Correlation and regression analysis for randomized response designs
Daniel W. Heck, dheck@mail.uni-mannheim.de
Last update: September 11th, 2014
The randomized response (RR) design was developed by Warner (1965) to allow for complete anonymity of the respondents in surveys and thus to reduce effects of social desirability. In general, some kind of random noise (e.g., throwing a pair of dices) is added to the true response. Whereas the prevalence of a senstive attribute (by convention termed \(\pi\)) can still be estimated at the group level, an individual response does not disclose the true state of the respondent. Accordingly, RR data require adapted models and formulas for multivariate analysis, implemented in the package RRreg.
The main functions provide the following functionality and are explained in more detail in Section 2:
Additionally, two functions can be used to generate data for robustness studies, bootstrap estimates and testing purposes:
RRuni,RRcor and RRlog, either for one RR and one continuous non-RR variable or for two RR variablesIn the following, cocaine abuse is used as a running example to demonstrate the RR analysis with RRreg. The following RR designs are included in the package: UQTknown, Mangat, Kuk, FR, Crosswise, UQTunknown, CDM, CDMsym, SLD
The original RR design proposed by Warner in 1965 works as follows: With randomization probability p, participants are supposed to answer to the question 'Have you ever used cocaine?', and with counter-probability 1-p, the question is reversed, i.e., 'Have you never used cocaine?'. As a randomization device, dices or coin flips with known probabilities may be used. The model is depicted in the following figure:
First, let's load the package and generate a single data frame using the function RRgen. We simulate data for 1000 participants, a proportion of 30% cocaine users (pi.true=.3; obviously, this rate is unrealistic, but it should suffice for the sake of example), and a randomization probability of p=.2:
library(RRreg)
data.W <- RRgen(n=1000, pi.true=.3, model="Warner", p=.2, complyRates = c(1, 1), sysBias = c(0,0))
head(data.W)
## true comply response
## 1 1 1 1
## 2 0 1 1
## 3 0 1 0
## 4 0 1 0
## 5 0 1 1
## 6 1 1 0
In the function RRgen, the argument complyRates gives the proportions of participants who do adhere to the RR instructions for carriers and non-carriers of the sensitive attribute, respectively. The argument sysBias gives the probability of yes responses in case of non-compliance for carriers and non-carriers, respectively. If sysBias=c(0,0), noncomplying individuals always respond with self-protective no responses (also known as SP-'no' responses). In comparison, sysBias=c(0,0.5) indicates that noncomplying non-carriers answer yes or no by chance. However, since we generated data assuming perfect compliance, this argument can be neglected. These options can be used to test how robust the results are against non-adherence.
Now that we have a data set, we can estimate the prevalence of cocaine users by means of RRuni:
warner <- RRuni(response=response, data=data.W, model="Warner", p=.2, MLest=T)
summary(warner)
## Call:
## Warner Model with p = 0.2
## Sample size: 1000
##
## Estimate StdErr z Pr(>|z|)
## pi 0.3033 0.0256 11.8 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
The response variable containing 1 for yes and 0 for no responses can either be given as a vector or as a column name in the data set data. Moreover, the argument model specifies the RR design with its randomization probability given by p (see Sections 4 and 5 for details about other RR models) . If MLest=FALSE is used, the prevalence estimate is derived as an analytical least-squares estimator, which can assume values outside of the interval [0,1]. However, using MLest=TRUE (default), estimates of pi are restricted to the interval [0,1].
First, we generate a continuous, non-RR variable. According to our simulation, respondents carrying the sensitive attribute (i.e., cocaine users) have higher scores on this covariate:
data.W$cov[data.W$true==1] <- rnorm(sum(data.W$true == 1),1)
data.W$cov[data.W$true==0] <- rnorm(sum(data.W$true == 0))
Now, we can estimate the bivariate (point-biserial) correlation between the dichotomous Warner RR variable and the continuous covariate:
RRcor(x=data.W$response, y=data.W$cov, models=c("Warner", "direct"), p.list=list(.2), bs.n=0, bs.type = c("n", "p"), nCPU=1)
## Call:
## Randomized response correlation
##
## Randomized response variables:
## Variable RRmodel p
## 1 data.W$response Warner 0.2
## 2 data.W$cov direct
##
## Sample size N = 1000
##
## Estimated correlation matrix:
## data.W$response data.W$cov
## data.W$response 1.000 0.426
## data.W$cov 0.426 1.000
The argument models defines which variables given by x and y are treated as RR variables. x and y can be vectors and/or data frames/matrices having one variable in each column (the order is then given as following: First, columns of x from left to right, and second, columns of y from left to right). The randomization probabilities p.list have to be in the same order as models. No values have to be specified for direct (nonRR) variables. To obtain bootstrapped standard errors, bs.n has to be set to a number of bootstrap replications larger than 0. Additionally, bs.type defines which kind of bootstrap should be performed (nonparametric and/or parametric). For faster processing, nCPUcan be increased to use parallel computation. In our example, we also know the true states of the participants in the data set and can check the results by:
cor(x=data.W$true, y=data.W$cov)
## [1] 0.4132
Alternatively, we can run a logistic regression to predict the probability of cocaine use. In the formula below, the Warner RR variable is defined as criterion on the left side, whereas the continuous covariate is used as predictor on the right. The interface is similar to that of the standard linear regression in R (i.e., lm()).
