prop.comb: prop.comb
In prop.comb.RR: Analyzing Combination of Proportions and Relative Risk
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Main function for making inferences (confidence intervals and tests) about a linear combination of proportions using optimal methods from the literature and score method.
Usage
1
2
Arguments
x
a vector of counts of successes.
n
a vector of counts of samples sizes.
p
hypothesized true value of the interest parameter.
a
coefficients of the linear combination of proportions.
alternative
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".
conf.level
confidence level of the returned confidence interval.
coverage
coverage probability of the confidence interval.
nrep
number of replays to calculate the estimated coverage probabilities
Details
Counts of successes must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers.
It checks how many samples there are (k). When k=1, it will be the case of one proporcion. When k=2 and a=(-1,1), we will have the difference of proportions. In another case, the parameter of interest will be a linear combination of k proportions.
When k=1, automatically a=1 and the main function will assign p=0.5 if p=NULL. If k=2 and a=NULL, automatically a=c(-1,+1) and the main function will assign p=0 if p=NULL, i.e. test of homogeneity of proportions.
If alternative is NULL, the main function will automatically assign alternative="two.sided".
If conf.level is NULL, the main function will automatically assign conf.level=0.95.
If coverage is NULL or FALSE, the main function does not calculate the estimated coverage probabilities. Whereas, if coverage is TRUE, the main function calculates the estimated coverage probabilities. In this case, if nrep is NULL, the main function will automatically assign nrep=1000.
Value
Returns a list with the following components:
estimate
a vector with the sample proportions x/n.
a
coefficients of the linear combination of proportions.
L
estimated value of the interest parameter. L=difference for the d case.
inference
confidence intervals (lower limit, upper limit) and p-values of the test (with z-values of statistics). The coverage probabilities are included when the user requests them.
alternative
a character string describing the alternative hypothesis.
lambda
hypothesized true value of the interest parameter.
k
number of samples.
x
number of successes.
n
number of trials.
p
same value of lambda
conf.level
confidence level of the confidence interval.
recommendation
recommended method by references.
Author(s)
Maria Alvarez Hernandez and Javier Roca Pardinas
References
Wilson, E. (1927). "Probable inference, the law of succession, and statistical inference." Journal of the American Statistical Association, 22, 209 - 212
Agresti, A. & Coull, B. A. (1998). "Approximate is better than "exact" for interval estimation of binomial proportions." The American Statistician, 52, 119 - 126
Herranz, I. & Martin, A. (2008). "A numerical comparison of several unconditional exact tests in problems of equivalence based on the difference of proportions." Journal of Statistical Computation and Simulation, 78, 969 - 981
Martin, A.; Alvarez, M. & Herranz, I. (2011). "Inferences about a linear combination of proportions." Statistical Methods in Medical Research, 2011, 20, 369 - 387
Martin, A.; Herranz, I. & Alvarez, M. (2012). "The optimal method to make inferences about a lineal combination of proportions." Journal of Statistical Computation and Simulation, 82, 123 - 135
Martin, A.; Alvarez, M. & Herranz, I. (2012). "Asymptotic two-tailed confidence intervals for the difference of proportions." Journal of Applied Statistics, 39, 1423 - 1435
Martin, A. & Alvarez, M. (2013). "Optimal method for realizing two-sided inferences about a linear combination of two proportions." Communications in Statistics - Simulation and Computation, 42, 327 - 343
Martin, A. & Alvarez, M. (2014). "Two-tailed asymptotic inferences for a proportion." Journal of Applied Statistics, 41, 1516 - 1529
Yu, W.; Guo, X. & Xu, W. (2014). "An improved score interval with a modified midpoint for a binomial proportion." Journal of Statistical Computation and Simulation, 84, 1022 - 1038
Alvarez, M. & Martin, A. (2015). "New asymptotic inferences about the difference, ratio and linear combination of two independent proportions." Communications in Statistics - Simulation and Computation (in press).
