Package 'rr'

R Package for the Randomized Response Technique

Description

rr implements methods developed by Blair, Imai, and Zhou (2015) such as multivariate regression and power analysis for the randomized response technique. Randomized response is a survey technique that introduces random noise to reduce potential bias from non-response and social desirability when asking questions about sensitive behaviors and beliefs. The current version of this package conducts multivariate regression analyses for the sensitive item under four standard randomized response designs: mirrored question, forced response, disguised response, and unrelated question. Second, it generates predicted probabilities of answering affirmatively to the sensitive item for each respondent. Third, it also allows users to use the sensitive item as a predictor in an outcome regression under the forced response design. Additionally, it implements power analyses to help improve research design. In future versions, this package will extend to new modified designs that are based on less stringent assumptions than those of the standard designs, specifically to allow for non-compliance and unknown distribution to the unrelated question under the unrelated question design.

Author(s)

Graeme Blair, Experiments in Governance and Politics, Columbia University [email protected], https://graemeblair.com

Kosuke Imai, Departments of Government and Statistics, Harvard University [email protected], https://imai.fas.harvard.edu

Yang-Yang Zhou, Department of Political Science, University of British Columbia [email protected], https://www.yangyangzhou.com

Maintainer: Graeme Blair <[email protected]>

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2015) "Design and Analysis of the Randomized Response Technique." Journal of the American Statistical Association. Available at https://graemeblair.com/papers/randresp.pdf.

---

Nigeria Randomized Response Survey Experiment on Social Connections to Armed Groups

Description

This data set is a subset of the data from the randomized response technique survey experiment conducted in Nigeria to study civilian contact with armed groups. The survey was implemented by Blair (2014).

Usage

data(nigeria)

Format

A data frame containing 2457 observations. The variables are:

• Quesid: Survey ID of civilian respondent.

• rr.q1: Randomized response survey item using the Forced Response Design asking the respondent whether they hold direct social connections with members of armed groups. 0 if no connection; 1 if connection.

• cov.age: Age of the respondent.

• cov.asset.index: The number of assets owned by the respondent from an index of nine assets including radio, T.V., motorbike, car, mobile phone, refrigerator, goat, chicken, and cow.

• cov.married: Marital status. 0 if single; 1 if married.

• cov.education: Education level of the respondent. 1 if no school; 2 if started primary school; 3 if finished primary school; 4 if started secondary school; 5 if finished secondary school; 6 if started polytechnic or college; 7 if finished polytechnic or college; 8 if started university; 9 if finished university; 10 if received graduate (masters or Ph.D) education.

• cov.female: Gender. 0 if male; 1 if female.

• civic: Whether or not the respondent is a member of a civic group in their communities, such as youth groups , women's groups, or community development committees.

Source

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) Replication data for: Design and Analysis of the Randomized Response Technique.

References

Blair, G. (2014). "Why do civilians hold bargaining power in state revenue conflicts? Evidence from Nigeria."

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

---

Nigeria Randomized Response Survey Experiment on Social Connections to Armed Groups

Description

This data set is a subset of the data from the randomized response technique survey experiment conducted in Nigeria to study civilian contact with armed groups. The survey was implemented by Blair (2014).

Usage

data(nigeria)

Format

A data frame containing 2457 observations. The variables are:

• Quesid: Survey ID of civilian respondent.

• rr.q1: Randomized response survey item using the Forced Response Design asking the respondent whether they hold direct social connections with members of armed groups. 0 if no connection; 1 if connection.

• cov.age: Age of the respondent.

• cov.asset.index: The number of assets owned by the respondent from an index of nine assets including radio, T.V., motorbike, car, mobile phone, refrigerator, goat, chicken, and cow.

• cov.married: Marital status. 0 if single; 1 if married.

• cov.education: Education level of the respondent. 1 if no school; 2 if started primary school; 3 if finished primary school; 4 if started secondary school; 5 if finished secondary school; 6 if started polytechnic or college; 7 if finished polytechnic or college; 8 if started university; 9 if finished university; 10 if received graduate (masters or Ph.D) education.

• cov.female: Gender. 0 if male; 1 if female.

• civic: Whether or not the respondent is a member of a civic group in their communities, such as youth groups , women's groups, or community development committees.

Source

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) Replication data for: Design and Analysis of the Randomized Response Technique.

References

Blair, G. (2014). "Why do civilians hold bargaining power in state revenue conflicts? Evidence from Nigeria."

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

---

Power Analysis Plot for Randomized Response

Description

power.rr.plot generates a power analysis plot for randomized response survey designs.