log1 <- RRlog(formula=response~cov, data=data.W, model="Warner", p=.2, LR.test=TRUE)
summary(log1)
## Call:
## RRlog.formula(formula = response ~ cov, data = data.W, model = "Warner",
## p = 0.2, LR.test = TRUE)
##
## Model:
## Warner with p = 0.2
##
## Model fit:
## n logLik
## 1000 -635.6
##
## Estimate StdErr Wald test Pr(>Chi2,df=1) deltaG2 Pr(>deltaG2)
## (Intercept) -1.300 0.185 49.642 0.000 89.8 <2e-16
## cov 0.968 0.164 34.818 0.000 59.0 <2e-16
##
## (Intercept) ***
## cov ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
To remove the intercept from the model, use formula=response~cov - 1. The argument LR.test determines whether the coefficients should be tested by means of a likelihood ratio test, that is, by removing one predictor at a time. Again, we can check the results by:
glm(formula=true~cov, data=data.W, family=binomial(link = "logit"))
##
## Call: glm(formula = true ~ cov, family = binomial(link = "logit"),
## data = data.W)
##
## Coefficients:
## (Intercept) cov
## -1.38 1.01
##
## Degrees of Freedom: 999 Total (i.e. Null); 998 Residual
## Null Deviance: 1220
## Residual Deviance: 1030 AIC: 1030
Third, we can use the Warner RR variable as predictor in a linear regression:
lin1 <- RRlin(formula=cov~response, data=data.W, models="Warner", p.list=.2, bs.n=0)
summary(lin1)
## Call:
## RRlin(formula = cov ~ response, data = data.W, models = "Warner",
## p.list = 0.2, bs.n = 0)
##
## Randomized response variables:
## Variable Model p
## 1 response Warner 0.2
##
## Coefficients (beta):
## Estimate StdErr Wald test Pr(>Chi2,df=1)
## (Intercept) 0.0280 0.0485 0.33 0.56
## response 0.9535 0.1159 67.66 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error (sigma): 0.997; N=1000
##
## Prevalence estimates for combinations of RR responses:
## Estimate StdErr
## 0 0.694 0.03
As in RRcor, bootstrapped standard errors can be obtained by setting bs.n to the number of bootstrap replications. The order of the RR models is defined by models and p.list. For instance, given three RR variables rr1, rr2, and rr3 and two non-RR/direct variables x1 and x2, the model could be specified by formula = y ~ rr1 + x1 + rr2 + rr3 + x2 and models=c('Warner', 'direct', 'Warner', 'Warner', 'direct'). Note that the predictors in formula can either be vectors or column names of variables in the data frame data.
