# Visualizing Cross-Sections of 3D Objects: Developing Efficient Measures Using Item Response Theory

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Cross-Sectioning

#### 1.2. This Present Study

#### 1.3. Study 1: Pilot with University Students

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Materials

#### 2.2.1. Santa Barbara Solids Test (SBST, Cohen and Hegarty 2012)

#### 2.2.2. Planes of Reference Test (PRT, Titus and Horsman 2009)

#### 2.2.3. Crystal Slicing Test (CST, Ormand et al. 2017)

#### 2.2.4. Geologic Block Cross-Sectioning Test (GBCST, Ormand et al. 2014)

#### 2.3. Procedure

## 3. Results

## 4. Discussion

#### Study 2: Large Representative Sample

## 5. Materials and Methods

#### 5.1. Participants

#### 5.2. Materials

#### 5.3. Procedure

## 6. Results

#### 6.1. Scoring

#### 6.2. Descriptive Statistics

#### 6.3. Reliability and Correlations

#### 6.4. Unidimensionality and Local Independence

^{2}statistic (Chen and Thissen 1997) and the Jackknife Slope Index (JSI; Edwards et al. 2018) were calculated for each possible combination of items in each individual test using the residuals function in the R package mirt (Chalmers 2012). In order to examine each model for possible locally dependent item pairs, the distribution of the standardized values of G

^{2}and JSI for all item pairs was inspected (see the range for each diagnostic in Table 6). There was no evidence of local dependent item pairs.

#### 6.5. Within Task Item Response Theory Analyses

_{2}statistic, which has been demonstrated to be less influenced by the sparsity of the contingency table of response patterns than chi-square (Maydeu-Olivares 2014), as well as Root Mean Square Error of Approximation (RMSEA), Comparative Fit Index (CFI), Tucker–Lewis Index (TLI), and Standardized Root Mean Squared Residual (SRMR). Acceptable fit was determined based on the following criteria: Root Mean Square Error of Approximation (RMSEA) below 0.05 (Maydeu-Olivares 2013), Comparative Fit Index (CFI) above 0.95, Tucker–Lewis Index (TLI) above 0.953, and SRMR less than 0.05 (Maydeu-Olivares 2013). While a significant M

_{2}indicates that we should reject the null hypothesis of the fitted model being true, this statistic was small relative to the degrees of freedom ($\frac{{M}_{2}}{df}$ < 3), indicating an acceptable model fit.

#### 6.6. Santa Barbara Solids Test

#### 6.7. Planes of Reference Test

#### 6.8. Crystal Slicing

_{2}significance tests indicated a poor fit (see Table 7). The consideration of individual items indicated that items 2, 10, and 14 showed significant misfits (p < .05). Item 10 had the highest RMSEA (0.081), while the other two were within acceptable bounds (≤0.05). While there were not any obvious problems with these items upon inspection, one possible source of error for non-geology students might have misinterpreted the symmetry of the shapes as they are unlikely to be familiar with these shapes.

#### 6.9. Combined Task Item Response Theory Analysis: Model Comparisons

#### 6.10. Individual Differences

^{2}= 0.19, F(6, 491) = 18.79, p < 0.01.

#### 6.11. Creating A Refined, Efficient Test

## 7. Discussion

#### Limitations and Future Directions

## 8. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | We did not include the original Mental Cutting test, because it is identical in format to the Planes of Reference test, or the Tooth tion test, as a previous study indicated that it depended on dentistry expertise (Hegarty et al. 2007). |

2 | The research was funded by the Office of Naval Research, which is interested in the spatial abilities of potential Navy recruits. The demographics of the US Navy are similar to that of the whole country, cf. Census Bereau (2022) and Department of Defense (2017). |

3 | Although Hu and Bentler’s (1999) recommended cutoffs for these indices may be too low for IRT, they are used as a rough heuristic here as there is not a universally agreed upon value. Cai et al. (2023) recommend 0.97 for the TLI. |

4 | Analysis of local independence diagnostics indicated no locally dependent item pairs for any of the models except for Model B (which had a right-skewed distribution of the standardized G2 statistic; min = −0.12, max = 0.40). It is possible that removing one item from each of the item pairs with an extreme G2 would improve model fit; however, other models having good fit without removing further items was taken as sufficient evidence that Model B was not the best fit. |

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**Figure 2.**Sample items from the four cross-section tests used in this research. (

**A**) Santa Barbara Solids Test (SBST) item (

**B**) Crystal Slicing Test (CST) item (

**C**) Planes of Reference Test (PRT) item (

**D**) Geologic Block Cross-Sectioning Test (GBCST) item.

**Figure 3.**Possible unidimensional, multidimensional, and hierarchical models of cross-sectioning ability.

**Figure 4.**Distribution of scores on each cross-section test. Santa Barbara Solids Tests (SBST), Planes of Reference (PRT), Crystal Slicing Tests (CST) frequency of scores. The vertical red line indicates chance performance.

