library(effects)    # for Arrests data
library(car)        # for Anova()
library(MASS)       # for stepAIC
data(Arrests)
Arrests$year <- as.factor(Arrests$year)

all main effects

arrests.mod1 <- glm(released ~ ., family=binomial, data=Arrests)
Anova(arrests.mod1)
## Analysis of Deviance Table (Type II tests)
## 
## Response: released
##          LR Chisq Df Pr(>Chisq)    
## colour     19.544  1   9.83e-06 ***
## year        6.712  5     0.2430    
## age         0.241  1     0.6237    
## sex         0.003  1     0.9545    
## employed   75.715  1  < 2.2e-16 ***
## citizen    23.853  1   1.04e-06 ***
## checks    203.562  1  < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

all two-way effects

arrests.mod2 <- glm(released ~ .^2, family=binomial, data=Arrests)
Anova(arrests.mod2)
## Analysis of Deviance Table (Type II tests)
## 
## Response: released
##                  LR Chisq Df Pr(>Chisq)    
## colour             18.690  1  1.538e-05 ***
## year                6.394  5  0.2697805    
## age                 1.262  1  0.2613526    
## sex                 0.084  1  0.7713700    
## employed           69.898  1  < 2.2e-16 ***
## citizen            23.641  1  1.161e-06 ***
## checks            195.890  1  < 2.2e-16 ***
## colour:year        21.643  5  0.0006121 ***
## colour:age          8.457  1  0.0036367 ** 
## colour:sex          0.289  1  0.5906163    
## colour:employed     0.005  1  0.9450466    
## colour:citizen      1.123  1  0.2892219    
## colour:checks       0.012  1  0.9112238    
## year:age           12.909  5  0.0242427 *  
## year:sex            5.547  5  0.3528332    
## year:employed       9.854  5  0.0794897 .  
## year:citizen        7.833  5  0.1656900    
## year:checks        17.079  5  0.0043521 ** 
## age:sex             1.060  1  0.3032908    
## age:employed        3.893  1  0.0484968 *  
## age:citizen         0.013  1  0.9095946    
## age:checks          0.324  1  0.5694023    
## sex:employed        5.514  1  0.0188645 *  
## sex:citizen         0.847  1  0.3574757    
## sex:checks          1.066  1  0.3019524    
## employed:citizen    0.132  1  0.7165898    
## employed:checks     0.159  1  0.6901753    
## citizen:checks      5.863  1  0.0154587 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

