Confidence intervals and experimental design for negative and positive predictive values in binary diagnostic tests.

Description

Computing asymptotic confidence intervals for negative and positive predictive values of binary diagnostic test assuming a case-control design. Experimental design based on asymptotic formulas and Monte Carlo simulation for hypothesis tests on predictive values, including some plot functions to explore various experimental designs.

Details

1) Computing confidence intervals: The function BDtest computes the asymptotic confidence intervals for negative and positive predictive value given in Mercaldo et al. (2007), assuming binomial sampling for obtaining estimates of sensitivity and specificity (leading to a 2x2 table with numbers of diseased and healthy fixed by design) and known prevalence. Alternatively, the functions CIpvBI and CIpvBII allow to simulate Bayesian intervals for negative and positive predictive values in case-control designs (Stamey and Holt, 2010), where prior knowledge concerning sensitivity, specificity may be included and external data and/or prior knowledge on prevalence may be included. By default, flat, non-informative priors are used, resulting in intervals with improved frequentist small sample performance (Stamey and Holt, 2010).

2) The function nPV uses the asymptotic formulas of Steinberg et al.(2009) to calculate the sample size necessary to reject tests witzh H0: PPV>=PPV0, H0NPV>=PNPV0, with a prespecified power in a case-control setting. Further necessary input arguments are sensitivity, specificity, prevalence, NPV0, PPV0, the range and number of steps of proportion of true positives in the trial. The results of this function can be plotted using plotnPV, plotnPV2 and be somewhat edited by as.data.frame.nPV.

3) Because the results of these functions may be misleading in small sample or extreme proportion situations, the simulation functions simPV and simPVmat allow to check power and coverage probability for given parameter stettings.

The remaining functions are meant for internal use.

Author(s)

Frank Schaarschmidt, on behalf of the Institute of Biostatistics, LUH, Hannover, Germany Maintainer: Frank Schaarschmidt <schaarschmidt@biostat.uni-hannover.de>

References

Steinberg DM, Fine J, Chappell R (2009). Sample size for positive and negative predictive value in diagnostic research using case-control designs. Biostatistics 10,1, 94-105.

Mercaldo ND, Lau KF, Zhou XH (2007). Confidence intervals for predictive values with an emphasis to case-control studies. Statistics in Medicine 26:2170-2183.

Stamey JD and Holt MM (2010). Bayesian interval estimation for predictive values for case-control studies. Communications in Statistics - Simulation and Computation. 39:1, 101-110.

Examples

# 1) Example data: Mercaldo et al.(2007), Table VIII:

Tab8<-matrix(c(240, 178, 87, 288), ncol=2)
colnames(Tab8)<-c("Case","Control")
rownames(Tab8)<-c("ApoEe4plus","ApoEe4minus")
Tab8

# Assuming prevalence=0.03
BDtest(xmat=Tab8, pr=0.03, conf.level = 0.95)

# Assuming prevalence=0.5
BDtest(xmat=Tab8, pr=0.5, conf.level = 0.95)

# 2) Experimental design acc. to Steinberg et al.(2009)

TEST<-nPV(se=c(0.76, 0.78, 0.80, 0.82, 0.84),
 sp=c(0.93, 0.94, 0.95, 0.96, 0.97),
 pr=0.0625, NPV0=0.98, PPV0=0.25, NPVpower = 0.8, PPVpower = 0.8,
 rangeP = c(0.10, 0.9), nsteps = 20, alpha = 0.05)

TEST

plotnPV(TEST, log="y", legpar=list(x=0.6))


# 3) Simulation of power and coverage probability

simPVmat(se=0.8, sp=0.95, pr=0.0625, n1=c(177, 181), 
 n0=c(554, 87), NPV0=0.98, PPV0=c(0.4, 0.25))

Computing confidence intervals for sensitivity, specificity and predictive values assuming a case-control study.

Description

This function computes confidence intervals for negative and positive predictive values. Confidence intervals for sensitivity, specificity are computed for completeness. All methods assume that data are obtained by binomial sampling, with the number of true positives and true negatives in the study fixed by design. The methods to compute negative and positive predictive values (NPV, PPV) assume that prevalence is a known quantity, based on external knowledge.

Usage

BDtest(xmat, pr, conf.level = 0.95)

Arguments

Details

The exact, conservative Clopper Pearson (1934) method is used to compute intervals for the sensitivty and specificity. The asymptotic standard logit intervals (Mercaldo et al. 2007) are used to compute intervals for the predictive values. In case that the table contains any 0, the adjusted logit intervals (Mercaldo et al. 2007) are returned instead to compute intervals for the predictive values.

