# COMMunicationRESearch.NET

### Site Tools

multiple_regression_examples

# E.g. 1

d.yyk <- read.csv("http://commres.net/wiki/_media/d.yyk.csv")
d.yyk
d.yyk <- subset(d.yyk, select = -c(1))
d.yyk
> d.yyk <- subset(d.yyk, select = -c(1))
> d.yyk
bmi stress happiness
1  15.1      2         4
2  15.3      2         4
3  16.4      1         5
4  16.3      2         4
5  17.5      2         3
6  18.8      2         4
7  19.2      2         3
8  20.3      1         4
9  21.3      1         4
10 21.3      2         4
11 22.4      2         5
12 23.5      2         5
13 23.7      2         4
14 24.2      3         3
15 24.3      3         3
16 25.6      2         3
17 26.4      3         3
18 26.4      3         2
19 26.4      3         2
20 27.5      3         3
21 28.6      3         2
22 28.2      4         2
23 31.3      3         2
24 32.1      4         1
25 33.1      4         1
26 33.2      5         1
27 34.4      5         1
28 35.8      5         1
29 36.1      5         1
30 38.1      5         1

우선 여기에서 종속변인인 (dv) happiness에 bmi와 stress를 리그레션 해본다.

attach(d.yyk)
lm.happiness.bmistress <- lm(happiness ~ bmi + stress, data=d.yyk)
summary(lm.happiness.bmistress)
anova(lm.happiness.bmistress)
> attach(d.yyk)
> lm.happiness.bmistress <- lm(happiness ~ bmi + stress, data=d.yyk)
> summary(lm.happiness.bmistress)

Call:
lm(formula = happiness ~ bmi + stress, data = d.yyk)

Residuals:
Min       1Q   Median       3Q      Max
-0.89293 -0.40909  0.08816  0.29844  1.46429

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  6.29098    0.50779  12.389 1.19e-12 ***
bmi         -0.05954    0.03626  -1.642  0.11222
stress      -0.67809    0.19273  -3.518  0.00156 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5869 on 27 degrees of freedom
Multiple R-squared:  0.8217,	Adjusted R-squared:  0.8085
F-statistic: 62.22 on 2 and 27 DF,  p-value: 7.76e-11
>
>
> anova(lm.happiness.bmistress)
Analysis of Variance Table

Response: happiness
Df Sum Sq Mean Sq F value    Pr(>F)
bmi        1 38.603  38.603 112.070 4.124e-11 ***
stress     1  4.264   4.264  12.378  0.001558 **
Residuals 27  9.300   0.344
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>


위의 분석을 보면
R2 = 0.8217, F (2, 27) = 62.77, p = 7.76e-11

그러나, coefficient 값을 보면 bmi는 significant 하지 않고, stress는 significant하다. 이는 R제곱에 영향을 주는 것으로 stress가 주이고 bmi의 영향력은 미미하다고 하다는 결론을 내리도록 해준다. 그러나, multiple regression에서 언급한 것처럼 독립변인이 두 개 이상일 때에는 무엇이 얼마나 종속변인에 영향을 주는지 그림을 그릴 수 있어야 하므로, 아래와 같이 bmi만을 가지고 regression을 다시 해본다.

lm.happiness.bmi <- lm(happiness ~ bmi, data=d.yyk)
summary(lm.happiness.bmi)
anova(lm.happiness.bmi)
> lm.happiness.bmi <- lm(happiness ~ bmi, data=d.yyk)
>
> summary(lm.happiness.bmi)

Call:
lm(formula = happiness ~ bmi, data = d.yyk)

Residuals:
Min       1Q   Median       3Q      Max
-1.20754 -0.49871 -0.03181  0.35669  1.83265

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  7.24143    0.50990  14.202 2.54e-14 ***
bmi         -0.17337    0.01942  -8.927 1.11e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.696 on 28 degrees of freedom
Multiple R-squared:   0.74,	Adjusted R-squared:  0.7307
F-statistic: 79.69 on 1 and 28 DF,  p-value: 1.109e-09

> anova(lm.happiness.bmi)
Analysis of Variance Table

Response: happiness
Df Sum Sq Mean Sq F value    Pr(>F)
bmi        1 38.603  38.603  79.687 1.109e-09 ***
Residuals 28 13.564   0.484
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 

놀랍게도 bmi 하나만을 가지고 regression을 했더니, R제곱 값이 .74이었다 (F(1,28) = 79.69, p = 1.109e-09). 이런 결과가 나올 수 있는 이유는 독립변인인 bmi와 stress 간의 상관관계가 높아서 처음 분석에서 그 영향력을 (설명력, R제곱에 기여하는 부분을) 하나의 독립변인이 모두 가졌갔기 때문이라고 생각할 수 있다. 이 경우에는 그 독립변인이 stress이다.

happiness에 stress 만을 regression 해본 결과는 아래와 같다.

lm.happiness.stress <- lm(happiness ~ stress, data = d.yyk)
summary(lm.happiness.stress)
anova(lm.happiness.stress)
> lm.happiness.stress <- lm(happiness ~ stress, data = d.yyk)
> summary(lm.happiness.stress)

Call:
lm(formula = happiness ~ stress, data = d.yyk)

Residuals:
Min      1Q  Median      3Q     Max
-0.7449 -0.6657  0.2155  0.3343  1.3343

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  5.58651    0.27965   19.98  < 2e-16 ***
stress      -0.96041    0.08964  -10.71 2.05e-11 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6044 on 28 degrees of freedom
Multiple R-squared:  0.8039,	Adjusted R-squared:  0.7969
F-statistic: 114.8 on 1 and 28 DF,  p-value: 2.053e-11

> anova(lm.happiness.stress)
Analysis of Variance Table

Response: happiness
Df Sum Sq Mean Sq F value    Pr(>F)
stress     1 41.938  41.938   114.8 2.053e-11 ***
Residuals 28 10.229   0.365
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> 

R제곱값은 0.80로 (F(1, 28) = 114.8, 2.053e-11) 스트레스만으로도 significant한 결과를 갖는다.

그렇다면 stress 와 bmi가 공통으로 기여하는 부분을 뺀 순수 기여분은 어떻게 될까? 즉, 위의 .80 부분 중 bmi와 공통으로 기여하는 부분을 제외한 나머지는 얼마일까? 보통 이와 같은 작업을 bmi의 (다른 독립변인의) 영향력을 제어하고 (control) 순수기여분만을 살펴본다고 이야기 한다.

## 방법 1

이를 위해서 아래를 계획, 수행해본다.

