International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2269

ISSN 2229-5518

Partial Mantel Analysis on Estimating the

Resembance of Students Performance

Aronu, C. O, Ebuh, G. U., Ogbogbo G. O., Bilesanmi, A. O.

Nnamdi Azikiwe University, Awka - Nigeria

Abstract

The partial Mantel analysis is a test statistic that is used to measure the resemblance between two distance matrices after controlling for the effect of the third distance matrix measured over the same objects. The method used in this study is; permute the objects in one of the vectors (or matrices). Association on student’s performance of three faculties (Physical Sciences, Biosciences and Engineering) in Nnamdi Azikiwe University, Awka-Nigeria, was used to illustrate the method, where interest is on estimating the resemblance between two faculties students’ performance while controlling for the effect of the third faculty student performance. From the result obtained in this study, we conclude that there exists a strong negative resemblance between the performance of students in faculty of Physical sciences and faculty of Biosciences while controlling for the effect of student performance of faculty of Engineering for 100,000 permutations and using the Canonical distance which is “method three” in the “dist.quant (distance quantity)” function. R 2.13.0 programming package was used to run the analysis for 100,000 permutations.

Key words: Faculty, Canonical distance, Matrices, Permutations, Vectors, Engineering, Biosciences

1INTRODUCTION

——————————  ——————————

ultivariate tables of observations are usually condensed into resemblance matrices among any sampling unit of interest computed using similar-
Considering proximity matrices A, B , and C com- puted for three univariate or multivariate data tables.
ity distance (also called dissimilarity). Suppose we wish to consider three matrices, [1] proposed an ex-
The partial Mantel statistic, rM

(AB⋅C ), estimating

tension of the Mantel test to carry out partial correla- tion analysis in population genetics. [2], showed how French financial elite friendship ties are correlated
with (dis) similarity on several attribute variables,
such as political preference, educational institute, and club membership. In this study we measured the re-
the correlation between matrices A and B while con-
trolling for the effect of C , is computed in the same
way as a partial correlation coefficient:
semblance of student performance in three faculties and on three courses of an institution where interest is

rM (AB.C )=

rM (AB)− rM (AC )rM (BC )

(1)
on ascertain the performance association level of the selected students on three courses. Other contributors on Mantel and partial Mantel test includes; [3]; [4]; [5];

1− rM

(AC )2 1− r

(BC )2

[6]; [7]; [8]; [9] and [10]. The R- programming pack- age was used to run the analysis because it has the

Permute the objects in Matrix Aas proposed by [11]

1. Compute the Mantel correlations measure
ability of running mantel and partial mantel statistic for large number of permutations.

rM (AB),

rM (AC )

and rM (BC ). Cal-

2 Material and methodology

2.1 Partial mantel statistic

A partial Mantel test is a first-order partial correlation
analysis conducted on three distance matrices [1].

culate the reference value of the of the test sta- tistic, rM (AB⋅C ), using Eq. 1.

2. Permute Aat random using matrix permutation algorithm to obtain A* .
http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2270

ISSN 2229-5518

3. Compute r

(A* B)

and r
 A*C  , using
39, 50, 53, 65, 37, 41, 38, 41, 57, 71, 70, 65, 66, 71,

M M  

83, 76, 62, 57, 81)
the value

rM (BC )

calculated in step 1,

R> MCB <-c(60, 84, 84, 85, 77, 69, 80, 80, 75, 87, 77,

78, 76, 62, 62, 59, 66, 67, 66, 59, 38, 53, 52, 34, 56,
compute rM

(A* B⋅C )using Eq.1 to obtain a

53, 33, 51, 34, 46)

R> BCH <-c(66, 66, 69, 60, 83, 89, 86, 71, 62, 80, 65,

value r*M of the partial correlation statistic un- der permutation.
4. Repeat step 2 and 3 a large number of times to
65, 75, 67, 66, 61, 73, 66, 62, 68, 32, 37, 47, 49, 49,
53, 34, 40, 36, 40)

R> ZOO <-c(56, 68, 89, 56, 86, 81, 57, 78, 89, 63, 64,

obtain the distribution of
* under permuta-

66, 67, 79, 68, 62, 60, 68, 78, 67, 51, 37, 31, 32, 43,
38, 55, 44, 42, 46)
tion. Add the reference value rM
to the distribution.