Value

A list containing:

Author(s)

Frank Schaarschmidt

References

Mercaldo ND, Lau KF, Zhou XH (2007). Confidence intervals for predictive values with an emphasis to case-control studies. Statistics in Medicine 26:2170-2183.

See Also

CInpvppv for the internally used methods to compute the intervals for predictive values,

Examples

# Reproduce the standard logit interval results in 
# Table IX, Mercaldo et al.(2007)

# 1) Example data: Mercaldo et al.(2007), Table VIII:

Tab8<-matrix(c(240, 178, 87, 288), ncol=2)
colnames(Tab8)<-c("Case","Control")
rownames(Tab8)<-c("ApoEe4plus","ApoEe4minus")
Tab8

# Assuming prevalence=0.03
BDtest(xmat=Tab8, pr=0.03, conf.level = 0.95)

# Assuming prevalence=0.5
BDtest(xmat=Tab8, pr=0.5, conf.level = 0.95)

Asymptotic confidence intervals for negative and positive predictive values.

Description

Computes asymptotic confidence intervals for negative and positive predictive values under the assumption of binomial sampling and known prevalence, according to Mercaldo et al. (2007). The standard logit intervals and and adjusted version are available, where the standard logit intervals are recommended.

Usage

CIlnpv(x0, x1, p, conf.level = 0.95, 
alternative = c("two.sided", "less", "greater"))

CIlppv(x0, x1, p, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"))

CIlnpvak(x0, x1, p, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"))

CIlppvak(x0, x1, p, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"))

CombCInpv(x0, x1, p, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"))

CombCIppv(x0, x1, p, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"))

Arguments

Details

CIlnpv and CIlppv implement the standard logit intervals for NPV and PPV, Section 2.2, Eq.(8)-Eq.(11) in Mercaldo et al. (2007). CIlnpvak and CIlppval implement the logit intervals for NPV and PPV with adjusted estimates according to Table II in Mercaldo et al. (2007). The standard logit intervals have better properties, but are not defined in a number of extreme outcomes. The adjusted logit methods do always produce intervals, but have worse frequentist properties (Mercaldo et al. 2007). The functions CombCInpv, CombCIppv combine both methods by computing the stdnard logit method when possible and computing the adjusted methods in those cases where the standard method is not defined. These functions are meant to facilitate simulation, e.g. in simPV, simPVmat.

Value

A list with elements

Note

These functions are meant for internal use. There is not much checking for the validity of input.

Author(s)

Frank Schaarschmidt

References

Mercaldo ND, Lau KF, Zhou XH (2007). Confidence intervals for predictive values with an emphasis to case-control studies. Statistics in Medicine 26: 2170-2183.

See Also

BDtest as a user level function

Examples

CIlnpv(x0=c(87,288), x1=c(240,178), p=0.03,
 conf.level = 0.95, alternative = "two.sided")

CIlppv(x0=c(87,288), x1=c(240,178), p=0.03,
 conf.level = 0.95, alternative = "two.sided")

CIlnpvak(x0=c(87,288), x1=c(240,178), p=0.03,
 conf.level = 0.95, alternative = "two.sided")

CIlppvak(x0=c(87,288), x1=c(240,178), p=0.03,
 conf.level = 0.95, alternative = "two.sided")

Confidence intervals for negative and positive predictive values in a case-control setting by simulation from the posterior distribution.

Description

Computes confidence intervals for negative and positive predictive values by simulation from the posterior beta-distribution (Stamey and Holt, 2010), assuming a case-control design to estimate sensitivity and specificity, while prevalence estimates of an external study and/or prior knowledge concerning prevalence may be introduced additionally.

Usage

CIpvBI(x1, x0, pr, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"),
 B=5000, shapes1=c(1,1), shapes0=c(1,1), ...)

CIpvBII(x1, x0, xpr, conf.level = 0.95,
 alternative = c("two.sided", "less", "greater"),
 B=5000, shapes1=c(1,1), shapes0=c(1,1), shapespr=c(1,1), ...)

Arguments

Details

CIpvBI implements the method refered to as Bayes I in Stamey and Holt (2010), CIpvBI implements the method refered to as Bayes II in Stamey and Holt (2010), Equation (2) and following description (p. 103-104).