1. 각각의 독립변인이 고유하게 미치는 영향력은 (설명력은) 무엇인지를 본다.
2. 공통설명력은 얼마나 되는지 본다.
1. 1을 위해서는 각 독립변인과 종속변인인 happiness의 semi-partial correlation값을 구해서 제곱해보면 되겠다.
2. 2를 위해서는 두 독립변인을 써서 구했던 r 제곱값에서 위의 1에서 구한 제곱값들을 제외한 나머지를 보면 된겠다.
• 결론을 내기 위한 계획을 세우고 실행한다.
• 이는 아래와 같이 정리할 수 있다

## 각각의 독립변인이 고유하게 미치는 영향력은 (설명력은) 무엇인지를 본다

> spcor(d.yyk)
$estimate bmi stress happiness bmi 1.0000000 0.2730799 -0.1360657 stress 0.2371411 1.0000000 -0.2532032 happiness -0.1334127 -0.2858909 1.0000000$p.value
bmi    stress happiness
bmi       0.0000000 0.1517715 0.4815643
stress    0.2154821 0.0000000 0.1850784
happiness 0.4902316 0.1327284 0.0000000

$statistic bmi stress happiness bmi 0.0000000 1.475028 -0.7136552 stress 1.2684024 0.000000 -1.3600004 happiness -0.6994855 -1.550236 0.0000000$n
[1] 30

$gp [1] 1$method
[1] "pearson"

>
> 

happiness에 영향을 주는 변인을 보는 것이므로

                 bmi    stress
happiness -0.1334127 -0.2858909  

를 본다. 그리고 이 값의 제곱값이 각 독립변인의 고유 설명력이다.

> (-0.1334127)^2
[1] 0.01779895
> (-0.2858909)^2
[1] 0.08173361
> 

즉,  stress: 8.1%  bmi: 1.78%  만이 독립변인의 고유영향력이고 이를 제외한  82.17 - (9.88) = 72.29  가 공통영향력이라고 하겠다.

이를 파티션을 하면서 직접 살펴보려면

• 우선 $\frac{b}{a+b+c+d}$ 를 보려고 한다.
• 그림에서 m.bmi ← lm1) 와 같이 한후에 r제곱값을 보고, sqrt 하면 r값을 알 수 있다.
• b+e를 구하려면 lm(bmi~stress)를 한후, 그 residual을 보면 된다.
• a+b+c+d 는 happiness 그 자체이다.
m.bmi <- lm(bmi ~ stress)
mod <- lm(happiness ~ resid(m.bmi))
summary(mod)
> m.bmi <- lm(bmi ~ stress)
> mod <- lm(happiness ~ resid(m.bmi))
> summary(mod)

Call:
lm(formula = happiness ~ resid(m.bmi))

Residuals:
Min       1Q   Median       3Q      Max
-1.97283 -0.94440  0.05897  0.97961  2.29664

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   2.83333    0.24698  11.472 4.27e-12 ***
resid(m.bmi) -0.05954    0.08358  -0.712    0.482
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.353 on 28 degrees of freedom
Multiple R-squared:  0.0178,	Adjusted R-squared:  -0.01728
F-statistic: 0.5074 on 1 and 28 DF,  p-value: 0.4822

위의 분석에서 R-square 값인 0.0178 이 bmi의 고유의 설명력이다. r값은 sqrt(0.0178)이다. 그리고, 위의 모델은 significant하지 않음을 주목한다.

다음으로 $\frac {d}{a+b+c+d}$을 구해서 stress 고유설명력을 본다. 이제는

m.stress <- lm(stress ~ bmi)
mod2 <- lm(happiness ~ resid(m.stress))
sumary(mod2)
> m.stress <- lm(stress ~ bmi)
> mod2 <- lm(happiness ~ resid(m.stress))
> summary(mod2)

Call:
lm(formula = happiness ~ resid(m.stress))

Residuals:
Min      1Q  Median      3Q     Max
-1.9383 -1.2297  0.2170  0.9804  1.9284

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)       2.8333     0.2388  11.865 1.95e-12 ***
resid(m.stress)  -0.6781     0.4295  -1.579    0.126
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.308 on 28 degrees of freedom
Multiple R-squared:  0.08173,	Adjusted R-squared:  0.04894
F-statistic: 2.492 on 1 and 28 DF,  p-value: 0.1256

> 

Multiple R-squared 인 0.08173 이 고유 설명력이고, 이 또한 significant 하지 않다.
0.08173 값과 0.0178을 더한 값을 제외한 lm(happiness~bmi+stress) 에서의 R-squared 값이 공통설명력이 된다. 아래의 분석 결과에서 Multiple R-squared: 0.8217 이 두 변인을 모두 합한 설명력이다.

m.both <- lm(happiness~bmi+stress)
summary(m.both)
> m.both <- lm(happiness~bmi+stress)
> summary(m.both)

Call:
lm(formula = happiness ~ bmi + stress)

Residuals:
Min       1Q   Median       3Q      Max
-0.89293 -0.40909  0.08816  0.29844  1.46429

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  6.29098    0.50779  12.389 1.19e-12 ***
bmi         -0.05954    0.03626  -1.642  0.11222
stress      -0.67809    0.19273  -3.518  0.00156 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5869 on 27 degrees of freedom
Multiple R-squared:  0.8217,	Adjusted R-squared:  0.8085
F-statistic: 62.22 on 2 and 27 DF,  p-value: 7.76e-11

이 값은 0.72217 이다.

> 0.8217- (0.08173 + 0.0178)
[1] 0.72217
> 

bmi나 stress 중 하나를 IV로 취하는 것이 좋다는 결론을 내린다.

# with Two Predictor Variables

• Y1 - A measure of success in graduate school.
• X1 - A measure of intellectual ability.
• X2 - A measure of “work ethic.”
• X3 - A second measure of intellectual ability.
• X4 - A measure of spatial ability.
• Y2 - Score on a major review paper.
Analyze
Descriptive statistics
Descriptives
모든 변수를 Variable(s)로 이동
OK 누르기
Descriptive Statistics
N	Minimum	Maximum	Mean	Std. Deviation
sucess in graduate school	20	125	230	169.45	24.517
score on a major review paper	20	105	135	120.50	8.918
intellectual ability I		20	11	64	37.05	14.891
work ethic			20	11	56	29.10	14.056
intellectual ability II		20	17	79	49.35	18.622
spatial ability			20	11	56	32.50	13.004
Valid N (listwise)		20				

## Correlation matrix 검사

Analyze
Correlate
Bivariate
모든 변수를 Variable(s)로 이동

Correlations
y1	y2	x1	x2	x3	x4
y1	Pearson Correlation	1	.310	.764**	.769**	.687**	.736**
Sig. (2-tailed)		.184	.000	.000	.001	.000
N			20	20	20	20	20	20
y2	Pearson Correlation	.310	1	.251	.334	.168	.018
Sig. (2-tailed)		.184		.286	.150	.479	.939
N			20	20	20	20	20	20
x1	Pearson Correlation	.764**	.251	1	.255	.940**	.904**
Sig. (2-tailed)		.000	.286		.278	.000	.000
N			20	20	20	20	20	20
x2	Pearson Correlation	.769**	.334	.255	1	.243	.266
Sig. (2-tailed)		.000	.150	.278		.302	.257
N			20	20	20	20	20	20
x3	Pearson Correlation	.687**	.168	.940**	.243	1	.847**
Sig. (2-tailed)		.001	.479	.000	.302		.000
N			20	20	20	20	20	20
x4	Pearson Correlation	.736**	.018	.904**	.266	.847**	1
Sig. (2-tailed)		.000	.939	.000	.257	.000
N			20	20	20	20	20	20
** Correlation is significant at the 0.01 level (2-tailed).

