# Conditional Dependence across Slow and Fast Item Responses: With a Latent Space Item Response Modeling Approach

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## Abstract

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## 1. Introduction

#### 1.1. Background

#### 1.2. Current Study

#### 1.3. Structure

## 2. Latent-Space Item-Response Theory

## 3. Our Approach

#### 3.1. Classification of Item Responses into Slow and Fast Responses

#### 3.2. Estimation of Separate Latent Positions of Slow and Fast Responses for an Item

- The constrained model—same item easiness across slow and fast responses for an item: We constrain the item parameter ${c}_{i}$ (in Equation (2)) to be equal across fast and slow conditions. That is, we constrain ${c}_{i}$ to be equal to ${c}_{I+i}$ so that each item has a single easiness parameter regardless of the response-speed conditions. Such a constraint will allow the interaction map to capture the heterogeneity in the item–respondent interactions across response times that may be due to either items or persons or both. For instance, the distances we observe from the interaction map may reflect how each item performs differently depending on whether it is answered faster or slower by respondents and how individuals behave differently on different items depending on response times.
- The unconstrained model—separate item easiness across slow and fast responses for an item: We allow for a separate estimation of the item-easiness parameters for slow and fast conditions to examine the remaining heterogeneity after accounting for the differences in item properties across slow and fast responses. This approach removes the heterogeneity across slow and fast responses that is consistent across respondents from the interaction map and helps us explore the heterogeneity that cannot be explained by the items themselves by inspecting the interaction map. Thus, we anticipate that the former model (i.e., the constrained model) captures the heterogeneity in item–person interactions due to the differences in item properties across fast and slow responses as a part of residual dependence, whereas the latter model (i.e., the unconstrained model) captures any potential heterogeneity that remains even after accounting for the item-difficulty differences across response times.

## 4. Empirical Analysis

#### 4.1. Data

#### 4.1.1. Verbal Analogies

#### 4.1.2. Pattern Matrices

#### 4.1.3. Amsterdam Chess Test

#### 4.2. Analytic Methods

#### 4.2.1. Model Estimation

#### 4.2.2. Model Identifiability

#### 4.2.3. Model Evaluation

## 5. Results

#### 5.1. Constrained Model: Heterogeneity of Item–Respondent Interactions across Slow and Fast Responses

- We present the interaction map for the slow- and fast-response conditions separately for ease of interpretation. Note that the positions from the two maps are obtained from a single model for the same dataset. Therefore, the positions of the respondents (represented as dots) are the same in the two maps.
- Items are represented as their original item number for both the slow and fast conditions (instead of the item numbers in the expanded data format) so that the item positions for slow and fast responses can be easily compared. Thus, for example, by comparing the positions for the same item number in the interaction maps for the slow- and fast-response conditions, we can observe how an item behaves differently depending on whether it was responded to slower or faster than expected.
- We color the respondents and items each based on their levels of ability estimates and the proportion that each item is correct (the overall item easiness). We define the respondents’ ability as low, medium, and high if the estimates are below $-.5$, between $-.5$ and $.5$, and above $.5$, respectively, and each group is colored differently. There are 162, 377, and 187 respondents each in the low, medium, and high groups for the verbal analogies data, respectively; 137, 240, and 126 respondents for the matrices data, respectively; and 83, 91, and 75 respondents for the ACT data, respectively. For the items, easier items (with a higher proportion correct) are colored purple while more difficult items (with a lower proportion correct) are colored orange.

**Items with slow responses are farther away from respondents.**One common observation across the three datasets is that item positions are generally more scattered under the slow-response condition than the fast-response condition. Such a result possibly indicates that items tend to show more heterogeneity in item–person interactions when responded to slower than expected, while the interaction is less heterogeneous when responded to faster. As items for slow responses are more spread out, we can naturally expect slow responses, on average, to have a greater distance from the positions of respondents compared to fast responses.

- Item 13 in the verbal analogies data (a relatively easy item located at the top left of plot (a)-1 and in the middle of plot (a)-2 in Figure 1), the proportion correct is .865 for the slow responses (of which the average item–respondent distance is .251) and .987 for faster responses (the average distance is .056).
- Likewise, Item 63 in the ACT data (located on the left side of plot (c)-1 and toward the center of plot (c)-2) produces a relatively lower proportion correct (.820) for the slow responses (average distance = 1.002) than for the fast responses (proportion correct = .949, average distance = .544).