See Also
prop.RR for inferences about the relative risk
Examples
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# ONE PROPORTION
# Yu et al (2014) use a data set, which describes characteristics of some
# American bridges. The objetive is to construct CIs for the proportion
# of wood bridges.
prop.comb(x=16, n=109, alternative="two.sided", conf.level=0.95)
# DIFFERENCE OF TWO PROPORTION
# Rodary et al. (1989) studied the response to chemotherapy and radiation
# therapy in a randomized clinical trial on nephroblastoma.
# The main objetive is to contruct CIs about the difference of the response
# and to contrast the homogeneity of proportions.
prop.comb(x=c(83,69), n=c(88,76), alternative="two.sided", conf.level=0.95)
# COMBINATION OF TWO PROPORTIONS
# Martin and Alvarez (2013) construct CIs about the parameter "loss associated
# with a diagnostic test" in the context of comparing two diagnostic tests
# (Bloch, 1997) and use a data set by Hanley and McNeil (1983).
prop.comb(x=c(44, 3), n=c(54, 58), a=c(-0.4, 0.6),
alternative="two.sided", conf.level=0.95)
# Tebbs and Roths (2008) refer to data about a multicenter clinical trial whose
# aim was to evaluate the efficacy of a lowered salt regim in the treatment of male
# infants with acute diarrhea. The objetive it to obtain a confidence interval about
# the proportion of fever cases in South American area.
prop.comb(x=c(32, 34), n=c(107, 92), a=c(107/199, 92/199),
alternative="two.sided", conf.level=0.95)
# LINEAR COMBINATION OF K>3 PROPORTIONS
# Price and Bonett (2004) refer to a study by Cohen et al. (1991) in which 120 rats
# were randomly assigned four diets (high or low fat and with or without fiber).
# The absence or presence of a tumor was recorded for each rat.
# The contrast of interest will evaluate the effects of study's variables.
# Data of diet and tumor study:
x <-c(20, 14, 27, 19); n <-c(30, 30, 30, 30)
a1 <-c(+1, -1, -1, +1); a2 <-c(+1, +1, -1, -1); a3 <-c(+1, -1, +1, -1)
prop.comb(x, n, a=a1); prop.comb(x, n, a=a2); prop.comb(x, n, a=a3)
Example output
Loading required package: rootSolve
1-sample proportion test
number of successes = 16 number of trials = 109
sample estimates
p
0.1468
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.0924 0.2252 7.3753 0
Score (with cc) 0.0888 0.2304 7.2795 0
Adjusted Arc Sine 0.0898 0.2224 8.1324 0
Adjusted Wald 0.0918 0.2268 9.8970 0
Modified Score 0.0915 0.2243 NA NA
Recommendation:
Adjusted Wald
difference of proportions
x: number of successes
n: number of trials
prop: proportion sample estimates
sample x n prop
1 83 88 0.9432
2 69 76 0.9079
difference of proportions: D=p2-p1
estimated D: -0.0353
alternative hypothesis: true difference D is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.1286 0.0481 -0.8653 0.3869
Score (with cc) -0.1287 0.0483 -0.8617 0.3888
Adjusted Wald -0.1205 0.0488 -0.8306 0.4062
Adjusted Arc Sine -0.1292 0.0484 -0.8503 0.3952
Adjusted-M Arc Sine -0.1228 0.0460 -0.8515 0.3945
Recommendation:
Adjusted Arc Sine
There were 50 or more warnings (use warnings() to see the first 50)
2 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 44 54 -0.4 0.8148
2 3 58 0.6 0.0517
combination of interest: L= -0.4*p1+0.6*p2
estimated L: -0.2949
alternative hypothesis: true combination L is not equal to 0.1
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.3373 -0.2274 -8.2478 0
Score (with cc) -0.3374 -0.2273 -8.2445 0
Adjusted Wald (a) -0.3377 -0.2252 -13.2851 0
Adjusted Wald (b) -0.3381 -0.2258 -13.3242 0
Adjusted likelihood ratio -0.3350 -0.2246 -8.8228 0
Recommendation:
