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summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

Including Plots

You can also embed plots, for example:

plot(cars)

mo=lm(dist~speed,data=cars)
summary(mo)
## 
## 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: 1.49e-12
anova(mo)
## Analysis of Variance Table
## 
## Response: dist
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## speed      1  21186 21185.5  89.567 1.49e-12 ***
## Residuals 48  11354   236.5                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

If we take a random sampling of n=25 with uniform distribution to split the data in class we shold not found any difference in group except 5% of the time (Well for the reader is it really true with an uniform sampling?)

set.seed(999)
A=sample(50,25,replace=F)
hist(A)#not looking Unif but it is n=50!!!

ds1=cars[A,]
ds1$cat="class1"
ds2=cars[-A,]
ds2$cat="class2"
new=rbind(ds1,ds2)
new
##    speed dist    cat
## 27    16   32 class1
## 4      7   22 class1
## 7     10   18 class1
## 41    20   52 class1
## 14    12   24 class1
## 1      4    2 class1
## 10    11   17 class1
## 22    14   60 class1
## 35    18   84 class1
## 23    14   80 class1
## 15    12   28 class1
## 5      8   16 class1
## 48    24   93 class1
## 19    13   46 class1
## 18    13   34 class1
## 45    23   54 class1
## 37    19   46 class1
## 12    12   14 class1
## 9     10   34 class1
## 31    17   50 class1
## 21    14   36 class1
## 47    24   92 class1
## 39    20   32 class1
## 25    15   26 class1
## 26    15   54 class1
## 2      4   10 class2
## 3      7    4 class2
## 6      9   10 class2
## 8     10   26 class2
## 11    11   28 class2
## 13    12   20 class2
## 16    13   26 class2
## 17    13   34 class2
## 20    14   26 class2
## 24    15   20 class2
## 28    16   40 class2
## 29    17   32 class2
## 30    17   40 class2
## 32    18   42 class2
## 33    18   56 class2
## 34    18   76 class2
## 36    19   36 class2
## 38    19   68 class2
## 40    20   48 class2
## 42    20   56 class2
## 43    20   64 class2
## 44    22   66 class2
## 46    24   70 class2
## 49    24  120 class2
## 50    25   85 class2
#check summary
summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00
summary(new)
##      speed           dist            cat           
##  Min.   : 4.0   Min.   :  2.00   Length:50         
##  1st Qu.:12.0   1st Qu.: 26.00   Class :character  
##  Median :15.0   Median : 36.00   Mode  :character  
##  Mean   :15.4   Mean   : 42.98                     
##  3rd Qu.:19.0   3rd Qu.: 56.00                     
##  Max.   :25.0   Max.   :120.00
new$cat=factor(new$cat)
boxplot(new$dist~new$cat)

boxplot(new$speed~new$cat)#num balanced not diff

interaction.plot(cut(new$dist,3),new$cat,new$speed)##Interaction dissmissed

t.test(new$dist~new$cat)
## 
##  Welch Two Sample t-test
## 
## data:  new$dist by new$cat
## t = -0.30991, df = 47.685, p-value = 0.758
## alternative hypothesis: true difference in means between group class1 and group class2 is not equal to 0
## 95 percent confidence interval:
##  -17.07448  12.51448
## sample estimates:
## mean in group class1 mean in group class2 
##                41.84                44.12
t.test(new$speed~new$cat)
## 
##  Welch Two Sample t-test
## 
## data:  new$speed by new$cat
## t = -1.0714, df = 47.91, p-value = 0.2893
## alternative hypothesis: true difference in means between group class1 and group class2 is not equal to 0
## 95 percent confidence interval:
##  -4.602664  1.402664
## sample estimates:
## mean in group class1 mean in group class2 
##                 14.6                 16.2
mo1=lm(new$speed~new$cat+new$dist)#set factor in 2pos
summary(mo1)
## 
## Call:
## lm(formula = new$speed ~ new$cat + new$dist)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.8771 -2.3332  0.3476  2.1304  7.0186 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    7.71754    0.95797   8.056 2.10e-10 ***
## new$catclass2  1.22495    0.88507   1.384    0.173    
## new$dist       0.16449    0.01735   9.482 1.73e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.126 on 47 degrees of freedom
## Multiple R-squared:  0.6647, Adjusted R-squared:  0.6505 
## F-statistic:  46.6 on 2 and 47 DF,  p-value: 7.021e-12
summary(aov((mo1)))
##             Df Sum Sq Mean Sq F value   Pr(>F)    
## new$cat      1   32.0    32.0   3.275   0.0768 .  
## new$dist     1  878.7   878.7  89.916 1.73e-12 ***
## Residuals   47  459.3     9.8                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

