In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Dunnett's test is a
multiple comparison
In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values.
The more inferences ...
procedure developed by Canadian statistician
Charles Dunnett to compare each of a number of treatments with a single control. Multiple comparisons to a control are also referred to as many-to-one comparisons.
History
Dunnett's test was developed in 1955;
an updated table of critical values was published in 1964.
[Dunnett C. W. (1964.) "New tables for multiple comparisons with a control", ''Biometrics'', 20:482491.]
Multiple Comparisons Problem
The multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. The major issue in any discussion of multiple-comparison procedures is the question of the probability of Type I errors. Most differences among alternative techniques result from different approaches to the question of how to control these errors. The problem is in part technical; but it is really much more a subjective question of how you want to define the error rate and how large you are willing to let the maximum possible error rate be.
[David C. Howell, "Statistical Methods for Psychology",8th ed.]
Dunnett's test are well known and widely used in multiple comparison procedure for simultaneously comparing, by interval estimation or hypothesis testing, all active treatments with a control when sampling from a distribution where the normality assumption is reasonable. Dunnett's test is designed to hold the
family-wise error rate
In statistics, family-wise error rate (FWER) is the probability of making one or more false discoveries, or type I errors when performing multiple hypotheses tests.
Familywise and Experimentwise Error Rates
Tukey (1953) developed the concept of a ...
at or below
when performing multiple comparisons of treatment group with control.
Uses of Dunnett’s test
The original work on Multiple Comparisons problem was made by
Tukey
John Wilder Tukey (; June 16, 1915 – July 26, 2000) was an American mathematician and statistician, best known for the development of the Cooley–Tukey FFT algorithm, fast Fourier Transform (FFT) algorithm and box plot. The Tukey's range test ...
and
Scheffé. Their method was a general one, which considered all kinds of pairwise comparisons.
Tukey's and Scheffé's methods allow any number of comparisons among a set of sample means. On the other hand, Dunnett's test only compares one group with the others, addressing a special case of multiple comparisons problem — pairwise comparisons of multiple treatment groups with a single control group. In the general case, where we compare each of the pairs, we make
comparisons (where k is the number of groups), but in the treatment vs. controls case we will make only
comparisons. If in the case of treatment and control groups we were to use the more general Tukey's and Scheffé's methods, they can result in unnecessarily wide confidence intervals. Dunnett's test takes into consideration the special structure of comparing treatment against control, yielding narrower confidence intervals.
It is very common to use Dunnett's test in medical experiments, for example comparing blood count measurements on three groups of animals, one of which served as a control while the other two were treated with two different drugs. Another common use of this method is among agronomists: agronomists may want to study the effect of certain chemicals added to the soil on crop yield, so they will leave some plots untreated (control plots) and compare them to the plots where chemicals were added to the soil (treatment plots).
Formal description of Dunnett's test
Dunnett's test is performed by computing a
Student's t-statistic
In statistics, the ''t''-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's ''t''-test. The ''t''-statistic is used in a ...
for each experimental, or treatment, group where the statistic compares the treatment group to a single control group.
Mechanics of Different Tests - Biostatistics BI 345
, Saint Anselm College Since each comparison has the same control in common, the procedure incorporates the dependencies between these comparisons. In particular, the t-statistics are all derived from the same estimate of the error variance which is obtained by pooling the sums of squares for error across all (treatment and control) groups. The formal test statistic for Dunnett's test is either the largest in absolute value of these t-statistics (if a two-tailed test is required), or the most negative or most positive of the t-statistics (if a one-tailed test is required).
In Dunnett's test we can use a common table of critical values, but more flexible options are nowadays readily available in many statistics packages. The critical values for any given percentage point depend on: whether a one- or- two-tailed test is performed; the number of groups being compared; the overall number of trials.
Assumptions
The analysis considers the case where the results of the experiment are numerical, and the experiment is performed to compare p treatments with a control group. The results can be summarized as a set of calculated means of the sets of observations, , while are referring to the treatment and is referring to the control set of observations, and is an independent estimate of the common standard deviation of all sets of observations. All of the sets of observations are assumed to be independently and normally distributed with a common variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
and means . There is also an assumption that there is an available estimate for .
Calculation
Dunnett's test's calculation is a procedure that is based on calculating confidence statements about the true or the expected values of the differences , thus the differences between treatment groups' mean and control group's mean. This procedure ensures that the probability of all statements being simultaneously correct is equal to a specified value,. When calculating one sided upper (or lower) Confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
for the true value of the difference between the mean of the treatment and the control group, constitutes the probability that this actual value will be less than the upper (or greater than the lower) limit of that interval. When calculating two-sided confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
, constitutes the probability that the true value will be between the upper and the lower limits.
First, we will denote the available N observations by when and and estimate the common variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
by, for example:
when is the mean of group and is the number of observations in group , and degrees of freedom. As mentioned before, we would like to obtain separate confidence limits for each of the differences such that the probability that all confidence intervals will contain the corresponding is equal to .
We will consider the general case where there are treatment groups and one control group. We will write:
we will also write: , which follows the Student's t-statistic
In statistics, the ''t''-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's ''t''-test. The ''t''-statistic is used in a ...
distribution with n degrees of freedom. The lower confidence limits with joint confidence coefficient for the treatment effects will be given by:
and the constants are chosen so that