frdHouseTest {PMCMRplus}

House Test

Description

Performs House nonparametric equivalent of William's test for contrasting increasing dose levels of a treatment in a complete randomized block design.

Usage

frdHouseTest(y, ...)

## Default S3 method:
frdHouseTest(y, groups, blocks, alternative = c("greater", "less"), ...)

Arguments

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 "x". Ignored with a warning if "x" is a list.

blocks

a vector or factor object giving the block for the corresponding elements of "x". Ignored with a warning if "x" is a list.

alternative

the alternative hypothesis. Defaults to greater.

...

further arguments to be passed to or from methods.

Details

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).

Value

A list with class "PMCMR" containing the following components:

method

a character string indicating what type of test was performed.

data.name

a character string giving the name(s) of the data.

statistic

lower-triangle matrix of the estimated quantiles of the pairwise test statistics.

p.value

lower-triangle matrix of the p-values for the pairwise tests.

alternative

a character string describing the alternative hypothesis.

p.adjust.method

a character string describing the method for p-value adjustment.

model

a data frame of the input data.

dist

a string that denotes the test distribution.

References

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.

See Also

friedmanTest, friedman.test, frdManyOneExactTest, frdManyOneDemsarTest

Examples

 ## 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")