Usage

power.rr.plot(p, p0, p1, q, design, n.seq, r, presp.seq, presp.null =
NULL, sig.level, prespT.seq, prespC.seq, prespT.null = NULL, prespC.null,
type = c("one.sample", "two.sample"), alternative = c("one.sided",
"two.sided"), solve.tolerance = .Machine$double.eps, legend = TRUE, legend.x
= "bottomright", legend.y, par = TRUE, ...)

Arguments

p	The probability of receiving the sensitive question (Mirrored Question Design, Unrelated Question Design); the probability of answering truthfully (Forced Response Design); the probability of selecting a red card from the 'yes' stack (Disguised Response Design).
p0	The probability of forced 'no' (Forced Response Design).
p1	The probability of forced 'yes' (Forced Response Design).
q	The probability of answering 'yes' to the unrelated question, which is assumed to be independent of covariates (Unrelated Question Design).
design	Call of design (including modified designs) used: "forced-known", "mirrored", "disguised", "unrelated-known", "forced-unknown", and "unrelated-unknown".
n.seq	A sequence of number of observations or sample sizes.
r	For the modified designs only (i.e. "forced-unknown" for Forced Response with Unknown Probability and "unrelated-unknown" for Unrelated Question with Unknown Probability), r is the proportion of respondents allocated to the first group, which is the group that is directed to answer the sensitive question truthfully with probability p as opposed to the second group which is directed to answer the sensitive question truthfully with probability 1-p.
presp.seq	For a one sample test, a sequence of probabilities of possessing the sensitive trait under the alternative hypothesis.
presp.null	For a one sample test, the probability of possessing the sensitive trait under the null hypothesis. The default is NULL meaning zero probability of possessing the sensitive trait.
sig.level	Significance level (Type I error probability).
prespT.seq	For a two sample test, a sequence of probabilities of the treated group possessing the sensitive trait under the alternative hypothesis.
prespC.seq	For a two sample test, a sequence of probabitilies of the control group possessing the sensitive trait under the alternative hypothesis.
prespT.null	For a two sample test, the probability of the treated group possessing the sensitive trait under the null hypothesis. The default is NULL meaning there is no difference between the treated and control groups, specifically that prespT.null is the same as prespC.null, the probability of the control group possessing the sensitive trait under the null hypothesis.
prespC.null	For a two sample test, the probability of the control group possessing the sensitive trait under the null hypothesis.
type	One or two sample test. For a two sample test, the alternative and null hypotheses refer to the difference between the two samples of the probabilities of possessing the sensitive trait.
alternative	One or two sided test.
solve.tolerance	When standard errors are calculated, this option specifies the tolerance of the matrix inversion operation solve.
legend	Indicator of whether to include a legend of sample sizes. The default is TRUE.
legend.x	Placement on the x-axis of the legend. The default is "bottomright".
legend.y	Placement on the y-axis of the legend.
par	Option to set or query graphical parameters within the function. The default is TRUE.
...	Additional arguments to be passed to par()

Details

This function generates a power analysis plot for randomized response survey designs, both for the standard designs ("forced-known", "mirrored", "disguised", "unrelated-known") and modified designs ("forced-unknown", and "unrelated -unknown"). The x-axis shows the population proportions with the sensitive trait; the y-axis shows the statistical power; and different sample sizes are shown as different lines in grayscale.

Value

Power curve plot

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

Examples

## Generate a power plot for the forced design with known 
## probabilities of 2/3 in truth-telling group, 1/6 forced to say "yes" 
## and 1/6 forced to say "no", varying the number of respondents from 
## 250 to 2500 and the population proportion of respondents 
## possessing the sensitive trait from 0 to .15.

presp.seq <- seq(from = 0, to = .15, by = .0025)
n.seq <- c(250, 500, 1000, 2000, 2500)
power.rr.plot(p = 2/3, p1 = 1/6, p0 = 1/6, n.seq = n.seq, 
              presp.seq = presp.seq, presp.null = 0,
              design = "forced-known", sig.level = .01, 
              type = "one.sample",
              alternative = "one.sided", legend = TRUE)
				       
    
## Replicates the results for Figure 2 in Blair, Imai, and Zhou (2014)

---

Power Analysis for Randomized Response

Description

power.rr.test is used to conduct power analysis for randomized response survey designs.

