getDesignUnorderedMultinom {lrstat} | R Documentation |
Power and Sample Size for Unordered Multi-Sample Multinomial Response
Description
Obtains the power given sample size or obtains the sample size given power for the chi-square test for unordered multi-sample multinomial response.
Usage
getDesignUnorderedMultinom(
beta = NA_real_,
n = NA_real_,
ngroups = NA_integer_,
ncats = NA_integer_,
pi = NA_real_,
allocationRatioPlanned = NA_integer_,
rounding = TRUE,
alpha = 0.05
)
Arguments
beta |
The type II error. |
n |
The total sample size. |
ngroups |
The number of treatment groups. |
ncats |
The number of categories of the multinomial response. |
pi |
The matrix of response probabilities for the treatment groups.
It should have |
allocationRatioPlanned |
Allocation ratio for the treatment groups. |
rounding |
Whether to round up sample size. Defaults to 1 for sample size rounding. |
alpha |
The two-sided significance level. Defaults to 0.05. |
Details
A multi-sample multinomial response design is used to test whether the
response probabilities differ among multiple treatment arms.
Let \pi_{gi}
denote the response probability for
category i = 1,\ldots,C
in group
g = 1,\ldots,G
, where G
is the total number of
treatment groups, and C
is the total number of categories
for the response variable.
The chi-square test statistic is given by
X^2 = \sum_{g=1}^{G} \sum_{i=1}^{C}
\frac{(n_{gi} - n_{g+}n_{+i}/n)^2}{n_{g+} n_{+i}/n}
where n_{gi}
is the number of subjects in category i
for group g
, n_{g+}
is the total number of subjects
in group g
, and n_{+i}
is the total number of subjects
in category i
across all groups, and
n
is the total sample size.
Let r_g
denote the randomization probability for group g
, and
define the weighted average response probability
for category i
across all groups as
\bar{\pi_i} = \sum_{g=1}^{G} r_g \pi_{gi}
Under the null hypothesis,
X^2
follows a chi-square distribution with(G-1)(C-1)
degrees of freedom.Under the alternative hypothesis,
X^2
follows a non-central chi-square distribution with non-centrality parameter\lambda = n \sum_{g=1}^{G} \sum_{i=1}^{C} \frac{r_g (\pi_{gi} - \bar{\pi_i})^2} {\bar{\pi_i}}
The sample size is chosen such that the power to reject the null
hypothesis is at least 1-\beta
for a given
significance level \alpha
.
Value
An S3 class designUnorderedMultinom
object with the
following components:
-
power
: The power to reject the null hypothesis. -
alpha
: The two-sided significance level. -
n
: The maximum number of subjects. -
ngroups
: The number of treatment groups. -
ncats
: The number of categories of the multinomial response. -
pi
: The response probabilities for the treatment groups. -
effectsize
: The effect size for the chi-square test. -
allocationRatioPlanned
: Allocation ratio for the treatment groups. -
rounding
: Whether to round up sample size.
Author(s)
Kaifeng Lu, kaifenglu@gmail.com
Examples
(design1 <- getDesignUnorderedMultinom(
beta = 0.1, ngroups = 3, ncats = 4,
pi = matrix(c(0.230, 0.320, 0.272,
0.358, 0.442, 0.154,
0.142, 0.036, 0.039),
3, 3, byrow = TRUE),
allocationRatioPlanned = c(2, 2, 1),
alpha = 0.05))