Value

A list with elements

Author(s)

Frank Schaarschmidt

References

Stamey JD and Holt MM (2010). Bayesian interval estimation for predictive values for case-control studies. Communications in Statistics - Simulation and Computation. 39:1, 101-110.

Examples

# example data: Stamey and Holt, Table 8 (page 108)
#	Diseased
# Test	D=1	D=0
# T=1	240	87
# T=0	178	288
#n1,n0:	418	375


# reproduce the results for the Bayes I method
# in Stamey and Holt (2010), Table 9, page 108

# assuming known prevalence 0.03 
# ppv 0.0591, 0.0860
# npv 0.9810, 0.9850
CIpvBI( x1=c(240,178), x0=c(87,288), pr=0.03)

# assuming known prevalence 0.04 
# ppv 0.0779, 0.1111
# npv 0.9745, 0.9800
CIpvBI( x1=c(240,178), x0=c(87,288), pr=0.04)

# compare with standard logit intervals
tab <- cbind( x1=c(240,178), x0=c(87,288))
tab
BDtest(tab, pr=0.03)
BDtest(tab, pr=0.04)


# reproduce the results for the Bayes II method 
# in Stamey and Holt (2010), Table 9, page 108

CIpvBII( x1=c(240,178), x0=c(87,288),  shapespr=c(16,486))

CIpvBII( x1=c(240,178), x0=c(87,288), shapespr=c(21,481))

Coerce results of "nPV" to a data.frame.

Description

Coerce the possibly long sample size tables resulting from calling "nPV" to a data.frame.

Usage

## S3 method for class 'nPV'
as.data.frame(x, ...)

Arguments

Details

The lengthy lists in the output nPV, item nlist are coerced to a data.frame with columns propP, and the different NPV/PPV sample sizes for each of the parameters settings following.

Asymptotic experimental design for inference on negative and positive predictive values in case-control studies.

Description

For internal use. Functions to compute sample size (to reach a pre-specified power) and optimal allocation of true positives and true negatives in case-control designs for binary diagnostic tests (Mercaldo et al. 2007).

Usage

nNPV(propP, se, sp, prev, NPV0,
 conf.level = 0.95, power = 0.8)

nPPV(propP, se, sp, prev, PPV0,
 conf.level = 0.95, power = 0.8)

AOppvnpv(se, sp)

Arguments

Details

The functions implement the methods described in section 3.2 of Steinberg et al.(2009), nPPV gives the solution to Eq.(3.6) and NA if necesarry conditions mentioned before are not fulfilled, nNPV gives the solution to Eq.(3.8) and NA if necesarry conditions mentioned before are not fulfilled, AOppvnpv gives the optimal proportion of true poistives as are solutions to Eq.(3.4) and Eq. (3.6) for PPV and NPV, respectively.

Value

For nNPV and nPPV: a list with first element

and further elements giving the input arguments

Author(s)

Frank Schaarschmidt

References

Steinberg DM, Fine J, Chappell R (2009). Sample size for positive and negative predictive value in diagnostic research using case-control designs. Biostatistics 10,1, 94-105.

See Also

For a combination of PPV and NPV experimental design see nPV and plotnPV; to validate small sample results of these asymptotic formulas, see simPVmat

Examples

 nPPV(propP=c(0.2,0.4,0.6,0.8), se=0.9, sp=0.9,
 prev=0.1, PPV0=0.4, conf.level=0.95, power=0.8)

 nNPV(propP=c(0.2,0.4,0.6,0.8), se=0.9, sp=0.9,
 prev=0.1, NPV0=0.95, conf.level=0.95, power=0.8)

 AOppvnpv(se=0.9, sp=0.9)

Asymptotic sample size calculation for inference on negative and positive predictive values in case-control designs.

Description

Functions to compute sample size (to reach a pre-specified power) and optimal allocation of true positives and true negatives in case-control designs (Steinberg et al., 2008) for binary diagnostic tests (Mercaldo et al. 2007).

Usage

nPV(se, sp, prev, NPV0, PPV0,
 NPVpower = 0.8, PPVpower = 0.8,
 rangeP = c(0.05, 0.95), nsteps = 20,
 alpha = 0.05, setnames = NULL)

Arguments

Details

The function uses nNPVPPV and implement the methods described in section 3.2 of Steinberg et al.(2009). The results for NPV are the smallest integers fulfilling Eq.(3.6) and NA if necesarry conditions mentioned before are not met, the results for PPV are the smallest integers fulfilling Eq.(3.8) and NA if necesarry conditions mentioned before are not met.

