Package 'qbaconfound'

Monte Carlo quantitative bias analysis for unmeasured confounding

Description

Conducts the flexible Monte Carlo quantitative bias analysis (QBA) for unmeasured confounding of Hughes et al. Given a naive analysis model (which omits one or more unmeasured confounders) and informative priors for a small number of bias parameters, the function returns a bias-adjusted estimate of the exposure effect together with an interval that accounts for both the unmeasured confounding and sampling variability.

Usage

qbaconfound(
  formula,
  data,
  exposure = NULL,
  confounders,
  family = stats::gaussian(),
  reps = 1000L,
  sampling_error = TRUE,
  seed = NULL
)

Arguments

formula	The naive analysis model, e.g. y ~ x + c1 + c2 for a GLM or Surv(time, status) ~ x + c1 for a Cox model. The unmeasured confounders are not included in formula.
data	A data frame containing the outcome, exposure, and measured confounders.
exposure	Character vector naming the exposure term(s) in formula whose effect is of interest. Defaults to the first term on the right-hand side.
confounders	A single u_continuous()/u_binary() object, or a list of them, describing the unmeasured confounder(s).
family	The outcome model family: a stats::glm() family object, a family name, or "cox" for a Cox proportional hazards model. Ignored (and inferred as Cox) when the response is a survival::Surv() call.
reps	Number of Monte Carlo replications.
sampling_error	Logical; if TRUE (the default) Monte Carlo sampling error is incorporated at step 4 above. Set to FALSE to obtain the distribution of bias-adjusted point estimates without sampling error.
seed	Optional integer seed for reproducibility.

Details

The substantive analysis may be a generalised linear model (any stats::glm() family) or a Cox proportional hazards model. Survival outcomes are detected automatically when the left-hand side of formula is a survival::Surv() call, or family can be set to "cox".

Each Monte Carlo replication (see Hughes et al., section 2.4):

1. draws a value for every bias parameter from its prior (the priors are specified through u_continuous() / u_binary());

2. simulates a proxy for each unmeasured confounder as a function of the exposure only (a continuous proxy from a normal model, a binary proxy from a Bernoulli model whose intercept reproduces the drawn prevalence);

3. refits the outcome model including the simulated proxies, with their coefficients fixed to the drawn values (implemented as a model offset), and reads off the exposure coefficient and its standard error;

4. adds Monte Carlo sampling error by drawing the bias-adjusted estimate from a normal distribution centred on that coefficient.

The point estimate is the median and the interval the 2.5th and 97.5th percentiles of the resulting distribution of bias-adjusted estimates.

Value

An object of class qbaconfound, a list with elements including estimates (a data frame of naive and bias-adjusted estimates per exposure term), draws (the matrix of bias-adjusted estimates across replications), and n_failed (the number of replications whose model fit failed).

References

Hughes RA, Kawabata E, Palmer TM, et al. A flexible Monte Carlo quantitative bias analysis for unmeasured confounding. Statistical Methods in Medical Research (under review).

See Also

u_continuous(), u_binary(), sim_confounding()

Examples

df <- sim_confounding(n = 500, beta_x = 0, seed = 1)

# Naive model y ~ x + c1 omits the confounder u; adjust for one continuous U.
fit <- qbaconfound(
  y ~ x + c1, data = df, exposure = "x",
  confounders = u_continuous(coef_out = c(0.8, 0.1),
                             coef_exp = c(0.6, 0.05),
                             resid_sd = c(0.9, 1.1)),
  reps = 200, seed = 1
)
fit

---

Simulate data with an unmeasured confounder

Description

A small helper that simulates a dataset with one measured confounder c1 and one unmeasured confounder u that jointly confound the exposure-outcome relationship. The naive model y ~ x + c1 (which omits u) is therefore biased for the exposure effect, making the data useful for examples and tests of qbaconfound().

Usage

sim_confounding(
  n = 1000L,
  beta_x = 0,
  family = c("gaussian", "binomial"),
  seed = NULL
)

Arguments

n	Number of observations.
beta_x	True exposure effect (on the linear-predictor scale).
family	Outcome type: "gaussian" (continuous outcome) or "binomial" (binary outcome).
seed	Optional integer seed for reproducibility.

Value

A data frame with columns y, x, c1, and u (the unmeasured confounder, included so it can be removed to mimic the unmeasured case). The true exposure effect is stored in attr(, "beta_x").

