Inference on the stress-strength model R = P(Y<X)

Description

Compute confidence intervals and point estimates for R, under parametric model assumptions for Y and X. Y and X are two independent continuous random variables from two different populations.

Details

The package can be used for computing accurate confidence intervals and point estimates for the stress-strength (reliability) model R = P(Y<X); maximum likelihood estimates, Wald statistic, signed log-likelihood ratio statistic and its modified version ca be computed.
The main function is Prob, which evaluates confidence intervals and point estimates under different approaches and parametric assumptions.

Author(s)

Giuliana Cortese

Maintainer: Giuliana Cortese gcortese@stat.unipd.it

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

Kotz S, Lumelskii Y, Pensky M. (2003). The Stress-Strength Model and its Generalizations. Theory and Applications. World Scientific, Singapore.

Examples

	 # data from the first population
	 Y <- rnorm(15, mean=5, sd=1)    
	 # data from the second population
	 X <- rnorm(10, mean=7, sd=1.5)      
     level <- 0.01        # \eqn{\alpha} level 
     # estimate and confidence interval under the assumption of two
     # normal variables with different variances.
	 Prob(Y, X, "norm_DV", "RPstar", level)   
	 # method has to be set equal to "RPstar".

Maximum likelihood estimates of the stress-strength model R = P(Y<X).

Description

Compute maximum likelihood estimates of R, considered as the parameter of interest. Maximum likelihood estimates of the nuisance parameter are also supplied.

Usage

MLEs(ydat, xdat, distr)

Arguments

Details

The two independent random variables Y and X with given distribution distr are measurements of a certain characteristics on two different populations. For the relationship of the parameter of interest (R) and nuisance parameters with the original parameters of distr, look at the details in loglik.

Value

Vector of estimetes of the nuisance parameters and the R quantity (parameter of interest), respectively.

Author(s)

Giuliana Cortese

References

Kotz S, Lumelskii Y, Pensky M. (2003). The Stress-Strength Model and its Generalizations. Theory and Applications. World Scientific, Singapore.

See Also

loglik, Prob

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1)        
	# data from the second population
    X <- rnorm(10, mean=7, sd=1.5)      
    # vector of MLEs for the nuisance parameters and the quantity R
    MLEs(Y, X, "norm_DV")  

Estimation of the stress-strength model R = P(Y<X)

Description

Compute confidence intervals and point estimates for the probability R, under parametric model assumptions for Y and X. Y and X are two independent continuous random variable from two different populations.

Usage

Prob(ydat, xdat, distr = "exp", method = "RPstar", level = 0.05)

Arguments

Value

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on R=P(Y<X). Computational Statistics, 28:1035-1059.

See Also

wald, rp, rpstar

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1)        
	# data from the second population
	X <- rnorm(10, mean=7, sd=1.5)     
    level <- 0.01                  ## \eqn{\alpha} level 
    # estimate and confidence interval under the assumption of two
    # normal variables with different variances.
	Prob(Y, X, "norm_DV", "RPstar", level)  
	# method has to be set equal to "RPstar".

Estimated ROC curves

Description

Plot of ROC curves estimated under parametric model assumptions on the continuous diagnostic marker.

Usage

ROC.plot(ydat, xdat, distr = "exp", method = "RPstar", mc = 1)

Arguments

Details

If mc is different from 1, method does not need to be specified.

Value

Plot of ROC curves

Note

The two independent random variables Y and X with given distribution distr are measurements of the diagnostic marker on the diseased and non-diseased subjects, respectively.
In "Wald" method, or equivalently "RP" method, MLEs for parameters of the Y and X distributions are computed and then used to estimate specificity and sensitivity. These measures are evaluated as P(Y<t) and P(X>t), respectively.
In "RPstar" method, parameters of the Y and X distributions are estimated from the r_p^*-based estimate of the AUC.

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

See Also

Prob

Examples

	# data from the non-diseased population
	Y <- rnorm(15, mean=5, sd=1)       
	# data from the diseased population
	X <- rnorm(10, mean=7, sd=1.5)      
 	ROC.plot(Y, X, "norm_DV", method = "RP", mc = 2)

Log-likelihood of the bivariate distribution of (Y,X)

Description

Computation of the log-likelihood function of the bivariate distribution (Y,X). The log-likelihood is reparametrized with the parameter of interest \psi, corresponding to the quantity R, and the nuisance parameter \lambda.