Adjusted Wald(b)
There were 50 or more warnings (use warnings() to see the first 50)
2 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 32 107 0.5377 0.2991
2 34 92 0.4623 0.3696
combination of interest: L= 0.5377*p1+0.4623*p2
estimated L: 0.3317
alternative hypothesis: true combination L is not equal to 0.5
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.2701 0.3995 -4.7612 0
Score (with cc) 0.2701 0.3996 -4.7598 0
Adjusted Wald (a) 0.2702 0.3997 -4.9971 0
Adjusted Wald (b) 0.2700 0.3996 -4.9995 0
Adjusted likelihood ratio 0.2706 0.4000 -4.7850 0
Recommendation:
Adjusted Wald(b)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 -1 0.4667
3 27 30 -1 0.9000
4 19 30 1 0.6333
combination of interest: L= 1*p1-1*p2-1*p3+1*p4
estimated L: -0.0667
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.3883 0.2445 -0.4119 0.6804
Score (with cc) -0.3883 0.2445 -0.4119 0.6804
Adjusted Wald -0.3808 0.2516 -0.4004 0.6889
Peskun -0.4167 0.2875 -0.3651 0.7150
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 1 0.4667
3 27 30 -1 0.9000
4 19 30 -1 0.6333
combination of interest: L= 1*p1+1*p2-1*p3-1*p4
estimated L: -0.4
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.7097 -0.0771 -2.4241 0.0153
Score (with cc) -0.7097 -0.0771 -2.4241 0.0153
Adjusted Wald -0.7038 -0.0714 -2.4023 0.0163
Peskun -0.7329 -0.0422 -2.1909 0.0285
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 -1 0.4667
3 27 30 1 0.9000
4 19 30 -1 0.6333
combination of interest: L= 1*p1-1*p2+1*p3-1*p4
estimated L: 0.4667
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.1421 0.7742 2.8033 0.0051
Score (with cc) 0.1420 0.7742 2.8033 0.0051
Adjusted Wald 0.1360 0.7684 2.8027 0.0051
Peskun 0.1094 0.7950 2.5560 0.0106
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
prop.comb.RR documentation built on May 2, 2019, 12:41 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Main function for making inferences (confidence intervals and tests) about a linear combination of proportions using optimal methods from the literature and score method.
Usage
1 2 |
Arguments
x |
a vector of counts of successes. |
n |
a vector of counts of samples sizes. |
p |
hypothesized true value of the interest parameter. |
a |
coefficients of the linear combination of proportions. |
alternative |
a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". |
conf.level |
confidence level of the returned confidence interval. |
coverage |
coverage probability of the confidence interval. |
nrep |
number of replays to calculate the estimated coverage probabilities |
Details
Counts of successes must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers.
It checks how many samples there are (k). When k=1, it will be the case of one proporcion. When k=2 and a=(-1,1), we will have the difference of proportions. In another case, the parameter of interest will be a linear combination of k proportions.
When k=1, automatically a=1 and the main function will assign p=0.5 if p=NULL. If k=2 and a=NULL, automatically a=c(-1,+1) and the main function will assign p=0 if p=NULL, i.e. test of homogeneity of proportions.
If alternative is NULL, the main function will automatically assign alternative="two.sided".
If conf.level is NULL, the main function will automatically assign conf.level=0.95.
If coverage is NULL or FALSE, the main function does not calculate the estimated coverage probabilities. Whereas, if coverage is TRUE, the main function calculates the estimated coverage probabilities. In this case, if nrep is NULL, the main function will automatically assign nrep=1000.
