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Posterior predictive model checks for assessing model fit

Source: R/zzz-methods-ppmc.R
fit_ppmc.Rd

For models estimated with method = "mcmc", use the posterior distributions to compute expected distributions for fit statistics and compare to values in the observed data.

Usage

fit_ppmc(
  x,
  ...,
  model_fit = NULL,
  item_fit = NULL,
  ndraws = NULL,
  probs = c(0.025, 0.975),
  return_draws = 0,
  force = FALSE
)

Arguments

x

An estimated model object (e.g., from dcm_estimate()).

...

Unused. For future extensions.

model_fit

The posterior predictive model checks to compute for an evaluation of model-level fit. If NULL, no model-level checks are computed. See details.

item_fit

The posterior predictive model checks to compute for an evaluation of item-level fit. If NULL, no item-level checks are computed. See details.

ndraws

The number of posterior draws to base the checks on. Must be less than or equal to the total number of posterior draws retained in the estimated model. If NULL (the default) the total number from the estimated model is used.

probs

The percentiles to be computed by the stats::quantile() function for summarizing the posterior distribution for each fit statistic.

return_draws

Number of posterior draws for each specified fit statistic to be returned. This does not affect the calculation of the posterior predictive checks, but can be useful for visualizing the fit statistics. Must be less than ndraws (or the total number of draws if ndraws = NULL). If 0 (the default), only summaries of the posterior are returned (no individual samples).

force

If all requested PPMCs have already been added to the model object using add_fit(), should they be recalculated. Default is FALSE.

Value

A list with two elements, "model_fit" and "item_fit". If either model_fit = NULL or item_fit = NULL in the function call, this will be a one-element list, with the null criteria excluded. Each list element, is itself a list with one element for each specified PPMC containing a tibble. For example if item_fit = c("conditional_prob", "odds_ratio"), the "item_fit" element will be a list of length two, where each element is a tibble containing the results of the PPMC. All tibbles follow the same general structure:

  • obs_{ppmc}: The value of the relevant statistic in the observed data.

  • ppmc_mean: The mean of the ndraws posterior samples calculated for the given statistic.

  • Quantile columns: 1 column for each value of probs, providing the corresponding quantiles of the ndraws posterior samples calculated for the given statistic.

  • samples: A list column, where each element contains a vector of length return_draws, representing samples from the posterior distribution of the calculated statistic. This column is excluded if return_draws = 0.

  • ppp: The posterior predictive p-value. This is the proportion of posterior samples for calculated statistic that are greater than the observed value. Values very close to 0 or 1 indicate incompatibility between the fitted model and the observed data.

Details

Posterior predictive model checks (PPMCs) use the posterior distribution of an estimated model to compute different statistics. This creates an expected distribution of the given statistic, if our estimated parameters are correct. We then compute the statistic in our observed data and compare the observed value to the expected distribution. Observed values that fall outside of the expected distributions indicate incompatibility between the estimated model and the observed data.

For DCMs, we currently support PPMCs at the model and item level. At the model level, we calculate the expected raw score distribution (model_fit = "raw_score") as described by Thompson (2019) and Park et al. (2015). At the item level, we can calculate the conditional probability that a respondent in each class provides a correct response (item_fit = "conditional_prob") as described by Thompson (2019) and Sinharay & Almond (2007) or the overall proportion correct for an item (item_fit = "pvalue"), as described by Thompson (2019). We can also calculate the odds ratio for each pair of items (item_fit = "odds_ratio") as described by Park et al. (2015) and Sinharay et al. (2006).

References

Park, J. Y., Johnson, M. S., Lee, Y-S. (2015). Posterior predictive model checks for cognitive diagnostic models. International Journal of Quantitative Research in Education, 2(3-4), 244-264. doi:10.1504/IJQRE.2015.071738

Sinharay, S., & Almond, R. G. (2007). Assessing fit of cognitive diagnostic models. Educational and Psychological Measurement, 67(2), 239-257. doi:10.1177/0013164406292025

Sinharay, S., Johnson, M. S., & Stern, H. S. (2006). Posterior predictive assessment of item response theory models. Applied Psychological Measurement, 30(4), 298-321. doi:10.1177/0146621605285517

Thompson, W. J. (2019). Bayesian psychometrics for diagnostic assessments: A proof of concept (Research Report No. 19-01). University of Kansas; Accessible Teaching, Learning, and Assessment Systems. doi:10.35542/osf.io/jzqs8

Examples

mdm_dina <- measr_dcm(
  data = mdm_data, missing = NA, qmatrix = mdm_qmatrix,
  resp_id = "respondent", item_id = "item", type = "dina",
  method = "mcmc", seed = 63277, backend = "rstan",
  iter = 700, warmup = 500, chains = 2, refresh = 0
)
#> Warning: `measr_dcm()` was deprecated in measr 2.0.0.
#> ℹ This is a limited version of dcm_estimate(); use it instead.
#> Error: object 'mdm_qmatrix' not found

fit_ppmc(mdm_dina, model_fit = "raw_score", item_fit = NULL)
#> Error: object 'mdm_dina' not found