# popower

##### Power and Sample Size for Ordinal Response

`popower`

computes the power for a two-tailed two sample comparison
of ordinal outcomes under the proportional odds ordinal logistic
model. The power is the same as that of the Wilcoxon test but with
ties handled properly. `posamsize`

computes the total sample size
needed to achieve a given power. Both functions compute the efficiency
of the design compared with a design in which the response variable
is continuous. `print`

methods exist for both functions. Any of the
input arguments may be vectors, in which case a vector of powers or
sample sizes is returned. These functions use the methods of
Whitehead (1993).

`pomodm`

is a function that assists in translating odds ratios to
differences in mean or median on the original scale.

##### Usage

```
popower(p, odds.ratio, n, n1, n2, alpha=0.05)
# S3 method for popower
print(x, …)
posamsize(p, odds.ratio, fraction=.5, alpha=0.05, power=0.8)
# S3 method for posamsize
print(x, …)
pomodm(x=NULL, p, odds.ratio=1)
```

##### Arguments

- p
a vector of marginal cell probabilities which must add up to one. The

`i`

th element specifies the probability that a patient will be in response level`i`

, averaged over the two treatment groups.- odds.ratio
the odds ratio to be able to detect. It doesn't matter which group is in the numerator.

- n
total sample size for

`popower`

. You must specify either`n`

or`n1`

and`n2`

. If you specify`n`

,`n1`

and`n2`

are set to`n/2`

.- n1
for

`popower`

, the number of subjects in treatment group 1- n2
for

`popower`

, the number of subjects in group 2- alpha
type I error

- x
an object created by

`popower`

or`posamsize`

, or a vector of data values given to`pomodm`

that corresponds to the vector`p`

of probabilities. If`x`

is omitted for`pomodm`

, the`odds.ratio`

will be applied and the new vector of individual probabilities will be returned. Otherwise if`x`

is given to`pomodm`

, a 2-vector with the mean and median`x`

after applying the odds ratio is returned.- fraction
for

`posamsize`

, the fraction of subjects that will be allocated to group 1- power
for

`posamsize`

, the desired power (default is 0.8)- …
unused

##### Value

a list containing `power`

and `eff`

(relative efficiency) for `popower`

,
or containing `n`

and `eff`

for `posamsize`

.

##### References

Whitehead J (1993): Sample size calculations for ordered categorical data. Stat in Med 12:2257--2271.

Julious SA, Campbell MJ (1996): Letter to the Editor. Stat in Med 15: 1065--1066. Shows accuracy of formula for binary response case.

##### See Also

##### Examples

```
# NOT RUN {
# For a study of back pain (none, mild, moderate, severe) here are the
# expected proportions (averaged over 2 treatments) that will be in
# each of the 4 categories:
p <- c(.1,.2,.4,.3)
popower(p, 1.2, 1000) # OR=1.2, total n=1000
posamsize(p, 1.2)
popower(p, 1.2, 3148)
# If p was the vector of probabilities for group 1, here's how to
# compute the average over the two groups:
# p2 <- pomodm(p=p, odds.ratio=1.2)
# pavg <- (p + p2) / 2
# Compare power to test for proportions for binary case,
# proportion of events in control group of 0.1
p <- 0.1; or <- 0.85; n <- 4000
popower(c(1 - p, p), or, n) # 0.338
bpower(p, odds.ratio=or, n=n) # 0.320
# Add more categories, starting with 0.1 in middle
p <- c(.8, .1, .1)
popower(p, or, n) # 0.543
p <- c(.7, .1, .1, .1)
popower(p, or, n) # 0.67
# Continuous scale with final level have prob. 0.1
p <- c(rep(1 / n, 0.9 * n), 0.1)
popower(p, or, n) # 0.843
# Compute the mean and median x after shifting the probability
# distribution by an odds ratio under the proportional odds model
x <- 1 : 5
p <- c(.05, .2, .2, .3, .25)
# For comparison make up a sample that looks like this
X <- rep(1 : 5, 20 * p)
c(mean=mean(X), median=median(X))
pomodm(x, p, odds.ratio=1) # still have to figure out the right median
pomodm(x, p, odds.ratio=0.5)
# }
```

*Documentation reproduced from package Hmisc, version 4.3-1, License: GPL (>= 2)*