propTestMdd {EnvStats} | R Documentation |

## Minimal Detectable Difference Associated with a One- or Two-Sample Proportion Test

### Description

Compute the minimal detectable difference associated with a one- or two-sample proportion test, given the sample size, power, and significance level.

### Usage

```
propTestMdd(n.or.n1, n2 = n.or.n1, p0.or.p2 = 0.5, alpha = 0.05, power = 0.95,
sample.type = "one.sample", alternative = "two.sided",
two.sided.direction = "greater", approx = TRUE,
correct = sample.type == "two.sample", warn = TRUE,
return.exact.list = TRUE, tol = 1e-07, maxiter = 1000)
```

### Arguments

`n.or.n1` |
numeric vector of sample sizes. When |

`n2` |
numeric vector of sample sizes for group 2. The default value is |

`p0.or.p2` |
numeric vector of proportions. When |

`alpha` |
numeric vector of numbers between 0 and 1 indicating the Type I error level
associated with the hypothesis test. The default value is |

`power` |
numeric vector of numbers between 0 and 1 indicating the power associated with
the hypothesis test. The default value is |

`sample.type` |
character string indicating whether to compute power based on a one-sample or
two-sample hypothesis test. When |

`alternative` |
character string indicating the kind of alternative hypothesis.
The possible values are |

`two.sided.direction` |
character string indicating the direction (positive or negative) for the
minimal detectable difference when |

`approx` |
logical scalar indicating whether to compute the power based on the normal
approximation to the binomial distribution. The default value is |

`correct` |
logical scalar indicating whether to use the continuity correction when |

`warn` |
logical scalar indicating whether to issue a warning. The default value is |

`return.exact.list` |
logical scalar relevant to the case when |

`tol` |
numeric scalar passed to the |

`maxiter` |
integer passed to the |

### Details

If the arguments `n.or.n1`

, `n2`

, `p0.or.p2`

, `alpha`

, and
`power`

are not all the same length, they are replicated to be the same
length as the length of the longest argument.

*One-Sample Case* (`sample.type="one.sample"`

)

The help file for `propTestPower`

gives references that explain
how the power of the one-sample proportion test is computed based on the values of
`p_0`

(the hypothesized value for `p`

, the probability of “success”),
`p`

(the true value of `p`

), the sample size `n`

, and the Type
I error level `\alpha`

. The function `propTestMdd`

computes the value
of the minimal detectable difference `p - p_0`

for specified values of
sample size, power, and Type I error level by calling the `uniroot`

function to perform a search.

*Two-Sample Case* (`sample.type="two.sample"`

)

The help file for `propTestPower`

gives references that explain
how the power of the two-sample proportion test is computed based on the values of
`p_1`

(the value of the probability of “success” for group 1),
`p_2`

(the value of the probability of “success” for group 2),
the sample sizes for groups 1 and 2 (`n_1`

and `n_2`

), and the Type
I error level `\alpha`

. The function `propTestMdd`

computes the value
of the minimal detectable difference `p_1 - p_2`

for specified values of
sample size, power, and Type I error level by calling the `uniroot`

function to perform a search.

### Value

**Approximate Test ( approx=TRUE)**.
numeric vector of minimal detectable differences.

**Exact Test ( approx=FALSE)**.
If

`return.exact.list=FALSE`

, `propTestMdd`

returns a numeric vector of
minimal detectable differences.
If `return.exact.list=TRUE`

, `propTestMdd`

returns a list with the
following components:

`delta` |
numeric vector of minimal detectable differences. |

`power` |
numeric vector of powers. |

`alpha` |
numeric vector containing the true significance levels.
Because of the discrete nature of the binomial distribution, the true significance
levels usually do not equal the significance level supplied by the user in the
argument |

`q.critical.lower` |
numeric vector of lower critical values for rejecting the null
hypothesis. If the observed number of "successes" is |

`q.critical.upper` |
numeric vector of upper critical values for rejecting the null
hypothesis. If the observed number of "successes" is |

### Note

See the help file for `propTestPower`

.

### Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

### References

See the help file for `propTestPower`

.

### See Also

`propTestPower`

, `propTestN`

,
`plotPropTestDesign`

, `prop.test`

, `binom.test`

.