## Regression

Analyze
Regression
Linear
Dependent: 	y1 (success in graduate school)
Independent:	x1 (intell. ability)
x2 (work ethic)
Click Statistics (오른쪽 상단 버튼)
Choose
Estimates (under Regression Coefficients)
Model fit
R squared change
Descriptives
Part and partial correlations
Collinearity diagnostics    
Model Summaryb
Change Statistics
Model	R	R	Adjusted	Std. Error	R  	F 	df1	df2	Sig. F
Square	R Square	of the 		Square	Change			Change
Estimate	Change
1	.968a	.936	.929		6.541		.936	124.979	2	17	.000
a Predictors: (Constant), work ethic, intellectual ability I
b Dependent Variable: success in graduate school
ANOVAa
Model			Sum of Squares	df	Mean Square	F		Sig.
1	Regression	10693.657	2	5346.828	124.979		.000b
Residual	727.293		17	42.782
Total		11420.950	19
a Dependent Variable: sucess in graduate school
b Predictors: (Constant), work ethic, intellectual ability I						
Coefficientsa
Model		Unstandardized 	Standardized 		Correlations		Collinearity
Coefficients	Coefficients					Statistics
-------------	-----	------------	--------------------	---------	----
B	Std. 	Beta	t	Sig.	Zero	Partial	Part	Tolerance	VIF
Error					-order
1 (Constant)	101.222	4.587		22.065	.000
x1		1.000	.104	.608	9.600	.000	.764	.919	.588	.935		1.070
x3		1.071	.110	.614	9.699	.000	.769	.920	.594	.935		1.070
a Dependent Variable: success in graduate school
x1  intellectual ability I
x3  work ethic

from the above output:

 x zero-order cor part cor squared zero-order cor squared part cor shared square cor x1 .764 .588 0.583696 0.345744 0.237952 x2 .769 .594 0.591361 0.352836 0.238525

note that the values of two raws at the last column are similar. The portion is the shared effects from both x1 and x2.

$$\hat{Y_{i}} = 101.222 + 1.000X1_{i} + 1.071X2_{i}$$
$$\hat{Y_{i}} = 101.222 + 1.000 \ \text{intell. ability}_{i} + 1.071 \ \text{work ethic}_{i}$$

# E.g. 2

?state.x77

US State Facts and Figures

Description

Data sets related to the 50 states of the United States of America.

Usage

state.abb
state.area
state.center
state.division
state.name
state.region
state.x77
Details

R currently contains the following “state” data sets. Note that all data are arranged according to alphabetical order of the state names.

state.abb:
character vector of 2-letter abbreviations for the state names.

state.area:
numeric vector of state areas (in square miles).

state.center:
list with components named x and y giving the approximate geographic center of each state in negative longitude and latitude. Alaska and Hawaii are placed just off the West Coast.

state.division:
factor giving state divisions (New England, Middle Atlantic, South Atlantic, East South Central, West South Central, East North Central, West North Central, Mountain, and Pacific).

state.name:
character vector giving the full state names.

state.region:
factor giving the region (Northeast, South, North Central, West) that each state belongs to.

state.x77:
matrix with 50 rows and 8 columns giving the following statistics in the respective columns.

Population:
population estimate as of July 1, 1975

Income:
per capita income (1974)

Illiteracy:
illiteracy (1970, percent of population)

Life Exp:
life expectancy in years (1969–71)

Murder:
murder and non-negligent manslaughter rate per 100,000 population (1976)

Frost:
mean number of days with minimum temperature below freezing (1931–1960) in capital or large city

Area:
land area in square miles

Source

U.S. Department of Commerce, Bureau of the Census (1977) Statistical Abstract of the United States.

U.S. Department of Commerce, Bureau of the Census (1977) County and City Data Book.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