as exepcted selecting a SRS and split equally the data n=50 into 2 groups show no change of effect in mean diff

Lets explore how nbalancing class affect model preformance

new10=new[new$cat=="class1",]
AA=sample(10,10,replace=FALSE)
new100=new10[AA,]
new200=new[new$cat=="class2", ]

df1=rbind(new100,new200)
df1
##    speed dist    cat
## 23    14   80 class1
## 41    20   52 class1
## 22    14   60 class1
## 27    16   32 class1
## 14    12   24 class1
## 7     10   18 class1
## 4      7   22 class1
## 10    11   17 class1
## 1      4    2 class1
## 35    18   84 class1
## 2      4   10 class2
## 3      7    4 class2
## 6      9   10 class2
## 8     10   26 class2
## 11    11   28 class2
## 13    12   20 class2
## 16    13   26 class2
## 17    13   34 class2
## 20    14   26 class2
## 24    15   20 class2
## 28    16   40 class2
## 29    17   32 class2
## 30    17   40 class2
## 32    18   42 class2
## 33    18   56 class2
## 34    18   76 class2
## 36    19   36 class2
## 38    19   68 class2
## 40    20   48 class2
## 42    20   56 class2
## 43    20   64 class2
## 44    22   66 class2
## 46    24   70 class2
## 49    24  120 class2
## 50    25   85 class2
dim(df1)
## [1] 35  3
t.test(df1$dist~df1$cat)
## 
##  Welch Two Sample t-test
## 
## data:  df1$dist by df1$cat
## t = -0.48092, df = 16.012, p-value = 0.6371
## alternative hypothesis: true difference in means between group class1 and group class2 is not equal to 0
## 95 percent confidence interval:
##  -27.14701  17.10701
## sample estimates:
## mean in group class1 mean in group class2 
##                39.10                44.12
mo2=lm(speed~dist+cat,data=df1)#set factor in 2pos
summary(mo2)
## 
## Call:
## lm(formula = speed ~ dist + cat, data = df1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.7848 -2.2793  0.4539  2.2242  5.3527 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.39448    1.20995   5.285 8.67e-06 ***
## dist         0.15871    0.01913   8.298 1.76e-09 ***
## catclass2    2.80328    1.12950   2.482   0.0185 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.008 on 32 degrees of freedom
## Multiple R-squared:  0.7119, Adjusted R-squared:  0.6939 
## F-statistic: 39.54 on 2 and 32 DF,  p-value: 2.249e-09
head(model.matrix(mo1),7)
##   (Intercept) new$catclass2 new$dist
## 1           1             0       32
## 2           1             0       22
## 3           1             0       18
## 4           1             0       52
## 5           1             0       24
## 6           1             0        2
## 7           1             0       17
anova(mo2)
## Analysis of Variance Table
## 
## Response: speed
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## dist       1 659.75  659.75 72.9256 9.286e-10 ***
