frdHouseTest {PMCMRplus} | 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}: θ_0 = θ_1 = … = θ_m, & \mathrm{A}_{m} = θ_0 ≤ θ_1 ≤ … θ_m, θ_0 < θ_m \\ \mathrm{H}_{m-1}: θ_0 = θ_1 = … = θ_{m-1}, & \mathrm{A}_{m-1} = θ_0 ≤ θ_1 ≤ … θ_{m-1}, θ_0 < θ_{m-1} \\ \vdots & \vdots \\ \mathrm{H}_{1}: θ_0 = θ_1, & \mathrm{A}_{1} = θ_0 < θ_1\\ \end{array}
Let Y_{ij} ~ (1 ≤q i ≤q n, 0 ≤q j ≤q k) be a i.i.d. random variable of at least ordinal scale. Further, let \bar{R}_0,~\bar{R}_1, …,~\bar{R}_k be Friedman's average ranks and set \bar{R}_0^*, ≤q … ≤q \bar{R}_k^* to be its isotonic regression estimators under the order restriction θ_0 ≤q … ≤q θ_k.
The statistics is
T_j = ≤ft(\bar{R}_j^* - \bar{R}_0 \right)~ ≤ft[ ≤ft(V_j - H_j \right) ≤ft(2 / n \right) \right]^{-1/2} \qquad (1 ≤q j ≤q k),
with
V_j = ≤ft(j + 1\right) ~ ≤ft(j + 2 \right) / 12
and
H_j = ≤ft(t^3 - t \right) / ≤ft(12 j n \right),
where t is the number of tied ranks.
The critical t'_{i,v,α}-values as given in the tables of Williams (1972) for α = 0.05 (one-sided) are looked up according to the degree of freedoms (v = ∞) 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. 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")