(A* B⋅C )

R> MECH <-c(47, 64, 59, 36, 31, 30, 56, 44, 24, 28,

58, 69, 80, 80, 72, 62, 55, 57, 77, 78, 64, 63, 85, 78,
83, 64, 83, 68, 68, 60 )
5. To determine the probability For a one – tailed test involving the upper tail, calculate the
probability as the proportion of values *

greater than or equal to rM . In the lower tail,
the probability is the proportion of

R> CIVIL <-c(21, 57, 63, 45, 22, 25, 43, 34, 35, 63,

71, 56, 86, 76, 76, 57, 54, 63, 57, 61, 75, 87, 63, 68,
89, 68, 61, 81, 72, 60)

R> ELECT <-c(40, 57, 64, 40, 32, 55, 67, 29, 46, 79,

81, 86, 80, 73, 56, 75, 74, 86, 84, 92, 63, 76, 64, 81,
81, 90, 86, 73, 65, 60)

R> FACULTYPHYSICALSCIENCES <-

ues *

smaller than or equal to rM .
matrix(c(STAT, MATHS, PHY), nrow = 3, byrow = TRUE)

R> FACULTYBIOSCIENCES <-matrix(c(MCB, BCH, ZOO), nrow = 3, byrow = TRUE)

2.2 Data presentation

The data for this study was presented as Appendix 1

3.0 Data analysis

Testing the hypothesis;

R> FACULTYENGINEERING <-matrix(c(MECH, CIVIL, ELECT), nrow = 3, byrow = TRUE)

It is important to note that the class distance of matrices FACULTYPHYSICALSCIENCES, FAC- ULTYBIOSCIENCES and FACULTYENGINEER-

H 1 +: Vs
H 2 +:

r (AB.C )= 0

r (AB.C )≠ 0

ING as defined above are based on canonical measure

(Method=1), labelled as FACULTYPHYSI- CALSCIENCESdist, FACULTYBIOSCIENCESdist

Inputting the data in Table 1on R 2.13.0 command
window, where STAT, MATHS and PHY are in FAC- ULTYPHYSICALSCIENCES matrix (matrix A), MCB, BCH and ZOO are in FACULTYBIOSCI- ENCES matrix (matrix B) while MECH, CIVIL and ELECT are in FACULTYENGINEERING matrix (matrix C) as given;

R>STAT <-c(78, 74, 68, 77, 78, 54, 75, 73, 56, 72, 61,

39, 55, 53, 50, 58, 48, 39, 64, 41, 79, 73, 67, 62, 71,
87, 70, 68, 69, 67)

R>MATHS <-c(53, 76, 69, 59, 78, 57, 76, 55, 57, 54,

66, 62, 39, 61, 38, 43, 65, 43, 55, 39, 72, 83, 77, 58,
57, 71, 80, 83, 81, 82 )

R> PHY <-c(66, 62, 69, 65, 78, 73, 70, 66, 66, 78, 67,

and FACULTYENGINEERINGdist respectively.

R> FACULTYPHYSICALSCIENCESdist <- dist.quant(FACULTYPHYSICALSCIENCES, method

= 1)

R> FACULTYBIOSCIENCESdist <- dist.quant(FACULTYBIOSCIENCES, method = 1)

R> FACULTYENGINEERINGdist <- dist.quant(FACULTYENGINEERING, method = 1)

Below is the elements of distance matrices FACUL- TYPHYSICALSCIENCESdist which contains objects
of matrix FACULTYPHYSICALSCIENCES on a class distances based on the canonical measure (meth-

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2271

ISSN 2229-5518

od =1). Where the result displayed by FACUL- TYPHYSICALSCIENCESdist expressed that the dis- tance between the performance of STAT and MATHS is 68.24222, STAT and PHY is 55.56078 and MATHS and PHY is 76.31514. R>FACULTYPHYSICALSCIENCESdist
STAT MATHS
MATHS 68.24222
PHY 55.56078 76.31514
Similarly, below is the elements of distance matrices FACULTYBIOSCIENCESdist which contains objects of matrix FACULTYBIOSCIENCES on a class dis- tances based on the canonical measure (method =1). Where the result displayed by FACULTYBIOSCI-
0)
Mantel statistic r: -1
Significance: 0.67327
Empirical upper confidence limits of r:
90% 95% 97.5% 99%
1 1 1 1
Based on 100 permutations, stratified within 0