## cat        1  55.73   55.73  6.1597   0.01851 *  
## Residuals 32 289.50    9.05                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(mo1)
## Analysis of Variance Table
## 
## Response: new$speed
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## new$cat    1   32.0   32.00  3.2745  0.07676 .  
## new$dist   1  878.7  878.70 89.9163 1.73e-12 ***
## Residuals 47  459.3    9.77                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
contrasts(df1$cat)##attention sum
##        class2
## class1      0
## class2      1
df1
##    speed dist    cat
## 23    14   80 class1
## 41    20   52 class1
## 22    14   60 class1
## 27    16   32 class1
## 14    12   24 class1
## 7     10   18 class1
## 4      7   22 class1
## 10    11   17 class1
## 1      4    2 class1
## 35    18   84 class1
## 2      4   10 class2
## 3      7    4 class2
## 6      9   10 class2
## 8     10   26 class2
## 11    11   28 class2
## 13    12   20 class2
## 16    13   26 class2
## 17    13   34 class2
## 20    14   26 class2
## 24    15   20 class2
## 28    16   40 class2
## 29    17   32 class2
## 30    17   40 class2
## 32    18   42 class2
## 33    18   56 class2
## 34    18   76 class2
## 36    19   36 class2
## 38    19   68 class2
## 40    20   48 class2
## 42    20   56 class2
## 43    20   64 class2
## 44    22   66 class2
## 46    24   70 class2
## 49    24  120 class2
## 50    25   85 class2
relevel(df1$cat,ref="class2")
##  [1] class1 class1 class1 class1 class1 class1 class1 class1 class1 class1
## [11] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## [21] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## [31] class2 class2 class2 class2 class2
## Levels: class2 class1
library(effects)
## Warning: le package 'effects' a été compilé avec la version R 4.5.3
## Le chargement a nécessité le package : carData
## lattice theme set by effectsTheme()
## See ?effectsTheme for details.
effect("cat",mo2)
## 
##  cat effect
## cat
##   class1   class2 
## 13.16909 15.97237
7.25+0.19*mean(df1$dist)
## [1] 15.36029
library(effsize)
## Warning: le package 'effsize' a été compilé avec la version R 4.5.3
cohen.d(speed~cat,data=df1)#to low effesize 
## 
## Cohen's d
## 
## d estimate: -0.6846472 (medium)
## 95 percent confidence interval:
##       lower       upper 
## -1.46388597  0.09459158
new10=new[new$cat=="class1",]
AAA=sample(10,5,replace=FALSE)
new1000=new10[AAA,]
new1000
##    speed dist    cat
## 4      7   22 class1
## 22    14   60 class1
## 35    18   84 class1
## 23    14   80 class1
## 41    20   52 class1
new200=new[new$cat=="class2", ]

