Honestly Significant Difference (HSD) Test

Notations

The data to be analyzed comprise A groups; a given group is denoted a. The number of observations of the a. The group is denoted Sa. If all groups have the same size, it is denoted S. The total number of observations is denoted N. The mean of Group a is denoted Ma+. Obtained from a preliminary ANOVA, the error source (i.e., within group) is denoted S(A), the effect (i.e., between group) is denoted A. The mean square of error is denoted MSS(A) and the mean square of effect is denoted MS A.

Least Significant Difference

The rationale behind the HSD technique comes from the observation that, when the null hypothesis is true, the value of the q statistics evaluating the difference between Groups a and a′ is equal to [Page 584]

and follows a Studentized range q distribution with a range of A and N − A degrees of freedom. The ratio t would therefore be declared significant at a given α level if the value of q is larger than the critical value for the α level obtained from the q distribution and denoted qA, α where v = N − A is the number of degrees of freedom of the error, and A is the range (i.e., the number of groups). This value can be obtained from a table of the Studentized range distribution. Rewriting Equation 1 shows that a difference between the means of Group a and a′ will be significant if

When there is an equal number of observation per group, Equation 2 can be simplified as

To evaluate the difference between the means of Groups a and a′, the absolute value of the difference between the means is taken and compared with the value of HSD. If then the comparison is declared significant at the chosen a level (usually .05 or .01). Then this procedure is repeated or all

Note that HSD has less power than almost all other post hoc comparison methods (e.g., Fisher's [Page 585]LSD or Newmann—Keuls) except the Scheffe approach and the Bonferonni method because the α level for each difference between means is set at the same level as the largest difference.

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