frdHouseTest | R Documentation |
Performs House nonparametric equivalent of William's test for contrasting increasing dose levels of a treatment in a complete randomized block design.
frdHouseTest(y, ...)
## Default S3 method:
frdHouseTest(y, groups, blocks, alternative = c("greater", "less"), ...)
y |
a numeric vector of data values, or a list of numeric data vectors. |
groups |
a vector or factor object giving the group for the
corresponding elements of |
blocks |
a vector or factor object giving the block for the
corresponding elements of |
alternative |
the alternative hypothesis. Defaults to |
... |
further arguments to be passed to or from methods. |
House test is a non-parametric step-down trend test for testing several treatment levels
with a zero control. Let there be k
groups including the control and let
the zero dose level be indicated with i = 0
and the highest
dose level with i = m
, then the following m = k - 1
hypotheses are tested:
\begin{array}{ll}
\mathrm{H}_{m}: \theta_0 = \theta_1 = \ldots = \theta_m, & \mathrm{A}_{m} = \theta_0 \le \theta_1 \le \ldots \theta_m, \theta_0 < \theta_m \\
\mathrm{H}_{m-1}: \theta_0 = \theta_1 = \ldots = \theta_{m-1}, & \mathrm{A}_{m-1} = \theta_0 \le \theta_1 \le \ldots \theta_{m-1}, \theta_0 < \theta_{m-1} \\
\vdots & \vdots \\
\mathrm{H}_{1}: \theta_0 = \theta_1, & \mathrm{A}_{1} = \theta_0 < \theta_1\\
\end{array}
Let Y_{ij} ~ (1 \leq i \leq n, 0 \leq j \leq k)
be a i.i.d. random variable
of at least ordinal scale. Further, let \bar{R}_0,~\bar{R}_1, \ldots,~\bar{R}_k
be Friedman's average ranks and set \bar{R}_0^*, \leq \ldots \leq \bar{R}_k^*
to be its isotonic regression estimators under the order restriction
\theta_0 \leq \ldots \leq \theta_k
.
The statistics is
T_j = \left(\bar{R}_j^* - \bar{R}_0 \right)~ \left[ \left(V_j - H_j \right)
\left(2 / n \right) \right]^{-1/2} \qquad (1 \leq j \leq k),
with
V_j = \left(j + 1\right) ~ \left(j + 2 \right) / 12
and
H_j = \left(t^3 - t \right) / \left(12 j n \right),
where t
is the number of tied ranks.
The critical t'_{i,v,\alpha}
-values
as given in the tables of Williams (1972) for \alpha = 0.05
(one-sided)
are looked up according to the degree of freedoms (v = \infty
) and the order number of the
dose level (j
).
For the comparison of the first dose level (j = 1)
with the control, the critical
z-value from the standard normal distribution is used (Normal
).
A list with class "PMCMR"
containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
Chen, Y.-I., 1999. Rank-Based Tests for Dose Finding in Nonmonotonic Dose–Response Settings. Biometrics 55, 1258–1262. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/j.0006-341X.1999.01258.x")}
House, D.E., 1986. A Nonparametric Version of Williams’ Test for Randomized Block Design. Biometrics 42, 187–190.
friedmanTest
, friedman.test
,
frdManyOneExactTest
, frdManyOneDemsarTest
## Sachs, 1997, p. 675
## Six persons (block) received six different diuretics
## (A to F, treatment).
## The responses are the Na-concentration (mval)
## in the urine measured 2 hours after each treatment.
## Assume A is the control.
y <- matrix(c(
3.88, 5.64, 5.76, 4.25, 5.91, 4.33, 30.58, 30.14, 16.92,
23.19, 26.74, 10.91, 25.24, 33.52, 25.45, 18.85, 20.45,
26.67, 4.44, 7.94, 4.04, 4.4, 4.23, 4.36, 29.41, 30.72,
32.92, 28.23, 23.35, 12, 38.87, 33.12, 39.15, 28.06, 38.23,
26.65),nrow=6, ncol=6,
dimnames=list(1:6, LETTERS[1:6]))
## Global Friedman test
friedmanTest(y)
## Demsar's many-one test
summary(frdManyOneDemsarTest(y=y, p.adjust = "bonferroni",
alternative = "greater"))
## Exact many-one test
summary(frdManyOneExactTest(y=y, p.adjust = "bonferroni",
alternative = "greater"))
## Nemenyi's many-one test
summary(frdManyOneNemenyiTest(y=y, alternative = "greater"))
## House test
frdHouseTest(y, alternative = "greater")
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