df2=rbind(new1000,new200)
df2
##    speed dist    cat
## 4      7   22 class1
## 22    14   60 class1
## 35    18   84 class1
## 23    14   80 class1
## 41    20   52 class1
## 2      4   10 class2
## 3      7    4 class2
## 6      9   10 class2
## 8     10   26 class2
## 11    11   28 class2
## 13    12   20 class2
## 16    13   26 class2
## 17    13   34 class2
## 20    14   26 class2
## 24    15   20 class2
## 28    16   40 class2
## 29    17   32 class2
## 30    17   40 class2
## 32    18   42 class2
## 33    18   56 class2
## 34    18   76 class2
## 36    19   36 class2
## 38    19   68 class2
## 40    20   48 class2
## 42    20   56 class2
## 43    20   64 class2
## 44    22   66 class2
## 46    24   70 class2
## 49    24  120 class2
## 50    25   85 class2
dim(df2)
## [1] 30  3
cohen.d(speed~cat,data=df2)
## 
## Cohen's d
## 
## d estimate: -0.2998578 (small)
## 95 percent confidence interval:
##      lower      upper 
## -1.3064964  0.7067808
table (df1$cat)
## 
## class1 class2 
##     10     25
table(df2$cat)
## 
## class1 class2 
##      5     25
t.test(df2$dist~df2$cat)
## 
##  Welch Two Sample t-test
## 
## data:  df2$dist by df2$cat
## t = 1.2498, df = 6.0515, p-value = 0.2575
## alternative hypothesis: true difference in means between group class1 and group class2 is not equal to 0
## 95 percent confidence interval:
##  -14.7658  45.7258
## sample estimates:
## mean in group class1 mean in group class2 
##                59.60                44.12
mo3=lm(df2$speed~df2$dist+df2$cat)#set factor in 2pos
#2 predictors 4 regressors
summary(mo3)
## 
## Call:
## lm(formula = df2$speed ~ df2$dist + df2$cat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5378 -1.5335 -0.1822  2.1179  6.6612 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    4.70941    1.85140   2.544   0.0170 *  
## df2$dist       0.16595    0.02129   7.794 2.22e-08 ***
## df2$catclass2  4.16890    1.51303   2.755   0.0104 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.014 on 27 degrees of freedom
## Multiple R-squared:  0.6963, Adjusted R-squared:  0.6738 
## F-statistic: 30.96 on 2 and 27 DF,  p-value: 1.029e-07
anova(mo3)
## Analysis of Variance Table
## 
## Response: df2$speed
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## df2$dist   1 493.57  493.57 54.3227 6.284e-08 ***
## df2$cat    1  68.98   68.98  7.5919   0.01037 *  
## Residuals 27 245.32    9.09                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(mo1)
## Analysis of Variance Table
## 
## Response: new$speed
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## new$cat    1   32.0   32.00  3.2745  0.07676 .  
## new$dist   1  878.7  878.70 89.9163 1.73e-12 ***
## Residuals 47  459.3    9.77                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
confint(mo2)
##                 2.5 %    97.5 %
## (Intercept) 3.9298981 8.8590599
## dist        0.1197491 0.1976688
## catclass2   0.5025584 5.1040035
confint(mo3)#factor contain zero value mean no diff in groups
##                   2.5 %   97.5 %
## (Intercept)   0.9106504 8.508178
## df2$dist      0.1222599 0.209639
## df2$catclass2 1.0644188 7.273376
#However for teh continuous variable the beta coef is unbiased