Usage

power.rr.test(p, p0, p1, q, design, n = NULL, r, presp, presp.null =
NULL, sig.level, prespT, prespC, prespT.null = NULL, prespC.null, power =
NULL, type = c("one.sample", "two.sample"), alternative = c("one.sided",
"two.sided"), solve.tolerance = .Machine$double.eps)

Arguments

p	The probability of receiving the sensitive question (Mirrored Question Design, Unrelated Question Design); the probability of answering truthfully (Forced Response Design); the probability of selecting a red card from the 'yes' stack (Disguised Response Design).
p0	The probability of forced 'no' (Forced Response Design).
p1	The probability of forced 'yes' (Forced Response Design).
q	The probability of answering 'yes' to the unrelated question, which is assumed to be independent of covariates (Unrelated Question Design).
design	Call of design (including modified designs) used: "forced-known", "mirrored", "disguised", "unrelated-known", "forced-unknown", and "unrelated-unknown".
n	Number of observations. Exactly one of 'n' or 'power' must be NULL.
r	For the modified designs only (i.e. "forced-unknown" for Forced Response with Unknown Probability and "unrelated-unknown" for Unrelated Question with Unknown Probability), r is the proportion of respondents allocated to the first group, which is the group that is directed to answer the sensitive question truthfully with probability p as opposed to the second group which is directed to answer the sensitive question truthfully with probability 1-p.
presp	For a one sample test, the probability of possessing the sensitive trait under the alternative hypothesis.
presp.null	For a one sample test, the probability of possessing the sensitive trait under the null hypothesis. The default is NULL meaning zero probability of possessing the sensitive trait.
sig.level	Significance level (Type I error probability).
prespT	For a two sample test, the probability of the treated group possessing the sensitive trait under the alternative hypothesis.
prespC	For a two sample test, the probability of the control group possessing the sensitive trait under the alternative hypothesis.
prespT.null	For a two sample test, the probability of the treated group possessing the sensitive trait under the null hypothesis. The default is NULL meaning there is no difference between the treated and control groups, specifically that prespT.null is the same as prespC.null, the probability of the control group possessing the sensitive trait under the null hypothesis.
prespC.null	For a two sample test, the probability of the control group possessing the sensitive trait under the null hypothesis.
power	Power of test (Type II error probability). Exactly one of 'n' or 'power' must be NULL.
type	One or two sample test. For a two sample test, the alternative and null hypotheses refer to the difference between the two samples of the probabilities of possessing the sensitive trait.
alternative	One or two sided test.
solve.tolerance	When standard errors are calculated, this option specifies the tolerance of the matrix inversion operation solve.

Details

This function allows users to conduct power analysis for randomized response survey designs, both for the standard designs ("forced-known", "mirrored", "disguised", "unrelated-known") and modified designs ("forced-unknown", and "unrelated -unknown").

Value

power.rr.test contains the following components (the inclusion of some components such as the design parameters are dependent upon the design used):

n	Point estimates for the effects of covariates on the randomized response item.
r	Standard errors for estimates of the effects of covariates on the randomized response item.
presp	For a one sample test, the probability of possessing the sensitive trait under the alternative hypothesis. For a two sample test, the difference between the probabilities of possessing the sensitive trait for the treated and control groups under the alternative hypothesis.
presp.null	For a one sample test, the probability of possessing the sensitive trait under the null hypothesis. For a two sample test, the difference between the probabilities of possessing the sensitive trait for the treated and control groups under the null hypothesis.
sig.level	Significance level (Type I error probability).
power	Power of test (Type II error probability).
type	One or two sample test.
alternative	One or two sided test.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2015) "Design and Analysis of the Randomized Response Technique." Journal of the American Statistical Association. Available at https://graemeblair.com/papers/randresp.pdf.

Examples

## Calculate the power to detect a sensitive item proportion of .2
## with the forced design with known probabilities of 2/3 in truth-telling group,
## 1/6 forced to say "yes" and 1/6 forced to say "no" and sample size of 200.

power.rr.test(p = 2/3, p1 = 1/6, p0 = 1/6, n = 200, 
             presp = .2, presp.null = 0,
             design = "forced-known", sig.level = .01,
             type = "one.sample", alternative = "one.sided")

---

Predicted Probabilities for Randomized Response Regression

Description

predict.rrreg is used to generate predicted probabilities from a multivariate regression object of survey data using randomized response methods.

Usage

## S3 method for class 'rrreg'
predict(object, given.y = FALSE, alpha = .05, n.sims =
1000, avg = FALSE, newdata = NULL, quasi.bayes = FALSE, keep.draws = FALSE,
...)

