Two-stage predictor substitution (TSPS) estimators

Terza et al. (2008) give an excellent description of TSPS estimators. They proceed by fitting a first stage model of the exposure regressed upon the instruments (and possibly any measured confounders). From this the predicted values of the exposure are obtained. A second stage model is then fitted of the outcome regressed upon the predicted values of the exposure (and possibly measured confounders).

Usage

tsps(
  formula,
  instruments,
  data,
  subset,
  na.action,
  contrasts = NULL,
  t0 = NULL,
  link = "identity",
  ...
)

Arguments

formula, instruments: formula specification(s) of the regression relationship and the instruments. Either instruments is missing and formula has three parts as in y ~ x1 + x2 | z1 + z2 + z3 (recommended) or formula is y ~ x1 + x2 and instruments is a one-sided formula ~ z1 + z2 + z3 (only for backward compatibility).
data: an optional data frame containing the variables in the model. By default the variables are taken from the environment of the formula.
subset: an optional vector specifying a subset of observations to be used in fitting the model.
na.action: a function that indicates what should happen when the data contain NAs. The default is set by the na.action option.
contrasts: an optional list. See the contrasts.arg of stats::model.matrix().
t0: A vector of starting values for the gmm optimizer. This should have length equal to the number of exposures plus 1.
link: character; one of "identity" (the default), "logadd", "logmult", "logit". This is the link function for the second stage model. "identity" corresponds to linear regression; "logadd" is log-additive and corresponds to Poisson / log-binomial regression; "logmult" is log-multiplicative and corresponds to gamma regression; "logit" corresponds to logistic regression.
...: further arguments passed to or from other methods.

Value

An object of class "tsps" with the following elements

fit: the fitted object of class "gmm" from the call to gmm::gmm().
estci: a matrix of the estimates with their corresponding confidence interval limits.
link: a character vector containing the specificed link function.

Details

tsps() performs GMM estimation to ensure appropriate standard errors on its estimates similar to the approach described in Clarke et al. (2015).

References

Burgess S, CRP CHD Genetics Collaboration. Identifying the odds ratio estimated by a two-stage instrumental variable analysis with a logistic regression model. Statistics in Medicine, 2013, 32, 27, 4726-4747. doi:10.1002/sim.5871

Clarke PS, Palmer TM, Windmeijer F. Estimating structural mean models with multiple instrumental variables using the Generalised Method of Moments. Statistical Science, 2015, 30, 1, 96-117. doi:10.1214/14-STS503

Dukes O, Vansteelandt S. A note on G-estimation of causal risk ratios. American Journal of Epidemiology, 2018, 187, 5, 1079-1084. doi:10.1093/aje/kwx347

Palmer TM, Sterne JAC, Harbord RM, Lawlor DA, Sheehan NA, Meng S, Granell R, Davey Smith G, Didelez V. Instrumental variable estimation of causal risk ratios and causal odds ratios in Mendelian randomization analyses. American Journal of Epidemiology, 2011, 173, 12, 1392-1403. doi:10.1093/aje/kwr026

Terza JV, Basu A, Rathouz PJ. Two-stage residual inclusion estimation: Addressing endogeneity in health econometric modeling. Journal of Health Economics, 2008, 27, 3, 531-543. doi:10.1016/j.jhealeco.2007.09.009

Examples

# Two-stage predictor substitution estimator
# with second stage logistic regression
set.seed(9)
n            <- 1000
psi0         <- 0.5
Z            <- rbinom(n, 1, 0.5)
X            <- rbinom(n, 1, 0.7*Z + 0.2*(1 - Z))
m0           <- plogis(1 + 0.8*X - 0.39*Z)
Y            <- rbinom(n, 1, plogis(psi0*X + log(m0/(1 - m0))))
dat          <- data.frame(Z, X, Y)
tspslogitfit <- tsps(Y ~ X | Z , data = dat, link = "logit")
summary(tspslogitfit)
#> 
#> GMM fit summary:
#> 
#> Call:
#> gmm::gmm(g = tspsLogitMoments, x = dat, t0 = t0, vcov = "iid")
#> 
#> 
#> Method:  twoStep 
#> 
#> Coefficients:
#>               Estimate    Std. Error  t value     Pr(>|t|)  
#> Z(Intercept)  1.7647e-01  1.7169e-02  1.0278e+01  8.8297e-25
#> ZZ            5.4740e-01  2.6249e-02  2.0854e+01  1.4138e-96
#> (Intercept)   9.7133e-01  1.4042e-01  6.9173e+00  4.6027e-12
#> xhat          6.1451e-01  2.7464e-01  2.2375e+00  2.5256e-02
#> 
#> J-Test: degrees of freedom is 0 
#>                 J-test               P-value            
#> Test E(g)=0:    8.6661572346798e-23  *******            
#> 
#> #############
#> Information related to the numerical optimization
#> Convergence code =  0 
#> Function eval. =  169 
#> Gradian eval. =  NA 
#> 
#> Estimates with 95% CI limits:
#>              Estimate   0.025  0.975
#> Z(Intercept)   0.1765 0.14282 0.2101
#> ZZ             0.5474 0.49595 0.5988
#> (Intercept)    0.9713 0.69611 1.2465
#> xhat           0.6145 0.07622 1.1528
#> 
#> Causal odds ratio with 95% CI limits:
#>             Estimate 0.025 0.975
#> (Intercept)    2.641 2.006 3.478
#> xhat           1.849 1.079 3.167
#>