library(car)
Anova(mo3)
## Anova Table (Type II tests)
## 
## Response: df2$speed
##           Sum Sq Df F value    Pr(>F)    
## df2$dist  551.88  1 60.7406 2.218e-08 ***
## df2$cat    68.98  1  7.5919   0.01037 *  
## Residuals 245.32 27                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mo2)
## Anova Table (Type II tests)
## 
## Response: speed
##           Sum Sq Df F value    Pr(>F)    
## dist      622.90  1 68.8529 1.764e-09 ***
## cat        55.73  1  6.1597   0.01851 *  
## Residuals 289.50 32                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
mo33=lm(df2$speed~df2$dist+df2$cat-1)#set factor in 2pos
summary(mo33)
## 
## Call:
## lm(formula = df2$speed ~ df2$dist + df2$cat - 1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5378 -1.5335 -0.1822  2.1179  6.6612 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## df2$dist       0.16595    0.02129   7.794 2.22e-08 ***
## df2$catclass1  4.70941    1.85140   2.544    0.017 *  
## df2$catclass2  8.87831    1.11624   7.954 1.50e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.014 on 27 degrees of freedom
## Multiple R-squared:  0.9709, Adjusted R-squared:  0.9676 
## F-statistic: 300.1 on 3 and 27 DF,  p-value: < 2.2e-16
mo22=lm(df1$speed~df1$dist+df1$cat-1)#set factor in 2pos
summary(mo22)
## 
## Call:
## lm(formula = df1$speed ~ df1$dist + df1$cat - 1)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.7848 -2.2793  0.4539  2.2242  5.3527 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## df1$dist       0.15871    0.01913   8.298 1.76e-09 ***
## df1$catclass1  6.39448    1.20995   5.285 8.67e-06 ***
## df1$catclass2  9.19776    1.03634   8.875 3.86e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.008 on 32 degrees of freedom
## Multiple R-squared:  0.968,  Adjusted R-squared:  0.9651 
## F-statistic: 323.2 on 3 and 32 DF,  p-value: < 2.2e-16
summary(df1)
##      speed            dist            cat    
##  Min.   : 4.00   Min.   :  2.00   class1:10  
##  1st Qu.:11.50   1st Qu.: 23.00   class2:25  
##  Median :16.00   Median : 36.00              
##  Mean   :15.17   Mean   : 42.69              
##  3rd Qu.:19.00   3rd Qu.: 62.00              
##  Max.   :25.00   Max.   :120.00
b333=bootCase(mo33,B=100)##boot the difference from group refs included in Intercept + ref base Factor
## Warning in bootCase(mo33, B = 100): 'bootCase' est obsolète.
## Utilisez plutôt ‘Boot’.
## Voir help("Deprecated") et help("car-deprecated").
b333
##         df2$dist df2$catclass1 df2$catclass2
##   [1,] 0.1794679    2.03547383      7.922805
##   [2,] 0.1525650    4.95861111      9.790170
##   [3,] 0.1597958    2.89674392      9.377126
##   [4,] 0.2316686    2.07990940      5.691982
##   [5,] 0.2122395    1.00174398      7.189193
##   [6,] 0.1974037   -1.00267948      7.401829
##   [7,] 0.2107547    1.12331900      6.933244
##   [8,] 0.1741874    6.39387772      8.725137
##   [9,] 0.1496360    7.42087092     10.478998
##  [10,] 0.1383398    6.26285680      9.492557
##  [11,] 0.1740466    3.52245643      8.668740
##  [12,] 0.1621869    4.34046383      9.410306
##  [13,] 0.1520176    5.25069731      9.694565
##  [14,] 0.2017440    0.07792311      7.718985
##  [15,] 0.1805615    4.36537644      8.930331
##  [16,] 0.1484325    5.43332338     10.292358
##  [17,] 0.2103240    2.89489077      7.795451
##  [18,] 0.1741322    3.33071729      7.832398
##  [19,] 0.1790671    5.01527278      8.047275
##  [20,] 0.2386351    3.35232057      5.460759
##  [21,] 0.1473025    2.95231426      8.745947
##  [22,] 0.1842429    1.38736404      7.748151
##  [23,] 0.1663476    4.01914498      8.876395
##  [24,] 0.1714748    3.63458255      9.365897
##  [25,] 0.1509925    5.92854162      9.573448
##  [26,] 0.1482048    