> state.x77                            # output not shown
> str(state.x77)                       # clearly not a data frame! (it's a matrix)
> st = as.data.frame(state.x77)        # so we'll make it one
> str(st)
'data.frame':   50 obs. of  8 variables:
$Population: num 3615 365 2212 2110 21198 ...$ Income    : num  3624 6315 4530 3378 5114 ...
$Illiteracy: num 2.1 1.5 1.8 1.9 1.1 0.7 1.1 0.9 1.3 2 ...$ Life Exp  : num  69.0 69.3 70.5 70.7 71.7 ...
$Murder : num 15.1 11.3 7.8 10.1 10.3 6.8 3.1 6.2 10.7 13.9 ...$ HS Grad   : num  41.3 66.7 58.1 39.9 62.6 63.9 56 54.6 52.6 40.6 ...
$Frost : num 20 152 15 65 20 166 139 103 11 60 ...$ Area      : num   50708 566432 113417  51945 156361 ...
> colnames(st)[4] = "Life.Exp"                   # no spaces in variable names, please
> colnames(st)[6] = "HS.Grad"                    # ditto
> st$Density = st$Population * 1000 / st$Area # create a new column in st > summary(st) Population Income Illiteracy Life.Exp Murder Min. : 365 Min. :3098 Min. :0.500 Min. :67.96 Min. : 1.400 1st Qu.: 1080 1st Qu.:3993 1st Qu.:0.625 1st Qu.:70.12 1st Qu.: 4.350 Median : 2838 Median :4519 Median :0.950 Median :70.67 Median : 6.850 Mean : 4246 Mean :4436 Mean :1.170 Mean :70.88 Mean : 7.378 3rd Qu.: 4968 3rd Qu.:4814 3rd Qu.:1.575 3rd Qu.:71.89 3rd Qu.:10.675 Max. :21198 Max. :6315 Max. :2.800 Max. :73.60 Max. :15.100 HS.Grad Frost Area Density Min. :37.80 Min. : 0.00 Min. : 1049 Min. : 0.6444 1st Qu.:48.05 1st Qu.: 66.25 1st Qu.: 36985 1st Qu.: 25.3352 Median :53.25 Median :114.50 Median : 54277 Median : 73.0154 Mean :53.11 Mean :104.46 Mean : 70736 Mean :149.2245 3rd Qu.:59.15 3rd Qu.:139.75 3rd Qu.: 81162 3rd Qu.:144.2828 Max. :67.30 Max. :188.00 Max. :566432 Max. :975.0033 > cor(st) # correlation matrix (not shown, yet) > round(cor(st), 3) # rounding makes it easier to look at Population Income Illiteracy Life.Exp Murder HS.Grad Frost Area Density Population 1.000 0.208 0.108 -0.068 0.344 -0.098 -0.332 0.023 0.246 Income 0.208 1.000 -0.437 0.340 -0.230 0.620 0.226 0.363 0.330 Illiteracy 0.108 -0.437 1.000 -0.588 0.703 -0.657 -0.672 0.077 0.009 Life.Exp -0.068 0.340 -0.588 1.000 -0.781 0.582 0.262 -0.107 0.091 Murder 0.344 -0.230 0.703 -0.781 1.000 -0.488 -0.539 0.228 -0.185 HS.Grad -0.098 0.620 -0.657 0.582 -0.488 1.000 0.367 0.334 -0.088 Frost -0.332 0.226 -0.672 0.262 -0.539 0.367 1.000 0.059 0.002 Area 0.023 0.363 0.077 -0.107 0.228 0.334 0.059 1.000 -0.341 Density 0.246 0.330 0.009 0.091 -0.185 -0.088 0.002 -0.341 1.000 > > pairs(st) # scatterplot matrix; plot(st) does the same thing > ### model1 = lm(Life.Exp ~ Population + Income + Illiteracy + Murder + + ### HS.Grad + Frost + Area + Density, data=st) > ### This is what we're going to accomplish, but more economically, by > ### simply placing a dot after the tilde, which means "everything else." > model1 = lm(Life.Exp ~ ., data=st) > summary(model1) Call: lm(formula = Life.Exp ~ ., data = st) Residuals: Min 1Q Median 3Q Max -1.47514 -0.45887 -0.06352 0.59362 1.21823 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.995e+01 1.843e+00 37.956 < 2e-16 *** Population 6.480e-05 3.001e-05 2.159 0.0367 * Income 2.701e-04 3.087e-04 0.875 0.3867 Illiteracy 3.029e-01 4.024e-01 0.753 0.4559 Murder -3.286e-01 4.941e-02 -6.652 5.12e-08 *** HS.Grad 4.291e-02 2.332e-02 1.840 0.0730 . Frost -4.580e-03 3.189e-03 -1.436 0.1585 Area -1.558e-06 1.914e-06 -0.814 0.4205 Density -1.105e-03 7.312e-04 -1.511 0.1385 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7337 on 41 degrees of freedom Multiple R-squared: 0.7501, Adjusted R-squared: 0.7013 F-statistic: 15.38 on 8 and 41 DF, p-value: 3.787e-10 > summary.aov(model1) Df Sum Sq Mean Sq F value Pr(>F) Population 1 0.409 0.409 0.760 0.38849 Income 1 11.595 11.595 21.541 3.53e-05 *** Illiteracy 1 19.421 19.421 36.081 4.23e-07 *** Murder 1 27.429 27.429 50.959 1.05e-08 *** HS.Grad 1 4.099 4.099 7.615 0.00861 ** Frost 1 2.049 2.049 3.806 0.05792 . Area 1 0.001 0.001 0.002 0.96438 Density 1 1.229 1.229 2.283 0.13847 Residuals 41 22.068 0.538 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 >  ## Model fitting > model2 = update(model1, .~. -Area) > summary(model2) Call: lm(formula = Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Density, data = st) Residuals: Min 1Q Median 3Q Max -1.50252 -0.40471 -0.06079 0.58682 1.43862 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.094e+01 1.378e+00 51.488 < 2e-16 *** Population 6.249e-05 2.976e-05 2.100 0.0418 * Income 1.485e-04 2.690e-04 0.552 0.5840 Illiteracy 1.452e-01 3.512e-01 0.413 0.6814 Murder -3.319e-01 4.904e-02 -6.768 3.12e-08 *** HS.Grad 3.746e-02 2.225e-02 1.684 0.0996 . Frost -5.533e-03 2.955e-03 -1.873 0.0681 . Density -7.995e-04 6.251e-04 -1.279 0.2079 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7307 on 42 degrees of freedom Multiple R-squared: 0.746, Adjusted R-squared: 0.7037 F-statistic: 17.63 on 7 and 42 DF, p-value: 1.173e-10 > anova(model2, model1) Analysis of Variance Table Model 1: Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Density Model 2: Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area + Density Res.Df RSS Df Sum of Sq F Pr(>F) 1 42 22.425 2 41 22.068 1 0.35639 0.6621 0.4205 > model3 = update(model2, .~. -Illiteracy) > summary(model3) Call: lm(formula = Life.Exp ~ Population + Income + Murder + HS.Grad + Frost + Density, data = st) Residuals: Min 1Q Median 3Q Max -1.49555 -0.41246 -0.05336 0.58399 1.50535 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.131e+01 1.042e+00 68.420 < 2e-16 *** Population 5.811e-05 2.753e-05 2.110 0.0407 * Income 1.324e-04 2.636e-04 0.502 0.6181 Murder -3.208e-01 4.054e-02 -7.912 6.32e-10 *** HS.Grad 3.499e-02 2.122e-02 1.649 0.1065 Frost -6.191e-03 2.465e-03 -2.512 0.0158 * Density -7.324e-04 5.978e-04 -1.225 0.2272 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7236 on 43 degrees of freedom Multiple R-squared: 0.745, Adjusted R-squared: 0.7094 F-statistic: 20.94 on 6 and 43 DF, p-value: 2.632e-11 >  > model4 = update(model3, .~. -Income) > summary(model4) Call: lm(formula = Life.Exp ~ Population + Murder + HS.Grad + Frost + Density, data = st) Residuals: Min 1Q Median 3Q Max -1.56877 -0.40951 -0.04554 0.57362 1.54752 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.142e+01 1.011e+00 70.665 < 2e-16 *** Population 6.083e-05 2.676e-05 2.273 0.02796 * Murder -3.160e-01 3.910e-02 -8.083 3.07e-10 *** HS.Grad 4.233e-02 1.525e-02 2.776 0.00805 ** Frost -5.999e-03 2.414e-03 -2.485 0.01682 * Density -5.864e-04 5.178e-04 -1.132 0.26360 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7174 on 44 degrees of freedom Multiple R-squared: 0.7435, Adjusted R-squared: 0.7144 F-statistic: 25.51 on 5 and 44 DF, p-value: 5.524e-12  > model5 = update(model4, .