backward selection, using AIC

arrests.step <- stepAIC(arrests.mod2, direction="backward")
## Start:  AIC=4294.08
## released ~ (colour + year + age + sex + employed + citizen + 
##     checks)^2
## 
##                    Df Deviance    AIC
## - year:sex          5   4185.6 4289.6
## - year:citizen      5   4187.9 4291.9
## - colour:employed   1   4180.1 4292.1
## - colour:checks     1   4180.1 4292.1
## - age:citizen       1   4180.1 4292.1
## - employed:citizen  1   4180.2 4292.2
## - employed:checks   1   4180.2 4292.2
## - colour:sex        1   4180.4 4292.4
## - age:checks        1   4180.4 4292.4
## - sex:citizen       1   4180.9 4292.9
## - age:sex           1   4181.1 4293.1
## - sex:checks        1   4181.1 4293.1
## - colour:citizen    1   4181.2 4293.2
## - year:employed     5   4189.9 4293.9
## <none>                  4180.1 4294.1
## - age:employed      1   4184.0 4296.0
## - year:age          5   4193.0 4297.0
## - sex:employed      1   4185.6 4297.6
## - citizen:checks    1   4185.9 4297.9
## - colour:age        1   4188.5 4300.5
## - year:checks       5   4197.2 4301.2
## - colour:year       5   4201.7 4305.7
## 
## Step:  AIC=4289.63
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:employed + 
##     colour:citizen + colour:checks + year:age + year:employed + 
##     year:citizen + year:checks + age:sex + age:employed + age:citizen + 
##     age:checks + sex:employed + sex:citizen + sex:checks + employed:citizen + 
##     employed:checks + citizen:checks
## 
##                    Df Deviance    AIC
## - year:citizen      5   4193.5 4287.5
## - colour:employed   1   4185.6 4287.6
## - colour:checks     1   4185.6 4287.6
## - age:citizen       1   4185.6 4287.6
## - employed:citizen  1   4185.7 4287.7
## - employed:checks   1   4185.8 4287.8
## - age:checks        1   4185.9 4287.9
## - age:sex           1   4186.2 4288.2
## - colour:sex        1   4186.2 4288.2
## - year:employed     5   4194.6 4288.6
## - colour:citizen    1   4186.7 4288.7
## - sex:checks        1   4186.8 4288.8
## <none>                  4185.6 4289.6
## - sex:citizen       1   4188.2 4290.2
## - age:employed      1   4189.3 4291.3
## - year:age          5   4199.0 4293.0
## - sex:employed      1   4191.7 4293.7
## - citizen:checks    1   4191.9 4293.9
## - year:checks       5   4201.3 4295.3
## - colour:age        1   4193.9 4295.9
## - colour:year       5   4208.3 4302.3
## 
## Step:  AIC=4287.54
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:employed + 
##     colour:citizen + colour:checks + year:age + year:employed + 
##     year:checks + age:sex + age:employed + age:citizen + age:checks + 
##     sex:employed + sex:citizen + sex:checks + employed:citizen + 
##     employed:checks + citizen:checks
## 
##                    Df Deviance    AIC
## - age:citizen       1   4193.5 4285.5
## - colour:employed   1   4193.5 4285.5
## - colour:checks     1   4193.6 4285.6
## - employed:checks   1   4193.7 4285.7
## - age:checks        1   4193.8 4285.8
## - employed:citizen  1   4193.9 4285.9
## - age:sex           1   4194.1 4286.1
## - colour:sex        1   4194.2 4286.2
## - sex:checks        1   4194.8 4286.8
## - year:employed     5   4202.9 4286.9
## - colour:citizen    1   4195.4 4287.4
## <none>                  4193.5 4287.5
## - sex:citizen       1   4195.9 4287.9
## - age:employed      1   4197.3 4289.3
## - year:age          5   4206.7 4290.7
## - year:checks       5   4207.2 4291.2
## - citizen:checks    1   4199.6 4291.6
## - sex:employed      1   4199.8 4291.8
## - colour:age        1   4202.1 4294.1
## - colour:year       5   4217.9 4301.9
## 
## Step:  AIC=4285.54
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:employed + 
##     colour:citizen + colour:checks + year:age + year:employed + 
##     year:checks + age:sex + age:employed + age:checks + sex:employed + 
##     sex:citizen + sex:checks + employed:citizen + employed:checks + 
##     citizen:checks
## 
##                    Df Deviance    AIC
## - colour:employed   1   4193.5 4283.5
## - colour:checks     1   4193.6 4283.6
## - employed:checks   1   4193.7 4283.7
## - age:checks        1   4193.8 4283.8
## - employed:citizen  1   4193.9 4283.9
## - age:sex           1   4194.1 4284.1
## - colour:sex        1   4194.2 4284.2
## - sex:checks        1   4194.8 4284.8
## - year:employed     5   4202.9 4284.9
## - colour:citizen    1   4195.4 4285.4
## <none>                  4193.5 4285.5
## - sex:citizen       1   4195.9 4285.9
## - age:employed      1   4197.3 4287.3
## - year:age          5   4207.0 4289.0
## - year:checks       5   4207.2 4289.2
## - citizen:checks    1   4199.7 4289.7
## - sex:employed      1   4199.8 4289.8
## - colour:age        1   4203.1 4293.1
## - colour:year       5   4218.0 4300.0
## 
## Step:  AIC=4283.55
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     colour:checks + year:age + year:employed + year:checks + 