6.22497521      9.821105
##  [27,] 0.1627985    5.05780032      9.026743
##  [28,] 0.2147943    4.97330139      7.505449
##  [29,] 0.1640591    5.81659997      8.911381
##  [30,] 0.1917272    3.21514677      7.830910
##  [31,] 0.1533464    7.82618354      9.634129
##  [32,] 0.2069990    3.09085690      7.575439
##  [33,] 0.1484405    4.93210023      9.736840
##  [34,] 0.1983542   -0.54597276      7.945242
##  [35,] 0.1736795    2.07609750      9.035166
##  [36,] 0.1442858    6.54428108     11.212204
##  [37,] 0.1522285    8.47520380     10.258273
##  [38,] 0.1681385    4.57894644      7.903836
##  [39,] 0.1947157    1.27640172      7.620605
##  [40,] 0.1485393    7.10224826     10.164132
##  [41,] 0.1762244    5.16966201      7.511630
##  [42,] 0.1690942    4.32468628      8.595320
##  [43,] 0.1418308    5.49015287      9.547000
##  [44,] 0.1627499    5.90363265      8.736140
##  [45,] 0.1781420    3.05100785      8.055174
##  [46,] 0.2104260    5.21614505      7.220641
##  [47,] 0.2000868    2.92708255      6.885641
##  [48,] 0.1916766    0.03828403      8.203846
##  [49,] 0.1596314    5.20971765      9.865702
##  [50,] 0.1947225    3.96238807      7.038784
##  [51,] 0.2401127   -1.97899540      6.265057
##  [52,] 0.1698117    6.64526460      8.585646
##  [53,] 0.1522932    6.25323237      9.612169
##  [54,] 0.1514379    3.88759349      8.731221
##  [55,] 0.1853041    3.83622870      7.484249
##  [56,] 0.2082741    1.46147147      6.706531
##  [57,] 0.2166798    4.09098773      6.837993
##  [58,] 0.1876232    4.68269350      7.486792
##  [59,] 0.1551044    4.05696223      9.262200
##  [60,] 0.1452890    7.41092759      9.237898
##  [61,] 0.2525874    3.76695381      5.665249
##  [62,] 0.1902535    2.01870539      8.249077
##  [63,] 0.1620395    2.91653105      8.791381
##  [64,] 0.1980050    4.57173784      6.609403
##  [65,] 0.1887779    7.68014139      7.704727
##  [66,] 0.2016667    3.38732897      7.845264
##  [67,] 0.1703523    5.40198801      9.341245
##  [68,] 0.1245849    6.05535734     10.734051
##  [69,] 0.1715019    5.92626384      9.636816
##  [70,] 0.1889394    6.41478887      8.353025
##  [71,] 0.1469563    3.94958480      9.768676
##  [72,] 0.1414941    4.81491931     10.494935
##  [73,] 0.1500589    6.67618048      9.375178
##  [74,] 0.2114311    0.77696136      7.510925
##  [75,] 0.2152703    2.39767656      6.997503
##  [76,] 0.1971218    0.91298527      6.637885
##  [77,] 0.1514869    5.58447425      9.106526
##  [78,] 0.2106376    0.30855759      7.149700
##  [79,] 0.1513224    5.33348680      9.499346
##  [80,] 0.2168309    1.96967499      6.613439
##  [81,] 0.1856111    1.64366518      8.788079
##  [82,] 0.1521435    4.40223633      9.364883
##  [83,] 0.1745550    0.03560392      8.403041
##  [84,] 0.1587288    3.09531108      9.849573
##  [85,] 0.1118585    6.57150187     11.989201
##  [86,] 0.2112594    6.34962595      7.368221
##  [87,] 0.1503880    6.65485507     10.398554
##  [88,] 0.2091000    5.22127933      7.638556
##  [89,] 0.1505082    6.65891120      9.086483
##  [90,] 0.1960783    9.80392722      8.025915
##  [91,] 0.1402217    7.78780533     10.464775
##  [92,] 0.1479972    5.54820840     10.268426
##  [93,] 0.1399171    9.48563656     10.507376
##  [94,] 0.1853554    1.65747480      9.043277
##  [95,] 0.1890063    0.32076168      7.038208
##  [96,] 0.2166389    1.92818563      7.022349
##  [97,] 0.2236875   -0.05287408      6.540347
##  [98,] 0.1795098    1.97313898      7.986795
##  [99,] 0.1741087    5.30572589      8.014695
## [100,] 0.1977385    2.66961217      7.579386
## attr(,"class")
## [1] "bootCase" "matrix"  
## attr(,"pointEstimate")
##      df2$dist df2$catclass1 df2$catclass2 
##     0.1659494     4.7094140     8.8783112
hist(b333[,3])