~. -Density) > summary(model5) Call: lm(formula = Life.Exp ~ Population + Murder + HS.Grad + Frost, data = st) Residuals: Min 1Q Median 3Q Max -1.47095 -0.53464 -0.03701 0.57621 1.50683 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.103e+01 9.529e-01 74.542 < 2e-16 *** Population 5.014e-05 2.512e-05 1.996 0.05201 . Murder -3.001e-01 3.661e-02 -8.199 1.77e-10 *** HS.Grad 4.658e-02 1.483e-02 3.142 0.00297 ** Frost -5.943e-03 2.421e-03 -2.455 0.01802 * --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.7197 on 45 degrees of freedom Multiple R-squared: 0.736, Adjusted R-squared: 0.7126 F-statistic: 31.37 on 4 and 45 DF, p-value: 1.696e-12 >  > anova(model5, model1) Analysis of Variance Table Model 1: Life.Exp ~ Population + Murder + HS.Grad + Frost Model 2: Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area + Density Res.Df RSS Df Sum of Sq F Pr(>F) 1 45 23.308 2 41 22.068 4 1.2397 0.5758 0.6818  ## Stepwise > step(model1, direction="backward") Start: AIC=-22.89 Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area + Density Df Sum of Sq RSS AIC - Illiteracy 1 0.3050 22.373 -24.208 - Area 1 0.3564 22.425 -24.093 - Income 1 0.4120 22.480 -23.969 <none> 22.068 -22.894 - Frost 1 1.1102 23.178 -22.440 - Density 1 1.2288 23.297 -22.185 - HS.Grad 1 1.8225 23.891 -20.926 - Population 1 2.5095 24.578 -19.509 - Murder 1 23.8173 45.886 11.707 Step: AIC=-24.21 Life.Exp ~ Population + Income + Murder + HS.Grad + Frost + Area + Density Df Sum of Sq RSS AIC - Area 1 0.1427 22.516 -25.890 - Income 1 0.2316 22.605 -25.693 <none> 22.373 -24.208 - Density 1 0.9286 23.302 -24.174 - HS.Grad 1 1.5218 23.895 -22.918 - Population 1 2.2047 24.578 -21.509 - Frost 1 3.1324 25.506 -19.656 - Murder 1 26.7071 49.080 13.072 Step: AIC=-25.89 Life.Exp ~ Population + Income + Murder + HS.Grad + Frost + Density Df Sum of Sq RSS AIC - Income 1 0.132 22.648 -27.598 - Density 1 0.786 23.302 -26.174 <none> 22.516 -25.890 - HS.Grad 1 1.424 23.940 -24.824 - Population 1 2.332 24.848 -22.962 - Frost 1 3.304 25.820 -21.043 - Murder 1 32.779 55.295 17.033 Step: AIC=-27.6 Life.Exp ~ Population + Murder + HS.Grad + Frost + Density Df Sum of Sq RSS AIC - Density 1 0.660 23.308 -28.161 <none> 22.648 -27.598 - Population 1 2.659 25.307 -24.046 - Frost 1 3.179 25.827 -23.030 - HS.Grad 1 3.966 26.614 -21.529 - Murder 1 33.626 56.274 15.910 Step: AIC=-28.16 Life.Exp ~ Population + Murder + HS.Grad + Frost Df Sum of Sq RSS AIC <none> 23.308 -28.161 - Population 1 2.064 25.372 -25.920 - Frost 1 3.122 26.430 -23.877 - HS.Grad 1 5.112 28.420 -20.246 - Murder 1 34.816 58.124 15.528 Call: lm(formula = Life.Exp ~ Population + Murder + HS.Grad + Frost, data = st) Coefficients: (Intercept) Population Murder HS.Grad Frost 7.103e+01 5.014e-05 -3.001e-01 4.658e-02 -5.943e-03 >  ## Prediction > predict(model5, list(Population=2000, Murder=10.5, HS.Grad=48, Frost=100)) 1 69.61746  ## Regression Diagnostics > par(mfrow=c(2,2)) # visualize four graphs at once > plot(model5) > par(mfrow=c(1,1)) # reset the graphics defaults ## Model objects > names(model5) [1] "coefficients" "residuals" "effects" "rank" "fitted.values" [6] "assign" "qr" "df.residual" "xlevels" "call" [11] "terms" "model" > coef(model5) # an extractor function (Intercept) Population Murder HS.Grad Frost 7.102713e+01 5.013998e-05 -3.001488e-01 4.658225e-02 -5.943290e-03 > model5$coefficients                  # list indexing
7.102713e+01  5.013998e-05 -3.001488e-01  4.658225e-02 -5.943290e-03
> model5[[1]]                          # recall by position in the list (double brackets for lists)
7.102713e+01  5.013998e-05 -3.001488e-01  4.658225e-02 -5.943290e-03
> model5$resid Alabama Alaska Arizona Arkansas California 0.56888134 -0.54740399 -0.86415671 1.08626119 -0.08564599 Colorado Connecticut Delaware Florida Georgia 0.95645816 0.44541028 -1.06646884 0.04460505 -0.09694227 Hawaii Idaho Illinois Indiana Iowa 1.50683146 0.37010714 -0.05244160 -0.02158526 0.16347124 Kansas Kentucky Louisiana Maine Maryland 0.67648037 0.85582067 -0.39044846 -1.47095411 -0.29851996 Massachusetts Michigan Minnesota Mississippi Missouri -0.61105391 0.76106640 0.69440380 -0.91535384 0.58389969 Montana Nebraska Nevada New Hampshire New Jersey -0.84024805 0.42967691 -0.49482393 -0.49635615 -0.66612086 New Mexico New York North Carolina North Dakota Ohio 0.28880945 -0.07937149 -0.07624179 0.90350550 -0.26548767 Oklahoma Oregon Pennsylvania Rhode Island South Carolina 0.26139958 -0.28445333 -0.95045527 0.13992982 -1.10109172 South Dakota Tennessee Texas Utah Vermont 0.06839119 0.64416651 0.92114057 0.84246817 0.57865019 Virginia Washington West Virginia Wisconsin Wyoming -0.06691392 -0.96272426 -0.96982588 0.47004324 -0.58678863 > sort(model5$resid)                   # extract residuals and sort them
Maine South Carolina       Delaware  West Virginia     Washington
-1.47095411    -1.10109172    -1.06646884    -0.96982588    -0.96272426
Pennsylvania    Mississippi        Arizona        Montana     New Jersey
-0.95045527    -0.91535384    -0.86415671    -0.84024805    -0.66612086
-0.61105391    -0.58678863    -0.54740399    -0.49635615    -0.49482393
Louisiana       Maryland         Oregon           Ohio        Georgia
-0.39044846    -0.29851996    -0.28445333    -0.26548767    -0.09694227
California       New York North Carolina       Virginia       Illinois
-0.08564599    -0.07937149    -0.07624179    -0.06691392    -0.05244160
Indiana        Florida   South Dakota   Rhode Island           Iowa
-0.02158526     0.04460505     0.06839119     0.13992982     0.16347124
Oklahoma     New Mexico          Idaho       Nebraska    Connecticut
0.26139958     0.28880945     0.37010714     0.42967691     0.44541028
Wisconsin        Alabama        Vermont       Missouri      Tennessee
0.47004324     0.56888134     0.57865019     0.58389969     0.64416651
Kansas      Minnesota       Michigan           Utah       Kentucky
0.67648037     0.69440380     0.76106640     0.84246817     0.85582067
North Dakota          Texas       Colorado       Arkansas         Hawaii
0.90350550     0.92114057     0.95645816     1.08626119     1.50683146


