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====== Regression with two categorical variables (IVs): Two way anova  ======
https://advstats.psychstat.org/book/mregression/catpredictor.php#example-4-regression-with-one-categorical-and-one-continuous-predictors-ancova
<code>
college <- read.csv("http://commres.net/wiki/_media/r/college.csv")
attach(college)
str(college)
head(college)

salary <- salary / 1000

public<-factor(public, c(0,1), labels=c('Private', 'Public'))
location<-factor(location, c(1,2,3,4), labels=c('S', 'MW','NE', 'W'))

m1 <- lm(salary~public+location)
m2 <- lm(salary~public*location)

summary(m1)
summary(m2)

a.m2 <- aov(salary~public*location)
summary(a.m2)

interaction.plot(x.factor = location,
                 trace.factor = public,
                 response = salary,
                 fun = median, 
                 ylab = "salary", 
                 xlab = "location", 
                 col=c("red", "blue"),
                 lty = 1, 
                 lwd=2, 
                 trace.label="public")
</code>
====== output ======
<code>
####################
> college <- read.csv("http://commres.net/wiki/_media/r/college.csv")
> attach(college)
The following object is masked from acne.re (pos = 36):

    id

The following object is masked from acne.re (pos = 37):

    id

> str(college)
'data.frame':	85 obs. of  6 variables:
 $ id      : int  1 2 3 4 5 6 7 8 9 10 ...
 $ name    : chr  "Massachusetts Institute of Technology (MIT)" "Harvard University" "Dartmouth College" "Princeton University" ...
 $ salary  : num  119000 121000 123000 123000 110000 112000 111000 117000 111000 104000 ...
 $ cost    : int  189300 189600 188400 188700 194200 181900 191300 187600 180400 184900 ...
 $ public  : int  0 0 0 0 0 0 0 0 0 0 ...
 $ location: int  3 3 3 3 3 2 3 1 3 3 ...
> head(college)
  id                                        name salary   cost public location
1  1 Massachusetts Institute of Technology (MIT) 119000 189300      0        3
2  2                          Harvard University 121000 189600      0        3
3  3                           Dartmouth College 123000 188400      0        3
4  4                        Princeton University 123000 188700      0        3
5  5                             Yale University 110000 194200      0        3
6  6                    University of Notre Dame 112000 181900      0        2
> salary <- salary / 1000
> public<-factor(public, c(0,1), labels=c('Private', 'Public'))
> location<-factor(location, c(1,2,3,4), labels=c('S', 'MW','NE', 'W'))
> m1 <- lm(salary~public+location)
> m2 <- lm(salary~public*location)
> summary(m1)

Call:
lm(formula = salary ~ public + location)

Residuals:
   Min     1Q Median     3Q    Max 
-17.15  -4.75  -0.35   2.85  31.67 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)     99.56       1.90   52.30  < 2e-16 ***
publicPublic    -7.30       1.83   -3.99  0.00014 ***
locationMW      -2.40       2.52   -0.95  0.34366    
locationNE       8.79       2.39    3.68  0.00042 ***
locationW      -10.93       2.49   -4.39  3.4e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.86 on 80 degrees of freedom
Multiple R-squared:  0.587,	Adjusted R-squared:  0.566 
F-statistic: 28.4 on 4 and 80 DF,  p-value: 1.06e-14

> summary(m2)

Call:
lm(formula = salary ~ public * location)

Residuals:
   Min     1Q Median     3Q    Max 
-11.19  -4.88  -0.65   2.49  27.55 

Coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)               99.492      2.006   49.60  < 2e-16 ***
publicPublic              -7.142      3.172   -2.25   0.0272 *  
locationMW                -1.982      2.975   -0.67   0.5074    
locationNE                11.393      2.537    4.49  2.5e-05 ***
locationW                -17.554      3.172   -5.53  4.1e-07 ***
publicPublic:locationMW   -0.913      4.500   -0.20   0.8398    
publicPublic:locationNE  -12.843      4.704   -2.73   0.0078 ** 
publicPublic:locationW    10.650      4.451    2.39   0.0192 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 6.95 on 77 degrees of freedom
Multiple R-squared:  0.689,	Adjusted R-squared:  0.661 
F-statistic: 24.4 on 7 and 77 DF,  p-value: <2e-16
>
>
> a.m2 <- aov(salary~public*location)
> summary(a.m2)
                Df Sum Sq Mean Sq F value  Pr(>F)    
public           1   2970    2970   61.50 2.0e-11 ***
location         3   4057    1352   28.01 2.4e-12 ***
public:location  3   1225     408    8.46 6.3e-05 ***
Residuals       77   3718      48                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
>
> interaction.plot(x.factor = location,
+                  trace.factor = public,
+                  response = salary,
+                  fun = median, 
+                  ylab = "salary", 
+                  xlab = "location", 
+                  col=c("red", "blue"),
+                  lty = 1, 
+                  lwd=2, 
+                  trace.label="public")
> 
</code>
{{:r:pasted:20230602-013149.png}}

