examples/IV_double_check.R

In gmonette/causalsim: Covariance Matrix of a Causal Graph

#' ---
#' geometry: width=5in
#' fontsize: 12pt
#' output:
#'   pdf_document
#' bibliography: bibliography.bib
#' ---
#' \newcommand{\N}{\mathrm{N}}
#' \newcommand{\Var}{\mathrm{Var}}
#' \newcommand{\E}{\mathrm{E}}
#' \raggedright
#' 
#' # Sanity check
#' 
#' ![](dag1.png)
#' 
#' __Check:__ Maybe I'm wrong. Maybe an IV only needs to be
#' orthogonal to __some__ blocking confounder, not to all!!
#' 
#' - __Test it with causalsim.__ Is it true that an IV will
#' be worse than the worst unconfounded regression model???? 
#' 
#' Let's build the variance matrix for Z, I, X, Y. We can scale $I$, $Z$ and
#' $X$ so they have unit variance and zero means. This eliminates irrelevant nuisance 
#' parameters.
#' 
#' __Mention:__ we assume multivariate normality
#' 
#' Taking
#' $$Var{
#' \begin{pmatrix}
#' Z \\ I \\ X
#' \end{pmatrix}
#' } =
#' \begin{pmatrix}
#' 1 & 0 & a \\ 0 & 1 & b \\ a & b & 1
#' \end{pmatrix}$$
#' with $a^2 + b^2 \le 1$.
#' 
#' Focus first on the _assignment model_, i.e. the model that determines
#' the value of $X$ from the values of $Z$, $I$ and $U_X$.
#' 
#' Letting $c^2 = 1 - a^2 - b^2$, $c^2$ represents the portion of the variance
#' in $X$ that is not attributed to the instrument, $I$, nor to the confounder,
#' $Z$. 
#' 
#' Define
#' $$\rho_I = \frac{b^2}{b^2 + c^2}$$ 
#' which is the proportion of variance in $X$ not
#' due to $Z$ that is 'explained' by $I$. 
#' 
#' For an instrument that captures
#' all of the variation not due to the confounder, $c^2 = 0$ and $\rho_I = 1$. 
#' 
#' Focusing next on the model generating $Y$, 
#' let 
#' $$Y = \alpha X + \beta Z + \gamma \varepsilon$$ 
#' with $\varepsilon \sim \N(0,1)$,
#' independent of
#' other variables.
#' 
#' The variance matrix is:
#' $$Var{
#' \begin{pmatrix}
#' Z \\ I \\ X \\ Y
#' \end{pmatrix}
#' } =
#' \begin{pmatrix}
#' 1 & 0 & a & a\alpha + \beta\\ 
#' 0 & 1 & b & b\alpha\\ 
#' a & b & 1 & \alpha + a \beta \\
#' a\alpha + \beta & b\alpha & \alpha + a \beta & v_{yy}
#' \end{pmatrix}$$
#' where $v_{yy} = \alpha^2 + \beta^2 + 2 a \alpha\beta + \sigma^2_\varepsilon$
#' 
#' We can verify the regression coefficients for the regression of $Y$ on
#' $X$ and $Z$ are
#' $$ \begin{pmatrix}
#' 1 & a \\
#' a & 1
#' \end{pmatrix} ^{-1}
#' \begin{pmatrix}
#' \alpha + a\beta \\
#' a \alpha + \beta
#' \end{pmatrix} =
#' \begin{pmatrix}
#' \alpha \\ \beta
#' \end{pmatrix}$$ 
#' 
#' The variance of the least-squares estimator of $\alpha$ based on a regression
#' on $X$ and the confounder $Z$ is:
#' $$\begin{aligned}
#' \Var(\hat{\alpha}) & \approx \frac{1}{n}\;\frac{\sigma^2_\epsilon}{1-a^2} \\
#'         & = \frac{1}{n}\;\frac{\gamma^2}{b^2 + c^2}
#' \end{aligned}$$
#' 
#' The asymptotic expectation of the instrumental variable estimator $\tilde{\alpha}$ is 
#' $$\sigma_{IX}^{-1}\sigma_{IY} = \frac{1}{b}\times b\alpha = \alpha$$
#' 
#' The variance of $\tilde{\alpha}$ is
#' [@fox2016applied,p.241]:
#' 
#' $$\begin{aligned}
#' \Var(\tilde{\alpha}) & \approx \frac{1}{n}\;\sigma^2_{\epsilon IV} \sigma_{IX}^{-1}\sigma_{II}\sigma_{XI}^{-1}
