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Explanations and References
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.
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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.
- 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. ## ReferencesMarascuilo,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
# 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$RowTot[i] * ColTot[j] / total o = df[i,j] chiSq = chiSq + (e - o)^2 / e; } } p = 1 - pchisq(chiSq, df=degFdm) df # data frame with probability added c(chiSq, degFdm, p) # Chi Square, degrees of freedom, and Probability of Type I Error pThe initial results are as follows. Prob is the proportion of positives > df # data frame with probability added NPos NNeg RowTot Prop 1 60 40 100 0.6000000 2 20 60 80 0.2500000 3 10 50 60 0.1666667 > c(chiSq, degFdm, p) # Chi Square, degrees of freedom, and Probability of Type I Error p [1] 3.804444e+01 2.000000e+00 5.479663e-09Part 3: Post hoc comparison between rows #Part 3: Post hoc comparison between rows GrpA <- vector() GrpB <- vector() Diff <- vector() ChiSq <- vector() P <- vector() ii = nrow(df) - 1 jj = nrow(df) c(ii,jj) i = 1 while(i<=ii) { j = i + 1 while(j<=jj) { print(c(i,j)) p1 <- df$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 resultsThe 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 > |