p_linreg_yerrors function

Linear Regression of Y vs. Covariates with Y Measured in Pools and (Potentially) Subject to Additive Normal Errors

Assumes outcome given covariates is a normal-errors linear regression. Pooled outcome measurements can be assumed precise or subject to additive normal processing error and/or measurement error. Replicates are supported.

p_linreg_yerrors(g, ytilde, x = NULL, errors = "processing",
  estimate_var = TRUE, start_nonvar_var = c(0.01, 1),
  lower_nonvar_var = c(-Inf, 1e-04), upper_nonvar_var = c(Inf, Inf),
  nlminb_list = list(control = list(trace = 1, eval.max = 500, iter.max =
  500)), hessian_list = list(method.args = list(r = 4)))

Arguments

• g: Numeric vector with pool sizes, i.e. number of members in each pool.

• ytilde: Numeric vector (or list of numeric vectors, if some pools have replicates) with poolwise sum Ytilde values.

• x: Numeric matrix with poolwise ‘X’ values (if any), with one row for each pool. Can be a vector if there is only 1 covariate.

• errors: Character string specifying the errors that Y is subject to. Choices are "neither", "processing" for processing error only, "measurement" for measurement error only, and "both".

• estimate_var: Logical value for whether to return variance-covariance matrix for parameter estimates.

• start_nonvar_var: Numeric vector of length 2 specifying starting value for non-variance terms and variance terms, respectively.

• lower_nonvar_var: Numeric vector of length 2 specifying lower bound for non-variance terms and variance terms, respectively.

• upper_nonvar_var: Numeric vector of length 2 specifying upper bound for non-variance terms and variance terms, respectively.

• nlminb_list: List of arguments to pass to nlminb

  for log-likelihood maximization.

• hessian_list: List of arguments to pass to hessian.

Returns

List containing:

1. Numeric vector of parameter estimates.
2. Variance-covariance matrix (if estimate_var = TRUE).
3. Returned nlminb object from maximizing the log-likelihood function.
4. Akaike information criterion (AIC).

Details

The individual-level model of interest for Y|X is:

Y = beta_0 + beta_x ^T X + e, e ~ N(0, sigsq)

The implied model for summed Y*|X* in a pool with g members is:

Y* = g beta_0 + beta_x ^T X* + e*, e* ~ N(0, g sigsq)

The assay targets Ybar, the mean Y value for each pool, from which the sum Y* can be calculated as Y* = g Ybar. But the Ybar's may be subject to processing error and/or measurement error. Suppose Ybartilde is the imprecise version of Ybar from the assay. If both errors are present, the assumed error structure is:

Ybartilde = Ybar + e_p I(g > 1) + e_m, e_p ~ N(0, sigsq_p), e_m ~ N(0, sigsq_m)

with the processing error e_p and measurement error e_m assumed independent of each other. This motivates a maximum likelihood analysis for estimating theta = (beta_0, beta_x ^T)^T based on observed (Ytilde*, X ) values, where Ytilde = g Ytildebar.

Examples

# Load dataset containing data frame with (g, X1*, X2*, Y*, Ytilde*) values
# for 500 pools each of size 1, 2, and 3, and list of Ytilde values where 20
# of the single-specimen pools have replicates. Ytilde values are affected by
# processing error and measurement error; true parameter values are
# beta_0 = 0.25, beta_x1 = 0.5, beta_x2 = 0.25, sigsq = 1.
data(dat_p_linreg_yerrors)
dat <- dat_p_linreg_yerrors$dat
reps <- dat_p_linreg_yerrors$reps

# Fit Ytilde* vs. (X1*, X2*) ignoring errors in Ytilde (leads to loss of
# precision and overestimated sigsq, but no bias).
fit.naive <- p_linreg_yerrors(
  g = dat$g,
  y = dat$y,
  x = dat[, c("x1", "x2")],
  errors = "neither"
)
fit.naive$theta.hat

# Account for errors in Ytilde*, without using replicates
fit.corrected.noreps <- p_linreg_yerrors(
  g = dat$g,
  y = dat$ytilde,
  x = dat[, c("x1", "x2")],
  errors = "both"
)
fit.corrected.noreps$theta.hat

# Account for errors in Ytilde*, incorporating the 20 replicates
fit.corrected.reps <- p_linreg_yerrors(
  g = dat$g,
  y = reps,
  x = dat[, c("x1", "x2")],
  errors = "both"
)
fit.corrected.reps$theta.hat

# In this trial, incorporating replicates resulted in much better estimates
# of sigsq (truly 1), sigsq_p (truly 0.4), and sigsq_m (truly = 0.2) but very
# similar regression coefficient estimates.
fit.corrected.noreps$theta.hat
fit.corrected.reps$theta.hat

References

Schisterman, E.F., Vexler, A., Mumford, S.L. and Perkins, N.J. (2010) "Hybrid pooled-unpooled design for cost-efficient measurement of biomarkers." Stat. Med. 29 (5): 597--613.