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Multiple comparisons. How we decide which means are truly different from one another?. The ANOVA only test the null hypothesis that the treatment means were all sampled from the same distribution. Two general approaches.

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Multiple comparisons
Multiple comparisons

How we decide which means are truly different from one another
How we decide which means are truly different from one another?

  • The ANOVA only test the null hypothesis that the treatment means were all sampled from the same distribution

Two general approaches
Two general approaches another?

  • A posteriori comparisons (unplanned; after the fact)

  • A priori comparisons (planned; before the fact)

Effects of early snowmelt on alpine plant growth
Effects of early snowmelt on alpine plant growth another?

Three treatment groups and 4 replicates per treatment:


Heated with permanent solar-powered heating coils that melt spring snow pack earlier in the year than normal

Controls, fitted with heating coils that are never activated

Multiple comparisons
Data another?

Anova table for one way layout
ANOVA table for one-way layout another?

P=tail of F-distribution with (a-1) and a(n-1) degrees of freedom

A posteriori comparisons
A posteriori comparisons another?

  • We will use Tukey’s “honestly significant differences (HSD)

  • It controls for the fact that we are carrying out many simultaneous comparisons.

  • the P-value is adjusted downward for each individual test to achieve an experiment-wise error rate α=0.05

The hsd
The HSD another?


q= is the value from a statistical table of the studentized range distribution

n = sample size

The hsd1
The HSD another?

2.25, NS

3.25, P<0.05

1, NS

Dunnett s test
Dunnett’s test another?

For each treatment vs. control pair, CV= d(m,df) SEt;

Where m (number of treatments) includes the control and d is found in tables for Dunnett’s test.

Use Dunnett’s test by comparing treatment furthest from control first, then next furthest from control, etc.

Dunnett s test1
Dunnett’s test another?


Multiple comparisons
But… another?

  • Occasionally posterior tests may indicate that none of the pairs of means are significantly different from one another, even if the overall F-ratio led to reject the null-hypothesis!

  • This inconsistence results because the pairwise test are not as powerful as the overall F-ratio itself.

A priori planned comparisons
A priori (planned) comparisons another?

  • They are more specific

  • Usually they are more powerful

  • It forces you to think clearly about which particular treatment differences are of interest

A priori planned comparisons1
A priori (planned) comparisons another?

  • The idea is to establish contrasts, or specified comparisons between particular sets of means that test specific hypothesis.

  • These test must be orthogonal or independent of one another

  • They should represent a mathematical partitioning of the among group sum of squares

To create a contrast
To create a contrast another?

  • Assign an number (positive, negative or 0) to each treatment group

  • The sum of the coefficients for a particular contrast must equal 0 (zero)

  • Groups of means that are to be averaged together are assigned the same coefficient

  • Means that are not included in the comparison of a particular contrast are assigned a coefficient of 0

Contrast i heated vs non heated
Contrast I another?(Heated vs. Non-Heated)

F-ratio= 20.16/2.17=9.2934

F-critical1,9 =5.12

With 1 df

Contrast i heated vs non heated using formula for non equal samples
Contrast I another?(Heated vs. Non-Heated)using formula for non equal samples

F-ratio= 20.16/2.17=9.2934

F-critical1,9 =5.12

With 1 df

More information on Sokal and Rohlf (2000) Biometry

In order to create additional orthogonal contrasts
In order to create additional Orthogonal contrasts another?

  • If there are “a” treatment groups, at most there can be (“a”-1) orthogonal contrasts created (although there are several possible sets of such orthogonal contrasts).

  • All of the pair-wise cross products must sum to zero. In other words, a pair of contrast Q and R is independent if the sum of the products of their coefficients CQi and CRi equals zero.

Contrast ii control vs unmanipulated
Contrast II another?(Control vs. Unmanipulated)

F-ratio= 2/2.17=0.9217

F-critical1,9 =5.12

With 1 df