Arguments

object	An object of class "rrreg" generated by the rrreg() function.
given.y	Indicator of whether to use "y" the response vector to calculate the posterior prediction of latent responses. Default is FALSE, which simply generates fitted values using the logistic regression.
alpha	Confidence level for the hypothesis test to generate upper and lower confidence intervals. Default is .05.
n.sims	Number of sampled draws for quasi-bayesian predicted probability estimation. Default is 1000.
avg	Whether to output the mean of the predicted probabilities and uncertainty estimates. Default is FALSE.
newdata	Optional new data frame of covariates provided by the user. Otherwise, the original data frame from the "rreg" object is used.
quasi.bayes	Option to use Monte Carlo simulations to generate uncertainty estimates for predicted probabilities. Default is FALSE.
keep.draws	Option to return the Monte Carlos draws of the quantity of interest, for use in calculating differences for example.
...	Further arguments to be passed to predict.rrreg() command.

Details

This function allows users to generate predicted probabilities for the randomized response item given an object of class "rrreg" from the rrreg() function. Four standard designs are accepted by this function: mirrored question, forced response, disguised response, and unrelated question. The design, already specified in the "rrreg" object, is then directly inputted into this function.

Value

predict.rrreg returns predicted probabilities either for each observation in the data frame or the average over all observations. The output is a list that contains the following components:

est	Predicted probabilities for the randomized response item generated either using fitted values, posterior predictions, or quasi-Bayesian simulations. If avg is set to TRUE, the output will only include the mean estimate.
se	Standard errors for the predicted probabilities of the randomized response item generated using Monte Carlo simulations. If quasi.bayes is set to FALSE, no standard errors will be outputted.
ci.lower	Estimates for the lower confidence interval. If quasi.bayes is set to FALSE, no confidence interval estimate will be outputted.
ci.upper	Estimates for the upper confidence interval. If quasi.bayes is set to FALSE, no confidence interval estimate will be outputted.
qoi.draws	Monte Carlos draws of the quantity of interest, returned only if keep.draws is set to TRUE.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

See Also

rrreg to conduct multivariate regression analyses in order to generate predicted probabilities for the randomized response item.

Examples

data(nigeria)

set.seed(1)

## Define design parameters
p <- 2/3  # probability of answering honestly in Forced Response Design
p1 <- 1/6 # probability of forced 'yes'
p0 <- 1/6 # probability of forced 'no'

## Fit linear regression on the randomized response item of 
## whether citizen respondents had direct social contacts to armed groups

rr.q1.reg.obj <- rrreg(rr.q1 ~ cov.asset.index + cov.married + I(cov.age/10) + 
                      I((cov.age/10)^2) + cov.education + cov.female,   
                      data = nigeria, p = p, p1 = p1, p0 = p0, 
                      design = "forced-known")

## Generate the mean predicted probability of having social contacts to 
## armed groups across respondents using quasi-Bayesian simulations. 

rr.q1.reg.pred <- predict(rr.q1.reg.obj, given.y = FALSE, 
                                avg = TRUE, quasi.bayes = TRUE, 
                                n.sims = 10000)

## Replicates Table 3 in Blair, Imai, and Zhou (2014)

---

Predicted Probabilities for Randomized Response as a Regression Predictor

Description

predict.rrreg.predictor is used to generate predicted probabilities from a multivariate regression object of survey data using the randomized response item as a predictor for an additional outcome.

Usage

## S3 method for class 'rrreg.predictor'
predict(object, fix.z = NULL, alpha = .05,
n.sims = 1000, avg = FALSE, newdata = NULL, quasi.bayes = FALSE, keep.draws
= FALSE, ...)

Arguments

object	An object of class "rrreg.predictor" generated by the rrreg.predictor() function.
fix.z	An optional value or vector of values between 0 and 1 that the user inputs as the proportion of respondents with the sensitive trait or probability that each respondent has the sensitive trait, respectively. If the user inputs a vector of values, the vector must be the length of the data from the "rrreg.predictor" object. Default is NULL in which case predicted probabilities are generated for the randomized response item.
alpha	Confidence level for the hypothesis test to generate upper and lower confidence intervals. Default is .05.
n.sims	Number of sampled draws for quasi-bayesian predicted probability estimation. Default is 1000.
avg	Whether to output the mean of the predicted probabilities and uncertainty estimates. Default is FALSE.
newdata	Optional new data frame of covariates provided by the user. Otherwise, the original data frame from the "rreg" object is used.
quasi.bayes	Option to use Monte Carlo simulations to generate uncertainty estimates for predicted probabilities. Default is FALSE meaning no uncertainty estimates are outputted.
keep.draws	Option to return the Monte Carlos draws of the quantity of interest, for use in calculating differences for example.
...	Further arguments to be passed to predict.rrreg.predictor() command.

Details

This function allows users to generate predicted probabilities for the additional outcome variables with the randomized response item as a covariate given an object of class "rrreg.predictor" from the rrreg.predictor() function. Four standard designs are accepted by this function: mirrored question, forced response, disguised response, and unrelated question. The design, already specified in the "rrreg.predictor" object, is then directly inputted into this function.