As a special case, if the number of predictors used in the formula interface is larger than the number of RR variables specified in models, the remaining predictors are treated as directly measured, non-RR variables:
data.W$pred <- rnorm(1000)
lin2 <- RRlin(formula=cov~response + pred, data=data.W, models="Warner", p.list=list(.2), bs.n=0)
summary(lin2)
## Call:
## RRlin(formula = cov ~ response + pred, data = data.W, models = "Warner",
## p.list = list(0.2), bs.n = 0)
##
## Randomized response variables:
## Variable Model p
## 1 response Warner 0.2
##
## Coefficients (beta):
## Estimate StdErr Wald test Pr(>Chi2,df=1)
## (Intercept) 0.02785 0.04854 0.33 0.57
## response 0.95413 0.11599 67.66 <2e-16 ***
## pred 0.00455 0.03382 0.02 0.89
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error (sigma): 0.997; N=1000
##
## Prevalence estimates for combinations of RR responses:
## Estimate StdErr
## 0 0.694 0.03
To check the results of the simulation, run:
lm(cov~true + pred, data=data.W)
##
## Call:
## lm(formula = cov ~ true + pred, data = data.W)
##
## Coefficients:
## (Intercept) true pred
## 0.0271 0.9860 -0.0135
It is also possible to include several RR variables as predictors if the randomization devices used are stochastically independent:
data.W2 <- RRgen(n=1000, pi.true=.45, model="Warner", p=.35)
data.W$cov <-2*data.W$true - 3*data.W2$true+ rnorm(1000,1,5)
data.W$response2 <- data.W2$response
lin3 <- RRlin(cov ~ response + response2, data=data.W, models=c("Warner", "Warner"), p.list=list(.2, .35))
summary(lin3)
## Call:
## RRlin(formula = cov ~ response + response2, data = data.W, models = c("Warner",
## "Warner"), p.list = list(0.2, 0.35))
##
## Randomized response variables:
## Variable Model p
## 1 response Warner 0.2
## 2 response2 Warner 0.35
##
## Coefficients (beta):
## Estimate StdErr Wald test Pr(>Chi2,df=1)
## (Intercept) -0.191 0.511 0.14 0.70793
## response 2.799 0.743 14.20 0.00016 ***
## response2 -1.641 1.226 1.79 0.18063
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error (sigma): 5.171; N=1000
##
## Prevalence estimates for combinations of RR responses:
## Estimate StdErr
## 0:0 0.434 0.06
## 0:1 0.264 0.06
## 1:0 0.168 0.05
Besides the regression output, the function summary also provides 'Prevalence estimates for combinations of RR responses'. In our example using two Warner variables, the combination 0:0 indicates the sub-group having neither of both sensitive attributes and shows the corresponding prevalence estimate. Similarly, the combination 0:1 indicates the sub-group having only the second sensitive attribute and so on. Note that the last combination 1:1 is not provided and can be calculated by summing up all other prevalence estimates and substracting this sum from 1.
To include many RR variables as predictors, very large sample sizes are required for reliable estimation. In such a case, it might be necessary to adjust the optimization parameters maxit (maximum number of iterations) and pibeta (relative ratio of proability scale for pi to regression coefficients beta; e.g., choose a smaller value for larger absolute beta values).
The method RRlog can be used to test whether an RR design leads to higher prevalence estimates than direct questioning (DQ). For one-groups designs (such as the Warner model), define a vector with indices 0 for all participants who were questioned by DQ and indices 1 for participants questioned by the RR design and include this dummy variable as predictor. For two-group designs (see below), define a dummy variable contrasting DQ and RR as before, and define a vector with indices 0 for participants questioned by DQ and indices 1 and 2 denoting the group of the RR design.
If the regression coefficient of the dummy predictor is significant, this indicates different prevalence rates between RR and DQ.
Furthermore, interactions between question format (RR vs. DQ) and other predictors can be tested. For instance, one might want to know whether the regression coefficient of a predictor (e.g., age) differs between question formats (e.g., whether the RR format serves as a more valid criterion). Interactions are included by formula= RR ~ format + age + format:age.
An example could be:
# generate data with different prevalence rates for RR and DQ
RR <- RRgen(n=500, pi=.4, model="Warner", p=.35)
DQ <- rbinom(500, 1, .2)
response <- c(RR$response, DQ)
# dummy variable for RR vs. DQ
group <- rep(1:0, each=500)
# include interaction of question format and age (here, the rescaled z-variable z.age)
z.age <- c(rnorm(500, mean=RR$true*2,sd=3), rnorm(500, mean=0,sd=3))
# fit full model (i.e., test difference in prevalence estimates and interaction)
fit <- RRlog(response ~ group * z.age, model="Warner", p=.35, group=group, LR.test=TRUE)
## Warning: The group vector contains 2 categories, which are interpreted as
## follows: 0=direct question format ; 1=randomized response format
summary(fit)