====== Regression with a continous + a categorical variables: ANCOVA ======
<code>
summary(mod<-lm(salary~cost + location))
anova(mod)
</code>

<code>
> summary(mod<-lm(salary~cost + location))

Call:
lm(formula = salary ~ cost + location)

Residuals:
    Min      1Q  Median      3Q     Max 
-20.244  -4.933  -0.572   3.162  29.939 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)  8.78e+01   3.10e+00   28.34  < 2e-16 ***
cost         6.05e-05   1.73e-05    3.50  0.00076 ***
locationMW  -2.80e+00   2.57e+00   -1.09  0.27885    
locationNE   9.23e+00   2.42e+00    3.81  0.00027 ***
locationW   -1.05e+01   2.57e+00   -4.10  0.00010 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.02 on 80 degrees of freedom
Multiple R-squared:  0.571,	Adjusted R-squared:  0.549 
F-statistic: 26.6 on 4 and 80 DF,  p-value: 4.96e-14

> anova(mod)
Analysis of Variance Table

Response: salary
          Df Sum Sq Mean Sq F value  Pr(>F)    
cost       1   2757    2757    42.9 5.1e-09 ***
location   3   4072    1357    21.1 3.6e-10 ***
Residuals 80   5141      64                    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
</code>

<code>
(Intercept)  8.78e+01   3.10e+00   28.34  < 2e-16 ***
cost         6.05e-05   1.73e-05    3.50  0.00076 ***
locationMW  -2.80e+00   2.57e+00   -1.09  0.27885    
locationNE   9.23e+00   2.42e+00    3.81  0.00027 ***
locationW   -1.05e+01   2.57e+00   -4.10  0.00010 ***

</code>

''y hat ~ 87.8 + .00006*cost - 2.8 MW + 9.23 NE - 10.5 E ''

  * S, MW, NE, W 중에서 S가 default
    * y hat ~ 87.8 + .00006*cost
  * MW: 
    * y hat ~ 87.8 - 2.8 + .00006*cost
    * y hat ~ 85 + .00006*cost
  * NE:
    * y hat ~ 87.8 - 9.23 + .00006*cost
    * y hat ~ 78.57 + .00006*cost
  * E:
    * y hat ~ 87.8 - 10.5 + .00006*cost
    * y hat ~ 77.3 + .00006*cost

<code>
> summary(mod2<-lm(salary~cost * location))

Call:
lm(formula = salary ~ cost * location)

Residuals:
     Min       1Q   Median       3Q      Max 
-17.1126  -5.6241  -0.8152   2.6401  28.4861 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      8.863e+01  4.995e+00  17.744   <2e-16 ***
cost             5.480e-05  3.211e-05   1.707   0.0919 .  
locationMW      -5.526e+00  7.213e+00  -0.766   0.4460    
locationNE      -2.866e+00  7.383e+00  -0.388   0.6989    
locationW       -1.876e+00  6.509e+00  -0.288   0.7740    
cost:locationMW  1.885e-05  4.681e-05   0.403   0.6882    
cost:locationNE  7.480e-05  4.533e-05   1.650   0.1030    
cost:locationW  -7.815e-05  4.665e-05  -1.675   0.0980 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.644 on 77 degrees of freedom
Multiple R-squared:  0.6241,	Adjusted R-squared:  0.5899 
F-statistic: 18.26 on 7 and 77 DF,  p-value: 4.313e-14

> anova(mod2)
Analysis of Variance Table

Response: salary
              Df Sum Sq Mean Sq F value    Pr(>F)    
cost           1 2757.0 2756.97  47.183 1.470e-09 ***
location       3 4071.8 1357.26  23.228 8.276e-11 ***
cost:location  3  641.6  213.86   3.660   0.01596 *  
Residuals     77 4499.2   58.43                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
</code>