The arguments se, sp, prev, NPV0, PPV0, NPVpower, PPVpower can be given as vectors or single values, where shorter values are recycled to the length of the longest. The proportion of true positives is varied over nstep equidistant values over the range specified in argument rangeP. On each resulting parameter set, the asymptotic sample size formulas of Steinberg et al.(2009) are applied.

The result of those calculations may be plot using plotnPV and plotnPV2.

Warnings are returned by the internal function nNPV and nPPV if the validity of asymptotic formulas under binomial sampling may be doubtable, namely when the asymptotic formulas return a total sample size n for given propP, se, sp, such that min(n*propP*se, n*propP*(1-se))<5 or min(n*(1-propP)*sp, n*(1-propP)*(1-sp))<5. That is, a warning is returned if the proposed design of the case-control study (n1, n0) = (n*propP, n*(1-propP)) leads to expected counts < 5 for any cell of the 2x2 table.

Value

A list with elements

Author(s)

Frank Schaarschmidt

References

Steinberg DM, Fine J, Chappell R (2009). Sample size for positive and negative predictive value in diagnostic research using case-control designs. Biostatistics 10,1, 94-105.

See Also

plotnPV for showing the results in one graphic, and plotnPV for showing the results in a set of subgraphics,

Examples

#Reproducing illustration in Section 3.4 and 4.2 of
#Steinberg et al. (2009)

FIG1<-nPV(se=0.8, sp=0.95, prev=1/16, NPV0=0.98, PPV0=0.4,
 NPVpower = 0.8, PPVpower = 0.8,
 rangeP = c(0.01, 0.99), nsteps = 100, alpha = 0.05)

FIG1

DFIG1<-as.data.frame(FIG1)

plot(x=DFIG1$propP, y=DFIG1[,2], ylim=c(0,2000), lty=1,  type="l",
 ylab="total sample size", xlab="proportion of true positives")
lines(x=DFIG1$propP, y=DFIG1[,3], lty=2 )

Plot experimental design for different setting in a single figure.

Description

The function creates a plot from the results of the function nPV.

Usage

plotnPV(x, NPVpar = NULL, PPVpar = NULL, legpar = NULL, ...)

Arguments

Details

Required sample sizes for different experimental settings and prevalences, needed to achieve a prespecified power can be calculated in dependence of the proportion of true negative and true positive compounds in the validation set, using function nPV. This function draws a plot with the proportion of positive on x and the total sample size on y, combining all parameter settings in one plot. Parameter settings my be distinguished bylty, lwd, col, pch in NPVpar and PPVpar. By default a legend is drawn which can be further modified in legpar.

Value

A plot.

Author(s)

Frank Schaarschmidt

References

Steinberg DM, Fine J, Chappell R (2009). Sample size for positive and negative predictive value in diagnostic research using case-control designs. Biostatistics 10, 1, 94-105.

See Also

plotnPV2 for a plot with separate subplots for each parameter setting

Examples

TEST<-nPV(se=c(0.9, 0.92, 0.94, 0.96, 0.98), sp=c(0.98, 0.96, 0.94, 0.92, 0.90),
 pr=0.12, NPV0=0.98, PPV0=0.4, NPVpower = 0.8, PPVpower = 0.8,
 rangeP = c(0.05, 0.95), nsteps = 100, alpha = 0.05)

plotnPV(TEST)

# plot parameters maybe introduced via ...
# the legend maybe modified via legpar:

plotnPV(TEST, log="y", legpar=list(x=0.6))

# own colour definitions
plotnPV(TEST, NPVpar=list(col=1:6, lwd=2, lty=1),
 PPVpar=list(col=1:6, lwd=2, lty=3))

Plot experimental design for different settings in a set of sub figure.

Description

The function creates a plot from the results of the function nPV.

Usage

plotnPV2(x, NPVlty = 1, PPVlty = 3, ...)

Arguments

Details

Required sample sizes for different experimental settings and prevalences, needed to achieve a prespecified power can be calculated in dependence of the proportion of true negative and true positive compounds in the validation set, using function nPV. This function draws a plot with the proportion of true positives on x and the total sample size on y, combining all parameter settings in one plot.

Note that for huge numbers of setting this should not work.

Value

A plot.