Value
Returns a list with the following components:
estimate |
a vector with the sample proportions x/n. |
a |
coefficients of the linear combination of proportions. |
L |
estimated value of the interest parameter. |
inference |
confidence intervals (lower limit, upper limit) and p-values of the test (with z-values of statistics). The coverage probabilities are included when the user requests them. |
alternative |
a character string describing the alternative hypothesis. |
lambda |
hypothesized true value of the interest parameter. |
k |
number of samples. |
x |
number of successes. |
n |
number of trials. |
p |
same value of |
conf.level |
confidence level of the confidence interval. |
recommendation |
recommended method by references. |
Author(s)
Maria Alvarez Hernandez and Javier Roca Pardinas
References
Wilson, E. (1927). "Probable inference, the law of succession, and statistical inference." Journal of the American Statistical Association, 22, 209 - 212
Agresti, A. & Coull, B. A. (1998). "Approximate is better than "exact" for interval estimation of binomial proportions." The American Statistician, 52, 119 - 126
Herranz, I. & Martin, A. (2008). "A numerical comparison of several unconditional exact tests in problems of equivalence based on the difference of proportions." Journal of Statistical Computation and Simulation, 78, 969 - 981
Martin, A.; Alvarez, M. & Herranz, I. (2011). "Inferences about a linear combination of proportions." Statistical Methods in Medical Research, 2011, 20, 369 - 387
Martin, A.; Herranz, I. & Alvarez, M. (2012). "The optimal method to make inferences about a lineal combination of proportions." Journal of Statistical Computation and Simulation, 82, 123 - 135
Martin, A.; Alvarez, M. & Herranz, I. (2012). "Asymptotic two-tailed confidence intervals for the difference of proportions." Journal of Applied Statistics, 39, 1423 - 1435
Martin, A. & Alvarez, M. (2013). "Optimal method for realizing two-sided inferences about a linear combination of two proportions." Communications in Statistics - Simulation and Computation, 42, 327 - 343
Martin, A. & Alvarez, M. (2014). "Two-tailed asymptotic inferences for a proportion." Journal of Applied Statistics, 41, 1516 - 1529
Yu, W.; Guo, X. & Xu, W. (2014). "An improved score interval with a modified midpoint for a binomial proportion." Journal of Statistical Computation and Simulation, 84, 1022 - 1038
Alvarez, M. & Martin, A. (2015). "New asymptotic inferences about the difference, ratio and linear combination of two independent proportions." Communications in Statistics - Simulation and Computation (in press).
See Also
prop.RR for inferences about the relative risk
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 | # ONE PROPORTION
# Yu et al (2014) use a data set, which describes characteristics of some
# American bridges. The objetive is to construct CIs for the proportion
# of wood bridges.
prop.comb(x=16, n=109, alternative="two.sided", conf.level=0.95)
# DIFFERENCE OF TWO PROPORTION
# Rodary et al. (1989) studied the response to chemotherapy and radiation
# therapy in a randomized clinical trial on nephroblastoma.
# The main objetive is to contruct CIs about the difference of the response
# and to contrast the homogeneity of proportions.
prop.comb(x=c(83,69), n=c(88,76), alternative="two.sided", conf.level=0.95)
# COMBINATION OF TWO PROPORTIONS
# Martin and Alvarez (2013) construct CIs about the parameter "loss associated
# with a diagnostic test" in the context of comparing two diagnostic tests
# (Bloch, 1997) and use a data set by Hanley and McNeil (1983).
prop.comb(x=c(44, 3), n=c(54, 58), a=c(-0.4, 0.6),
alternative="two.sided", conf.level=0.95)
# Tebbs and Roths (2008) refer to data about a multicenter clinical trial whose
# aim was to evaluate the efficacy of a lowered salt regim in the treatment of male
# infants with acute diarrhea. The objetive it to obtain a confidence interval about
# the proportion of fever cases in South American area.
prop.comb(x=c(32, 34), n=c(107, 92), a=c(107/199, 92/199),
alternative="two.sided", conf.level=0.95)
