Sample Selection

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Course sections are listed by location. Internet sections were placed near their respective institution location. The number of students in the full data set is 5443, the number of removed students is 4696 and the model data set is 745.

Data Selection

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Note. Data for this study was collected by the Wyoming Community College Commission (WCCC) from 2012 to 2023, encompassing 5448 students who enrolled in CCC. Several criteria were applied to create the analytic sample. Only students’ first CCC attempt was included, concurrent enrollment sections taught in high schools, all students in high school, and students with missing data were listwise deleted. The analysis focused on CCC sections from the 2013-2014 academic year onward, allowing summer 2012 through summer 2014 to serve as a lookback period for capturing students’ precalculus coursework history. Sections with fewer than three students were removed to allow for adequate sample sizes for multilevel modeling (Clarke, 2008). The final analytic sample consisted of 745 students in 91 course sections across 5 of Wyoming’s 7 CCs. The smallest section size was 3, and the largest was 22 students.

Model Specification

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Model Diagram

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Level 1 (student) intercept is represented by the \(1_{1}\) triangle on the left, with level 1 predictors below in rectangles; Level 2 predictors are arranged horizontally along the top in rectangles with the level 2 intercept represented with the \(1_{2}\) triangle. Cross-level interaction term edges and their coefficient labels are color coded with their β-nodes. The intercept varies randomly across sections, indicated by the random effect \(u_{0j}\). Variables in bold font are centered within section (cwc), variables bold-italic font are grand mean centered (gmc). Parameter estimates without parentheses are statistically significant (p≤0.05) and within parentheses are non-statistically significant (p>0.05).

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Multilevel Logistic Regression Model - Hierarchical Form

The model can be written in hierarchical form with the student \(\beta\) level coefficients, which are then written in terms of section \(\gamma\) level coefficients. The random effect for the intercept is \(u_{0j}\) and there are no random slopes. There are two student-level interaction terms and two cross-level interaction terms.

Level 1 Model:

\[ \begin{aligned} \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) &= \beta_{0j}+\beta_{1j} (\text{HSPGA})_{cwc}+\beta_{2j} (\text{ACT Math})_{cwc} \\ &\quad + \beta_{3j} (\text{Years Since HS})_{cwc} + \beta_{4j} (\text{math prerequisites})_{cwc} \\ &\quad + \beta_{5j} (\text{female})+\beta_{6j} (\text{concurrent prereq})_{cwc} + \beta_{7j} (\text{URM}) \\ &\quad + \beta_{8j} (\text{HSGPA})_{cwc}×(\text{Years Since HS})_{cwc} \\ &\quad +\beta_{9j} (\text{concurrent prereq})_{cwc} ×(\text{max ACT Math})_{cwc} \end{aligned} \]

Level 2 Model:

\[ \begin{aligned} &\quad \beta_{0j} = γ_{00} + \gamma_{01} (\text{HSGPA})_{gmc}+\gamma_{02} (\text{coreqcollege}) +u_{0j} \\ &\quad \beta_{1j} =\gamma_{10} \\ &\quad \beta_{2j} =\gamma_{20} \\ &\quad \beta_{3j} =\gamma_{30} \\ &\quad \beta_{4j} =\gamma_{40}+\gamma_{41} (\text{coreq college}) \\ &\quad \beta_{5j}=\gamma_{50} \\ &\quad \beta_{6j}=\gamma_{60} \\ &\quad \beta_{7j}=\gamma_{70}+\gamma_{71} (\text{HSGPA})_{gmc}\\ &\quad \beta_{8j}=\gamma_{80} \\ &\quad \beta_{9j}=\gamma_{90} \end{aligned} \]

Multilevel Logistic Regression Model - Combined Form

Substituting the section-level \(\beta\) coefficient equations into the student-level equation results in a representation of the model in combined form:

\[\begin{aligned} \log\left(\frac{P(Y_{ij} = \text{Pass})}{P(Y_{ij} = \text{Fail})}\right) &= \gamma_{00} + \gamma_{01}(\text{HS GPA})_{gmc} + \gamma_{02}(\text{CoreqCollege}) \\ &\quad + \gamma_{10}(\text{HSGPA})_{cwc} + \gamma_{20}(\text{ACT Math})_{cwc} + \gamma_{30}(\text{YearsCalc})_{cwc} \\ &\quad + \gamma_{40}(\text{MathCourses})_{cwc} + \gamma_{41}(\text{MathCourses})_{cwc} \times (\text{CoreqCollege}) \\ &\quad + \gamma_{50}(\text{Female}) + \gamma_{60}(\text{ConcurrentPrereq})_{cwc} + \gamma_{70}(\text{URM}) \\ &\quad + \gamma_{71}(\text{URM}) \times (\text{HS GPA})_{gmc} + \gamma_{80}(\text{HS GPA})_{cwc} \times (\text{YearsCalc})_{cwc} \\ &\quad + \gamma_{90}(\text{ConcurrentPrereq})_{cwc} \times (\text{ACT Math})_{cwc} + u_{0j} \end{aligned}\]