##     age:sex + age:employed + age:checks + sex:employed + sex:citizen + 
##     sex:checks + employed:citizen + employed:checks + citizen:checks
## 
##                    Df Deviance    AIC
## - colour:checks     1   4193.6 4281.6
## - employed:checks   1   4193.7 4281.7
## - age:checks        1   4193.8 4281.8
## - employed:citizen  1   4193.9 4281.9
## - age:sex           1   4194.1 4282.1
## - colour:sex        1   4194.3 4282.3
## - sex:checks        1   4194.9 4282.9
## - year:employed     5   4202.9 4282.9
## - colour:citizen    1   4195.4 4283.4
## <none>                  4193.5 4283.5
## - sex:citizen       1   4196.0 4284.0
## - age:employed      1   4197.3 4285.3
## - year:age          5   4207.0 4287.0
## - year:checks       5   4207.3 4287.3
## - citizen:checks    1   4199.7 4287.7
## - sex:employed      1   4199.8 4287.8
## - colour:age        1   4203.1 4291.1
## - colour:year       5   4218.0 4298.0
## 
## Step:  AIC=4281.57
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     year:age + year:employed + year:checks + age:sex + age:employed + 
##     age:checks + sex:employed + sex:citizen + sex:checks + employed:citizen + 
##     employed:checks + citizen:checks
## 
##                    Df Deviance    AIC
## - employed:checks   1   4193.7 4279.7
## - age:checks        1   4193.9 4279.9
## - employed:citizen  1   4193.9 4279.9
## - age:sex           1   4194.1 4280.1
## - colour:sex        1   4194.3 4280.3
## - sex:checks        1   4194.9 4280.9
## - year:employed     5   4203.0 4281.0
## <none>                  4193.6 4281.6
## - colour:citizen    1   4195.6 4281.6
## - sex:citizen       1   4196.0 4282.0
## - age:employed      1   4197.3 4283.3
## - year:age          5   4207.1 4285.1
## - year:checks       5   4207.3 4285.3
## - sex:employed      1   4199.8 4285.8
## - citizen:checks    1   4200.5 4286.5
## - colour:age        1   4203.1 4289.1
## - colour:year       5   4218.0 4296.0
## 
## Step:  AIC=4279.73
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     year:age + year:employed + year:checks + age:sex + age:employed + 
##     age:checks + sex:employed + sex:citizen + sex:checks + employed:citizen + 
##     citizen:checks
## 
##                    Df Deviance    AIC
## - employed:citizen  1   4194.0 4278.0
## - age:checks        1   4194.0 4278.0
## - age:sex           1   4194.3 4278.3
## - colour:sex        1   4194.4 4278.4
## - sex:checks        1   4195.2 4279.2
## - year:employed     5   4203.3 4279.3
## <none>                  4193.7 4279.7
## - colour:citizen    1   4195.7 4279.7
## - sex:citizen       1   4196.2 4280.2
## - age:employed      1   4197.5 4281.5
## - year:age          5   4207.3 4283.3
## - year:checks       5   4207.6 4283.6
## - sex:employed      1   4200.4 4284.4
## - citizen:checks    1   4200.7 4284.7
## - colour:age        1   4203.3 4287.3
## - colour:year       5   4218.3 4294.3
## 
## Step:  AIC=4278.03
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     year:age + year:employed + year:checks + age:sex + age:employed + 
##     age:checks + sex:employed + sex:citizen + sex:checks + citizen:checks
## 
##                  Df Deviance    AIC
## - age:checks      1   4194.3 4276.3
## - age:sex         1   4194.6 4276.6
## - colour:sex      1   4194.7 4276.7
## - year:employed   5   4203.4 4277.4
## - sex:checks      1   4195.5 4277.5
## - colour:citizen  1   4196.0 4278.0
## <none>                4194.0 4278.0
## - sex:citizen     1   4196.7 4278.7
## - age:employed    1   4198.0 4280.0
## - year:age        5   4207.5 4281.5
## - year:checks     5   4207.8 4281.8
## - sex:employed    1   4200.9 4282.9
## - citizen:checks  1   4201.6 4283.6
## - colour:age      1   4203.6 4285.6
## - colour:year     5   4218.5 4292.5
## 
## Step:  AIC=4276.33
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     year:age + year:employed + year:checks + age:sex + age:employed + 