**The pattern of association varies by the overall item difficulty.**The observed pattern above is particularly apparent and consistent for easier items (purple-colored items) while some relatively difficult items (orange-colored items) show a weak or even an opposite pattern. For example,

- Item 29 with a moderate difficulty (overall proportion correct = .404) in the pattern matrices data is located inside the item cluster around the middle of plot (b)-1 and in the upper right of plot (b)-2 in Figure 1, showing a higher proportion correct (.519) for the slow responses (average distance = 0.033) and a lower proportion correct (.250) for the fast responses (average distance = .086).
- Item 76 in the ACT data, which is relatively difficult (overall proportion correct = .378) and located toward the center of plot (c)-1 and at the top of plot (c)-2 in Figure 1, appears to be easier for the slow responses (proportion correct = .566, average distance = .527) and more difficult for the fast responses (proportion correct = .150, average distance = 1.434).

**Item–respondent interactions vary across respondents.**While we observe differences in item difficulty across response times, it is important to highlight that such a pattern only explains a part of the residual dependency between responses and response times captured in the interaction map. Every individual interacts with items in different ways, and thus the relationship between responses and response times may not be the same for all respondents and items. For instance, an item that is easier when responded to faster for some respondents can be easier when responded to slower for other respondents and vice versa. Such heterogeneity can be examined by looking at the positions of individual respondents and items in the interaction map across fast and slow responses. The heterogeneity is expected to be more apparently demonstrated by items exhibiting very different positions in the interaction map across fast and slow responses. For example,

- Although Item 63 in the ACT data is shown to be generally easier for fast responses (as described above), respondents who are located around the position of Item 63 in plot (c)-1 of Figure 1 would show a lower accuracy for fast responses (as manifested by larger distances between these individuals and Item 63 in plot (c)-2).
- Item 27 in the pattern matrices data is located at the lower left part of plot (b)-1 and the upper right part of plot (b)-2 in Figure 1, which possibly suggests that respondents with positions around the lower left are more likely to get the item correct when responding slower than faster whereas those located around the upper right tend to have a higher probability of getting the item correct when responding faster.

#### 5.2. Unconstrained Model: Residual Dependency between Item Responses and RT after Accounting for the Heterogeneity in Item Difficulty across Fast and Slow Responses

**More difficult items tend to be more spread out.**As a result of removing the overall item differences in difficulty across the response times from the interaction map, we observe in Figure 4 that the items are now relatively equally spread out across the fast and slow responses compared to Figure 1. In addition, for both the fast and slow responses, the more difficult items (orange-colored) appear generally more spread out in the interaction map than the easier items (purple-colored) that are more clustered. This indicates that the more difficult items tend to show more heterogeneous interactions with respondents that cannot be explained by the person and item parameters, involving a greater amount of residual variation (e.g., item–respondent specificities). A related observation is a larger distance between the slow and fast responses for more difficult items. The distance between the slow and fast responses for an item tends to negatively correlate with the overall item difficulty in all three examples (ranging from $-.52$ to $-.26$). This suggests that more difficult items may exhibit a greater heterogeneity in item–respondent interactions across response times.

**The conditional dependence between responses and response times is heterogeneous.**As implied above, some items are located very closely across the fast- and slow-response conditions (e.g., Item 30 in the verbal analogies data) while some other items show a larger distance between the positions for the slow- and fast-response conditions. Such variability suggests that the amount of residual dependency between the responses and response times is different across items. A shorter distance implies that there is not much conditional dependence remaining between the responses and response times; thus, respondents would behave similarly for the slow and fast responses (in a way captured by the item-difficulty differences between the slow and fast responses). In contrast, a larger distance between the fast and slow responses indicates a greater residual dependency unexplained by item-difficulty differences. Thus, these items may involve more person and item specificities in the conditional dependence and consequently show a more heterogeneous conditional dependence between the responses and response times.

- Item 23 in the verbal analogies data has item-easiness estimates of 1.250 for the slow and 2.930 for the fast responses, indicating that the item is generally easier when responded to faster. However, we observe the opposite pattern for the respondents who are located around the slow-response position in plot (a) of Figure 5; the response accuracy is higher when the response is given slower (proportion correct = .57) than faster (proportion correct = .00).
- The item-easiness estimates for Item 22 in the pattern matrices data are 1.315 and 1.272 for the slow and fast responses, respectively, and .643 and .714 for Item 58 in the ACT data. Although the item-easiness estimates do not have a large difference between the slow and fast responses, we can see from plots (b) and (c) in Figure 5 that the respondents are showing a disproportionately higher correct proportion for the slow or fast responses depending on their distances from the item’s slow- and fast-response positions in the interaction map.
- Item 70 in the ACT data has item-easiness estimates of 1.546 for the slow and 1.103 for the fast responses, suggesting that the item is in general slightly easier when responded to slower. We, however, observe that the respondents located near the fast-response position are performing better when they respond to the item faster than slower.

