planScheffe {DOS2}

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. The function is the same as the planScheffe() function in the sensitivitymult package, but the examples are different. The method is discussed in Section 18.3 of "Design of Observational Studies", second edition, and in Rosenbaum (2019).

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.

Details

This method is discussed in section 18.3 of the second edition of "Design of Observational Studies".

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 (2019). The example below reproduces some of the comparisons in that manuscript.

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.

Note

The method is based on Rosenbaum (2019).

The functions "cohere" may be used to calculate the standardized deviates that are compared to the critical values from 'planScheffe'. The function "cohere" has options for an a priori comparison or consideration of all possible comparisons with a Scheffe correction. The function "planScheffe" provides a third option: one planned comparison plus all possible comparisons.

See also 'planScheffe' in the 'sensitivitymult' package.

Author(s)

Paul R. Rosenbaum.

References

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

Rosenbaum, P. R. (2019) <doi:10.1093/biostatistics/kxy055> "Combining planned and discovered comparisons in observational studies". Biostatistics, to appear.

Scheffe, H. (1953) <doi:10.1093/biomet/40.1-2.87> "A method for judging all contrasts in the analysis of variance". Biometrika, 40, 87-104.

Examples

data(teeth)
attach(teeth)

planScheffe(2,alpha=0.05)

# Planned comparison w=c(1,1)
cohere(cbind(either4up,either4low),smoker,mset,
             w=c(1,1),gamma=2,m=8,m1=6,m2=8)

# Discovered comparison emphasizing upper teeth
cohere(cbind(either4up,either4low),smoker,mset,
        w=c(1,3),gamma=2,m=8,m1=6,m2=8)
3.465038^2 #squared deviate

# Both deviates lead to rejection, because
# 3.291909 >= 1.894915
# and 3.465038^2 = 12.00649 >= 7.077349

detach(teeth)