Approaches for generating synthetic survey data with dependent answers?

Question

I would like to produce synthetic survey data. At the moment I produce independent answers between questions according to an arbitrary discrete distribution as in this question.

I want to generate randomly and independently answers to 2 different questions with categorical responses.

I want to then generate an answer to a third question which depends on the first two answers.

How can this be done for a continuous valued case?

How can this be done for a categorical case?

I am more interested in how to do the discrete case where a new (dependent) categorical results is produced.

I am interested in any type of dependency which would show up when measuring the mutual information between question answers. Having maybe 2 or 3 category numbers.

Answer 1

Here I use a latent variable approach. This readily extends to the continuous/categorical case.

The idea is to treat a continuous variable (the latent variable) as laying behind the ordered categories that are actually observed (by splitting up the continuous variable at breakpoints).

So for the two variables that are independent, we define breakpoints that give the desired proportions in each category. Then the third continuous variable, correlated with the other two, is also split up in similar fashion. It's common to use standardized normal variables for the latent variables, but other distributions could be used.

The example below is in R but I have annotated it to help conversion to other platforms.

set.seed(10345)    # just to make sure if you run this we have the same results
xu=rnorm(50)       # draw 50 observations from continuous latent variables
yu=rnorm(50)       #
zu= 0.8*xu+0.6*yu  #  the latent variables have correlations 0 between x and y,                
                   #  0.8 between x and z, and 0.6 between y and z

cor(cbind(xu,yu,zu)) # sample correlations will be similar to those population values

px=c(.3,.2,.5)     # our selected population proportions in the marginal categories
py=c(.1,.2,.4,.3)
pz=c(.1,.2,.4,.2,.1)

xc=cut(xu,qnorm(cumsum(c(0,px))),labels=c("AI","AII","AIII")) # convert to ord. categ.
yc=cut(yu,qnorm(cumsum(c(0,py))),labels=LETTERS[1:4])
zc=cut(zu,qnorm(cumsum(c(0,pz))),labels=letters[1:5])

Now let's see the relationships between variables:

table(xc,yc) #examine the resulting data. xc,yc populations are independent
      yc
xc      A  B  C  D
  AI    1  7  9  2
  AII   0  4 11  7
  AIII  2  5 18 14

> table(xc,zc) #xc,zc dependent
      zc
xc      a  b  c  d  e
  AI    4 11  4  0  0
  AII   0  2 19  1  0
  AIII  0  1 18 12  8

> table(yc,zc) #yc,zc dependent
   zc
yc   a  b  c  d  e
  A  1  1  1  0  0
  B  2  7  5  1  1
  C  1  5 27  5  0
  D  0  1  8  7  7

---

How correlations between the latent variables work.

I chose $X_u$ and $Y_u$ ($u$ for "underlying"; I'd have put $l$ for "latent", but it tends to look like a "1") to be two independent standard normal variates. You can make them correlated with a third variate, $Z_u$, by making $Z_u$ a linear combination of $X_u$, $Y_u$, and an independent noise variate $\epsilon$, which we'll also take to be standard normal here.

If we write $Z^*=aX_u+bY_u+c\epsilon$ then $Z^*$ is normal, but not standard normal.

$\text{Cov}(Z^*,X_u)=\text{Cov}(aX_u+bY_u+c\epsilon,X)=a\,\sigma^2_X=a$

Similarly $\text{Cov}(Z^*,Y_u)=b$ and $\text{Cov}(Z^*,\epsilon)=c$.

$\text{Var}(Z^*)=a^2+b^2+c^2$

So $\text{Cor}(Z^*,X_u)=\frac{a}{\sqrt{a^2+b^2+c^2}}$ and So $\text{Cor}(Z^*,Y_u)=\frac{b}{\sqrt{a^2+b^2+c^2}}$.

But I want $Z_u$ to have variance $1$, so if we define $Z_u=\frac{Z^*}{\sqrt{a^2+b^2+c^2}}$ then $\text{Var}(Z_u)=\frac{a^2+b^2+c^2}{a^2+b^2+c^2}=1$

In the example, I chose $a=0.8,b=0.6,c=0$, which has $a^2+b^2+c^2=1$ and in that case $Z_u=Z^*$, and we have $\text{Cor}(Z_u,X_u)=a=0.8$ and $\text{Cor}(Z_u,Y_u)=b$.

If you choose to have $\text{Cor}(Z_u,X_u)=\rho\,,$ then $-\sqrt{1-\rho^2}\leq\text{Cor}(Z_u,Y_u)\leq \sqrt{1-\rho^2}$ (with the limits being achieved when $c=0$).

Note that these are population correlations, not sample correlations.

In the example you mention in comments, $a=b=\frac{1}{2}$, and $c=0$ which gives $\text{Cor}(Z^*,X_u)=\frac{a}{\sqrt{a^2+b^2+c^2}}=\frac{1/2}{\sqrt{(1/2)^2+(1/2)^2}}=\sqrt{\frac{1}{2}}\approx 0.7071$

-- but now to make $Z_u$ standard normal we need to divide through by

$\sqrt{a^2+b^2+c^2}=\sqrt{\frac{1}{2}}$, i.e.

$Z_u=Z^*/\sqrt{\frac{1}{2}}=\sqrt{2}Z^*$.

Answer 2

Are your variables quantitative or categorical variables ?

In an article we recently wrote, we wanted to simulate three quantitative anwers to a survey : $z$ and $u$ had to be independant, and $X$ had to be correlated to both $z$ and $u$, so we generated them like this :

$\begin{align*} u &\sim \mathcal{U}[a,b] \\ z &\sim \mathcal{U}[a,b] \\ \forall k, X_k &= \alpha \cdot z_k + \beta \cdot u_k + \sigma \cdot \epsilon \\ \end{align*}$

with : $\epsilon \sim \mathcal{N}(0,1)$ and $\alpha, \beta, \sigma \in \mathbb{R}$. I believe it is very common way to proceed, I can think of plenty of papers where people did comparable things.

For categorical variables, I'd suggest a very similar approach :

$\begin{align*} z &\sim \mathcal{B}(n,p)~~~\text{(or whatever distribution suits your problem best)} \\ \forall k, X_k &= \lfloor z_k + \sigma \cdot \epsilon \rfloor \\ \end{align*}$

Parameters $\alpha, \beta, \sigma $ can be fine-tuned to match real survey answers in case you have data at your disposal.