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Model Structure

Component	Description
Level 1 Structure	\(\log\left(\frac{P(Pass)}{P(Fail)}\right) = \beta_{0j} + \sum\beta_{qj}X_{qij}\)
Level 2 Structure	\(\beta_{qj}=\gamma_{q0}+\sum\gamma_{qs}W_{sj}+u_{qj}\)
Level 1 Predictors	HSGPA, ACT Math, Years since HS, Math Courses, Female, Concurrent Prereq, URM
Level 2 Predictors	HSGPA (gmc), Coreq College
Level 1 Interactions	HSGPA×Years, Concurrent×ACTMath
Cross Level Interactions	Math Courses×Coreq, URM×HSGPA(gmc)
Random Effect	\(u_{0j} \sim N(0, \tau_{00}^2)\)

Fixed Effects

Effect	Parameter	Level	Centering
Intercept	\(\gamma_{00}\)	–	–
HSGPA (gmc)	\(\gamma_{01}\)	Section	Grand-Mean
Coreq College	\(\gamma_{02}\)	Section	None
HSGPA (cwc)	\(\gamma_{10}\)	Student	Section-Mean
ACT Math (cwc)	\(\gamma_{20}\)	Student	Section-Mean
Years since HS (cwc)	\(\gamma_{30}\)	Student	Section-Mean
Math Courses (cwc)	\(\gamma_{40}\)	Student	Section-Mean
Female	\(\gamma_{50}\)	Student	None
Concurrent Prereq (cwc)	\(\gamma_{60}\)	Student	Section-Mean
URM	\(\gamma_{70}\)	Student	None

Interactions

Interaction	Parameter	Level
Math Courses (cwc) × Coreq College	\(\gamma_{41}\)	Cross-Level
URM × HSGPA (gmc)	\(\gamma_{71}\)	Cross-Level
HSGPA (cwc) × Years since HS (cwc)	\(\gamma_{80}\)	Student-Level
Concurrent Prereq (cwc) × ACT Math (cwc)	\(\gamma_{90}\)	Student-level

Student-Level Descriptive Statistics

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Continuous Variables

Variable	N	M	SD	Min	Max
HS.GPA	745	3.49	0.44	1.80	4.00
ACT.Math	745	24.10	3.57	13.00	34.00
Years.since.HS	745	1.26	1.28	0.21	8.21
Math.prereqs	745	1.44	1.16	0.00	7.00

Categorical Variables

Variable	Yes	No	Percent_Yes	Percent_No
Female	288	457	38.7	61.3
URM	139	606	18.7	81.3
Concurrent prereq	148	597	19.9	80.1
Coreq college	278	467	37.3	62.7
Pass	537	208	72.1	27.9

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Gender Distribution

URM Status Distribution

Concurrent Prerequisite Distribution

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HS GPA Distribution

ACT Math Distribution

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Years Since HS Distribution

Math Prerequisites Distribution

Section-Level Statistics

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Overall Section Statistics

Statistic	Value
Total Sections	91.0
Total Students	745.0
Mean Section Size	8.2
SD Section Size	4.6
Min Section Size	3.0
Max Section Size	22.0

Section Mean Distributions

Variable	M	SD	Min	Max
Mean HS GPA	3.460	0.210	2.84	3.94
Mean ACT Math	24.060	1.860	18.00	27.80
Mean Years Since HS	1.410	0.840	0.21	4.75
Mean Math Prerequisites	1.440	0.620	0.00	3.00
Prop. Concurrent Prereq	0.178	0.253	0.00	1.00

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Group Sizes

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Section-Level Pass Rates Distribution

Section Mean High School GPA

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Section Mean Mean ACT Math

Section Mean Years Since HS

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Section Mean Number of Math Prerequisites

Section Mean Proportion Concurrent Prerequisite

Comparative Analysis

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Pass/Fail Distribution

Pass Rate by Time & Prerequisites

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Pass Rate by Academic Preparation

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Pass Rate by Gender and URM Status

Pass Rate by Categorical Variables

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Pass Rate Differences by Categorical Variables

Statistically Significant Main Effects and Interactions

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Model Diagram with Statistically Significant Effects

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Level 1 (student) intercept is represented by the \(1_{1}\) triangle on the left, with level 1 predictors below in rectangles; Level 2 predictors are arranged horizontally along the top in rectangles with the level 2 intercept represented with the \(1_{2}\) triangle. Cross-level interaction term edges and their coefficient labels are color coded with their β-nodes. The intercept varies randomly across sections, indicated by the random effect \(u_{0j}\). Variables in bold font are centered within section (cwc), variables bold-italic font are grand mean centered (gmc). Parameter estimates without parentheses are statistically significant (p≤0.05) and within parentheses are non-statistically significant (p>0.05). There were three main effects and four interactions that were statistically significant in the model. Shaded ribbon around predicted probability represents a 95% confidence interval. Horizontal axis spans range of data for variable.