Value

predict.rrreg.predictor returns predicted probabilities either for each observation in the data frame or the average over all observations. The output is a list that contains the following components:

est	Predicted probabilities of the additional outcome variable given the randomized response item as a predictor generated either using fitted values or quasi-Bayesian simulations. If avg is set to TRUE, the output will only include the mean estimate.
se	Standard errors for the predicted probabilities of the additional outcome variable given the randomized response item as a predictor generated using Monte Carlo simulations. If quasi.bayes is set to FALSE, no standard errors will be outputted.
ci.lower	Estimates for the lower confidence interval. If quasi.bayes is set to FALSE, no confidence interval estimate will be outputted.
ci.upper	Estimates for the upper confidence interval. If quasi.bayes is set to FALSE, no confidence interval estimate will be outputted.
qoi.draws	Monte Carlos draws of the quantity of interest, returned only if keep.draws is set to TRUE.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

See Also

rrreg.predictor to conduct multivariate regression analyses with the randomized response as predictor in order to generate predicted probabilities.

Examples

data(nigeria)

## Define design parameters


set.seed(44)

p <- 2/3  # probability of answering honestly in Forced Response Design
p1 <- 1/6 # probability of forced 'yes'
p0 <- 1/6 # probability of forced 'no'

## Fit joint model of responses to an outcome regression of joining a civic 
## group and the randomized response item of having a militant social connection

rr.q1.pred.obj <- 
    rrreg.predictor(civic ~ cov.asset.index + cov.married + I(cov.age/10) + 
              I((cov.age/10)^2) + cov.education + cov.female 
              + rr.q1, rr.item = "rr.q1", parstart = FALSE, estconv = TRUE,
              data = nigeria, verbose = FALSE, optim = TRUE,
              p = p, p1 = p1, p0 = p0, design = "forced-known")

## Generate predicted probabilities for the likelihood of joining 
## a civic group across respondents using quasi-Bayesian simulations. 

rr.q1.rrreg.predictor.pred <- predict(rr.q1.pred.obj, 
                                 avg = TRUE, quasi.bayes = TRUE, 
                                 n.sims = 1000)

---

Randomized Response Regression

Description

rrreg is used to conduct multivariate regression analyses of survey data using randomized response methods.

Usage

rrreg(formula, p, p0, p1, q, design, data, start = NULL, 
h = NULL, group = NULL, matrixMethod = "efficient",
maxIter = 10000, verbose = FALSE, optim = FALSE, em.converge = 10^(-8), 
glmMaxIter = 10000, solve.tolerance = .Machine$double.eps)

Arguments

formula	An object of class "formula": a symbolic description of the model to be fitted.
p	The probability of receiving the sensitive question (Mirrored Question Design, Unrelated Question Design); the probability of answering truthfully (Forced Response Design); the probability of selecting a red card from the 'yes' stack (Disguised Response Design). For "mirrored" and "disguised" designs, p cannot equal .5.
p0	The probability of forced 'no' (Forced Response Design).
p1	The probability of forced 'yes' (Forced Response Design).
q	The probability of answering 'yes' to the unrelated question, which is assumed to be independent of covariates (Unrelated Question Design).
design	One of the four standard designs: "forced-known", "mirrored", "disguised", or "unrelated-known".
data	A data frame containing the variables in the model.
start	Optional starting values of coefficient estimates for the Expectation-Maximization (EM) algorithm.
h	Auxiliary data functionality. Optional named numeric vector with length equal to number of groups. Names correspond to group labels and values correspond to auxiliary moments.
group	Auxiliary data functionality. Optional character vector of group labels with length equal to number of observations.
matrixMethod	Auxiliary data functionality. Procedure for estimating optimal weighting matrix for generalized method of moments. One of "efficient" for two-step feasible and "cue" for continuously updating. Default is "efficient". Only relevant if h and group are specified.
maxIter	Maximum number of iterations for the Expectation-Maximization algorithm. The default is 10000.
verbose	A logical value indicating whether model diagnostics counting the number of EM iterations are printed out. The default is FALSE.
optim	A logical value indicating whether to use the quasi-Newton "BFGS" method to calculate the variance-covariance matrix and standard errors. The default is FALSE.
em.converge	A value specifying the satisfactory degree of convergence under the EM algorithm. The default is 10^(-8).
glmMaxIter	A value specifying the maximum number of iterations to run the EM algorithm. The default is 10000.
solve.tolerance	When standard errors are calculated, this option specifies the tolerance of the matrix inversion operation solve.