Usage

loglik(ydat, xdat, lambda, psi, distr = "exp")

Arguments

Details

For further information on the random variables Y and X, see help on Prob.
Reparameterisation in order to determine \psi and \lambda depends on the assumed distribution. Here the following relashonships have been used:

Exponential models:

\psi= \frac{\alpha}{(\alpha + \beta)} and \lambda = \alpha + \beta, with Y \sim e^{\alpha} and X \sim e^{\beta};

Gaussian models with equal variances:

\psi = \Phi \left( \frac{\mu_2-\mu_1}{\sqrt{2 \sigma^2}} \right) and \lambda = (\lambda_1,\lambda_2) = ( \frac{\mu_1}{\sqrt{2 \sigma^2}}, \sqrt{2 \sigma^2} ), with Y \sim N(\mu_1, \sigma^2) and X \sim N(\mu_2, \sigma^2);

Gaussian models with unequal variances:

\psi = \Phi \left( \frac{\mu_2-\mu_1}{\sqrt{\sigma_1^2 + \sigma_2^2}} \right) and \lambda = (\lambda_1, \lambda_2, \lambda_3) = (\mu_1, \sigma_1^2, \sigma_2^2), with Y \sim N(\mu_1, \sigma_1^2) and X \sim N(\mu_2, \sigma_2^2).

The Standard Normal cumulative distribution function is indicated with \Phi.

Value

Value of the log-likelihood function computed in \psi=psi and \lambda=lambda.

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

See Also

MLEs

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1)                  
    # data from the second population      
	X <- rnorm(10, mean=7, sd=1)                        
    mu1 <- 5                                           
    mu2 <- 7
    sigma <- 1
    # parameter of interest, the R probability
    interest <- pnorm((mu2-mu1)/(sigma*sqrt(2)))         
    # nuisance parameters
    nuisance <- c(mu1/(sigma*sqrt(2)), sigma*sqrt(2))    
    # log-likelihood value 
    loglik(Y, X, nuisance, interest, "norm_EV")        

Signed log-likelihood ratio statistic

Description

Compute the signed log-likelihood ratio statistic (r_p) for a given value of the stress strength R = P(Y<X), that is the parameter of interest, under given parametric model assumptions.

Usage

rp(ydat, xdat, psi, distr = "exp")

Arguments

Details

The two independent random variables Y and X with given distribution distr are measurements of the diagnostic marker on the diseased and non-diseased subjects, respectively. For the relationship of the parameter of interest (R) and nuisance parameters with the original parameters of distr, look at the details in loglik.

Value

Value of the signed log-likelihood ratio statistic r_p.

Note

The r_p values can be also used for testing statistical hypotheses on the probability R.

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

Severini TA. (2000). Likelihood Methods in Statistics. Oxford University Press, New York.

Brazzale AR., Davison AC., Reid N. (2007). Applied Asymptotics. Case-Studies in Small Sample Statistics. Cambridge University Press, Cambridge.

See Also

wald, rpstar, MLEs, Prob

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1)        
	# data from the second population
	X <- rnorm(10, mean=7, sd=1.5)      
	# value of \eqn{r_p} for \code{psi=0.9}
	rp(Y, X, 0.9,"norm_DV")             

Modified signed log-likelihood ratio statistic

Description

Compute the modified signed log-likelihood ratio statistic (r_p^*) for a given value of the stress strength R = P(Y<X), that is the parameter of interest, under given parametric model assumptions.

Usage

rpstar(ydat, xdat, psi, distr = "exp")

Arguments

Details

The two independent random variables Y and X with given distribution distr are measurements from two different populations. For the relationship of the parameter of interest (R) and nuisance parameters with the original parameters of distr, look at the details in loglik.

Value

Note

The statistic r_p^* is a modified version of r_p which provides more statistically accurate estimates. The r_p^* values can be also used for testing statistical hypotheses on the probability R.

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

Severini TA. (2000). Likelihood Methods in Statistics. Oxford University Press, New York.

Brazzale AR., Davison AC., Reid N. (2007). Applied Asymptotics. Case-Studies in Small Sample Statistics. Cambridge University Press, Cambridge.

See Also

wald, rp, MLEs, Prob

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1)      
	# data from the second population  
	X <- rnorm(10, mean=7, sd=1.5) 
	# value of \eqn{r_p^*} for \code{psi=0.9}     
	rpstar(Y, X, 0.9,"norm_DV")        
	# method has be set equal to "RPstar".

Wald statistic

Description

Compute the Wald statistic for a given value of the stress-strength R = P(Y<X), that is the parameter of interest, under given parametric model assumptions.

Usage

wald(ydat, xdat, psi, distr = "exp")

Arguments

Details

The two independent random variables Y and X with given distribution distr are measurements from two different populations. For the relationship of the parameter of interest (R) and nuisance parameters with the original parameters of distr, look at the details in loglik.

Value

Note

Values of the Wald statistic can be also used for testing statistical hypotheses on the probability R.

Author(s)

Giuliana Cortese

References

Cortese G., Ventura L. (2013). Accurate higher-order likelihood inference on P(Y<X). Computational Statistics, 28:1035-1059.

Brazzale AR., Davison AC., Reid N. (2007). Applied Asymptotics. Case-Studies in Small Sample Statistics. Cambridge University Press, Cambridge.

See Also

rp, rpstar, MLEs, Prob

Examples

	# data from the first population
	Y <- rnorm(15, mean=5, sd=1) 
	# data from the second population       
	X <- rnorm(10, mean=7, sd=1.5) 
	# value of Wald for \code{psi=0.9}     
	wald(Y, X, 0.9,"norm_DV")