#'         & = \frac{1}{n}\;(\beta^2 + \gamma^2)\frac{1}{b^2}
#' \end{aligned}$$
#' 
#' Thus the variance inflation factor -- which is the same as the 'sample size inflation
#' factor to achieve the same power -- using IV estimation instead of controlling for
#' a confounder (assuming that both approaches are available) is:
#' 
#' $$\begin{aligned}
#' IVVIF & = \frac{\Var(\tilde{\alpha})}{\Var(\hat{\alpha})} \\
#' & =  \frac{\beta^2 + \gamma^2}{b^2} / \frac{\gamma^2}{b^2 + c^2} \\
#' & =   \frac{\beta^2 + \gamma^2}{\gamma^2} / \frac{b^2}{b^2 + c^2} \\
#' & =   \left(1 + \frac{\beta^2}{\gamma^2}\right) \times \left(1 + \frac{c^2}{b^2} \right) \\
#' & = \frac{1}{1 - R^2_{Y,Z|X}} \times \frac{1}{R^2_{X,I|Z}}
#' \end{aligned}$$
#' 
#' The first term is structural in the sense that it is a consequence
#' of the problem, specifically the degree of confounding relative
#' to the residual error variance in the model. For a given problem,
#' the IV has no impact on this, so it represents a lower bound
#' for the IVVIF. The second term clarifies that it is not the _correlation
#' of the IV with X_ directly that affects the IVVIF, but its __partial
#' correlation__ adjusted for the relationship of X with confounders.
#' 
##' # PLAN ------
#'
#' - Continue, express factors as partial $R^2$ thereby clearly showing
#'   scaling invariance.
#' - Generate some example.
#' - Discuss how this shows that it isn't specifically correlation with of I with X
#'   that matters but the partial $R^2$ having adjusted for $Z$ although we
#'   can't compute it not having $Z$.
#' - See also:
#'   - 
#'   
##' # NOTES ------
##' 
##' - Partial $R^2$ for I in X ~ Z + I is $\frac{b^2}{b^2 + c^2}$
##' - Partial $R^2$ for Z in Y ~ Z + X is $...$
##' - $\begin{aligned}
##'   R^2_{y,Z|X} &= \frac{SSR(X,Z) - SSR(X)}{SSE(X)}\\
##'                &= \frac{SSE(X) -SSE(X,Z) }{SSE(X)}
##'   \end{aligned}$ 
##' - Conclusions: In any situation where you have a choice,
##'   controlling for confounders will do better than using an IV,
##'   even with the worst confounder model, i.e. one that 'overpredicts'
##'   X (not in the sense of overfitting but in the sense of predicting
##'   better than necessary to unconfound).  __Fitting with IVs does
##'   not take the same advantage of a model with a small error variance 
##'   that a regression model does. The lower bound for error variance
##'   created by the confounder would usually swamp the benefit of
##'   small residual variance in the generating model. In contrast,
##'   a regression model takes full proportional advantage of a
##'   reduction in residual error.
##' - In practice, of course, we don't have Z. That's why we're
##'   using an IV. 
##' 
##'  
#'  
#' # Is this I an IV?
#' 
#' ![](dag2.png)
#' 
#' We can block the backdoor path by conditioning on $Z_Y$. $I_2$ would
#' seem to be an IV for the model for which $Z_Y$ is a confounder.
#' 
#' Are thee factors favoring using $I_2$ over $I_1$? Actually, won't
#' satisfy exclusion restrictions.
#' 
#' 
#' # References 
#'