Details

This function allows users to perform multivariate regression analysis on data from the randomized response technique. Four standard designs are accepted by this function: mirrored question, forced response, disguised response, and unrelated question. The method implemented by this function is the Maximum Likelihood (ML) estimation for the Expectation-Maximization (EM) algorithm.

Value

rrreg returns an object of class "rrreg". The function summary is used to obtain a table of the results. The object rrreg is a list that contains the following components (the inclusion of some components such as the design parameters are dependent upon the design used):

est	Point estimates for the effects of covariates on the randomized response item.
vcov	Variance-covariance matrix for the effects of covariates on the randomized response item.
se	Standard errors for estimates of the effects of covariates on the randomized response item.
data	The data argument.
coef.names	Variable names as defined in the data frame.
x	The model matrix of covariates.
y	The randomized response vector.
design	Call of standard design used: "forced-known", "mirrored", "disguised", or "unrelated-known".
p	The p argument.
p0	The p0 argument.
p1	The p1 argument.
q	The q argument.
call	The matched call.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

See Also

predict.rrreg for predicted probabilities.

Examples

data(nigeria)

set.seed(1)

## Define design parameters
p <- 2/3  # probability of answering honestly in Forced Response Design
p1 <- 1/6 # probability of forced 'yes'
p0 <- 1/6 # probability of forced 'no'

## Fit linear regression on the randomized response item of whether 
## citizen respondents had direct social contacts to armed groups

rr.q1.reg.obj <- rrreg(rr.q1 ~ cov.asset.index + cov.married + 
                    I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female,   
                    data = nigeria, p = p, p1 = p1, p0 = p0, 
                    design = "forced-known")
  
summary(rr.q1.reg.obj)

## Replicates Table 3 in Blair, Imai, and Zhou (2014)

---

Bayesian Randomized Response Regression

Description

Function to conduct multivariate regression analyses of survey data with the randomized response technique using Bayesian MCMC.

Usage

rrreg.bayes(formula, p, p0, p1, design, data, group.mixed,
formula.mixed = ~1, verbose = FALSE, n.draws = 10000, burnin = 5000, thin =
1, beta.start, beta.mu0, beta.A0, beta.tune, Psi.start, Psi.df, Psi.scale,
Psi.tune)

Arguments

formula	An object of class "formula": a symbolic description of the model to be fitted.
p	The probability of receiving the sensitive question (Mirrored Question Design, Unrelated Question Design); the probability of answering truthfully (Forced Response Design); the probability of selecting a red card from the 'yes' stack (Disguised Response Design).
p0	The probability of forced 'no' (Forced Response Design).
p1	The probability of forced 'yes' (Forced Response Design).
design	Character indicating the design. Currently only "forced-known" is supported.
data	A data frame containing the variables in the model.
group.mixed	A string indicating the variable name of a numerical group indicator specifying which group each individual belongs to for a mixed effects model.
formula.mixed	To specify a mixed effects model, include this formula object for the group-level fit. ~1 allows intercepts to vary, and including covariates in the formula allows the slopes to vary also.
verbose	A logical value indicating whether model diagnostics are printed out during fitting.
n.draws	Number of MCMC iterations.
burnin	The number of initial MCMC iterations that are discarded.
thin	The interval of thinning between consecutive retained iterations (1 for no thinning).
beta.start	Optional starting values for the sensitive item fit. This should be a vector of length the number of covariates.
beta.mu0	Optional vector of prior means for the sensitive item fit parameters, a vector of length the number of covariates.
beta.A0	Optional matrix of prior precisions for the sensitive item fit parameters, a matrix of dimension the number of covariates.
beta.tune	A required vector of tuning parameters for the Metropolis algorithm for the sensitive item fit. This must be set and refined by the user until the acceptance ratios are approximately .4 (reported in the output).
Psi.start	Optional starting values for the variance of the random effects in the mixed effects models. This should be a scalar.
Psi.df	Optional prior degrees of freedom parameter for the variance of the random effects in the mixed effects models.
Psi.scale	Optional prior scale parameter for the variance of the random effects in the mixed effects models.
Psi.tune	A required vector of tuning parameters for the Metropolis algorithm for variance of the random effects in the mixed effects models. This must be set and refined by the user until the acceptance ratios are approximately .4 (reported in the output).

Details

This function allows the user to perform regression analysis on data from the randomized response technique using a Bayesian MCMC algorithm.

The Metropolis algorithm for the Bayesian MCMC estimators in this function must be tuned to work correctly. The beta.tune and, for the mixed effects model Psi.tune, are required, and the values, one for each estimated parameter, will need to be manipulated. The output of the rrreg.bayes function displays the acceptance ratios from the Metropolis algorithm. If these values are far from 0.4, the tuning parameters should be changed until the ratios approach 0.4.

