Duncan Multiple Range - Post Hoc Test

Duncan’s Multiple Range Test

Duncan's test is based on a set of significant difference values that are increasing in size, depending on the distance between the ranks of the two compared mean values.  It can be used to test the differences between all possible treatment pairs regardless of the number of treatments.

Calculation steps:

1. Sort the mean value of the treatment (usually ascending order)
2. Calculate the shortest significant region for the region of various intermediate values using the following formula:
   $$\begin{matrix}R_p=r_{\alpha,p,\nu}s_{\bar{Y}}\\R_p=r_{\alpha,p,\nu}\sqrt{\frac{MSE}{r}}\\\end{matrix}$$
   Where:
   
   • MSE     =  Mean Squared Error
   • r          =  replication
   • _ar,p,n    =  Duncan significant region value     
   • a         =  significant level              
   • P         =  relative distance between a particular treatment and subsequent ratings (2, 3, ..., t)
   • n         =  error-degree of freedoms.
3. Test criteria:
   • Compare the absolute value of the second difference in the mean that we will see the difference with the shortest significant region value (R_p) with the following test criteria:
     $ if\ \ \left|\mu_i-\mu_j\right|\ \ \left\langle\ \ \begin{matrix}>R_p\ reject\ H_0,\ two\ means\ are\ significantly\ different\\\le R_p\ accept\ H_0,\ two\ means\ are\ not\ significantly\ different.\\\end{matrix}\right.$

Definition of rank (p) in mean comparison using multiple range test:"

1. p = the relative distance between one mean value and the treatment mean in the next rank after the mean has been sorted, which is calculated from that mean value (number 1) to the next p mean.
2. Example:
   • p = 2, means the distance to 1 mean value of the next rank (a comparison between two neighboring mean values), for example: 
     • the distance of rank 1 to rank 2 (3Dok13 vs 3Dok4); or
     • the distance of rank 2 with rank 3 (3Dok4 vs Combined).
     • the distance between rank 5 and rank 6 (3Dok5 vs 3Dok1), etc.
   • p = 3, means the distance with 2 mean values of the next rank, for example:
     • the distance of rank 1 with rank 3 (3Dok13 vs Combined),
     • the distance of rank 2 with rank 4 (3Dok4 vs 3Dok7),
     • the distance of the n rank to the n+ rank (p-1)
   • p = t, means the distance with (t-1) the mean value of the next rank, e.g. for the example above:
     • p = 6 means the ratio between the mean of the smallest treatment (3Dok13) and the mean of the largest treatment (3Dok1)

Examples of Duncan Test

There are various ways of composing letter notation to test the differences between the means of the treatments.  This depends on the algorithm or logic of each one, but the point is to compare the difference between the mean treatment and the corresponding comparison value.  If the absolute difference value is less than or equal to the comparison value, it means that the two means are not significantly different.  Unlike the preparation of notation in the use of the Tukey and LSD tests which only use one comparison value, this time we will use the mean difference value of the treatment which we then compare with the corresponding comparison value.  The full stages are as follows: 

As an illustration, we reuse Red Clover's experimental data.

1. Step 1: Calculate the shortest significant region value (R_p):
   • Determine the value of the MSE and its degree of freedom obtained from the Analysis of variance Table.
     • MSE = 11.7887
     • ν = df = 24
   • Determine its critical value from the student's significant region table based on the error-degree of freedom and the number of treatments to be compared. 
     • There are three parameters needed to determine the value of q, namely the significant level (α), p = the number of treatments to be compared, and the error-degree of freedom (df).  In this example, p = 2, 3, 4, 5, 6, the value of df = 24 (see df error in its Analysis of variance table) and α = 0.05. Next, determine the value of r_{0.05(6, 24)}.
     • To find the value of r_{0.05(6, 24)} we can look at it in the table Significant Ranges for Duncan's Multiple Range Test at a significant level α = 0.05 with p = 6 and degree of freedom (v) = 24.  Consider the following image to specify the q-table.
     • From the table we get the values of _ar,p,n which are 2.92; 3.07; 3.15; 3.22; and 3.28
     • Calculate the shortest significant region (R_p):
       
