 53.204263
       33        34        35        36        37        38        39        40
53.204263 53.204263 53.204263 57.136672 57.136672 57.136672 61.069080 61.069080
       41        42        43        44        45        46        47        48
61.069080 61.069080 61.069080 68.933898 72.866307 76.798715 76.798715 76.798715
       49        50
76.798715 80.731124
> yhat=B[1]+B[2]*cars$speed
> yhat
 [1] -1.849460 -1.849460  9.947766  9.947766 13.880175 17.812584 21.744993 21.744993
 [9] 21.744993 25.677401 25.677401 29.609810 29.609810 29.609810 29.609810 33.542219
[17] 33.542219 33.542219 33.542219 37.474628 37.474628 37.474628 37.474628 41.407036
[25] 41.407036 41.407036 45.339445 45.339445 49.271854 49.271854 49.271854 53.204263
[33] 53.204263 53.204263 53.204263 57.136672 57.136672 57.136672 61.069080 61.069080
[41] 61.069080 61.069080 61.069080 68.933898 72.866307 76.798715 76.798715 76.798715
[49] 76.798715 80.731124

> resid(out)
         1          2          3          4          5          6          7
  3.849460  11.849460  -5.947766  12.052234   2.119825  -7.812584  -3.744993
         8          9         10         11         12         13         14
  4.255007  12.255007  -8.677401   2.322599 -15.609810  -9.609810  -5.609810
        15         16         17         18         19         20         21
 -1.609810  -7.542219   0.457781   0.457781  12.457781 -11.474628  -1.474628
        22         23         24         25         26         27         28
 22.525372  42.525372 -21.407036 -15.407036  12.592964 -13.339445  -5.339445
        29         30         31         32         33         34         35
-17.271854  -9.271854   0.728146 -11.204263   2.795737  22.795737  30.795737
        36         37         38         39         40         41         42
-21.136672 -11.136672  10.863328 -29.069080 -13.069080  -9.069080  -5.069080
        43         44         45         46         47         48         49
  2.930920  -2.933898 -18.866307  -6.798715  15.201285  16.201285  43.201285
        50
  4.268876
> cars$dist - yhat
 [1]   3.849460  11.849460  -5.947766  12.052234   2.119825  -7.812584  -3.744993
 [8]   4.255007  12.255007  -8.677401   2.322599 -15.609810  -9.609810  -5.609810
[15]  -1.609810  -7.542219   0.457781   0.457781  12.457781 -11.474628  -1.474628
[22]  22.525372  42.525372 -21.407036 -15.407036  12.592964 -13.339445  -5.339445
[29] -17.271854  -9.271854   0.728146 -11.204263   2.795737  22.795737  30.795737
[36] -21.136672 -11.136672  10.863328 -29.069080 -13.069080  -9.069080  -5.069080
[43]   2.930920  -2.933898 -18.866307  -6.798715  15.201285  16.201285  43.201285
[50]   4.268876

# Residual standard error

> summary(out)$sigma
[1] 15.37959
> sqrt(sum(resid(out)^2)/(length(cars$dist)-2))
[1] 15.37959

### Simulation

# cars

> out = lm(sqrt(dist) ~ speed-1, data=cars)
> dist = rnorm(n=nrow(cars),mean=fitted(out),sd=summary(out)$sigma)^2

# one.comp

> out = lm(log(conc)~time, data=one.comp)
> conc = exp(rnorm(n=nrow(one.comp),mean=fitted(out),sd=summary(out)$sigma))

### 연습

> out=lm(log(conc)~time, data=one.comp)
> par(mfow=c(2,2))
> par(mfrow=c(2,2))
> plot(out)