To test if the means of the harvest in kilograms of tomatoes in three different greenhouses differ. The dataset contains observations from five fields in each greenhouse (dataset in Table A.12).

SAS code

proc anova data = crop;
 class house
 model kg = house;
run;
quit;

Source    DF     Anova SS   Mean Square  F Value   Pr> F
house      2   0.16329333    0.08164667     0.33   0.7262

• The SAS procedure proc anova is the standard procedure for the analysis of variance with a balanced design as given in this example. For an unbalanced design the procedure proc glm should be used (see below).
• By using the class statement, SAS treats the variable house as a categorical variable.
• The code model dependent variable= independent variable defines the model.
• The quit; statement is used to terminate the procedure; proc anova is an interactive procedure and SAS then knows not to expect any further input.
• The program code for proc glm is similar:

  proc glm data = crop;
   class house
   model kg = house;
  run;
  quit;

R code

summary(aov(crop$kg∼factor(crop$house)))

                   Df Sum Sq Mean Sq F value Pr(>F)
factor(crop$house)  2 0.1633 0.08165   0.329  0.726
Residuals          12 2.9815 0.24846

• The function aov() performs an analysis of variance in R. The response variable is placed on the left-hand side of the symbol and the independent variables which define the groups on the right-hand side.
• We use the R function factor() to tell R that house is a categorical variable.
• The summary function gets R to return the sum of squares, degrees of freedom, p-values, etc.

To test if the means of the harvest in kilograms of tomatoes in three different greenhouses and using five different fertilizers differ. The dataset contains observations from five fields with each fertilizer in each greenhouse (dataset in Table A.12).

SAS code

proc anova data= crop;
      class house fertilizer;
      model kg =  house fertilizer;
run;
quit;

                    The ANOVA Procedure
Dependent Variable: kg
Source       DF    Anova SS  Mean Square  F Value  Pr> F
house         2  0.16329333   0.08164667     0.50  0.6268
fertilizer    4  1.66337333   0.41584333     2.52  0.1235

• The SAS procedure proc anova is the standard procedure for an ANOVA with a balanced design. For an unbalanced design the procedure proc glm should be used.
• By using the class statement, SAS treats the variables house and fertilizer as categorical variables.
• The code model dependent variable= independent variables defines the model. To incorporate an interaction term a star is used, for example, variable1variable2.
• The quit; statement is used to terminate the procedure; proc anova is an interactive procedure and SAS then knows not to expect any further input.
• The program code for proc glm is similar:

  proc glm data = crop;
   class house fertilizer
   model kg = house fertilizer;
  run;
  quit;

R code

kg<-crop$kg
field<-crop$house
fertilizer<-crop$fertilizer
summary(aov(kg∼factor(field)+factor(fertilizer)))

                   Df Sum Sq Mean Sq F value Pr(>F)
factor(house)       2 0.1633  0.0816   0.496  0.627
factor(fertilizer)  4 1.6634  0.4158   2.524  0.123
Residuals           8 1.3181  0.1648

• The function aov() performs an ANOVA in R. The response variable is placed on the left-hand side of the symbol and the independent variables which define the groups on the right-hand side separated by a plus (+). To incorporate an interaction term a star is used, for example, variable1variable2.
• We use the R function factor() to tell R that house is a categorical variable.
• The summary function gets R to return the sum of squares, degrees of freedom, p-values, etc.

To test if the variances of the harvest in kilograms of tomatoes in three different greenhouses are the same (dataset in Table A.12).

SAS code

proc glm data = crop;
 class house;
 model kg = house;
 means house /hovtest=BARTLETT ;
run;
quit;

                The GLM Procedure
 Bartlett's Test for Homogeneity of kg Variance
  Source        DF    Chi-Square    Pr > ChiSq
  house          2        2.1346        0.3439

Remarks:

• The SAS procedure proc glm provides the Bartlett test.
• The first lines of code are enabling an ANOVA (see Test 16.2.1).
• The code means house /hovtest=BARTLETT lets SAS conduct the Bartlett test.

R code

bartlett.test(crop$kg∼crop$house)

 Bartlett test of homogeneity of variances
data:  crop$kg by crop$field
Bartlett's K-squared = 2.1346, df = 2, p-value = 0.3439

• The function bartlett.test() conducts the Bartlett test.
• The analysis variable is coded on the left-hand side of the and the group variable on the right-hand side.

To test if the variances of the harvest in kilograms of tomatoes in three different greenhouses are the same (dataset in Table A.12).

SAS code

proc glm data = crop;
 class house;
 model kg = house;
 means house /hovtest=levene (TYPE=ABS) ;
run;
quit;

      Levene's Test for Homogeneity of kg Variance
      ANOVA of Absolute Deviations from Group Means
                  Sum of        Mean
Source     DF     Squares      Square    F Value    Pr> F
house       2      0.2675      0.1337       2.79    0.1012
Error      12      0.5753      0.0479

• The SAS procedure proc glm provides the Levene test.
• The first lines of code are enabling an ANOVA (see Test 16.2.1).
• The code means house /hovtest=levene (TYPE=ABS) lets SAS do the Levene test. In SAS it is also possible to choose the option (TYPE=SQUARE) which uses the squared differences.
• The Brown–Forsythe test can be conducted with the option /hovtest=BF.

R code

# Calculate group means for each field
m<-tapply(crop$kg,crop$house,mean)
# Calculate the Z's
z<-abs(crop$kg-m[crop$house])
# Overall mean of the Z's
z_mean=mean(z)
# Group mean of the Z's
z_gm<-tapply(z,crop$house,mean)
# Make a matrix of the Z's (group in the rows)
z_matrix<-rbind(z[crop$house==1],z[crop$house==2],
                z[crop$house==3])
# Calculate the numerator
nu<-0
for (i in 1:3)
 {
  u<-5*(z_gm[i]-z_mean)∧2
  nu<-nu+u
 }
# Calculate the denominator
de<-0
for (j in 1:3)
{
 for (i in 1:5)
  {
   e<-(z_matrix[j,i]-z_gm[j])∧2
   de<-de+e
  }
}
# Calculate test statistic and p-value
l<-(12*nu)/(2*de)
p_value<-1-pf(l,2,12)
# Output results
“Levene Test”
l
p_value

[1] “Levene Test”
> l
       1
2.789499
> p_value
        1
0.1011865
>

• There is no basic R function to calculate Levene's test directly.
• In this example we have and . The respective parts must be adopted if other data are used.
• To apply the Brown–Forsythe test just change the first line of code to m<-tapply(crop$kg,crop$house,median).