# e.g.,

library(ISLR)
str(Carseats)

lm.full <- lm(Sales ~ . , data = Carseats)
lm.null <- lm(Sales ~ 1 , data = Carseats)

> stepAIC(lm.full)
Start:  AIC=26.82
Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Population   1      0.33  403.16  25.15
- Education    1      1.19  404.02  26.00
- Urban        1      1.23  404.06  26.04
- US           1      1.57  404.40  26.38
<none>                      402.83  26.82
- Income       1     76.16  478.99  94.09
- Advertising  1    127.14  529.97 134.54
- Age          1    217.44  620.27 197.48
- CompPrice    1    519.91  922.74 356.35
- ShelveLoc    2   1053.20 1456.03 536.80
- Price        1   1323.23 1726.06 606.85

Step:  AIC=25.15
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Urban        1      1.15  404.31  24.29
- Education    1      1.36  404.52  24.49
- US           1      1.89  405.05  25.02
<none>                      403.16  25.15
- Income       1     75.94  479.10  92.18
- Advertising  1    145.38  548.54 146.32
- Age          1    218.52  621.68 196.38
- CompPrice    1    521.69  924.85 355.27
- ShelveLoc    2   1053.18 1456.34 534.89
- Price        1   1323.51 1726.67 605.00

Step:  AIC=24.29
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + US

Df Sum of Sq     RSS    AIC
- Education    1      1.44  405.76  23.72
- US           1      1.85  406.16  24.12
<none>                      404.31  24.29
- Income       1     76.64  480.96  91.73
- Advertising  1    146.03  550.34 145.63
- Age          1    217.59  621.91 194.53
- CompPrice    1    526.17  930.48 355.69
- ShelveLoc    2   1053.93 1458.25 533.41
- Price        1   1322.80 1727.11 603.10

Step:  AIC=23.72
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + US

Df Sum of Sq     RSS    AIC
- US           1      1.63  407.39  23.32
<none>                      405.76  23.72
- Income       1     77.87  483.62  91.94
- Advertising  1    145.30  551.06 144.15
- Age          1    217.97  623.73 193.70
- CompPrice    1    525.25  931.00 353.92
- ShelveLoc    2   1056.88 1462.64 532.61
- Price        1   1322.83 1728.58 601.44

Step:  AIC=23.32
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age

Df Sum of Sq     RSS    AIC
<none>                      407.39  23.32
- Income       1     76.68  484.07  90.30
- Age          1    219.12  626.51 193.48
- Advertising  1    234.03  641.42 202.89
- CompPrice    1    523.83  931.22 352.01
- ShelveLoc    2   1055.51 1462.90 530.68
- Price        1   1324.42 1731.81 600.18

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Coefficients:
5.47523          0.09257          0.01578          0.11590
Price    ShelveLocGood  ShelveLocMedium              Age
-0.09532          4.83567          1.95199         -0.04613

> 
> step(lm.full, direction="both")
Start:  AIC=26.82
Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Population   1      0.33  403.16  25.15
- Education    1      1.19  404.02  26.00
- Urban        1      1.23  404.06  26.04
- US           1      1.57  404.40  26.38
<none>                      402.83  26.82
- Income       1     76.16  478.99  94.09
- Advertising  1    127.14  529.97 134.54
- Age          1    217.44  620.27 197.48
- CompPrice    1    519.91  922.74 356.35
- ShelveLoc    2   1053.20 1456.03 536.80
- Price        1   1323.23 1726.06 606.85

Step:  AIC=25.15
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Urban        1      1.15  404.31  24.29
- Education    1      1.36  404.52  24.49
- US           1      1.89  405.05  25.02
<none>                      403.16  25.15
+ Population   1      0.33  402.83  26.82
- Income       1     75.94  479.10  92.18
- Advertising  1    145.38  548.54 146.32
- Age          1    218.52  621.68 196.38
- CompPrice    1    521.69  924.85 355.27
- ShelveLoc    2   1053.18 1456.34 534.89
- Price        1   1323.51 1726.67 605.00

Step:  AIC=24.29
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + US

Df Sum of Sq     RSS    AIC
- Education    1      1.44  405.76  23.72
- US           1      1.85  406.16  24.12
<none>                      404.31  24.29
+ Urban        1      1.15  403.16  25.15
+ Population   1      0.25  404.06  26.04
- Income       1     76.64  480.96  91.73
- Advertising  1    146.03  550.34 145.63
- Age          1    217.59  621.91 194.53
- CompPrice    1    526.17  930.48 355.69
- ShelveLoc    2   1053.93 1458.25 533.41
- Price        1   1322.80 1727.11 603.10

Step:  AIC=23.72
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + US

Df Sum of Sq     RSS    AIC
- US           1      1.63  407.39  23.32
<none>                      405.76  23.72
+ Education    1      1.44  404.31  24.29
+ Urban        1      1.24  404.52  24.49
+ Population   1      0.41  405.35  25.32
- Income       1     77.87  483.62  91.94
- Advertising  1    145.30  551.06 144.15
- Age          1    217.97  623.73 193.70
- CompPrice    1    525.25  931.00 353.92
- ShelveLoc    2   1056.88 1462.64 532.61
- Price        1   1322.83 1728.58 601.44

Step:  AIC=23.32
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age

Df Sum of Sq     RSS    AIC
<none>                      407.39  23.32
+ US           1      1.63  405.76  23.72
+ Education    1      1.22  406.16  24.12
+ Urban        1      1.19  406.20  24.15
+ Population   1      0.72  406.67  24.62
- Income       1     76.68  484.07  90.30
- Age          1    219.12  626.51 193.48
- Advertising  1    234.03  641.42 202.89
- CompPrice    1    523.83  931.22 352.01
- ShelveLoc    2   1055.51 1462.90 530.68
- Price        1   1324.42 1731.81 600.18

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Coefficients:
5.47523          0.09257          0.01578          0.11590
Price    ShelveLocGood  ShelveLocMedium              Age
-0.09532          4.83567          1.95199         -0.04613