**Figure 5.**Scree plots for the Santa Barbara Solids Test, the Planes of Reference Test, and the Crystal Slicing Test.

**Figure 6.**Item characteristic curves for the Santa Barbara Solids Test (SBST), Planes of Reference (PRT), Crystal Slicing Test, and a combined test consisting of the 38 discriminating items on the SBST and PRT. P(Theta) is the probability of getting each item (represented by different colored lines) correct at a given ability level (Theta). Items with low discriminability are labeled for SBST, PRT, and CST. The combined test is the refined SBST and PRT.

**Figure 7.**Test information functions for refined versions of SBST, PRT, and the 38-item test with SBST and PRT Items. I(θ) is the information that the test gives (i.e., how precise the θ estimate is) at a given θ (level of ability).

Test | Possible Range | Score | Response Time | McDonald’s Omega | Spearman- Brown | Cronbach’s Alpha | ||
---|---|---|---|---|---|---|---|---|

M | SD | M | SD | |||||

Santa Barbara Solids | 0–30 | 19.32 | 6.71 | 11.49 | 6.04 | 0.91 | 0.88 | 0.89 |

Planes of Reference | 0–15 | 7.30 | 3.00 | 13.76 | 11.41 | 0.74 | 0.63 | 0.66 |

Crystal Slicing | 0–15 | 8.26 | 2.82 | 12.13 | 7.50 | 0.73 | 0.57 | 0.64 |

Geologic Block Cross-Sectioning | 0–16 | 5.77 | 2.90 | 16.63 | 10.77 | 0.74 | 0.60 | 0.64 |

**Table 2.**Correlation matrix for the measures used in the pilot study. Disattenuated correlations (corrected for reliability of each measure) are shown in parentheses; 95% confidence intervals of the correlations are shown above the diagonal.

Santa Barbara Solids | Planes of Reference | Crystal Slicing | Geologic Block Cross-Sectioning | |
---|---|---|---|---|

Santa Barbara Solids Test | 1 | [0.60, 0.87] | [0.69, 0.90] | [0.31, 0.74] |

Planes of Reference Test | 0.76 (1) | 1 | [0.52, 0.83] | [0.31, 0.74] |

Crystal Slicing Test | 0.82 (1) | 0.71 (1) | 1 | [0.24, 0.70] |

Geologic Block Cross-Sectioning Test | 0.56 (0.77) | 0.56 (0.91) | 0.50 (0.86) | 1 |

Race | Ethnicity | Education Status | Parents’ Education | ||||
---|---|---|---|---|---|---|---|

White | 334 | Hispanic | 132 | In High School | 78 | High School Diploma | 226 |

Black/African American | 106 | Not Hispanic | 366 | In 2Year College | 129 | Associate’s Degree | 52 |

Asian/Asian American | 13 | In 4-Year College | 151 | College Degree | 110 | ||

American Indian/Alaska Native | 7 | Not in an Educational Institution | 140 | Graduate/ Professional Degree | 90 | ||

Not Stated | 38 |

Test | Possible Range | Score | Response Time | McDonald’s Omega | Spearman- Brown | Cronbach’s Alpha | ||
---|---|---|---|---|---|---|---|---|

M | SD | M | SD | |||||

Santa Barbara Solids Test | 0–30 | 12.11 | 5.53 | 9.42 | 1.96 | 0.84 | 0.79 | 0.82 |

Planes of Reference Test | 0–15 | 5.24 | 2.44 | 9.96 | 2.65 | 0.55 | 0.46 | 0.50 |

Crystal Slicing Test | 0–15 | 5.28 | 2.55 | 9.16 | 2.45 | 0.63 | 0.51 | 0.56 |

Word Sum Test | 0–14 | 6.21 | 2.87 | - | - | 0.74 | 0.63 | 0.72 |

**Table 5.**Correlations between the measures. Numbers in parentheses indicate partial correlations between the cross-section tests after controlling for the Word Sum test. Above the diagonal (in italics) are disattenuated correlations.

Test | SBST | PRT | CST | WST |
---|---|---|---|---|

Santa Barbara Solids Test (SBST) | 1 | 0.90 | 0.92 | 0.48 |

Planes of Reference Test (PRT) | 0.54 * (0.48 *) | 1 | 1 | 0.67 |

Crystal Slicing Test (CST) | 0.58 * (0.53 *) | 0.50 * (0.43 *) | 1 | 0.61 |

Word Sum Test (WST) | 0.34 * | 0.36 * | 0.35 * | 1 |

**Table 6.**Standardized local dependence diagnostics for all models. Extreme values falling far outside the normal distribution indicate possible item pairs showing local dependence.