Author(s)

Frank Schaarschmidt

References

Steinberg DM, Fine J, Chappell R (2009). Sample size for positive and negaitive predictiove value in diagnostic research using case-control designs. Biostatoistics 10, 1, 94-105.

See Also

plotnPV, for sample sizes for several settings in one figure

Examples

TEST<-nPV(se=c(0.9, 0.92, 0.94, 0.96, 0.98), sp=c(0.98, 0.96, 0.94, 0.92, 0.90),
 pr=0.12, NPV0=0.98, PPV0=0.4, NPVpower = 0.8, PPVpower = 0.8,
 rangeP = c(0.05, 0.95), nsteps = 20, alpha = 0.05)

plotnPV2(TEST, log="x")

Detailed print out for BDtest

Description

Print details of the results of the function BDtest on the screen

Usage

## S3 method for class 'BDtest'
print(x, ...)

Arguments

Detailed print out for nPV

Description

Print details of the results of the experimental design function nPV on the screen

Usage

## S3 method for class 'nPV'
print(x, ...)

Arguments

See Also

plotnPV, plotnPV2 to plot the results of nPV

Simulate performance of confidence intervals for predictive values in a case-control design

Description

The function draws data under the binomial assumption and computes the asymptotic confidence bounds (lower bounds only!) for the positive and negative predictive values. Output are the power (probability to exclude NPV0/PPV0), the realized coverage probability, 0.1,0.2, and 0.5-quantiles of the realized distribution of confidence bounds.

Usage

simPV(se, sp, pr, n1, n0, NPV0, PPV0,
 conf.level = 0.95, NSIM = 500)

Arguments

Details

The function draws data under the binomial assumption in a case-control design (Mercaldo et al. 2007), where the binomial doistributions are defined by n1, n0, se, sp. Then, for each drawn data set, the asymptotic lower confidence bounds (with confidence level=1-alpha, i.e. as suitable for a one-sided test at level alpha) for the positive and negative predictive values ar computed. (Note, that the standard logit interval is replaced by the adjusted logit interval of Mercaldo et al. 2007, if the standard logit interval is not defined.) Output are the etsimated power (observed probability that NPV0/PPV0 are excluded by the lower confidence bound), the realized coverage probability (observed probability that the true NPV/PPV are included in their interval), as well as the 0.1, 0.2, and 0.5-quantiles of the realized distribution of confidence bounds.

Value

A (2x7) matrix with results for NPV and PPV in rows 1,2 respectively, and the columns giving estimates of the power to reject H0: NPV>=NPV0 / PPV>=PPV0 (pow), coverage probability (cov), the values which are excluded with 10, 20 and 50 percent probability (q10, q20, q50), as well as the true predictive values and the marginn of H0 used to calculated power.

Author(s)

Frank Schaarschmidt

See Also

simPVmat for the same function, allowing vector input for se, sp, pr, n1, n0, NPV0 and PPV0.

Examples

simPV(se=0.8, sp=0.95, pr=1/16, n1=177, n0=554, NPV0=0.98, PPV0=0.4)

simPV(se=0.8, sp=0.95, pr=1/16, n1=181, n0=87, NPV0=0.98, PPV0=0.25)

Simulate performance of confidence intervals for predictive values in case-control design

Description

Simulate the power (probability to exclude NPV0/PPV0), the coverage probability, and 0.1, 0.2, and 0.5-quantiles of the distribution of (lower!) asymptotic confidence bounds for predictive values. Different experimental setups may be compared. The function draws data under the binomial assumption and computes the asymptotic confidence bounds (lower bounds only!) for the positive and negative predictive values.

Usage

simPVmat(se, sp, pr, n1, n0, NPV0, PPV0,
 conf.level = 0.95, NSIM = 500, setnames = NULL)

Arguments

Details

The vector or single values in se, sp, pr, n1, n0, NPV0, PPV0 are put together (shorter vectors recycled to the length of longest vectors). Then each of the resulting parameter settings is simulated as described in simPV

Value

A list with elements

Author(s)

Frank Schaarschmidt

See Also

This function is meantb to check small sample results obtained by the asymptotci formulas for experimental design from nPV, nNPV, nPPV

Examples

simPVmat(se=0.8, sp=0.95, pr=1/16,
 n1=c(177, 181), n0=c(554, 87), NPV0=0.98, PPV0=c(0.4, 0.25))


simPVmat(se=0.8, sp=0.95, pr=c(0.05,0.0625, 0.075, 0.1),
 n1=177, n0=554, NPV0=0.98, PPV0=0.4)