> 
> stepAIC(lm.full, direction="backward")
Start:  AIC=26.82
Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Population   1      0.33  403.16  25.15
- Education    1      1.19  404.02  26.00
- Urban        1      1.23  404.06  26.04
- US           1      1.57  404.40  26.38
<none>                      402.83  26.82
- Income       1     76.16  478.99  94.09
- Advertising  1    127.14  529.97 134.54
- Age          1    217.44  620.27 197.48
- CompPrice    1    519.91  922.74 356.35
- ShelveLoc    2   1053.20 1456.03 536.80
- Price        1   1323.23 1726.06 606.85

Step:  AIC=25.15
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Urban        1      1.15  404.31  24.29
- Education    1      1.36  404.52  24.49
- US           1      1.89  405.05  25.02
<none>                      403.16  25.15
- Income       1     75.94  479.10  92.18
- Advertising  1    145.38  548.54 146.32
- Age          1    218.52  621.68 196.38
- CompPrice    1    521.69  924.85 355.27
- ShelveLoc    2   1053.18 1456.34 534.89
- Price        1   1323.51 1726.67 605.00

Step:  AIC=24.29
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + US

Df Sum of Sq     RSS    AIC
- Education    1      1.44  405.76  23.72
- US           1      1.85  406.16  24.12
<none>                      404.31  24.29
- Income       1     76.64  480.96  91.73
- Advertising  1    146.03  550.34 145.63
- Age          1    217.59  621.91 194.53
- CompPrice    1    526.17  930.48 355.69
- ShelveLoc    2   1053.93 1458.25 533.41
- Price        1   1322.80 1727.11 603.10

Step:  AIC=23.72
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + US

Df Sum of Sq     RSS    AIC
- US           1      1.63  407.39  23.32
<none>                      405.76  23.72
- Income       1     77.87  483.62  91.94
- Advertising  1    145.30  551.06 144.15
- Age          1    217.97  623.73 193.70
- CompPrice    1    525.25  931.00 353.92
- ShelveLoc    2   1056.88 1462.64 532.61
- Price        1   1322.83 1728.58 601.44

Step:  AIC=23.32
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age

Df Sum of Sq     RSS    AIC
<none>                      407.39  23.32
- Income       1     76.68  484.07  90.30
- Age          1    219.12  626.51 193.48
- Advertising  1    234.03  641.42 202.89
- CompPrice    1    523.83  931.22 352.01
- ShelveLoc    2   1055.51 1462.90 530.68
- Price        1   1324.42 1731.81 600.18

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Coefficients:
5.47523          0.09257          0.01578          0.11590         -0.09532
ShelveLocGood  ShelveLocMedium              Age
4.83567          1.95199         -0.04613

> 
> step(lm.full, direction="backward")
Start:  AIC=26.82
Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Population   1      0.33  403.16  25.15
- Education    1      1.19  404.02  26.00
- Urban        1      1.23  404.06  26.04
- US           1      1.57  404.40  26.38
<none>                      402.83  26.82
- Income       1     76.16  478.99  94.09
- Advertising  1    127.14  529.97 134.54
- Age          1    217.44  620.27 197.48
- CompPrice    1    519.91  922.74 356.35
- ShelveLoc    2   1053.20 1456.03 536.80
- Price        1   1323.23 1726.06 606.85

Step:  AIC=25.15
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + Urban + US

Df Sum of Sq     RSS    AIC
- Urban        1      1.15  404.31  24.29
- Education    1      1.36  404.52  24.49
- US           1      1.89  405.05  25.02
<none>                      403.16  25.15
- Income       1     75.94  479.10  92.18
- Advertising  1    145.38  548.54 146.32
- Age          1    218.52  621.68 196.38
- CompPrice    1    521.69  924.85 355.27
- ShelveLoc    2   1053.18 1456.34 534.89
- Price        1   1323.51 1726.67 605.00

Step:  AIC=24.29
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + US

Df Sum of Sq     RSS    AIC
- Education    1      1.44  405.76  23.72
- US           1      1.85  406.16  24.12
<none>                      404.31  24.29
- Income       1     76.64  480.96  91.73
- Advertising  1    146.03  550.34 145.63
- Age          1    217.59  621.91 194.53
- CompPrice    1    526.17  930.48 355.69
- ShelveLoc    2   1053.93 1458.25 533.41
- Price        1   1322.80 1727.11 603.10

Step:  AIC=23.72
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + US

Df Sum of Sq     RSS    AIC
- US           1      1.63  407.39  23.32
<none>                      405.76  23.72
- Income       1     77.87  483.62  91.94
- Advertising  1    145.30  551.06 144.15
- Age          1    217.97  623.73 193.70
- CompPrice    1    525.25  931.00 353.92
- ShelveLoc    2   1056.88 1462.64 532.61
- Price        1   1322.83 1728.58 601.44

Step:  AIC=23.32
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age

Df Sum of Sq     RSS    AIC
<none>                      407.39  23.32
- Income       1     76.68  484.07  90.30
- Age          1    219.12  626.51 193.48
- Advertising  1    234.03  641.42 202.89
- CompPrice    1    523.83  931.22 352.01
- ShelveLoc    2   1055.51 1462.90 530.68
- Price        1   1324.42 1731.81 600.18

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Coefficients:
5.47523          0.09257          0.01578          0.11590         -0.09532
ShelveLocGood  ShelveLocMedium              Age
4.83567          1.95199         -0.04613  

## Pick the model from stepAIC

lm.fit.01 <- lm(formula = Sales ~
CompPrice + Income +
ShelveLoc + Age,
data = Carseats)
summary(lm.fit.01)
> lm.fit.01 <- lm(formula = Sales ~ CompPrice +
Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)
> summary(lm.fit.01)

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Residuals:
Min      1Q  Median      3Q     Max
-2.7728 -0.6954  0.0282  0.6732  3.3292

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
CompPrice        0.092571   0.004123   22.45   <2e-16 ***
Income           0.015785   0.001838    8.59   <2e-16 ***
Advertising      0.115903   0.007724   15.01   <2e-16 ***
Price           -0.095319   0.002670  -35.70   <2e-16 ***
ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
Age             -0.046128   0.003177  -14.52   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.019 on 392 degrees of freedom
Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16

Compare the fitted model to full model

anova(lm.full, lm.fit.01)
> anova(lm.full, lm.fit.01)
Analysis of Variance Table

Model 1: Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US
Model 2: Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age
Res.Df    RSS Df Sum of Sq      F Pr(>F)
1    388 402.83
2    392 407.39 -4   -4.5533 1.0964  0.358
> 

