sensitivitymv (version 1.2)

senmv: Sensitivity analysis in observational studies using Huber-Maritz M-statistics.

Description

Computes the large sample approximation to the upper bound on the one sided P-value testing the null hypothesis of no treatment effect in a matched observational study with one or more controls matched to each treated subject. Uses Huber's M-statistics as test statistics in the sense proposed by Maritz (1979). The one-sided alternative hypothesis is that treatment increases the level of response.

The senmv function has as its default the use of Huber's (1981) psi function, which is similar in certain respects to a trimmed mean. By default, the trimming occurs (i.e., Huber's psi function becomes horizonal) at trim=2.5 times the median (i.e., lambda = 1/2) of the absolute pair differences; see Maritz (1979) for the paired case and see Rosenbaum (2007, section 4) for matched sets with one or more controls. With trim = 2.5, Huber's psi function is somewhat analogous to a trimmed mean that trims five percent from each tail. If trim is changed to 1, then the psi function is horizonal at 1 times the median (i.e., lambda=1/2) of the absolute pair differences and is somewhat analogous to a midmean.

method="t" performs the permutational t-test, the permutation distribution of a type of mean; see, for instance, Pitman (1937).

The senmv function can be used in the calculation of approximate evidence factors, as discussed and illustrated in the documentation for the truncatedP and truncatedPbg functions.

Usage

senmv(y, gamma = 1, method = NULL, inner = 0, trim = 2.5, 
     
	 lambda = 1/2, tau = 0, TonT=FALSE)

Arguments

y

If y is a vector, then y is the vector of treated-minus-control pair differences in outcomes in n=length(y) matched pairs. If y is an n by J matrix, then: (i) the rows are n matched sets, (ii) the first column is the treated response in a set, columns 2

gamma

gamma is the sensitivity parameter, gamma=1 for a randomization test, gamma>1 for sensitivity bounds. Use of gamma

method

If method is NULL, then the method is determined by the parameters, namely inner, trim, lambda, and TonT. If method is not NULL, then these parameters are set according to the selected method and stated values of the parameters are ignored. The default

inner

Inner trimming to increase design sensitivity. See the discussion of lambda. Use of inner<0 or="" inner="">trim will generate an error.

trim

Outer trimming for resistance to outliers. Setting trim = Inf does no trimming. See the discussion of lambda.

lambda

Observations are scaled by the lambda quantile of the absolute pair differences, defaulting to the median of all paired absolute differences; see Rosenbaum (2007, section 4.2) for a precise definition in the case of multiple controls. If the lambda quant

tau

If tau=0, senmv tests the null hypothesis of no treatment effect. If tau is not 0, senmv tests the null hypothesis that the treatment effect is an additive shift of tau against the alternative that the effect is larger than tau.

TonT

TonT refers to the effect of the treatment on the treated. The default is TonT=FALSE. For all methods except method="t", TonT=FALSE. TonT=TRUE is mostly relevant to the permutational t-test, method="t", but it remains an option whenever is.null(method)

Value

• pval
  Approximate upper bound on the one-sided P-value.
• deviate
  Deviate that is compared to the upper tail of the standard Normal distribution to obtain the P-value.
• statistic
  Value of the test statistic.
• expectation
  Maximum null expectation of the test statistic for the given value of gamma.
• variance
  Among null distributions that yield the maximum expectation, variance is the maximum possible variance for the given value of gamma. See Rosenbaum (2007, Section 4) and Gastwirth, Krieger and Rosenbaum (2000).

References

Main references:

Rosenbaum, P. R. (2007) Sensitivity analysis for m-estimates, tests and confidence intervals in matched observational studies. Biometrics, 2007, 63, 456-464. This paper is the main reference for the senmv function when weights are not used, m1=m2=m=1.

Rosenbaum, P. R. (2013) Impact of multiple matched controls on design sensitivity in observational studies. Biometrics, 2013, 69, 118-127. Evaluates the performance of the methods in the paper above, and in particular provides a basis for selecting the parameter values for senmv. In particular, this paper compares methods h, i, and t.

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Additional references:

Cox, D. R. and Reid, N. Theory of the Design of Experiments. New York: Chapman and Hall/CRC. Chapter 2 discusses randomization inference, in particular an unmatched version of the permutation distributiom of the treated minus control difference in mean responses, or the two-sample (unmatched) permutational t-test. Although there is a large old literature on tests that permute the observations, this recent discussion is written in a modern style.

