This is a simulated data set modeled after the DTMR study described by Bradshaw et al. (2014) and Izsák et al. (2019). The data was simulated from the loglinear cognitive diagnostic model (LCDM), which is the model that was used to analyze the data in the referenced articles. The data set consists of 990 responses to the 27 items included in the final version of the DTMR data, matching the sample that was collected by the original authors. Each respondent was randomly assigned a mastery profile using the profile proportions reported in Figure 10 of Izsák et al. (2019). Item responses were then generated for each respondent using their assigned mastery profile and the item parameters reported in Table 1 of Bradshaw et al. (2014). Reproducible code for generating the simulated data is available in the GitHub repository for this package.
Format
dtmr_data is a tibble containing
simulated DTMR response data with 990 rows and 28 variables.
id: Respondent identifier.1-22: Simulated dichotomous item responses to the 27 DTMR items.
dtmr_qmatrix is a tibble that identifies
which skills are measured by each DTMR item, as reported in Bradshaw et al.
(2014). The DTMR assessment contains 27 items measuring 4 skills.
The dtmr_qmatrix correspondingly is made up of 27 rows
and 5 variables.
item: Item identifier, corresponds to1-22indtmr_data.referent_units,partitioning_iterating,appropriateness, andmultiplicative_comparison: Dichotomous indicator for whether or not the skill is measured by each item. A value of1indicates the skill is measured by the item and a value of0indicates the skill is not measured by the item.
Simulation values
In addition to the simulated data sets, the true values used to simulate the data are included for reference. This may be useful if, for example, you want to estimate a model and then check how well the estimated parameters match values that were used to create the data.
To simulate the data, we first need dtmr_true_structural. This is a
tibble that contains the structural parameters
reported in Figure 10 of Izsák et al. (2019). The structural parameters
define the probability of observing each possible profile in the population
of respondents. Each row represents one possible mastery profile. Therefore,
there are 16 rows and
5 variables.
referent_units,partitioning_iterating,appropriateness,multiplicative_comparison: Integer values indicating whether each attribute has been mastered by respondents with the given profile.class_probability: The proportion of respondents estimated to demonstrate the given pattern of mastery.
Using the dtmr_true_structural values, we randomly sampled a mastery
profile for each of the 990 respondents. The true profiles for each
respondent are available in dtmr_true_profiles. There are a total of
990 rows and 5 variables.
id: Respondent identifier, corresponds toidindtmr_data.referent_units,partitioning_iterating,appropriateness,multiplicative_comparison: Integer values indicating whether each attribute has been mastered by the respondent.
We use the dtmr_true_profiles and the dtmr_qmatrix to identify whether
each respondent possess the attributes required by each item. Based on which
attributes are required and possessed, we use the dtmr_true_items to
calculate the log odds of each respondent providing a correct response to
each item. dtmr_true_items contains the estimated item parameters reported
in Table 1 of Bradshaw et al. (2014). This a tibble
with 27 rows and 7 columns.
item: Item identifier, corresponds to1-22indtmr_data.intercept: The LCDM intercept parameter for each item.referent_units: The LCDM main effect parameter for items measuring the referent units attribute.partitioning_iterating: The LCDM main effect parameter for items measuring the partitioning and iterating attribute.appropriateness: The LCDM main effect parameter for items measuring the appropriateness attribute.multiplicative_comparison: The LCDM main effect parameter for items measuring the multiplicative comparisons attribute.referent_units__partitioning_iterating: The LCDM interaction parameter for items measuring both referent units and partitioning and iterating attributes.
Finally, we convert the log odds values to probabilities and draw a random
Bernoulli variable using the probabilities of a correct response. The drawn
Bernoulli values are the simulated item scores that make up the dtmr_data.
Details
The skills correspond to knowledge of:
Referent units
Partitioning and iterating
Appropriateness
Multiplicative comparisons
References
Bradshaw, L., Izsák, A., Templin, J., & Jacobson, E. (2014). Diagnosing teachers' understandings of rational numbers: Building a multidimensional test within the diagnostic classification framework. Educational Measurement: Issues and Practice, 33(1), 2-14. doi:10.1111/emip.12020
Izsák, A., Jacobson, E., & Bradshaw, L. (2019). Surveying middle-grads teachers' reasoning about fraction arithmetic in terms of measured quantities. Journal for Research in Mathematics Education, 50(2), 156-209. doi:10.5951/jresematheduc.50.2.0156