# Test the AFT model Rank Regression estimator by Empirical Likelihood

### Description

Use the empirical likelihood ratio and Wilks theorem to test if the regression coefficient is equal to beta, based on the rank estimator for the AFT model.

The log empirical likelihood been maximized is

∑_{d=1} \log Δ F(e_i) + ∑_{d=0} \log [1-F(e_i)];

where e_i are the residuals.

### Usage

 1 RankRegTest(y, d, x, beta, type="Gehan") 

### Arguments

 y a vector of length N, containing the censored responses. d a vector (length N) of either 1's or 0's. d=1 means y is uncensored; d=0 means y is right censored. x a matrix of size N by q. beta a vector of length q. the value of the regression coefficient to be tested in the model y_i = β x_i + ε_i

.

 type default to Gehan type. The other option is Logrank type.

### Details

The estimator of beta can be obtained by function rankaft( ) in the package rankreg. But here you may test other values of beta. If you test the beta value that is obtained from the rankaft( ), then the -2LLR should be 0 and the p-value should be 1.

The above likelihood should be understood as the likelihood of the error term, so in the regression model the error e_i should be iid.

The estimation equation used when maximize the empirical likelihood is

0 = ∑_i φ (e_i) d_i Δ F(e_i) (x_i - \bar x_i )/(n w_i)

which was described in detail in the references below.

See also the function RankRegTestH, which is based on the hazard likelihood.

### Value

A list with the following components:

 "-2LLR" the -2 loglikelihood ratio; should have approximate chisq distribution under H_o. logel2 the log empirical likelihood, under estimating equation. logel the log empirical likelihood of the Kaplan-Meier of e's. prob the probabilities that max the empirical likelihood under rank estimating equation constraint.

Mai Zhou.

### References

Kalbfleisch, J. and Prentice, R. (2002) The Statistical Analysis of Failure Time Data. 2nd Ed. Wiley, New York. (Chapter 7)

Jin, Z., Lin, D.Y., Wei, L. J. and Ying, Z. (2003). Rank-based inference for the accelerated failure time model. Biometrika, 90, 341-53.

Zhou, M. (2005). Empirical likelihood analysis of the rank estimator for the censored accelerated failure time model. Biometrika, 92, 492-98.

### Examples

 1 2 3 4 data(myeloma) RankRegTest(y=myeloma[,1], d=myeloma[,2], x=myeloma[,3], beta= -15.50147) # you should get "-2LLR" = 9.050426e-05 (practically zero) # The beta value, -15.50147, was obtained by rankaft() from the rankreg package. 

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