### Examples

```
# Look at how the minimal detectable difference of the one-sample
# proportion test increases with increasing required power:
seq(0.5, 0.9, by = 0.1)
#[1] 0.5 0.6 0.7 0.8 0.9
mdd <- propTestMdd(n.or.n1 = 50, power = seq(0.5, 0.9, by=0.1))
round(mdd, 2)
#[1] 0.14 0.16 0.17 0.19 0.22
#----------
# Repeat the last example, but compute the minimal detectable difference
# based on the exact test instead of the approximation. Note that with a
# sample size of 50, the largest significance level less than or equal to
# 0.05 for the two-sided alternative is 0.03.
mdd.list <- propTestMdd(n.or.n1 = 50, power = seq(0.5, 0.9, by = 0.1),
approx = FALSE)
lapply(mdd.list, round, 2)
#$delta
#[1] 0.15 0.17 0.18 0.20 0.23
#
#$power
#[1] 0.5 0.6 0.7 0.8 0.9
#
#$alpha
#[1] 0.03 0.03 0.03 0.03 0.03
#
#$q.critical.lower
#[1] 17 17 17 17 17
#
#$q.critical.upper
#[1] 32 32 32 32 32
#==========
# Look at how the minimal detectable difference for the two-sample
# proportion test decreases with increasing sample sizes. Note that for
# the specified significance level, power, and true proportion in group 2,
# no minimal detectable difference is attainable for a sample size of 10 in
# each group.
seq(10, 50, by=10)
#[1] 10 20 30 40 50
propTestMdd(n.or.n1 = seq(10, 50, by = 10), p0.or.p2 = 0.5,
sample.type = "two", alternative="greater")
#[1] NA 0.4726348 0.4023564 0.3557916 0.3221412
#Warning messages:
#1: In propTestMdd(n.or.n1 = seq(10, 50, by = 10), p0.or.p2 = 0.5,
# sample.type = "two", :
# Elements with a missing value (NA) indicate no attainable minimal detectable
# difference for the given values of 'n1', 'n2', 'p2', 'alpha', and 'power'
#2: In propTestMdd(n.or.n1 = seq(10, 50, by = 10), p0.or.p2 = 0.5,
# sample.type = "two", :
# The sample sizes 'n1' and 'n2' are too small, relative to the computed value
# of 'p1' and the given value of 'p2', for the normal approximation to work
# well for the following element indices:
# 2 3
#----------
# Look at how the minimal detectable difference for the two-sample proportion
# test decreases with increasing values of Type I error:
mdd <- propTestMdd(n.or.n1 = 100, n2 = 120, p0.or.p2 = 0.4, sample.type = "two",
alpha = c(0.01, 0.05, 0.1, 0.2))
round(mdd, 2)
#[1] 0.29 0.25 0.23 0.20
#----------
# Clean up
#---------
rm(mdd, mdd.list)
#==========
# Modifying the example on pages 8-5 to 8-7 of USEPA (1989b), determine the
# minimal detectable difference to detect a difference in the proportion of
# detects of cadmium between the background and compliance wells. Set the
# compliance well to "group 1" and the background well to "group 2". Assume
# the true probability of a "detect" at the background well is 1/3, use a
# 5% significance level, use 80%, 90%, and 95% power, use the given sample
# sizes of 64 observations at the compliance well and 24 observations at the
# background well, and use the upper one-sided alternative (probability of a
# "detect" at the compliance well is greater than the probability of a "detect"
# at the background well).
# (The data are stored in EPA.89b.cadmium.df.)
#
# Note that the minimal detectable difference increases from 0.32 to 0.37 to 0.40 as
# the required power increases from 80% to 90% to 95%. Thus, in order to detect a
# difference in probability of detection between the compliance and background
# wells, the probability of detection at the compliance well must be 0.65, 0.70,
# or 0.74 (depending on the required power).
EPA.89b.cadmium.df
# Cadmium.orig Cadmium Censored Well.type
#1 0.1 0.100 FALSE Background
#2 0.12 0.120 FALSE Background
#3 BDL 0.000 TRUE Background
# ..........................................
#86 BDL 0.000 TRUE Compliance
#87 BDL 0.000 TRUE Compliance
#88 BDL 0.000 TRUE Compliance
p.hat.back <- with(EPA.89b.cadmium.df,
mean(!Censored[Well.type=="Background"]))
p.hat.back
#[1] 0.3333333
p.hat.comp <- with(EPA.89b.cadmium.df,
mean(!Censored[Well.type=="Compliance"]))
p.hat.comp
#[1] 0.375
n.back <- with(EPA.89b.cadmium.df,
sum(Well.type == "Background"))
n.back
#[1] 24
n.comp <- with(EPA.89b.cadmium.df,
sum(Well.type == "Compliance"))
n.comp
#[1] 64
mdd <- propTestMdd(n.or.n1 = n.comp, n2 = n.back,
p0.or.p2 = p.hat.back, power = c(.80, .90, .95),
sample.type = "two", alternative = "greater")
round(mdd, 2)
#[1] 0.32 0.37 0.40
round(mdd + p.hat.back, 2)
#[1] 0.65 0.70 0.73
#----------
# Clean up
#---------
rm(p.hat.back, p.hat.comp, n.back, n.comp, mdd)
```

*EnvStats*version 2.8.1 Index]