frdHouseTest() R function from [PMCMRplus]

House Test

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"), ...)

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.

Returns

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.

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).

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")

References

Chen, Y.-I., 1999. Rank-Based Tests for Dose Finding in Nonmonotonic Dose–Response Settings. Biometrics 55 , 1258--1262. 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.

frdHouseTest function