## Call:
## RRlog.formula(formula = response ~ group * z.age, model = "Warner",
## p = 0.35, group = group, LR.test = TRUE)
##
## Model:
## Warner with p = 0.35 (n=500) combined with DQ (n=500)
##
## Model fit:
## n logLik
## 1000 -605.7
##
## Estimate StdErr Wald test Pr(>Chi2,df=1) deltaG2
## (Intercept) -1.28777 0.10888 139.88287 0.00000 170.23
## group 0.73764 0.36501 4.08399 0.04329 2.96
## z.age 0.00668 0.03575 0.03486 0.85189 0.03
## group:z.age 0.11424 0.11173 1.04554 0.30654 1.12
## Pr(>deltaG2)
## (Intercept) <2e-16 ***
## group 0.085 .
## z.age 0.852
## group:z.age 0.289
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# get prevalence estimate for RR and DQ
logit1 <- coef(fit) %*% c(1, 1, mean(z.age), mean(z.age))
cat(exp(logit1) / (1+ exp(logit1)))
## 0.3749
logit2 <- coef(fit) %*% c(1, 0, mean(z.age), 0)
cat(exp(logit2) / (1+ exp(logit2)))
## 0.2166
In the following, several other available RR models are listed, which can be used throughout the package functions. For each model, the randomization probability p must be given as defined below.
model = 'UQTknown'
In this RR design, participants either respond to the sensitive question or to a second irrelevant question. Importantly, the interviewer is unaware which question was answered. The randomization probability p[1] determines the proportion of participants that is directed to answer the sensitive question, whereas the remaining participants answer the irrelevant question. In the UQTknown design, the prevalence rate of yes responses for this irrelevant question must be known beforehand (e.g., “Is your birthday in April?”, p[2]=1/12) and provided together with the randomization probability by the vector p=c(p[1], p[2]) (see UQTunknown if the prevalence is unknown).
model = 'Mangat'
In Mangat's model, carriers of the critical attribute are always provided with the sensitive statement. In contrast, non-carriers of the critical attribute are to respond to the sensitive statement with probability p and to its negation otherwise.
model = 'Kuk'
Kuk's method uses two decks of playing cards to ensure anonymity of the respondents. Whereas carriers of the sensitive attribute are told to report the color of a card drawn from the first deck, non-carriers should report the color of a card from the second deck. Importantly, the interviewer does not know, from which deck a card was drawn.
Each deck contains red and black cards, coded with 1 and 0, respectively. However, the proportion of red cards differs in the two decks, defined by the vector p=c(p[1], p[2]) (i.e., proportion of red cards for carriers and non-carriers of the sensitive attribute, respectively).
Note that the procedure can be applied repeatedly to obtain more reliable estimates (with replacement of the drawn cards). For instance, with 5 repetitions, the observed RR variable can assume values from 0 to 5, i.e., the reported number of red cards. The functions RRgen (data generation), RRuni (univariate analysis), and RRlin (linear regression with RR predictors) can accomodate this extended design, using the additional argument Kukrep. However, at the moment, this option is not available in RRcor and RRlog.
model = 'FR'
In the forced response (FR) model (also known as directed-answers model), a known proportion of the participants is prompted to respond yes, whereas another proportion is prompted to respond no, independent of their true status. Only the remaining participants answer truthfully. The randomization probability p is now a vector, containing two values, i.e., being prompted to respond no and yes, respectively. The model is shown in the following figure:
The FR model can easily be extended to account for more than two response categories. For instance, responses can be given on a 5-point Likert scale from 0 to 4. For these categories, the randomization probabilities, determining the proportion of forced responses, are provided in a vector p=c(p[1], p[2],...,p[5]). The sum of these probabilities must be smaller than one. This polytomous RR model can be used with the functions RRgen, RRuni, RRcor, and RRlin. Only the logistic RR regression RRlog requires a dichotomous response format for the FR method.
The following code snippets illustrate, how the FR method is used with multiple response categories:
RRgen(n=1000, pi.true=c(.3,.1,.4,.2), model="FR", p=c(.1,.1,.2,.05))RRlin(criterion~response, data=dat, models="FR",p.list=list(c(.1,.1,.2,.05)))For data generation of a polytomous FR model using RRgen, the arguments complyRates and sysBias have the following definition:
complyRates=c(c0, c1, ...) gives the probability of complying to the RR instructions separately for all possible true statessysBias = c(s1, s2, ...) defines the multinomial probability distribution which is used in case of noncompliance (the values have to sum up to 1). For instance, having the categories 0,1,2,3,4 and complyRates=c(1,1,1,.8,.5), sysBias=c(.5,.3,.2,0,0) would simulate the scenario that participants with true states 3 and 4 would comply less and instead prefer responses 0, 1, or 2.
model = 'Crosswise'
Concerning the model equations, the Crosswise model is technically identical to the Warner model. However, instead of answering to the sensitive question directly, respondents are instructed to respond to the sensitive and an irrelevant question simultaneously, according to the following scheme:
For the irrelevant question, the prevalence of yes responses has to be known beforehand and is specified in RRreg by p (e.g., for the irrelevant question 'Is Your birthday in March or April?', p = 2/12).
