The Marascuilo Procedure is used to compare proportions when there are more than two groups with binary outcomes (yes/no). It firstly performs a test of overall homogeneity for a large contingency table, using the standard Chi Square Test. This is followed by multiple post hoc comparisons between pairs of groups in the data.

The Mascuilo's procedure is simpler and replaces using multiple comparisons of two groups with a Bonferroni's correction for multiple comparisons, so it is preferred when there are multiple groups to be compared at the same time.

An Example : We wish to study academic achievements in 3 social groups of students.

• Group 1 consists of 100 students from single parent family living in a deprived suburb. 60 of these students (0.6 or 60%) are judged to be under achievers.
• Group 2 consists of 80 students, also from deprived suburbs, but coming from stable two parent families. 20 of these students (0.25 or 25%) are judged to be under achievers.
• Group 3 consists of 60 students from intact families living in middle class suburbs. 10 of these students (0.17 or 17%) are judged to be under achievers.

An initial analysis using the Chi Squares Test indicates that the rates of under achievement are significantly different in the 3 groups (Chi Sq = 38.04, df=2, α<0.001).

Using the Marascuilo procedure to perform post hoc analysis, we found that group 1 is significantly different to group 2 (difference = 35%, α<0.001), and to group 3 (difference = 43.3%, α<0.001).

However, the difference between groups 2 and 3 (difference = 8.3%, α=0.47) is not statistically significant.

References

Marascuilo,L. A. 1966: Large-sample multiple comparison. Psychological Bulletin 65: p. 280 - 290.

Daniel,W. W. 1990: Applied nonparametric statistics. 2nd ed. Boston PWS Kent

Zwick,R. and L.A.Marascuilo. (1984) Selection of pairwise multiple comparison procedures for parametric and nonparametric analysis of variance models. Psychological Bulletin, 95(1): 148-155.

https://www.itl.nist.gov/div898/handbook/prc/section4/prc464.htm Algoritm by NIST

Only one calculation is performed each time. The data is in 2 column, separated by white spaces (space or tab)   - Each row is data from a group   - Column 1 is the number with positive attribute   - Column 2 is the number with negative attribute

#         MultiProp.R Marascuilo's Procedure
# Part 1 : Data Input 
dat = ("
NPos   NNeg
60	   40
20	   60
10	   50
       ")
df <- read.table(textConnection(dat),header=TRUE)  # conversion to data frame    
#df   # optional display of input data

# Part 2: initial overall chi sq Test
df$RowTot <- df$NPos + df$NNeg   # row total
df$Prop <- df$NPos / df$RowTot   # proportion positives
ColTot <- c(sum(df[, 1]),sum(df[, 2]))  # col total
total = sum(arColTot)
degFdm = nrow(df) - 1
chiSq = 0
for(i in 1 : nrow(df))
{
  for(j in 1:2)
  {
    e = df$RowTot[i] * ColTot[j] / total
    o = df[i,j]
    chiSq = chiSq + (e - o)^2 / e;
  }
}
p = 1 - pchisq(chiSq, df=degFdm)
df   # data frame with probability added
c(chiSq, degFdm, p)  # Chi Square, degrees of freedom, and Probability of Type I Error p

The initial results are as follows. Prob is the proportion of positives

> df   # data frame with probability added
  NPos NNeg RowTot      Prop
1   60   40    100 0.6000000
2   20   60     80 0.2500000
3   10   50     60 0.1666667
> c(chiSq, degFdm, p)  # Chi Square, degrees of freedom, and Probability of Type I Error p
[1] 3.804444e+01 2.000000e+00 5.479663e-09

Part 3: Post hoc comparison between rows

#Part 3: Post hoc comparison between rows

GrpA <- vector()
GrpB <- vector()
Diff <- vector()
ChiSq <- vector()
P <- vector()
ii = nrow(df) - 1
jj = nrow(df)
c(ii,jj)
i = 1
while(i<=ii)
{
  j = i + 1
  while(j<=jj)
  {
    print(c(i,j))
    p1 <- df$Prop[i]
    GrpA <- append(GrpA,i)
    p2 <- df$Prop[j]
    GrpB <- append(GrpB,j)
    diff = p1 - p2
    Diff <- append(Diff,diff)
    chiSq = diff^2 / (p1*(1 - p1) / df$RowTot[i] + p2 * (1 - p2) / df$RowTot[j])
    ChiSq <- append(ChiSq,chiSq)
    p = 1 - pchisq(chiSq, df=degFdm)
    P <- append(P,p)
    j = j + 1
  }
  i = i + 1
}
dfR <- data.frame(GrpA, GrpB, Diff, ChiSq, P)
dfR # display results

The results are as follows

• Grp A and B are the 2 rows being compared
• Diff = difference in proportions of positives in the 2 row
• Chi Sq = chi sqyuare test, with degrees of freedom = row-1
• P = Probability of Type I Error (p, α)

> dfR # display results
  GrpA GrpB       Diff     ChiSq            P
1    1    2 0.35000000 25.823452 2.468929e-06
2    1    3 0.43333333 39.827180 2.247180e-09
3    2    3 0.08333333  1.490683 4.745722e-01
>