##     sex:employed + sex:citizen + sex:checks + citizen:checks
## 
##                  Df Deviance    AIC
## - age:sex         1   4194.8 4274.8
## - colour:sex      1   4195.0 4275.0
## - year:employed   5   4203.7 4275.7
## - sex:checks      1   4195.8 4275.8
## - colour:citizen  1   4196.3 4276.3
## <none>                4194.3 4276.3
## - sex:citizen     1   4197.1 4277.1
## - age:employed    1   4199.0 4279.0
## - year:age        5   4207.7 4279.7
## - year:checks     5   4208.0 4280.0
## - sex:employed    1   4201.2 4281.2
## - citizen:checks  1   4202.2 4282.2
## - colour:age      1   4204.0 4284.0
## - colour:year     5   4218.7 4290.7
## 
## Step:  AIC=4274.78
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:sex + colour:citizen + 
##     year:age + year:employed + year:checks + age:employed + sex:employed + 
##     sex:citizen + sex:checks + citizen:checks
## 
##                  Df Deviance    AIC
## - colour:sex      1   4195.4 4273.4
## - year:employed   5   4204.3 4274.3
## - sex:checks      1   4196.4 4274.4
## - colour:citizen  1   4196.7 4274.7
## <none>                4194.8 4274.8
## - sex:citizen     1   4197.6 4275.6
## - age:employed    1   4200.2 4278.2
## - year:checks     5   4208.4 4278.4
## - year:age        5   4208.7 4278.7
## - sex:employed    1   4201.2 4279.2
## - citizen:checks  1   4202.6 4280.6
## - colour:age      1   4204.1 4282.1
## - colour:year     5   4219.3 4289.3
## 
## Step:  AIC=4273.43
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:citizen + year:age + year:employed + 
##     year:checks + age:employed + sex:employed + sex:citizen + 
##     sex:checks + citizen:checks
## 
##                  Df Deviance    AIC
## - year:employed   5   4204.7 4272.7
## - sex:checks      1   4197.0 4273.0
## <none>                4195.4 4273.4
## - colour:citizen  1   4197.5 4273.5
## - sex:citizen     1   4199.0 4275.0
## - age:employed    1   4200.9 4276.9
## - year:checks     5   4209.0 4277.0
## - year:age        5   4209.1 4277.1
## - sex:employed    1   4201.7 4277.7
## - citizen:checks  1   4203.5 4279.5
## - colour:age      1   4204.8 4280.8
## - colour:year     5   4219.5 4287.5
## 
## Step:  AIC=4272.73
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:citizen + year:age + year:checks + 
##     age:employed + sex:employed + sex:citizen + sex:checks + 
##     citizen:checks
## 
##                  Df Deviance    AIC
## - sex:checks      1   4206.4 4272.4
## - colour:citizen  1   4206.7 4272.7
## <none>                4204.7 4272.7
## - year:checks     5   4215.6 4273.6
## - sex:citizen     1   4208.2 4274.2
## - year:age        5   4218.0 4276.0
## - sex:employed    1   4210.9 4276.9
## - age:employed    1   4211.2 4277.2
## - citizen:checks  1   4212.3 4278.3
## - colour:age      1   4214.1 4280.1
## - colour:year     5   4229.0 4287.0
## 
## Step:  AIC=4272.41
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + colour:citizen + year:age + year:checks + 
##     age:employed + sex:employed + sex:citizen + citizen:checks
## 
##                  Df Deviance    AIC
## - colour:citizen  1   4208.2 4272.2
## <none>                4206.4 4272.4
## - sex:citizen     1   4209.1 4273.1
## - year:checks     5   4217.4 4273.4
## - sex:employed    1   4211.1 4275.1
## - year:age        5   4219.4 4275.4
## - age:employed    1   4213.0 4277.0
## - citizen:checks  1   4213.5 4277.5
## - colour:age      1   4215.9 4279.9
## - colour:year     5   4230.8 4286.8
## 
## Step:  AIC=4272.25
## released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + year:age + year:checks + age:employed + 
##     sex:employed + sex:citizen + citizen:checks
## 
##                  Df Deviance    AIC
## <none>                4208.2 4272.2
## - sex:citizen     1   4211.1 4273.1
## - year:checks     5   4219.4 4273.4
## - sex:employed    1   4212.9 4274.9
## - year:age        5   4221.1 4275.1
## - citizen:checks  1   4214.5 4276.5
## - age:employed    1   4214.7 4276.7
## - colour:age      1   4217.1 4279.1
## - colour:year     5   4232.1 4286.1
anova(arrests.mod1, arrests.step, arrests.mod2, test="Chisq")
## Analysis of Deviance Table
## 
## Model 1: released ~ colour + year + age + sex + employed + citizen + checks
## Model 2: released ~ colour + year + age + sex + employed + citizen + checks + 
##     colour:year + colour:age + year:age + year:checks + age:employed + 
##     sex:employed + sex:citizen + citizen:checks
## Model 3: released ~ (colour + year + age + sex + employed + citizen + 
##     checks)^2
##   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
## 1      5214     4292.4                          
## 2      5194     4208.2 20   84.127 7.738e-10 ***
## 3      5169     4180.1 25   28.165    0.3003    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

