In the analysis of
designed experiment
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
s, the
Friedman test
The Friedman test is a non-parametric statistical test developed by Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking ...
is the most common
non-parametric test
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distr ...
for complete
block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of b ...
s. The Durbin test is a nonparametric test for balanced incomplete designs that reduces to the Friedman test in the case of a complete block design.
Background
In a randomized block design, ''k'' treatments are applied to ''b'' blocks. In a complete block design, every treatment is run for every block and the data are arranged as follows:
For some experiments, it may not be realistic to run all treatments in all blocks, so one may need to run an
incomplete block design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of b ...
. In this case, it is strongly recommended to run a
balanced incomplete design. A balanced incomplete block design has the following properties:
#Every block contains ''k'' experimental units.
#Every treatment appears in ''r'' blocks.
#Every treatment appears with every other treatment an equal number of times.
Test assumptions
The Durbin test is based on the following assumptions:
#The ''b'' blocks are mutually independent. That means the results within one block do not affect the results within other blocks.
#The data can be meaningfully ranked (i.e., the data have at least an
ordinal scale
Ordinal data is a categorical, statistical data type where the variables have natural, ordered categories and the distances between the categories are not known. These data exist on an ordinal scale, one of four levels of measurement described ...
).
Test definition
Let ''R''(''X
ij'') be the rank assigned to ''X
ij'' within block ''i'' (i.e., ranks within a given row). Average ranks are used in the case of ties. The ranks are summed to obtain
:
The Durbin test is then
:H
0: The treatment effects have identical effects
:H
a: At least one treatment is different from at least one other treatment
The test statistic is
:
where
:
:
:
where ''t'' is the number of treatments, ''k'' is the number of treatments per block, ''b'' is the number of blocks, and ''r'' is the number of times each treatment appears.
For
significance level
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
α, the critical region is given by
:
where ''F''
α, ''k'' − 1, ''bk'' − ''b'' − ''t'' + 1 denotes the α-
quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile ...
of the
F distribution
In probability theory and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution ...
with ''k'' − 1 numerator degrees of freedom and ''bk'' − ''b'' − ''t'' + 1 denominator degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the hypothesis of identical treatment effects is rejected, it is often desirable to determine which treatments are different (i.e.,
multiple comparisons
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 inferenc ...
). Treatments ''i'' and ''j'' are considered different if
:
where ''R
j'' and ''R
i'' are the column sum of ranks within the blocks, ''t''
1 − α/2, ''bk'' − ''b'' − ''t'' + 1 denotes the 1 − α/2 quantile of the
t-distribution with ''bk'' − ''b'' − ''t'' + 1 degrees of freedom.
Historical note
''T
1'' was the original statistic proposed by
James Durbin, which would have an approximate null distribution of
(that is,
chi-squared with
degrees of freedom). The ''T
2'' statistic has slightly more accurate critical regions, so it is now the preferred statistic. The ''T
2'' statistic is the two-way analysis of variance statistic computed on the ranks ''R''(''X
ij'').
Related tests
Cochran's Q test In statistics, in the analysis of two-way randomized block designs where the response variable can take only two possible outcomes (coded as 0 and 1), Cochran's Q test is a non-parametric statistical test to verify whether ''k'' treatments have ide ...
is applied for the special case of a binary response variable (i.e., one that can have only one of two possible outcomes). Cochran's Q test is valid for complete block designs only.
See also
*
Analysis of variance
Analysis of variance (ANOVA) is a collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...
*
Friedman test
The Friedman test is a non-parametric statistical test developed by Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking ...
*
Kruskal-Wallis one-way analysis of variance
*
Van der Waerden test
Named after the Dutch mathematician Bartel Leendert van der Waerden, the Van der Waerden test is a statistical test that ''k'' population distribution functions are equal. The Van der Waerden test converts the ranks from a standard Kruskal-Wallis ...
References
*
{{NIST-PD
Statistical tests
Nonparametric statistics