Convergence is at times difficult to achieve, so we recommend running multiple chains from overdispersed starting values by, for example, running an MLE using the rrreg() function, and then generating a set of overdispersed starting values using those estimates and their estimated variance-covariance matrix. An example is provided below for each of the possible designs. Running summary() after such a procedure will output the Gelman-Rubin convergence statistics in addition to the estimates. If the G-R statistics are all below 1.1, the model is said to have converged.

Value

rrreg.bayes returns an object of class "rrreg.bayes". The function summary is used to obtain a table of the results.

beta	The coefficients for the sensitive item fit. An object of class "mcmc" that can be analyzed using the coda package.
data	The data argument.
coef.names	Variable names as defined in the data frame.
x	The model matrix of covariates.
y	The randomized response vector.
design	Call of standard design used: "forced-known", "mirrored", "disguised", or "unrelated-known".
p	The p argument.
p0	The p0 argument.
p1	The p1 argument.
beta.tune	The beta.tune argument.
mixed	Indicator for whether a mixed effects model was run.
call	the matched call.

If a mixed-effects model is used, then several additional objects are included:

Psi	The coefficients for the group-level fit. An object of class "mcmc" that can be analyzed using the coda package.
gamma	The random effects estimates. An object of class "mcmc" that can be analyzed using the coda package.
coef.names.mixed	Variable names for the predictors for the second-level model
z	The predictors for the second-level model.
groups	A vector of group indicators.
Psi.tune	The Psi.tune argument.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

Examples

data(nigeria)
 
## Define design parameters
p <- 2/3  # probability of answering honestly in Forced Response Design
p1 <- 1/6 # probability of forced 'yes'
p0 <- 1/6 # probability of forced 'no'

## run three chains with overdispersed starting values

set.seed(1)

## starting values constructed from MLE model
mle.estimates <- rrreg(rr.q1 ~ cov.asset.index + cov.married + 
                         I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female, 
                         data = nigeria, 
                      p = p, p1 = p1, p0 = p0,
                      design = "forced-known")
                      


library(MASS)
draws <- mvrnorm(n = 3, mu = coef(mle.estimates), 
  Sigma = vcov(mle.estimates) * 9)

## run three chains
bayes.1 <- rrreg.bayes(rr.q1 ~ cov.asset.index + cov.married + 
                         I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female,   
                      data = nigeria, p = p, p1 = p1, p0 = p0,
                      beta.tune = .0001, beta.start = draws[1,],
                      design = "forced-known")
bayes.2 <- rrreg.bayes(rr.q1 ~ cov.asset.index + cov.married + 
                         I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female,   
                      data = nigeria, p = p, p1 = p1, p0 = p0,
                      beta.tune = .0001, beta.start = draws[2,],
                      design = "forced-known")

bayes.3 <- rrreg.bayes(rr.q1 ~ cov.asset.index + cov.married + 
                         I(cov.age/10) + I((cov.age/10)^2) + cov.education + cov.female,   
                      data = nigeria, p = p, p1 = p1, p0 = p0,
                      beta.tune = .0001, beta.start = draws[3,],
                      design = "forced-known")
                      
bayes <- as.list(bayes.1, bayes.2, bayes.3)

summary(bayes)

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Randomized Response as a Regression Predictor

Description

rrreg.predictor is used to jointly model the randomized response item as both outcome and predictor for an additional outcome given a set of covariates.

Usage

rrreg.predictor(formula, p, p0, p1, q, design, data, rr.item,
model.outcome = "logistic", fit.sens = "bayesglm", fit.outcome = "bayesglm",
bstart = NULL, tstart = NULL, parstart = TRUE, maxIter = 10000, verbose =
FALSE, optim = FALSE, em.converge = 10^(-4), glmMaxIter = 20000, estconv =
TRUE, solve.tolerance = .Machine$double.eps)

