Detect Influential Observations in GMM — diagnose

Identifies observations that have substantial influence on parameter estimates using survey-weighted influence metrics including Cook's distance, DFBETAS, and leverage.

Usage

diagnose_influence(
  object,
  measure = c("cooks", "dfbetas", "leverage", "all"),
  threshold = NULL
)

Arguments

object: Fitted SurveyMixr object (must be fitted with keep_data = TRUE)
measure: Character string specifying influence measure: "cooks" (Cook's distance), "dfbetas" (parameter-specific influence), "leverage", or "all"
threshold: Numeric threshold for flagging influential cases. If NULL, uses conventional cutoffs (e.g., 4/n for Cook's D)

Value

An S4 object of class InfluenceDiagnostics with slots:

influence_measures: Data frame with influence statistics per observation
influential_ids: IDs flagged as influential
threshold: Threshold used for flagging
summary: Summary statistics

Details

**Note:** This function requires the original data to be stored in the fitted object. Make sure to fit your model with keep_data = TRUE.

**Cook's Distance (Survey-Weighted):**

Measures the influence of each observation on all parameter estimates simultaneously. The survey-weighted version accounts for sampling weights:

$$D_i = \frac{(\hat{\beta} - \hat{\beta}_{(-i)})' X'WX (\hat{\beta} - \hat{\beta}_{(-i)})}{p \hat{\sigma}^2}$$

where W is the diagonal weight matrix.

**DFBETAS:**

Measures the influence of each observation on individual parameters:

$$DFBETAS_{ij} = \frac{\hat{\beta}_j - \hat{\beta}_{j(-i)}}{SE(\hat{\beta}_{j(-i)})}$$

**Leverage:**

Identifies observations with unusual predictor patterns:

$$h_i = X_i (X'WX)^{-1} X_i' w_i$$

References

Cook, R. D. (1977). Detection of influential observation in linear regression. *Technometrics*, 19(1), 15-18.

Pfeffermann, D., & Sverchkov, M. (1999). Parametric and semi-parametric estimation of regression models fitted to survey data. *Sankhyā: The Indian Journal of Statistics, Series B*, 61(1), 166-186.

Examples

if (FALSE) { # \dontrun{
# Full example (requires model fitted with keep_data = TRUE)
set.seed(123)
mcs_simulated <- simulate_gmm_survey(
  n_individuals = 500,
  n_times = 6,
  n_classes = 3,
  time_scores = c(3, 5, 7, 11, 14, 17),
  design = "stratified_cluster",
  n_strata = 4,
  n_clusters = 50,
  missing_rate = 0.12,
  seed = 123
)

fit <- gmm_survey(
  data = mcs_simulated,
  id = "id", time = "time", outcome = "outcome",
  n_classes = 3, weights = "weight", starts = 20,
  keep_data = TRUE
)

# Detect influential observations
influence <- diagnose_influence(fit, measure = "all")
print(influence)

# Get IDs of influential cases
influential_ids <- influence@influential_ids

# Examine influential cases
mcs_simulated[mcs_simulated$id %in% influential_ids, ]
} # }