For several reasons, RR designs were proposed that require two independent random samples (groups). Historically, the second group was used to estimate the prevalence of the irrelevant question in the UQT. Alternatively, two groups allow for a second parameter, quantifying cheating or compliance with the RR procedure.
For these two-group models, the functions RRuni, RRcor, RRlog, and RRlin additionally require a vector group with values in 1 and 2, indicating the group membership of each respondent. If more than one RR variable is in the model, group is specified by a matrix containing the corresponding group vectors (one in each column) in the same order as the RR variables appear in the model. Moreover, the randomization probability p often specifies different values for the two groups. Code examples are given in Section 5.5.
Note that in the functions RRcor and RRlin, the second parameter of these two-group models is estimated first and then treated as a fixed constant in further computations. Ideally, the second parameter should be treated as an additional, free parameter in the optimization routine (this is the case for the function RRlog). However, simulations show that the results are still adequate. Use RRsimu to run such a simulation for RRcor yourself. Furthermore, RRcor uses only the most informative subgroup of a two-group RR variable (e.g., for SLD, the group with the larger randomization probability p). According to simulations, this provides better results than averaging estimated correlations across subgroups.
model = 'UQTunknown'
In this extension of the UQTknown RR design, the prevalence of the second, irrelevant question is unknown beforehand and estimated from the data. Note that this second question has to be stochastically independent of the sensitive attribute.
The randomization probability p is defined as
p[1] - probability to respond to the sensitive question in group 1p[2] - probability to respond to the sensitive question in group 2The randomization probabilities must differ across groups to render the model identifiable (i.e., p[1]!=p[2]). When generating data by means of RRgen, the prevalence rates of yes responses can be specified by pi=c(pi.sensitive, pi.irrelevant). If only a single value is provided, the prevalence rate for the irrelevant question is randomly drawn from a uniform distribution.
Note that Moor's variant of the UQT with unknown prevalence can be used by setting p[2]=0. Thereby, the second group is only used to estimate the prevalence of responding yes to the irrelevant question.
Greenberg, B. G., Abul-Ela, A.-L. A., Simmons, W. R., & Horvitz, D. G. (1969). The Unrelated Question Randomized Response Model: Theoretical Framework. Journal of the American Statistical Association, 64, 520-539.
Moors, J. J. A. (1971). Optimization of the Unrelated Question Randomized Response Model. Journal of the American Statistical Association, 66, 627-629.
model = 'CDM'
The CDM is based on the forced-response design and prompts a proportion p of participants to respond yes regardless of their true state. In contrast to the FR design, the cheating detection model (CDM) divides the population into three distinct groups: Compliant carriers of the sensitive attribute (pi), compliant noncarriers (beta), and cheating respondents who always respond no (gamma). This last group of cheaters is not further divided into carriers and noncarriers of the sensitive attribute. To estimate the two parameters pi and gamma, the sample is randomly divided into two parts using different randomization probabilities defined as:
p[1] - probability to be prompted to say yes in group 1p[2] - probability to be prompted to say yes in group 2The randomization probabilities must differ across groups to render the model identifiable (i.e., p[1]!=p[2]).
Because of the assumption of three distinct groups, the CDM is not available in the functions RRcor and RRlin. However, the function RRlog performs a logistic regression to predict the probability of being in the first group (i.e., having the sensitive attribute and complying with the RR procedure). Note that conclusions are restricted to this group, only.
model = 'CDMsym'
In contrast to the standard cheating detection model (CDM) discussed in Section 5.2, in the symmetric CDM (CDMsym), some respondents are also prompted to respond no with a given probability, not only yes. Thereby, anonymity and compliance can be increased (Ostapczuk & al., 2009), because participants directly see that both, yes and no responses are uninformative in respect to the sensitive attribute. Similarly as with the CDM, using CDMsym in RRlog predicts the joint probability of having the sensitive attribute and complying with the RR procedure. CDMsym cannot be used in the functions RRcorand RRlin.