Arguments

formula	An object of class "formula": a symbolic description of the model to be fitted with the randomized response item as one of the covariates.
p	The probability of receiving the sensitive question (Mirrored Question Design, Unrelated Question Design); the probability of answering truthfully (Forced Response Design); the probability of selecting a red card from the 'yes' stack (Disguised Response Design).
p0	The probability of forced 'no' (Forced Response Design).
p1	The probability of forced 'yes' (Forced Response Design).
q	The probability of answering 'yes' to the unrelated question, which is assumed to be independent of covariates (Unrelated Question Design).
design	One of the four standard designs: "forced-known", "mirrored", "disguised", or "unrelated-known".
data	A data frame containing the variables in the model. Observations with missingness are list-wise deleted.
rr.item	A string containing the name of the randomized response item variable in the data frame.
model.outcome	Currently the function only allows for logistic regression, meaning the outcome variable must be binary.
fit.sens	Indicator for whether to use Bayesian generalized linear modeling (bayesglm) in the Maximization step for the Expectation-Maximization (EM) algorithm to generate coefficients for the randomized response item as the outcome. Default is "bayesglm"; otherwise input "glm".
fit.outcome	Indicator for whether to use Bayesian generalized linear modeling (bayesglm) in the Maximization step for the EM algorithm to generate coefficients for the outcome variable given in the formula with the randomized response item as a covariate. Default is "bayesglm"; otherwise input "glm".
bstart	Optional starting values of coefficient estimates for the randomized response item as outcome for the EM algorithm.
tstart	Optional starting values of coefficient estimates for the outcome variable given in the formula for the EM algorithm.
parstart	Option to use the function rrreg to generate starting values of coefficient estimates for the randomized response item as outcome for the EM algorithm. The default is TRUE, but if starting estimates are inputted by the user in bstart, this option is overidden.
maxIter	Maximum number of iterations for the Expectation-Maximization algorithm. The default is 10000.
verbose	A logical value indicating whether model diagnostics counting the number of EM iterations are printed out. The default is FALSE.
optim	A logical value indicating whether to use the quasi-Newton "BFGS" method to calculate the variance-covariance matrix and standard errors. The default is FALSE.
em.converge	A value specifying the satisfactory degree of convergence under the EM algorithm. The default is 10^(-4).
glmMaxIter	A value specifying the maximum number of iterations to run the EM algorithm. The default is 20000 .
estconv	Option to base convergence on the absolute value of the difference between subsequent coefficients generated through the EM algorithm rather than the subsequent log-likelihoods. The default is TRUE.
solve.tolerance	When standard errors are calculated, this option specifies the tolerance of the matrix inversion operation solve.

Details

This function allows users to perform multivariate regression analysis with the randomized response item as a predictor for a separate outcome of interest. It does so by jointly modeling the randomized response item as both outcome and predictor for an additional outcome given the same set of covariates. Four standard designs are accepted by this function: mirrored question, forced response, disguised response, and unrelated question.

Value

rrreg.predictor returns an object of class "rrpredreg" associated with the randomized response item as predictor. The object rrpredreg is a list that contains the following components (the inclusion of some components such as the design parameters are dependent upon the design used):

est.t	Point estimates for the effects of the randomized response item as predictor and other covariates on the separate outcome variable specified in the formula.
se.t	Standard errors for estimates of the effects of the randomized response item as predictor and other covariates on the separate outcome variable specified in formula.
est.b	Point estimates for the effects of covariates on the randomized response item.
vcov	Variance-covariance matrix for estimates of the effects of the randomized response item as predictor and other covariates on the separate outcome variable specified in formula as well as for estimates of the effects of covariates on the randomized response item.
se.b	Standard errors for estimates of the effects of covariates on the randomized response item.
data	The data argument.
coef.names	Variable names as defined in the data frame.
x	The model matrix of covariates.
y	The randomized response vector.
o	The separate outcome of interest vector.
design	Call of standard design used: "forced-known", "mirrored", "disguised", or "unrelated-known".
p	The p argument.
p0	The p0 argument.
p1	The p1 argument.
q	The q argument.
call	The matched call.

References

Blair, Graeme, Kosuke Imai and Yang-Yang Zhou. (2014) "Design and Analysis of the Randomized Response Technique." Working Paper. Available at http://imai.princeton.edu/research/randresp.html.

See Also

rrreg for multivariate regression.

Examples

data(nigeria)

## Define design parameters

set.seed(44)

p <- 2/3  # probability of answering honestly in Forced Response Design
p1 <- 1/6 # probability of forced 'yes'
p0 <- 1/6 # probability of forced 'no'

## Fit joint model of responses to an outcome regression of joining a civic 
## group and the randomized response item of having a militant social connection
rr.q1.pred.obj <- 
    rrreg.predictor(civic ~ cov.asset.index + cov.married + I(cov.age/10) + 
              I((cov.age/10)^2) + cov.education + cov.female 
              + rr.q1, rr.item = "rr.q1", parstart = FALSE, estconv = TRUE,
              data = nigeria, verbose = FALSE, optim = TRUE,
              p = p, p1 = p1, p0 = p0, design = "forced-known")

summary(rr.q1.pred.obj)

## Replicates Table 4 in Blair, Imai, and Zhou (2014)

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