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Course-Level Effect: HSGPA (gmc)

The main effect of HSGPA (gmc) was statistically significant in the model. This represents the influence of mean course-section HSGPA compared to grand mean HSGPA.

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Student-Level Effect: HSGPA (cwc)

The student-level main effect of HSGPA (cwc) was statistically significant in the model. This represents the influence of the difference between a student’s HSGPA and their course-section mean.

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Student-Level Effect: math prereqs (cwc)

The main effect of math prereqs (cwc) was statistically significant in the model. This represents the influence of the difference between the number of prerequisite math courses a student took and their course section mean.

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Student-Level Interaction: concurrent prereq (cwc) x ACT Math (cwc)

The student-level interaction concurrent prereq (cwc) x ACT Math (cwc) was statistically significant in the model. Slider bar goes two standard deviations on each side of the mean of ACT Math (cwc).

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Student-Level Interaction: HSGPA (cwc) x years since HS (cwc)

The student-level interaction HSGPA (cwc) x years since HS (cwc) was statistically significant in the model. Slider bar goes two standard deviations on each side of the mean of HSGPA (cwc).

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Cross-Level Interaction: coreq college x math prereqs (cwc)

The the cross-level interaction coreq college x math prereqs (cwc) was statistically significant. Button toggles coreq college status.

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Cross-Level Interaction: URM x HSGPA (gmc)

The the cross-level interaction URM x HSGPA (gmc) was statistically significant. Button toggles URM status.

Model Summary and Assumption Testing

Testing assumptions for multilevel logistic regression: multicollinearity, the log odds of the dependent variable needs to be linearly related to the independent variables, and sample sizes must be sufficient to have a minimum number of cases in each outcome.

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ICC

Intraclass Correlation Coefficients
	Value
Component	ICC
ICC_adjusted	0.114
ICC_conditional	0.114
ICC_unadjusted	0.114

Null Model Summary

Null Model Summary
Effect	Group	Term	Estimate	SE	z	p
Fixed Effects
fixed	NA	(Intercept)	1.017	0.115	8.823	< 0.001
Random Effects
ran_pars	Location_Term_Institution	sd__(Intercept)	0.650	NA	NA

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Full model summary

Full Model Fixed Effects
Term	Estimate	SE	z	p
(Intercept)	1.174	0.177	6.613	< 0.001
HSGPA_cwc	1.608	0.255	6.302	< 0.001
years_calc_since_HS_cwc	-0.054	0.093	-0.585	0.559
URM	-0.215	0.236	-0.911	0.362
math_courses_below_2200_cwc	0.305	0.136	2.237	< 0.05
concurrent_prereq_cwc	-0.474	0.314	-1.508	0.131
max_actmath_cwc	0.053	0.035	1.524	0.128
female	-0.003	0.203	-0.014	0.989
coreqcollege	-0.189	0.240	-0.789	0.43
HSGPA_gmc	2.912	0.642	4.537	< 0.001
concurrent_prereq_cwc:max_actmath_cwc	0.203	0.100	2.035	< 0.05
HSGPA_cwc:years_calc_since_HS_cwc	-0.798	0.226	-3.526	< 0.001
URM:HSGPA_gmc	-2.738	1.200	-2.281	< 0.05
math_courses_below_2200_cwc:coreqcollege	-0.552	0.189	-2.922	< 0.01

Compare Null vs Full Model

Model Comparison
Model	npar	AIC	BIC	logLik	deviance	Chisq	df	Pr..Chisq.
Null Model	2	873.61	882.83	-434.8	869.61	NA	NA	NA
Full Model	15	802.80	872.00	-386.4	772.80	96.81	13	< 0.001

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VIF

Variance Inflation Factors
Variable	VIF
HSGPA_cwc	1.209
years_calc_since_HS_cwc	1.270
URM	1.081
math_courses_below_2200_cwc	2.284
concurrent_prereq_cwc	1.126
max_actmath_cwc	1.499
female	1.100
coreqcollege	1.052
HSGPA_gmc	1.227
concurrent_prereq_cwc:max_actmath_cwc	1.061
HSGPA_cwc:years_calc_since_HS_cwc	1.179
URM:HSGPA_gmc	1.249
math_courses_below_2200_cwc:coreqcollege	1.671