## Backward elimination

drop1(lm.full, test = "F")
> drop1(lm.full, test = "F")
Single term deletions

Model:
Sales ~ CompPrice + Income + Advertising + Population + Price +
ShelveLoc + Age + Education + Urban + US
Df Sum of Sq     RSS    AIC   F value    Pr(>F)
<none>                    402.83  26.82
CompPrice    1    519.91  922.74 356.35  500.7659 < 2.2e-16 ***
Income       1     76.16  478.99  94.09   73.3537  2.58e-16 ***
Advertising  1    127.14  529.97 134.54  122.4571 < 2.2e-16 ***
Population   1      0.33  403.16  25.15    0.3149    0.5750
Price        1   1323.23 1726.06 606.85 1274.5022 < 2.2e-16 ***
ShelveLoc    2   1053.20 1456.03 536.80  507.2079 < 2.2e-16 ***
Age          1    217.44  620.27 197.48  209.4333 < 2.2e-16 ***
Education    1      1.19  404.02  26.00    1.1450    0.2853
Urban        1      1.23  404.06  26.04    1.1831    0.2774
US           1      1.57  404.40  26.38    1.5094    0.2200
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> 
drop1(update(lm.full, ~ . -Population), test = "F")
> drop1(update(lm.full, ~ . -Population), test = "F")
Single term deletions

Model:
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + Urban + US
Df Sum of Sq     RSS    AIC   F value    Pr(>F)
<none>                    403.16  25.15
CompPrice    1    521.69  924.85 355.27  503.3686 < 2.2e-16 ***
Income       1     75.94  479.10  92.18   73.2717 2.652e-16 ***
Advertising  1    145.38  548.54 146.32  140.2694 < 2.2e-16 ***
Price        1   1323.51 1726.67 605.00 1277.0276 < 2.2e-16 ***
ShelveLoc    2   1053.18 1456.34 534.89  508.0927 < 2.2e-16 ***
Age          1    218.52  621.68 196.38  210.8411 < 2.2e-16 ***
Education    1      1.36  404.52  24.49    1.3122    0.2527
Urban        1      1.15  404.31  24.29    1.1132    0.2920
US           1      1.89  405.05  25.02    1.8262    0.1774
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> 
drop1(update(lm.full, ~. - Population -Urban), test="F")
> drop1(update(lm.full, ~. - Population -Urban), test="F")
Single term deletions

Model:
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + Education + US
Df Sum of Sq     RSS    AIC   F value Pr(>F)
<none>                    404.31  24.29
CompPrice    1    526.17  930.48 355.69  507.5374 <2e-16 ***
Income       1     76.64  480.96  91.73   73.9307 <2e-16 ***
Advertising  1    146.03  550.34 145.63  140.8593 <2e-16 ***
Price        1   1322.80 1727.11 603.10 1275.9661 <2e-16 ***
ShelveLoc    2   1053.93 1458.25 533.41  508.3098 <2e-16 ***
Age          1    217.59  621.91 194.53  209.8897 <2e-16 ***
Education    1      1.44  405.76  23.72    1.3930 0.2386
US           1      1.85  406.16  24.12    1.7848 0.1823
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> 
drop1(update(lm.full, ~. - Population -Urban -Education), test="F")
> drop1(update(lm.full, ~. - Population -Urban -Education), test="F")
Single term deletions

Model:
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age + US
Df Sum of Sq     RSS    AIC   F value Pr(>F)
<none>                    405.76  23.72
CompPrice    1    525.25  931.00 353.92  506.1421 <2e-16 ***
Income       1     77.87  483.62  91.94   75.0336 <2e-16 ***
Advertising  1    145.30  551.06 144.15  140.0181 <2e-16 ***
Price        1   1322.83 1728.58 601.44 1274.7123 <2e-16 ***
ShelveLoc    2   1056.88 1462.64 532.61  509.2202 <2e-16 ***
Age          1    217.97  623.73 193.70  210.0409 <2e-16 ***
US           1      1.63  407.39  23.32    1.5693 0.2111
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
drop1(update(lm.full, ~. - Population -Urban -Education -US), test="F")
> drop1(update(lm.full, ~. - Population -Urban -Education -US), test="F")
Single term deletions

Model:
Sales ~ CompPrice + Income + Advertising + Price + ShelveLoc +
Age
Df Sum of Sq     RSS    AIC  F value    Pr(>F)
<none>                    407.39  23.32
CompPrice    1    523.83  931.22 352.01  504.047 < 2.2e-16 ***
Income       1     76.68  484.07  90.30   73.784 < 2.2e-16 ***
Advertising  1    234.03  641.42 202.89  225.192 < 2.2e-16 ***
Price        1   1324.42 1731.81 600.18 1274.400 < 2.2e-16 ***
ShelveLoc    2   1055.51 1462.90 530.68  507.822 < 2.2e-16 ***
Age          1    219.12  626.51 193.48  210.848 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
lm.fit.be <- lm(Sales ~
CompPrice + Income +
ShelveLoc + Age, data = Carseats)

summary(lm.fit.be)
> lm.fit.be <- lm(Sales ~ CompPrice +
Price + ShelveLoc +
Age, data = Carseats)
> summary(lm.fit.be)

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Residuals:
Min      1Q  Median      3Q     Max
-2.7728 -0.6954  0.0282  0.6732  3.3292

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
CompPrice        0.092571   0.004123   22.45   <2e-16 ***
Income           0.015785   0.001838    8.59   <2e-16 ***
Advertising      0.115903   0.007724   15.01   <2e-16 ***
Price           -0.095319   0.002670  -35.70   <2e-16 ***
ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
Age             -0.046128   0.003177  -14.52   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.019 on 392 degrees of freedom
Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16

> 
lm.fit.01 <- lm(formula = Sales ~
CompPrice + Income +
ShelveLoc + Age,
data = Carseats)
summary(lm.fit.01)
> lm.fit.01 <- lm(formula = Sales ~ CompPrice +
Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)
> summary(lm.fit.01)

Call:
lm(formula = Sales ~ CompPrice + Income + Advertising + Price +
ShelveLoc + Age, data = Carseats)

Residuals:
Min      1Q  Median      3Q     Max
-2.7728 -0.6954  0.0282  0.6732  3.3292

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)      5.475226   0.505005   10.84   <2e-16 ***
CompPrice        0.092571   0.004123   22.45   <2e-16 ***
Income           0.015785   0.001838    8.59   <2e-16 ***
Advertising      0.115903   0.007724   15.01   <2e-16 ***
Price           -0.095319   0.002670  -35.70   <2e-16 ***
ShelveLocGood    4.835675   0.152499   31.71   <2e-16 ***
ShelveLocMedium  1.951993   0.125375   15.57   <2e-16 ***
Age             -0.046128   0.003177  -14.52   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.019 on 392 degrees of freedom
Multiple R-squared:  0.872,	Adjusted R-squared:  0.8697
F-statistic: 381.4 on 7 and 392 DF,  p-value: < 2.2e-16

1)
a+b+c+d)~(b+e