Fisher, R. A. (1935) Design of Experiments. Edinburgh: Oliver and Boyd. Chapter 3 contains an early, conceptual discussion of the permutation distribution of the mean or the permutational t-test.

Huber, P. (1981) Robust Statistics. New York: Wiley, 1981. Huber first proposed the use of m-statistics in 1964 in a paper in the Annals.

Maritz, J. S. (1979) Exact robust confidence intervals for location. Biometrika 1979, 66, 163-166. Proposed exact permutation tests using m-statistics that Maritz inverts to obtain exact confidence limits. The subtle aspect is the scaling which must be invariant to treatment assignment under the null hypothesis, so it differs from the scaling used by Huber.

Gastwirth, J. L., Krieger, A. M., and Rosenbaum, P. R. (2000) Asymptotic separability in sensitivity analysis. Journal of the Royal Statistical Society B 2000, 62, 545-556. Provides a general large sample approximation when matching with multiple controls, as used in Rosenbaum (2007, Section 4).

Pitman, E. J. G. (1937) Significance tests which may be applied to samples from any populations. JRSS-supplement (later called series B), 4, 119-130. An early technical discussion of the permutation distribution of the sample mean, or the permutational t-test.

Rosenbaum, P. R. (2010) Design of Observational Studies. New York: Springer 2010. Section 2.9 contains an elementary textbook discussion of Maritz's permutation distribution for m-statistics.

Rosenbaum, P. R. (2011) Some approximate evidence factors in observational studies. Journal of the American Statistical Association, 2011, 106, 285-295. The method described in this paper may be implemented using senmv. To do this, one uses senmv several times, combining the resulting one-sided P-value bounds, perhaps using Fisher's method for combining P-values. In the example of this paper, y is n x 3 for three groups in matched triples, and the paper uses Fisher's method to combine the P-value bounds from senmv(y) and senmv(y[,2:3]) for an appropriately defined y. See the mtm example in the documentation for truncatedP and truncatedPbg.

Welch, B. L. (1937) On the z-test in randomized blocks. Biometrika 29, 21-52. As in Pitman (1937) above, discusses permutation inference in which the responses are permuted, essentially a permutational F-test. Expresses causal effects as comparisons of potential responses under alternative treatments.

Examples

# This example reproduces parts of Table 1 in Rosenbaum (2007).
data(erpcp)
senmv(erpcp,gamma=3,trim=1)
senmv(erpcp,gamma=2,trim=1)
senmv(erpcp,gamma=2,trim=1,tau=0.34)
senmv(erpcp,gamma=2,trim=1,tau=0.18)
senmv(erpcp,gamma=2,trim=1,tau=0.185)

# Example reproduces parts of sect. 4.3 in Rosenbaum (2007)
data(tbmetaphase)
senmv(tbmetaphase,gamma=2,trim=1)
senmv(tbmetaphase,gamma=2,trim=1,tau=0.94)
senmv(tbmetaphase,gamma=2,trim=1,tau=0.945)

# Example reproduces part of Table 1 in Rosenbaum (2013)
data(lead150)
senmv(lead150,gamma=2,trim=2)
data(lead250)
senmv(lead250,gamma=2,trim=2)

# Example reproduces parts of of Rosenbaum (2011).  See documentation for truncatedP.
data(mtm)
senmv(-mtm,gamma=11.7,trim=1)
senmv(-mtm[,2:3],gamma=2.1,trim=1)

# Illustrates method = "i"
data(mercury)
senmv(mercury,gamma=15)
senmv(mercury,gamma=15,method="i")  

# Illustrates TonT=TRUE as in method="t".  See note above.
data(mercury)
mean(mercury[,1])-(mean(mercury[,2])+mean(mercury[,3]))/2
senmv(mercury,gamma=15,trim=Inf,TonT=TRUE)$statistic
senmv(mercury,gamma=15,method="t")$statistic
senmv(mercury,gamma=15,method="t",tau=1)$statistic
senmv(mercury,gamma=15,method="t",tau=2)$statistic
senmv(mercury,gamma=15,trim=Inf,TonT=FALSE)$statistic