sensitivitymult (version 1.0.2)

planScheffe: Combining One Planned Comparison and a Scheffe Correction For All Comparisons.

Description

The function planScheffe() computes the critical values for a level alpha test that combines one planned linear combination of a K-dimensional multivariate Normal outcome and consideration of all possible combinations correcting for multiple testing using a Scheffe projection.

Usage

planScheffe(K, alpha = 0.05)

Arguments

K

An integer >=2 giving the number of outcomes to be compared.

alpha

The level of the test, with 0 < alpha < 1.

Value

critical

critical is a vector with two elements, a and c. The null hypothesis is rejected at level alpha if either the Normal deviate for the planned comparison is >= a or if the square of the Normal deviate for any comparison is >= b. Then the probability of a false rejection is <= alpha.

alpha

alpha is a vector with three elements, a, c and joint. The value of joint should equal the input value of alpha aside from numerical errors of computation: it is the probability of a false rejection using the joint test that rejects if either of the two critical values in critical is exceeded. In contrast, a is the probability that the planned deviate will be >= critical[1] when the null hypothesis is true. Also, c is the probability that at least one comparison will have a squared deviate >= critical[2] when the null hypothesis is true.

Details

Although the calculation uses the multivariate Normal distribution, a typical application uses K test statistics that are asymptotically Normal.

The method is based on Rosenbaum (2017). The example below reproduces some of the comparisons in that manuscript.

References

Miller, R. G., Jr. (1981) Simultaneous Statistical Inference (2nd edition). New York: Springer. Section 2.2, pp. 48-67 discusses Scheffe projections.

Rosenbaum, P. R. (2016) Using Scheffe projections for multiple outcomes in an observational study of smoking and periondontal disease. Annals of Applied Statistics, 10, 1447-1471. <doi:10.1214/16-AOAS942>

Rosenbaum, P. R. (2017) Combining planned and discovered comparisons in observational studies. Manuscript.

Scheffe, H. (1953) A method for judging all contrasts in the analysis of variance. Biometrika, 40, 87-104.

Examples

# NOT RUN {
# Please READ the documentation for artcog, and in particular
# the distinction between simulated and actual data.
# The dontrun section refers to the acutal data and
# reproduces results in Rosenbaum (2017).

planScheffe(2,alpha=0.05)
# Interpretation of this output follows.
# Suppose there is a bivariate Normal outcome.  We specify
# one a priori linear combination of its two coordinates.
# We test that the expectation is (0,0) with known covariance
# matrix.  We compute the standardized difference for the
# a priori contrast, rejecting if it is >=1.895.  We also
# reject if we can find any linear combination of the two
# coordinates whose squared standardized difference is
# >=7.077.  The chance that we falsely reject a true
# null hypothesis is 0.05.  The chance of a false rejection
# using the a priori comparison is 0.029.  The chance of
# false rejection using any linear combination is 0.029.
#
# The a priori comparison could be the first principal
# component.  Using the principal() function with
# w=c(1,0) gives the deviate for the first principal
# component.  Exploring every w=c(w1,w2) gives the
# deviates that are squared for comparison with 7.077.

# }
# NOT RUN {
# For this illustration, obtain the actual data,
# as described in the documentation for artcog.
# An illustration from Rosenbaum (2017) follows.
data(artcog)
attach(artcog)
# The comparison using the first principal component:
principal(cbind(words,wordsdelay,animals),arthritis,mset,
     w=c(1,0),gamma=1.396,detail=TRUE)
# The resulting deviate, 1.900 is slightly greated than 1.895,
# so the hypothesis of no effect would be rejected at 0.05 even if
# we allow for a bias in treatment assignment of gamma=1.396.
principal(cbind(words,wordsdelay,animals),arthritis,mset,
     w=c(1,-.075),gamma=1.396,detail=TRUE)
# The comparison w=c(1,-.075) yields a slightly larger
# deviate, 1.907, but 1.907^2 < 7.077, so this ad hoc
# comparison would not lead to rejection.
detach(artcog)
# Interpret gamma:
amplify(1.396,c(2,3))
amplify(1.4,c(2,3))
  
# }