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\begin{document}
\begin{center}
{\textsclarge{Conditioning on a Collider}}
\end{center}


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The graph
\begin{figure}[ht]
\centering
\begin{tikzpicture}[scale=1.5]
    % x node set with absolute coordinates
    \node[state] (x) at (-1,0) {${X}$};
    \node[state] (y) at (1,0){${Y}$};
    \node[state] (z) at (0,-1) {${Z}$};
    \node[state] (u) at (0,0) {${U_Z}$};

    \path (x) edge (z);
    \path (y) edge (z);
    \path (u) edge (z);
 \end{tikzpicture}
\end{figure}
encodes the dependencies between the four random variables $(X,Y,Z,U_Z)$: the joint density can be represented
\[
f_{X,Y,Z,U_Z}(x,y,z,u) = f_X(x) f_Y(y) f_{U_Z}(u) f_{Z|X,Y,U_Z}(z|x,y,u)
\]
that is, $X,Y,U_Z$ are mutually independent.  Suppose that $X$ and $Y$ are distributed as standard Normal random variables, $U_Z \sim \text{Normal}(0,0.1^2)$, and
\[
Z= X + Y + U_Z
\]

<<coll0,comment='+', fig.width=5.25, fig.height=5.25, fig.align='center'>>=
set.seed(2101)                         #Set the random number generator seed value
n<-10000                               #Set the sample size
X<-rnorm(n,0,1)                        #Generate the X random variables
Y<-rnorm(n,0,1)                        #Generate the Y random variables
UZ<-rnorm(n,0,0.1)                     #Generate the UZ random variables
Z<-X+Y+UZ
par(mar=c(3,2,1,0))                    #Set up the plotting margins
pairs(cbind(X,Y,Z),pch=19,cex=0.7)
@

If we condition on the value of $Z$, and inspect the joint density of $X$ and $Y$ given $Z$, we see that $X$ and $Y$ are conditionally \textbf{dependent}.

<<coll1,comment='+', fig.width=5.25, fig.height=5.25, fig.align='center'>>=
par(mar=c(3,2,1,0))                                 #Set up the plotting margins
X1<-X[Z>-2.5 & Z < -1.5];Y1<-Y[Z>-2.5 & Z < -1.5];  #First subset analysis
X2<-X[Z>-0.5 & Z < 0.5];Y2<-Y[Z>-0.5 & Z < 0.5];    #Second subset analysis
X3<-X[Z>0.5 & Z < 1.5];Y3<-Y[Z>0.5 & Z < 1.5];      #Third subset analysis
par(mar=c(4,3,1,0),pty='s',mfrow=c(2,2))            #Set up the plotting margins
plot(X1,Y1,pch=19,cex=0.7)
plot(X2,Y2,pch=19,cex=0.7)
plot(X3,Y3,pch=19,cex=0.7)
cor(X1,Y1)
cor(X2,Y2)
cor(X3,Y3)
@



\end{document}