**Figure 4.**Item–respondent interaction maps for items under slow- and fast-response conditions, derived from latent-space item-response models allowing for a separate estimation of item parameters for slow and fast responses for the same item for (

**a**) verbal analogies, (

**b**) pattern matrices, and (

**c**) Amsterdam chess test data.

**Figure 5.**Example items exhibiting heterogeneous item–respondent interactions across slow and fast responses. Two groups of thirty respondents each located near the item positions for slow and fast responses (among those who responded to the item) are displayed in the interaction map. Bar graphs present the proportions of correct items for slow and fast responses calculated from each group.

## 6. Conclusions

#### 6.1. Summary and Discussion

#### 6.2. Limitations and Future Studies

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Posterior Predictive Checks (PPCs) Plots

**Figure A1.**Posterior predictive checks (PPCs) plots for verbal analogies data: (

**a**) constrained and (

**b**) unconstrained models. For total score distribution plots, solid circles represent the observed data, solid gray lines indicate the 50th percentile, and dotted gray lines represent the 5th and 95th percentiles of the generated samples. For proportion of correct items plots, solid circles indicate the observed data, and error bars show the 5th and 95th percentiles of the generated samples.

**Figure A2.**Posterior predictive checks (PPCs) plots for pattern matrices data: (

**a**) constrained and (

**b**) unconstrained models. For total score distribution plots, solid circles represent the observed data, solid gray lines indicate the 50th percentile, and dotted gray lines represent the 5th and 95th percentiles of the generated samples. For proportion of correct items plots, solid circles indicate the observed data and error bars show the 5th and 95th percentiles of the generated samples.

**Figure A3.**Posterior predictive checks (PPCs) plots for ACT data: (

**a**) constrained and (

**b**) unconstrained models. For total score distribution plots, solid circles represent the observed data, solid gray lines indicate the 50th percentile, and dotted gray lines represent the 5th and 95th percentiles of the generated samples. For proportion of correct items plots, solid circles indicate the observed data and error bars show the 5th and 95th percentiles of the generated samples.

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**Figure 1.**Item–respondent interaction maps for items under slow- and fast-response conditions, derived from latent-space item-response models constraining item parameters to be equal across slow and fast responses for the same item for (

**a**) verbal analogies, (

**b**) pattern matrices, and (

**c**) Amsterdam chess test data.

**Figure 2.**Boxplots of items’ average distance from individuals for slow- and fast-response conditions for (

**a**) verbal analogies, (

**b**) pattern matrices, and (

**c**) Amsterdam chess test data.

**Figure 3.**Scatter plots of relative item easiness (under the unconstrained model) against item distance difference (under the constrained model) for (

**a**) verbal analogies, (

**b**) pattern matrices, and (

**c**) Amsterdam chess test data.

**Table 1.**An exemplary illustration of an expanded item-response matrix, treating slow and fast responses to the same item as different items.

Person | Slow Responses | Fast Responses | ||||||
---|---|---|---|---|---|---|---|---|

Item 1 | Item 2 | ⋯ | Item I | Item $\mathit{I}+1$ | Item $\mathit{I}+2$ | ⋯ | Item $\mathit{I}+\mathit{I}$ | |

$p=1$ | 1 | 0 | NA | NA | NA | 1 | ||

$p=2$ | 0 | NA | 0 | NA | 1 | NA | ||

$p=3$ | NA | 1 | NA | 1 | NA | 1 | ||

⋮ | ||||||||

$p=P$ | NA | NA | 1 | 0 | 1 | NA |

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## Share and Cite

**MDPI and ACS Style**

Kim, N.; Jeon, M.; Partchev, I.
Conditional Dependence across Slow and Fast Item Responses: With a Latent Space Item Response Modeling Approach. *J. Intell.* **2024**, *12*, 23.
https://doi.org/10.3390/jintelligence12020023

**AMA Style**

Kim N, Jeon M, Partchev I.
Conditional Dependence across Slow and Fast Item Responses: With a Latent Space Item Response Modeling Approach. *Journal of Intelligence*. 2024; 12(2):23.
https://doi.org/10.3390/jintelligence12020023

**Chicago/Turabian Style**

Kim, Nana, Minjeong Jeon, and Ivailo Partchev.
2024. "Conditional Dependence across Slow and Fast Item Responses: With a Latent Space Item Response Modeling Approach" *Journal of Intelligence* 12, no. 2: 23.
https://doi.org/10.3390/jintelligence12020023