Dunnett's test

Reminder on ANOVA

Let E₁, E₂, …, E_k be k samples drawn from normal distributions with identical variances, but possibly different means. ANOVA will test the null hypothesis H₀ according to which the means of these k distributions are indeed equal. So :

• The null hypothesis is H₀ :   µ₁ =  µ₂ =  µ₃ = … =  µ_k,    against
• The alternative hypothesis H₁ : at least one of the means is different from the others.

If H₀ is rejected, the only conclusion of ANOVA is : "At least one of the k groups has its mean significantly different from the mean of the total set of observations  (to the chosen significance level). No clue is given as to which group(s) might have its (their) mean(s) significantly different from the general mean.

Many "multiple comparisons" tests have been developed for the purpose of analyzing the reasons that made ANOVA reject the null hypothesis. These tests are globally known as « a posteriori » or « post-hoc » tests.

The Dunnett's test is one of these tests.

Dunnett's test

Dunnett's test compares group means. Its is specifically designed for situations where all groups are to be pitted against one "Reference" group. It is commonly used after ANOVA has rejected the hypothesis of equality of the means of the distributions (although this is not necessary from a strictly technical standpoint).

Its goal is to identify groups whose means are significantly different from the mean of this reference group. It tests the null hypothesis that no group has its mean significantly different from the mean of the reference group.

For each pair (Reference, Group_i), the Dunnett's test calculates the value of a statistic « t_observed ». This value is compared to a critical value read in a "Dunnett's table". This critical value depends on the group sizes, the number of groups to be compared to the reference group, and the chosen significance level of the test.

For a given pair (Reference, Group_i), if t_{i observed} is larger than this critical value t_critical, the mean of Group i is declared significantly different from the mean of the reference group.

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Dunnett's test demands that all groups have equal sizes. Yet, it is possible to somewhat depart from this condition (see tutorial).

An example

Therapeutical tests are meant to decide whether new treatments are effective. For example, suppose that three new molecules are to be tested, together with a placebo. Each molecule is given to all the members of a group of patients. After the treatment, one ANOVA is run to decide if the means of a certain quantity (e.g. blood pressure in the case of high blood pressure treatment) measured on groups are significantly different.

If ANOVA rejects the "equality" hypothesis, it is appropriate to ask which group has its mean significantly different from that of the placebo group. Dunnett's test will then successively compare :

• Reference group vs Group 1 (H₀ : µ_t = µ₁ / H₁ : µ_t  µ₁),
• Reference group vs Group 2 (H₀ : µ_t = µ₂ / H₁ : µ_t  µ₂),
• Reference group vs Group 3 (H₀ : µ_t = µ₃ / H₁ : µ_t  µ₃).

If t_observed > t_critical for a certain group, then the H₀ pertaining to this group will be rejected, and the group mean will be declared significantly different from the mean of the reference group (at the chosen significance level).

This example is developed as a small but realistic case study in the tutorial.

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This Tutorial goes over the mechanism of the Dunnett's test, and is followed by a case study.

DUNNET'S TEST

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