# LINEAR COMBINATION OF K>3 PROPORTIONS
# Price and Bonett (2004) refer to a study by Cohen et al. (1991) in which 120 rats
# were randomly assigned four diets (high or low fat and with or without fiber).
# The absence or presence of a tumor was recorded for each rat.
# The contrast of interest will evaluate the effects of study's variables.
# Data of diet and tumor study:
x <-c(20, 14, 27, 19); n <-c(30, 30, 30, 30)
a1 <-c(+1, -1, -1, +1); a2 <-c(+1, +1, -1, -1); a3 <-c(+1, -1, +1, -1)
prop.comb(x, n, a=a1); prop.comb(x, n, a=a2); prop.comb(x, n, a=a3)
|
Example output
Loading required package: rootSolve
1-sample proportion test
number of successes = 16 number of trials = 109
sample estimates
p
0.1468
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.0924 0.2252 7.3753 0
Score (with cc) 0.0888 0.2304 7.2795 0
Adjusted Arc Sine 0.0898 0.2224 8.1324 0
Adjusted Wald 0.0918 0.2268 9.8970 0
Modified Score 0.0915 0.2243 NA NA
Recommendation:
Adjusted Wald
difference of proportions
x: number of successes
n: number of trials
prop: proportion sample estimates
sample x n prop
1 83 88 0.9432
2 69 76 0.9079
difference of proportions: D=p2-p1
estimated D: -0.0353
alternative hypothesis: true difference D is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.1286 0.0481 -0.8653 0.3869
Score (with cc) -0.1287 0.0483 -0.8617 0.3888
Adjusted Wald -0.1205 0.0488 -0.8306 0.4062
Adjusted Arc Sine -0.1292 0.0484 -0.8503 0.3952
Adjusted-M Arc Sine -0.1228 0.0460 -0.8515 0.3945
Recommendation:
Adjusted Arc Sine
There were 50 or more warnings (use warnings() to see the first 50)
2 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 44 54 -0.4 0.8148
2 3 58 0.6 0.0517
combination of interest: L= -0.4*p1+0.6*p2
estimated L: -0.2949
alternative hypothesis: true combination L is not equal to 0.1
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.3373 -0.2274 -8.2478 0
Score (with cc) -0.3374 -0.2273 -8.2445 0
Adjusted Wald (a) -0.3377 -0.2252 -13.2851 0
Adjusted Wald (b) -0.3381 -0.2258 -13.3242 0
Adjusted likelihood ratio -0.3350 -0.2246 -8.8228 0
Recommendation:
Adjusted Wald(b)
There were 50 or more warnings (use warnings() to see the first 50)
2 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 32 107 0.5377 0.2991
2 34 92 0.4623 0.3696
combination of interest: L= 0.5377*p1+0.4623*p2
estimated L: 0.3317
alternative hypothesis: true combination L is not equal to 0.5
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.2701 0.3995 -4.7612 0
Score (with cc) 0.2701 0.3996 -4.7598 0
Adjusted Wald (a) 0.2702 0.3997 -4.9971 0
Adjusted Wald (b) 0.2700 0.3996 -4.9995 0
Adjusted likelihood ratio 0.2706 0.4000 -4.7850 0
Recommendation:
Adjusted Wald(b)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 -1 0.4667
3 27 30 -1 0.9000
4 19 30 1 0.6333
combination of interest: L= 1*p1-1*p2-1*p3+1*p4
estimated L: -0.0667
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.3883 0.2445 -0.4119 0.6804
Score (with cc) -0.3883 0.2445 -0.4119 0.6804
Adjusted Wald -0.3808 0.2516 -0.4004 0.6889
Peskun -0.4167 0.2875 -0.3651 0.7150
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 1 0.4667
3 27 30 -1 0.9000
4 19 30 -1 0.6333
combination of interest: L= 1*p1+1*p2-1*p3-1*p4
estimated L: -0.4
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) -0.7097 -0.0771 -2.4241 0.0153
Score (with cc) -0.7097 -0.0771 -2.4241 0.0153
Adjusted Wald -0.7038 -0.0714 -2.4023 0.0163
Peskun -0.7329 -0.0422 -2.1909 0.0285
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
4 linear combination of proportions
x: number of successes
n: number of trials
beta: coefficients of the combination
prop: proportion sample estimates
sample x n beta prop
1 20 30 1 0.6667
2 14 30 -1 0.4667
3 27 30 1 0.9000
4 19 30 -1 0.6333
combination of interest: L= 1*p1-1*p2+1*p3-1*p4
estimated L: 0.4667
alternative hypothesis: true combination L is not equal to 0
95 percent confidence interval
lower limit upper limit z value p-value
Score (without cc) 0.1421 0.7742 2.8033 0.0051
Score (with cc) 0.1420 0.7742 2.8033 0.0051
Adjusted Wald 0.1360 0.7684 2.8027 0.0051
Peskun 0.1094 0.7950 2.5560 0.0106
Recommendation:
Score (without cc)
There were 50 or more warnings (use warnings() to see the first 50)
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