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.

## Arguments

- model
A measrfit object.

- 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 distributions of the specified fit statistics.- return_draws
Proportion 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. For example, if

`ndraws = 500`

,`return_draws = 0.2`

, and`model_fit = "raw_score"`

, then the raw score chi-square will be computed 500 times (once for each draw) and 100 of those values (0.2 * 500) will be returned. If`0`

(the default), only summaries of the posterior are returned (no individual samples).- 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. Multiple checks can be provided in order to calculate more than one check simultaneously (e.g.,`item_fit = c("conditional_prob", "odds_ratio")`

). See details.

## 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`(ndraws * 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.

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). 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

```
if (FALSE) { # interactive()
cmds_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
)
fit_ppmc(cmds_mdm_dina, model_fit = "raw_score", item_fit = NULL)
}
```