propTestMdd: 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 sample.type="one.sample", this argument denotes $n$, the number of observations in the single sample. When sample.type="two.sample", this argument denotes $n_1$, the number of ob

numeric vector of sample sizes for group 2. The default value is n2=n.or.n1. This argument is ignored when sample.type="one.sample". Missing (NA), undefined (NaN), and infinite (Inf

p0.or.p2

numeric vector of proportions. When sample.type="one.sample", this argument denotes the hypothesized value of $p$, the probability of dQuote{success}. When sample.type="two.sample", this argument denotes the value of $p

alpha

numeric vector of numbers between 0 and 1 indicating the Type I error level associated with the hypothesis test. The default value is alpha=0.05.

power

numeric vector of numbers between 0 and 1 indicating the power associated with the hypothesis test. The default value is power=0.95.

sample.type

character string indicating whether to compute power based on a one-sample or two-sample hypothesis test. When sample.type="one.sample", the computed power is based on a hypothesis test for a single proportion. When sampl

alternative

character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "less", and "greater".

two.sided.direction

character string indicating the direction (positive or negative) for the minimal detectable difference when alternative="two.sided". When two.sided.direction="greater" (the default), the minimal detectable difference

approx

logical scalar indicating whether to compute the power based on the normal approximation to the binomial distribution. The default value is approx=TRUE. Currently, the exact method (approx=FALSE) is only available for t

correct

logical scalar indicating whether to use the continuity correction when approx=TRUE. The default value is approx=TRUE when sample.type="two.sample" and approx=FALSE when sample.type="one.sample

warn

logical scalar indicating whether to issue a warning. The default value is warn=TRUE. When approx=TRUE (power based on the normal approximation) and warn=TRUE, a warning is issued for cases when the normal app

return.exact.list

logical scalar relevant to the case when approx=FALSE (i.e., when the power is based on the exact test). This argument indicates whether to return a list containing extra information about the exact test in addition to the power

tol

numeric scalar passed to the uniroot function that indicates the tolerance to use in the search algorithm. The default value is tol=1e-7.

maxiter

integer passed to the uniroot function that indicates the maximum number of iterations to use in the search algorithm. The default value is maxiter=1000.

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:
deltanumeric vector of minimal detectable differences.
powernumeric vector of powers.
alphanumeric 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 alpha.
q.critical.lowernumeric vector of lower critical values for rejecting the null hypothesis. If the observed number of "successes" is less than or equal to these values, the null hypothesis is rejected. (Not present if alternative="greater".)
q.critical.uppernumeric vector of upper critical values for rejecting the null hypothesis. If the observed number of "successes" is greater than these values, the null hypothesis is rejected. (Not present if alternative="less".)

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.

References

See the help file for propTestPower.

Examples

Run this code

# 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)

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