Allowing course compensation in higher education: a latent class regression analysis to evaluate performance on a follow-up course

Figures & data

Table 1. Descriptive statistics continuous variables.

Study program	Course	Year	Variable	Mean	Median	SD	Min	Max
Psychology		1	Yearly average^a	6.77	6.60	0.65	5.95	9.25
		1	Yearly SD	0.89	0.88	0.24	0.28	1.67
	Statistics I	1	Course grade	6.40	6.40	1.28	4	10
	Statistics II	2	Course grade^b	6.66	6.80	1.49	1	10
Law		1	Yearly average^a	6.82	6.75	0.61	6	9
		1	Yearly SD	0.80	0.79	0.22	0	1.51
	Intro constitutional and administrative law	1	Course grade	6.54	7	1.03	5	10
	Constitutional law	2	Course grade^c	6.26	6	1.08	3	10

^aThe first year average was only computed for students who received a grade for all first-year courses in their study program.

^bNote that second-year courses do not have the requirements in the academic dismissal policy as in the first year, so grades run from 1 to 10, NA = 77.

^cNA = 192.

Table 2. Descriptive statistics categorical variables.

Study program	Course	Year	Variable	0	1	2	3	4	5
Psychology		1	Number of compensated courses^a (< 6.0)	25.5% (275)	22.9% (247)	20.3% (219)	20.3%(219)	8.6% (93)	2.2% (24)
		1	Number of resits	71.7% (772)	11% (119)	17.3%(186)
	Statistics I	1	Course compensated	62.1% (669)	37.9% (408)
		1	Course retaken	90.3% (972)	9.7% (105)
Law		1	Number of compensated courses^a	52.3% (586)	27.1% (304)	16.8% (188)	3.4% (38)	0.4% (4)
		1	Number of resits	50.7% (568)	34.6% (387)	14.7% (165)
	Intro constitutional and administrative law	1	Course compensated	83.2% (932)	16.8% (188)
		1	Course retaken	89% (995)	11.2% (125)

^aCompensated courses are courses on which the grade was below the required average of 6.0. This thus differs from the number of insufficient grades (below the Dutch cut-off of 5.5).

Table 3. Validation information criteria and classification errors for different LC models for psychology.

Model	LL^a	BIC	AIC	AIC(3)	Number of parameters	Proportion of classification errors
1- class	−1732.44	3485.61	3470.87	3473.87	3	0
2- class	−1632.06	3340.16	3286.12	3297.12	11	0.12
3- class	−1605.28	3341.91	3248.57	3267.57	19	0.2
4- class	−1616.03	3418.71	3286.07	3313.07	27	0.25
5- class	−1611.22	3464.39	3292.45	3327.45	35	0.22
6- class	−1615.40	3528.05	3316.80	3359.80	43	0.22

^aLL = Log-Likelihood. Note that the Log-Likelihood slightly increases at the 4- and 6-class model, this is possible because of the holdout validation procedure used to estimate these values.

Table 4. Descriptive statistics for the three-class model for psychology.

		Class
Variable		1	2	3
First-year average^a	Average	6.20	6.69	7.61
	SD	0.20	0.43	0.62
	Max	7	8	9
First-year standard deviation^b	Average	0.88	0.91	0.86
	SD	0.22	0.23	0.25
Number of compensations^c	Average	2.86	1.61	0.63
	SD	1.07	1.22	1.09
Number of retaken tests^d	Average	1.35	0.21	0.09
	SD	0.87	0.51	0.33
First-year precursor course grade^e	Average	5.64	6.35	7.32
	SD	1.03	1.14	1.25
	Max	9	9	10
Second-year sequel course grade	Average	4.98	6.68	8.47
	SD	1.26	0.90	0.61
	Min	1	4	7
	Max	9	9	10
Class size		0.23	0.56	0.21
N^f		234	563	208

^aThe minimum first-year average is six for all classes.

^bThe range of the standard deviation in first-year grades is similar across classes: from 0 to 2.

^cThe number of compensated courses ranges between 0 and five for each class.

^dThe number of retaken tests ranges between 0 and two for each class.

^eThe minimum grade is four across all classes.

^fSample size here is smaller as reported in the Results section as there were 72 missing values on the second-year grade.

Table 5. Validation information criteria and classification errors for different LC models for law.

Mod	LL^a	BIC	AIC	AIC(3)	Number of parameters	Proportion of classification errors
1- class	−1321.90	2698.47	2659.81	2667.81	8	0
2- class	−1268.43	2680.35	2578.86	2599.86	21	0.11
3- class	−1269.22	2770.77	2606.44	2640.44	34	0.23
4- class	−1270.50	2862.15	2635.00	2682.00	47	0.17
5- class	−1268.05	2946.08	2656.10	2716.10	60	0.23
6- class	−1272.71	3044.23	2691.42	2764.42	73	0.19

^aLL = Log-Likelihood. Note that the Log-Likelihood slightly increases at the 4- and 6-class model, this is possible because of the holdout validation procedure used to estimate these values.

Table 6. Descriptive statistics for the two-class model for law.

		Class
Variable		1	2
First-year average^a	Average	6.57	7.5
	SD	0.41	0.52
	Max	8	9
First-year standard deviation^b	Average	0.78	0.87
	SD	0.20	0.22
	Max	1	2
Number of compensations^c	Average	0.91	0.23
	SD	0.93	0.49
	Max	4	2
Number of retaken tests^d	Average	0.79	0.25
	SD	0.74	0.50
First-year precursor course grade^e	Average	6.30	7.22
	SD	0.90	1.07
	Max	9	10
Second-year sequel course grade	Average	5.78	7.32
	SD	0.84	0.76
	Min	3	6
	Max	8	10
Class size		0.69	0.31
N^f		286	642

^aThe minimum first-year average is six for both classes.

^bThe minimum of the first-year standard deviation is 0 in both classes.

^cThe minimum number of compensations is 0 for both classes.

^dThe range of the number of retaken tests ranges between 0 and two for both classes.

^eThe minimum of the first-year precursor course grade is 5 in both classes.

^fSample size here is smaller as there were 192 students for who second year grade was missing.

Table 7. Descriptive statistics for the three-class model for law.

		Class
Variable		1	2	3
First-year average^a	Average	6.42	6.71	7.25
	SD	0.37	0.46	0.63
	Max	8	8	9
First-year standard deviation^b	Average	0.88	0.70	0.87
	SD	0.18	0.17	0.21
	Max	1	1	2
Number of compensations^c	Average	1.5	0.43	0.45
	SD	0.93	0.63	0.72
	Max	4	2	3
Number of retaken tests^d	Average	0.68	0.86	0.39
	SD	0.70	0.78	0.60
First-year precursor course grade^e	Average	6.04	6.54	6.96
	SD	0.95	0.84	1.12
	Max	9	9	10
Second-year sequel course grade	Average	5.2	5.86	7.26
	SD	0.61	0.61	0.70
	Min	4	3	6
	Max	6	7	10
Class size		0.25	0.35	0.40
N^f		230	327	371

^aThe minimum first-year average was six for all classes.

^bThe minimum first-year standard deviation was 0 for all classes.

^cThe minimum number of compensations was 0 for all classes.

^dThe number of retaken tests ranges between 0 and two for each class.

^eThe minimum first-year precursor grade was five for all classes.

^fSample size here is smaller as there were 192 students for who second year grade was missing.