quantile(b333[,3],c(0.05,0.95))
##       5%      95% 
##  6.60595 10.47979

Intercept: The intercept shifts to reflect the mean of the larger baseline group.Standard Errors: Small groups get high standard errors. The estimates for rare classes have high uncertainty.Statistical Power: You lose power to find true effects in the small class.Unbiased Slopes: The estimated slope for the effect remains mathematically correct on average.Statistical ConsequencesHigh Variance: Estimates bounce around a lot across different samples.Wide Confidence Intervals: You cannot be precise about the true value for the rare group.P-values: False negatives increase because tests lack the power to cross the significance threshold.Would you like me to explain how to fix this using re-sampling methods, weighted regression, or changing the reference category This can lead to issues where the model might be biased towards predicting the majority class, thus neglecting the minority classes. In such cases, the performance metrics can be misleading, especially if the minority classes are of particular interest.

A better way to see is to set the factor at SUM contrast otherwise the effects is captured by the intercept with ctrst treatment:

contrasts(df2$cat)=contr.sum(2)
df2$cat
##  [1] class1 class1 class1 class1 class1 class2 class2 class2 class2 class2
## [11] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## [21] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## attr(,"contrasts")
##        [,1]
## class1    1
## class2   -1
## Levels: class1 class2
mo333=lm(df2$speed~df2$dist+df2$cat)#set factor in 2pos
summary(mo333)#class1 is -075 and class 2 is +0.7534 mean is 7.35
## 
## Call:
## lm(formula = df2$speed ~ df2$dist + df2$cat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.5378 -1.5335 -0.1822  2.1179  6.6612 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  6.79386    1.32836   5.114 2.24e-05 ***
## df2$dist     0.16595    0.02129   7.794 2.22e-08 ***
## df2$cat1    -2.08445    0.75651  -2.755   0.0104 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.014 on 27 degrees of freedom
## Multiple R-squared:  0.6963, Adjusted R-squared:  0.6738 
## F-statistic: 30.96 on 2 and 27 DF,  p-value: 1.029e-07
contrasts(df1$cat)=contr.sum(2)
df1$cat
##  [1] class1 class1 class1 class1 class1 class1 class1 class1 class1 class1
## [11] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## [21] class2 class2 class2 class2 class2 class2 class2 class2 class2 class2
## [31] class2 class2 class2 class2 class2
## attr(,"contrasts")
##        [,1]
## class1    1
## class2   -1
## Levels: class1 class2
mo222=lm(df1$speed~df1$dist+df1$cat)#set factor in 2pos
head(model.matrix(mo222),18)
##    (Intercept) df1$dist df1$cat1
## 1            1       80        1
## 2            1       52        1
## 3            1       60        1
## 4            1       32        1
## 5            1       24        1
## 6            1       18        1
## 7            1       22        1
## 8            1       17        1
## 9            1        2        1
## 10           1       84        1
## 11           1       10       -1
## 12           1        4       -1
## 13           1       10       -1
## 14           1       26       -1
## 15           1       28       -1
## 16           1       20       -1
## 17           1       26       -1
## 18           1       34       -1
summary(mo222)#class1 is -075 and class 2 is +0.7534 from mean distanceis 7.35
## 
## Call:
## lm(formula = df1$speed ~ df1$dist + df1$cat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.7848 -2.2793  0.4539  2.2242  5.3527 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.79612    0.97470   7.998 3.94e-09 ***
## df1$dist     0.15871    0.01913   8.298 1.76e-09 ***
## df1$cat1    -1.40164    0.56475  -2.482   0.0185 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.008 on 32 degrees of freedom
## Multiple R-squared:  0.7119, Adjusted R-squared:  0.6939 
## F-statistic: 39.54 on 2 and 32 DF,  p-value: 2.249e-09
Anova(mo333,type="III")
## Anova Table (Type III tests)
## 
## Response: df2$speed
##             Sum Sq Df F value    Pr(>F)    
## (Intercept) 237.67  1 26.1580 2.239e-05 ***
## df2$dist    551.88  1 60.7406 2.218e-08 ***
## df2$cat      68.98  1  7.5919   0.01037 *  
## Residuals   245.32 27                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Anova(mo222,type="III")
## Anova Table (Type III tests)
## 
## Response: df1$speed
##             Sum Sq Df F value    Pr(>F)    
## (Intercept) 578.78  1 63.9759 3.942e-09 ***
## df1$dist    622.90  1 68.8529 1.764e-09 ***
## df1$cat      55.73  1  6.1597   0.01851 *  
## Residuals   289.50 32                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

INFERENCE TREES

library(partykit)
## Warning: le package 'partykit' a été compilé avec la version R 4.5.3
## Le chargement a nécessité le package : grid
## Le chargement a nécessité le package : libcoin
## Warning: le package 'libcoin' a été compilé avec la version R 4.5.3
## Le chargement a nécessité le package : mvtnorm
X=ctree(df1$speed~df1$dist+df1$cat)
tapply(df1$dist,df1$cat,mean)
## class1 class2 
##  39.10  44.12
plot(X)

XX=ctree(df2$speed~df2$dist+df2$cat)
plot(XX)

tapply(df2$dist,df2$cat,mean)
## class1 class2 
##  59.60  44.12
##tree should be avoided for too simple regression pattern like ANCOVA
tapply(df1$dist,df1$cat,mean)
## class1 class2 
##  39.10  44.12
#trees act a litle bit as seqeuential ANova