Model | G^{2} | JSI | ||
---|---|---|---|---|

Min | Max | Min | Max | |

Santa Barbara Solids Test | −0.13 | 0.14 | −1.39 | 1.39 |

Planes of Reference Test | −0.08 | 0.13 | −1.41 | 1.93 |

Crystal Slicing Test | −0.11 | 0.16 | −1.36 | 2.28 |

**Table 7.**Fit of 2PL unidimensional model for the Santa Barbara Solids Test, Planes of Reference Test, and Crystal Slicing Test.

Test | M_{2} Statistic | df(M_{2}) | p | RMSEA | TLI | CFI | SRMR |
---|---|---|---|---|---|---|---|

Santa Barbara Solids Test | 697.52 | 405 | <.001 | 0.04 | 0.93 | 0.94 | 0.05 |

Planes of Reference Test | 111.21 | 90 | =.06 | 0.02 | 0.94 | 0.95 | 0.04 |

Crystal Slicing Test | 220.13 | 90 | <.001 | 0.05 | 0.78 | 0.81 | 0.06 |

Model | M_{2} | df(M_{2}) | p | RMSEA | TLI | CFI | SRMR | AIC | BIC |
---|---|---|---|---|---|---|---|---|---|

A (Unidimensional 2PL) | 1104.01 | 665 | <.001 | 0.036 | 0.937 | 0.941 | 0.051 | 21,153.29 | 21,473.29 |

B (Multidimensional 2PL) | 1275.63 | 663 | <.001 | 0.043 | 0.912 | 0.917 | 0.094 | 21,359.02 | 21,687.45 |

C (Hierarchical Bifactor, 4 Sub-factors) | 860.90 | 627 | <.001 | 0.027 | 0.965 | 0.968 | 0.046 | 21,047.96 | 21,527.97 |

D (Hierarchical Bifactor, 2 Sub-factors, Orientation) | 775.37 | 627 | <.001 | 0.022 | 0.978 | 0.980 | 0.042 | 20,967.59 | 21,447.60 |

E (Hierarchical Bifactor, 2 Sub-factors, Complexity) | 923.84 | 627 | <.001 | 0.031 | 0.955 | 0.960 | 0.047 | 21,073.25 | 21,553.25 |

20-D (20-item version of D) | 207.49 | 150 | =.001 | 0.028 | 0.981 | 0.985 | 0.039 | 10,810.84 | 11,063.47 |

**Table 9.**Correlation between demographic variables, education, verbal ability, total score on the 38-item test, and theta scores from Model D.

Sex | Age | Education | Parents’ Education | Math Courses | Word Sum | Total Test Score | |
---|---|---|---|---|---|---|---|

Age | 0.07 | ||||||

Education | 0.09 | 0.11 ** | |||||

Parents’ Education | −0.03 | 0.02 | 0.36 ** | ||||

Math Courses | −0.01 | 0.01 | 0.17 ** | 0.20 ** | |||

Word Sum | −0.02 | 0.02 | 0.09 * | 0.17 ** | 0.28 ** | ||

Total Test Score | −0.09 | −0.07 | 0.07 | 0.16 ** | 0.24 ** | 0.39 ** | |

Theta | −0.09* | −0.08 | 0.07 | 0.16 ** | 0.25 ** | 0.39 ** | 0.96 ** |

**Table 10.**Regression coefficients for linear model with demographic variables as predictors of score.

Estimate | SE | t Value | p Value | |
---|---|---|---|---|

Intercept | 16.60 | 0.89 | 18.61 | <0.01 |

Sex | −0.98 | 0.56 | −1.75 | 0.08 |

Age | −0.50 | 0.28 | −1.78 | 0.08 |

Math Courses | 0.90 | 0.29 | 3.04 | <0.01 |

Parents’ Education | 0.48 | 0.30 | 1.58 | 0.12 |

Education Status | 0.04 | 0.30 | 0.15 | 0.88 |

Word Sum Score | 2.32 | 0.29 | 7.94 | <0.01 |

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**MDPI and ACS Style**

Munns, M.E.; He, C.; Topete, A.; Hegarty, M.
Visualizing Cross-Sections of 3D Objects: Developing Efficient Measures Using Item Response Theory. *J. Intell.* **2023**, *11*, 205.
https://doi.org/10.3390/jintelligence11110205

**AMA Style**

Munns ME, He C, Topete A, Hegarty M.
Visualizing Cross-Sections of 3D Objects: Developing Efficient Measures Using Item Response Theory. *Journal of Intelligence*. 2023; 11(11):205.
https://doi.org/10.3390/jintelligence11110205

**Chicago/Turabian Style**

Munns, Mitchell E., Chuanxiuyue He, Alexis Topete, and Mary Hegarty.
2023. "Visualizing Cross-Sections of 3D Objects: Developing Efficient Measures Using Item Response Theory" *Journal of Intelligence* 11, no. 11: 205.
https://doi.org/10.3390/jintelligence11110205