   • Test criteria:
     • Compare the absolute value of the second difference in the mean that we will see the difference with the shortest significant region value (R_p) that corresponds to the following test criteria:
       $ if\ \ \left|\mu_i-\mu_j\right|\ \ \left\langle\ \ \begin{matrix}>R_p\ reject\ H_0,\ two\ means\ are\ significantly\ different\\\le R_p\ accept\ H_0,\ two\ means\ are\ not\ significantly\ different.\\\end{matrix}\right.$
2. Step 2: Sort the treatment mean value from small to large or vice versa. In this example, the mean treatment is sorted from small to large.  The next step is to calculate the difference between the means of the treatments. 

First Method

Create a Matrix Table (Crosstabulation) of the mean difference between all combinations of treatment pairs.  Since it is a handlebar, it is enough to create a lower triangle matrix table as presented in the Table below.

Matrix Table of mean differences of treatment

Comparisons Table (Rp)

Information: 

1. The number on the body of the Table is the value of the difference between the means of the treatment
2. Superscript (2); (3); ...; (6) = p, that is, the relative distance (rank) between the mean of one treatment and another treatment
3. Compare the mean difference with the comparison that matches the rank (Rp). If it is smaller than the value of Rp (ns), give the same line to the right of it.  The provision of the same line is stopped if the mean difference value > the value of Rp. Proceed to the next treatment comparison.
   • For example, if we compare 3Dok13 with 3Dok4, compare the difference (1.38) with the appropriate Rp rating, Rp(2) = 4.50. Since 1.38 ≤ 4.50 which indicates no difference, then we give the same line to the two means.
   • 3Dok13 vs Combined: Compare the difference (5.44) with Rp(3) = 4.73. Because 5.44 ≥ 4.73 (*); stop! The same line is not given again. Continue with the comparison of 3Dok4 vs others.
   • 3Dok4 vs Combined. Compare the difference (4.06) with Rp(2) = 4.50. Since 4.06 ≤ 4.50 (ns) so give the same line to the column 3Dok4.
   • 3Dok4 vs 3Dok7: 5.28 ≥ 4.73 (*); stop! the same line is not given again. Continue with the Combined vs other comparison. etc. comes to the comparison in the 9th column.
4. Ignore (discard) the red lines, as they are already represented by the lines found in the Merge!
5. Finally, provide the letter code for the line.  The line that is notated is only the line that is black.  The first black line is given the letter "a", the second letter "b", the third letter "c", the fourth letter "d", ignore the code of the letter on the red line (the line already represented by the other line, in this case the letter "d"), finally the 5th line of the letter "e".

Second Method

1. Step Forward #1:

• Give the letter "a" to the mean value of the first smallest treatment, which is 13.26.
• Compare the difference between 3Dok13 (13.26) and the next rank treatment using the appropriate Rp value.
• Give the same letter (in this case "a") if the difference is ≤ the corresponding Rp
• p = 2: 3Dok13 vs 3Dok4: |13.26 – 14.64|= 1.38. Compare this value with Rp(2) = 4.50.  Since 1.38 ≤ 4.50, which means no different, then 3Dok4 is given the letter "a" 
• p = 3: 3Dok13 vs Combined: |13.26 – 18.70| = 5.44. Compare this value with Rp(3) = 4.73.  Since 5.44 > 4.73, which means a significant difference, then the Combine is given a different letter, which is "b". 
• The comparison between 3Dok13 and the next treatment does not need to be done because it must be different.  Thus, at a step forward, subsequent comparison for the same group is stopped if the next treatment has been given a different notation.

2. Step Back:

• We've given the letter b to the Combined treatment, but we don't yet know if the combined is different from the other treatments located between 3Dok13 ("a") and The Combined ("b")!  In this example only one treatment, which is 3Dok4.  A more practical way is to check the step back, which is to compare the Combined with the previous treatment whose value is smaller than the Combined but greater than 3Dok13.
• p = 2: Combined vs. 3Dok4: |18.70 – 14.64| = 4.06. Compare this value with Rp(2) = 4.50.  Since 4.06 ≤ 4.50, which means no different, 3Dok4 besides being given the letter "a", is also given the letter "b" because it is not different from the combination. 