Examples

df <- sim_confounding(n = 1000, beta_x = 0.5, seed = 42)
# Naive (biased) versus full (unbiased) model:
coef(lm(y ~ x + c1, df))["x"]
coef(lm(y ~ x + c1 + u, df))["x"]

---

Summarise a Monte Carlo QBA

Description

Summarise a Monte Carlo QBA

Usage

## S3 method for class 'qbaconfound'
summary(object, ...)

Arguments

object	A qbaconfound object returned by qbaconfound().
...	Unused.

Value

A data frame of the naive and bias-adjusted estimates for each exposure term, with columns term, naive, naive_se, estimate, conf.low, and conf.high.

---

Specify a binary unmeasured confounder

Description

Describes the prior distributions of the bias parameters for a single binary unmeasured confounder, for use in qbaconfound().

Usage

u_binary(coef_out, coef_exp, prevalence, prev_dist = c("uniform", "beta"))

Arguments

coef_out	Length-2 numeric c(mean, sd) giving the normal prior for the coefficient of the unmeasured confounder in the outcome model ($\beta_U$). Set sd = 0 for a point-mass (fixed value) prior.
coef_exp	Normal prior for the coefficient(s) of the exposure in the model for the unmeasured confounder ($\alpha_X$). For a single exposure term, a length-2 numeric c(mean, sd). For a categorical exposure with several terms, a matrix with one row c(mean, sd) per exposure term, or a list of such length-2 vectors.
prevalence	Length-2 numeric giving the prior for the marginal prevalence $\pi$ of the binary unmeasured confounder. If prev_dist = "uniform" (the default) this is c(min, max) of a uniform prior; if prev_dist = "beta" this is c(a, b) of a beta prior.
prev_dist	Prior family for the prevalence: "uniform" or "beta".

Details

As for u_continuous() the coefficient of U in the outcome model (coef_out) and the coefficient(s) of the exposure in the model for U (coef_exp) are bias parameters. For a binary confounder the third bias parameter is the marginal prevalence of U (prevalence, i.e. $\pi$) rather than a residual standard deviation. A prevalence is usually easier to elicit and more readily reported in the literature than the intercept of a logistic model; the intercept needed to reproduce the drawn prevalence is derived internally.

Value

An object of class qba_u describing a binary unmeasured confounder.

See Also

u_continuous(), qbaconfound()

Examples

u_binary(coef_out = c(0.7, 0.1), coef_exp = c(0.4, 0.05),
         prevalence = c(0.15, 0.25))

---

Specify a continuous unmeasured confounder

Description

Describes the prior distributions of the bias parameters for a single continuous unmeasured confounder, for use in qbaconfound().

Usage

u_continuous(coef_out, coef_exp, resid_sd, resid_dist = c("uniform", "gamma"))

Arguments

coef_out	Length-2 numeric c(mean, sd) giving the normal prior for the coefficient of the unmeasured confounder in the outcome model ($\beta_U$). Set sd = 0 for a point-mass (fixed value) prior.
coef_exp	Normal prior for the coefficient(s) of the exposure in the model for the unmeasured confounder ($\alpha_X$). For a single exposure term, a length-2 numeric c(mean, sd). For a categorical exposure with several terms, a matrix with one row c(mean, sd) per exposure term, or a list of such length-2 vectors.
resid_sd	Length-2 numeric giving the prior for the residual standard deviation $\eta$. If resid_dist = "uniform" (the default) this is c(min, max) of a uniform prior on $\eta$. If resid_dist = "gamma" this is c(shape, scale) of a gamma prior on the residual precision $1/\eta^2$.
resid_dist	Prior family for the residual standard deviation: either "uniform" (on $\eta$) or "gamma" (on the precision $1/\eta^2$).

Details

The bias model relates the unmeasured confounder U to the study data through three bias parameters: the coefficient of U in the outcome model (coef_out, i.e. $\beta_U$), the coefficient(s) of the exposure in the model for U (coef_exp, i.e. $\alpha_X$), and the residual standard deviation of U given the exposure (resid_sd, i.e. $\eta$). Values for these parameters cannot be estimated from the data and so are drawn from the prior distributions specified here.

Value

An object of class qba_u describing a continuous unmeasured confounder.

See Also

u_binary(), qbaconfound()

Examples

u_continuous(coef_out = c(0.8, 0.1), coef_exp = c(0.3, 0.05),
             resid_sd = c(0.9, 1.1))

---