Applications of discrete factor analysis

Figures & data

Table

x1	Feathers (feather injury)	0, 1, 2
x2	Hock burns (injury marks)	0, 1
x3	FPD (footpad dermatitis)	0, 1, 2
x4	Plumage cleanliness	0, 1, 2

Table 1. Veterinary data, means and variances.

Variable	1	2	3	4
Mean	0.23	0.12	0.26	0.54
Variance	0.18	0.10	0.24	0.36

Table 2. Veterinary data, empirical correlations.

Variable number	1	2	3	4
1	1	0.047	–0.059	0.064
2	.	1	0.107	0.297
3	.	.	1	0.224
4	.	.	.	1

Table 3. Veterinary data, estimated parameters.

$\widehat{\lambda }$	${\widehat{\mu }}_1$	${\widehat{\mu }}_2$	${\widehat{\mu }}_3$	${\widehat{\mu }}_4$
0.079	0.230	0.038	0.257	0.458

Table 4. Veterinary data, estimated correlations from the factor model.

Variable number	1	2	3	4
variable 1	1	0	0	0
variable 2	$·$	1	0	0.31
variable 3	$·$	$·$	1	0
variable 4	$·$	$·$	$·$	1

Table 5. Veterinary data, estimated principal components.

	x1	x2	x3	x4
z1	0.097	0.580	0.471	0.658
z2	–0.868	–0.170	0.464	–0.055
z3	0.470	–0.586	0.659	–0.025

Table

x1	Length, expressed in number of notes written
x2	Range (in semitones)
x3	Number of unique pitch classes
x4	Initial interval (in semitones)
x5	Number of unique intervals between successive notes
x6	Maximum interval between successive notes (in semitones)

Figure 1. J.S. Bach: subject from Fugue in C minor (BWV 537).

Table

x1	x2	x3	x4	x5	x6
17	9	7	7	7	9

Table 6. Music data, means and variances.

Variable number	1	2	3	4	5	6
Mean	15.9	6.9	3.6	2.9	5.0	7.3
Variance	143.5	10.3	2.6	6.4	4.2	7.7

Table 7. Music data, empirical correlations.

Variable number	1	2	3	4	5	6
variable 1	1	0.55	0.35	0.12	0.67	0.40
variable 2	$·$	1	0.41	0.21	0.58	0.59
variable 3	$·$	$·$	1	−0.004	0.34	0.28
variable 4	$·$	$·$	$·$	1	0.20	0.31
variable 5	$·$	$·$	$·$	$·$	1	0.68
variable 6	$·$	$·$	$·$	$·$	$·$	1

Table 8. Music data, estimated parameters, Poisson model.

$\widehat{\lambda }$	${\widehat{\mu }}_1$	${\widehat{\mu }}_2$	${\widehat{\mu }}_3$	${\widehat{\mu }}_4$	${\widehat{\mu }}_5$	${\widehat{\mu }}_6$
2.7	13.2	4.2	3.6	2.9	2.3	4.6

Table 9. Music data, estimated correlations from the Poisson factor model.

Variable number	1	2	3	4	5	6
variable 1	1	0.26	0	0	0.30	0.25
variable 2	$·$	1	0	0	0.46	0.38
variable 3	$·$	$·$	1	0	0	0
variable 4	$·$	$·$	$·$	1	0	0
variable 5	$·$	$·$	$·$	$·$	1	0.44
variable 6	$·$	$·$	$·$	$·$	$·$	1

Table 10. Music data, estimated parameters, negative binomial model.

${\widehat{r}}_0$	2.5	${\widehat{p}}_0$	0.44
${\widehat{r}}_1$	1.5	${\widehat{p}}_1$	0.11
${\widehat{r}}_2$	9.2	${\widehat{p}}_2$	0.71
${\widehat{r}}_4$	4.7	${\widehat{p}}_4$	0.57
${\widehat{r}}_5$	$1.4·{10}^7$	${\widehat{p}}_5$	0.99999986
${\widehat{r}}_6$	273	${\widehat{p}}_6$	0.985

Table 11. Music data, estimated variances from the negative binomial model.

Variable number	1	2	3	4	5	6
Variance	127.4	12.3	3.6	5.9	9.0	11.3

Table 12. Music data, estimated correlations from the negative binomial model.

Variable number	1	2	3	4	5	6
variable 1	1	0.18	0	0	0.21	0.19
variable 2	$·$	1	0	0	0.68	0.60
variable 3	$·$	$·$	1	0	0	0
variable 4	$·$	$·$	$·$	1	0	0
variable 5	$·$	$·$	$·$	$·$	1	0.71
variable 6	$·$	$·$	$·$	$·$	$·$	1

Table 13. Music data, estimated principal components.

	x1	x2	x3	x4	x5	x6
z1	0.44	0.47	0.31	0.19	0.49	0.46
z2	–0.19	–0.040	–0.49	0.82	–0.0067	0.22
z3	0.33	–0.055	–0.78	–0.41	0.32	0.08

Table

x1	Energy factor (1, 2)
x2	Weight class (1–3)
x3	Damage category (1–4)
x4	Count of tubers in each combination of categories (integer)

Table 14. Potato data, means and variances.

Variable number	1	2	3	4
Mean	0.50	1.00	1.50	4.68
Variance	0.25	0.67	1.25	31.71

Table 15. Potato data, empirical correlations.

Variable number	1	2	3	4
variable 1	1	0	0	0.03
variable 2	$·$	1	0	0.04
variable 3	$·$	$·$	1	0.58
variable 4	$·$	$·$	$·$	1

Table 16. Potato data, estimated parameters, Poisson model.

$\widehat{\lambda }$	${\widehat{\mu }}_1$	${\widehat{\mu }}_2$	${\widehat{\mu }}_3$	${\widehat{\mu }}_4$
0.63	0.50	1.00	0.87	4.04

Table 17. Potato data, estimated parameters, negative binomial model.

${\widehat{r}}_0$	0.51	${\widehat{p}}_0$	0.61
${\widehat{r}}_3$	$7.2·{10}^6$	${\widehat{p}}_3$	0.99999983
${\widehat{r}}_4$	0.42	${\widehat{p}}_4$	0.09

Table 18. Potato data, estimated variances from the negative binomial model.

Variable number	1	2	3	4
Variance	0.25	0.67	1.75	49.97

Table 19. Potato data, estimated principal components.

	y1	y2	y3	y4
z1	0.00	0.01	0.12	0.99
z2	–0.02	–0.14	0.98	–0.12