The randomization probability p is a vector with 4 values:
p[1] - probability to be prompted to say yes in group 1 p[2] - probability to be prompted to say no in group 1 p[3] - probability to be prompted to say yes in group 2 p[4] - probability to be prompted to say no in group 2 The randomization probabilities must meet these restrictions:
p[1]!=p[3]and p[2]!=p[4])p[1]+p[2]<1 and p[3]+p[4]<1)
model = 'SLD'
The stochastic lie detector (SLD) is based on Mangat's model (Section 4.2) and adds a second parameter t, which gives the probability of honest responding within the group carrying the sensitive attribute. Note that in contrast to the CDM, the probability of having the sensitive attribute pi is corrected for cheating.
The randomization probability p is a vector with 2 values:
p[1] - probability for noncarriers to reply with no in group 1 p[2] - probability for noncarriers to reply with no in group 2 The randomization probabilities must differ across groups to render the model identifiable (i.e., p[1]!=p[2]).
### generate 2 two-group RR variables and a continuous covariate
RR1 <- RRgen(1000, pi=.4, model="SLD", p=c(.2,.8), complyRates=c(.8,1))
RR2 <- RRgen(1000, pi=.6, model="SLD", p=c(.3 ,.7), complyRates=c(.9,1))
cov <- RR1$true + RR2$true + rnorm(1000, 0, 3)
### logistic regression (dependent RR variable)
logmod <- RRlog(RR1$response ~ cov, model="SLD", p=c(.2,.8), group=RR1$group)
summary(logmod)
## Call:
## RRlog.formula(formula = RR1$response ~ cov, model = "SLD", p = c(0.2,
## 0.8), group = RR1$group)
##
## Model:
## SLD with p1 = 0.2; p2 = 0.8
##
## Model fit:
## n logLik
## 1000 -589.1
##
## Estimate StdErr Wald test Pr(>Chi2,df=1) deltaG2
## (Intercept) -0.61557 0.21313 8.34146 0.00388 8.99
## cov 0.14344 0.05709 6.31360 0.01198 7.67
## t 0.78655 0.04314 24.47754 0.00000 14.20
## Pr(>deltaG2)
## (Intercept) 0.00272 **
## cov 0.00562 **
## t 0.00016 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Note that the parameter t is tested against the null hypothesis that all participants answered truthfully (i.e., H0: t=1).
### group matrix for RRlin and RRcor: use multiple group vectors in one matrix
group <- cbind(RR1$group, RR2$group)
# bivariate correlation between 2 two-group variables and a continuous covariate
# note that only the most informative subsets of the data are used (see ?RRcor)
responses <- cbind(RR1$response, RR2$response, cov)
colnames(responses) <- c("RR1", "RR2", "cov")
RRcor( responses, models=c("SLD","SLD","direct"), p.list=list(c(.2,.8), c(.3, .7)), group=group)
## Call:
## Randomized response correlation
##
## Randomized response variables:
## Variable RRmodel p
## 1 RR1 SLD 0.2 0.8
## 2 RR2 SLD 0.3 0.7
## 3 cov direct
##
## Sample size N:
## RR1 RR2 cov
## RR1 1000 500 500
## RR2 500 1000 500
## cov 500 500 1000
##
## Estimated correlation matrix:
## RR1 RR2 cov
## RR1 1.0000 0.10574 0.20638
## RR2 0.1057 1.00000 0.09835
## cov 0.2064 0.09835 1.00000
# linear model with 2 RR predictors
linmod <- RRlin(cov ~ RR1$response+RR2$response, models=c("SLD","SLD"), p.list=list(c(.2,.8), c(.3, .7)), group=group)
summary(linmod)
## Call:
## RRlin(formula = cov ~ RR1$response + RR2$response, models = c("SLD",
## "SLD"), p.list = list(c(0.2, 0.8), c(0.3, 0.7)), group = group)
##
## Randomized response variables:
## Variable Model p
## 1 RR1$response SLD c(0.2, 0.8)
## 2 RR2$response SLD c(0.3, 0.7)
##
## Coefficients (beta):
## Estimate StdErr Wald test Pr(>Chi2,df=1)
## (Intercept) 0.275 0.262 1.10 0.2946
## RR1$response 1.257 0.482 6.79 0.0092 **
## RR2$response 0.523 0.353 2.20 0.1380
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error (sigma): 3.08; N=1000
##
## Prevalence estimates for combinations of RR responses:
## Estimate StdErr
## 0:0 0.310 0.04
## 0:1 0.305 0.04
## 1:0 0.149 0.03