3. Step Forward #2:

• The starting point now starts from the smallest treatment that gets the letter "b", in this case on the 3Dok4 (14.64) treatment.
• Compare the difference between 3Dok4 (14.64) and the next rank treatment that has not been given letter notation, namely 3Dok7 etc. using the appropriate Rp value.   Thus, p = 2:3Dok4 vs Combined: no need to do! because it had previously been given the notation "b"!
• Give the same letter (in this case "b") if the difference is ≤ the corresponding Rp
• p = 3: 3Dok4 vs 3Dok7: |14.64 – 19.92| = 5.28. Compare this value with Rp(3) = 4.73.  Since 5.28 > 4.73, which means a significant difference, then 3Dok4 is given a different letter, which is "c".  The comparison between 3Dok4 and the next treatment does not need to be done because it must be different.

4. Step Back:

• We have already given the letter c to the 3Dok7 treatment, but we do not yet know whether 3Dok7 is different from the previous treatment whose value lies between 3Dok7 and 3Dok4!  In this example only Merge. 
• p = 2: 3Dok7 vs Combined: |19.92 – 18.70| = 1.22. Compare this value with Rp(2) = 4.50.  Since 1.22 ≤ 4.50, which means no different, the Combined, besides earlier, has been given the letter "b", it is also given the letter "c" because it is no different from 3Dok7. 

5. Step Forward #3:

• The starting point now starts from the smallest treatment that gets the letter "c", in this case on the Combined treatment (18.70).
• Compare the difference between Combined (18.70) and the next ranked treatment that has not been given letter notation, namely 3Dok5 etc. using the appropriate Rp value. Thus p = 2: Combined vs 3Dok7 need not be done again!
• Give the same letter (in this case "c") if the difference is ≤ the corresponding Rp
• p = 3: Combined vs 3Dok5: |18.70 – 23.98| = 5.28. Compare this value with Rp(3) = 4.73.  Since 5.28 > 4.73, which means a significant difference, then 3Dok5 is given a different letter, i.e. "d".  The comparison between the Combined and the next treatment does not need to be done because it must be different.

6. Step Back:

• Now we compare the 3Dok5 treatment with the previous treatment located between 3Dok5 and the Combined.
• p = 2: 3Dok5 vs 3Dok7: |23.98 – 19.92| = 4.06 Compare this value with Rp(2) = 4.50.  Because 4.06 ≤ 4.50, which means it is no different, 3Dok5 besides being given the letter "c", is also given the letter "c" because it is not different from 3Dok5.

7. Step Forward #4:

• The starting point now starts from the smallest treatment that gets the letter "d", in this case on the 3Dok7 (19.92) treatment.
• p = 2: 3Dok7 vs 3Dok5. Already!
• p = 3: 3Dok7 vs 3Dok1: |19.92 – 28.82| = 8.90. Compare this value with Rp(3) = 4.73.  Since 8.9 > 4.73, which means a significant difference, then 3Dok1 is given a different letter, namely "e".

8. Step Back:

• Finally, we check again whether the treatment of 3Dok1 is different from the previous treatment, which is located between 3Dok1 and 3Dok7, namely with 3Dok5.
• p = 2: 3Dok1 vs 3Dok5: |28.82 – 23.98| = 4.84.  Compare this value with Rp(2) = 4.50.  Since 4.84 > 4.50, which means different, then 3Dok5 is not given the letter "e".

•  The result is as follows:

If we are used to it, the procedure for compiling the letter notation above can actually be summarized in the form of groupings on the same column for mean values that do not differ markedly, as in the following table example: 

Information:

↓ Comparison with next rank (step forward)

→ the granting of a new notation (move on to the next column)

↑ Comparison with previous ratings (step back)

Calculation using SmartstatXL Excel Add-In