one reduced model

arrests.mod <- glm(released ~ employed + citizen + checks + colour*year + colour*age,
 family=binomial, data=Arrests)

#summary(arrests.mod)
Anova(arrests.mod)
## Analysis of Deviance Table (Type II tests)
## 
## Response: released
##             LR Chisq Df Pr(>Chisq)    
## employed      72.673  1  < 2.2e-16 ***
## citizen       25.783  1  3.820e-07 ***
## checks       205.211  1  < 2.2e-16 ***
## colour        19.572  1  9.687e-06 ***
## year           6.087  5  0.2978477    
## age            0.459  1  0.4982736    
## colour:year   21.720  5  0.0005917 ***
## colour:age    13.886  1  0.0001942 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(arrests.mod1, arrests.mod, arrests.mod2)
## Analysis of Deviance Table
## 
## Model 1: released ~ colour + year + age + sex + employed + citizen + checks
## Model 2: released ~ employed + citizen + checks + colour * year + colour * 
##     age
## Model 3: released ~ (colour + year + age + sex + employed + citizen + 
##     checks)^2
##   Resid. Df Resid. Dev Df Deviance
## 1      5214     4292.4            
## 2      5209     4257.1  5   35.316
## 3      5169     4180.1 40   76.976
anova(arrests.mod1, arrests.mod, arrests.mod2, test="Chisq")
## Analysis of Deviance Table
## 
## Model 1: released ~ colour + year + age + sex + employed + citizen + checks
## Model 2: released ~ employed + citizen + checks + colour * year + colour * 
##     age
## Model 3: released ~ (colour + year + age + sex + employed + citizen + 
##     checks)^2
##   Resid. Df Resid. Dev Df Deviance  Pr(>Chi)    
## 1      5214     4292.4                          
## 2      5209     4257.1  5   35.316 1.301e-06 ***
## 3      5169     4180.1 40   76.976 0.0003966 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1