4. Interpretation:

From the result obtained we observe that the partial
mantel measure of FACULTYPHYSICALSCIENC- ESdist, FACULTYBIOSCIENCESdist, while control- ling for the effect of FACULTYENGINEERINGdist
= -1 and a significance value = 0.67327 for 100,000
permutations. This expression can equally be ex-
ENCESdist expressed that the distance between the
performance of MCB and BCH is 56.92100, MCB and
pressed as given

r (AB.C )= −1

and 67.33% risk of
ZOO is 74.17547 and BCH and ZOO is 69.58448.

R> FACULTYBIOSCIENCESdist

MCB BCH
BCH 56.92100
ZOO 74.17547 69.58448
Similarly, below is the elements of distance matrices FACULTYENGINEERINGdist which contains ob- jects of matrix FACULTYENGINEERING on a class distances based on the canonical measure (method
=1). Where the result displayed by FACULTYENGI- NEERINGdist expressed that the distance between the performance of MECH and CIVIL is 75.51159, MECH and ELECT is 90.98351 and CIVIL and ELECT is 92.05433.

R> FACULTYENGINEERINGdist

MECH CIVIL CIVIL 75.51159
ELECT 90.98351 92.05433
The mantel.partial function was used to perform the
partial mantel test for 100,000 permutations, where “permutation” represents the number of permutations; R>mantel.partial(FACULTYPHYSICALSCIENCESdi
st, FACULTYBIOSCIENCESdist, FACULTYENGI-
NEERINGdist, method ="pearson", permutations =
100,000)
Partial Mantel statistic based on Pearson's product- moment correlation
Call:
mantel.partial(xdis = FACULTYPHYSI-
CALSCIENCESdist, ydis = FACULTYBIOSCIENC- ESdist, zdis = FACULTYENGINEERINGdist, method = "pearson", permutations = 100, strata =
rejecting the null hypothesis while it is true, which
fall’s on the acceptance region assuming α=0.05.
Where,
A=FACUTYPHYSICALSCIENCES, B=FACULTYBIOSCIENCESdist and C=FACULTYENGINEERINGdist.

5.0 Conclusion

From the interpretation we can conclude that there ex-
ists a strong negative resemblance between the per- formance of students in faculty of Physical science and faculty of Biosciences while controlling for the effect of performance of Faculty of Engineering for
100,000 permutations and using the canonical distance which is “method =1” in the “dist.quant” function. This implies that the class distance measures of the control which is Faculty of Engineering is far better than the measures of Faculty of Physical Sciences and Faculty of Biosciences as can be observed that in the Analysis section 3.0; hence the performance of the department in Faculty of Engineering is more associ- ated than that of other departments.

REFERENCES

[1] Manly, B. J. F. Randomization and Regression Methods for Testing for Associations with Geograph- ical, Environmental and Biological Distances between Populations. Res. Popul. Ecol., 28, 201 – 218; 1986.
[2] Kadushin, C. Friendships among the French Fi- nancial Elite. American Sociological Review, (1995),

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2272

ISSN 2229-5518

60(2), 201–221.
[3] Aronu, C. O, Ebuh, G. U. Application of Mantel’s Permutation Technique on Asphalt Production in Ni- geria. International Journal of Statistics and Applica- tions 2013, 3(3): 81–85.
[4] Anderson, M. J. and Robinson, J. Permutation
Tests for Linear Models. Aust. N. Z. J.Stat, (2001)
43(1) 75-88.
[5] Baker, F. & Hubert, L. “The Analysis of Social Interaction Data” Sociological Methods and Research, (1981); 9, 339–361.
[6] Jackson, D. A. and Somers, K. M. “Are Probability Estimates from the Permutation Model of Mantel's Test Stable?” Canadian Journal of Zoology, (1989);
67(766-769).
[7] Krackhardt, D. Predicting with Networks: Nonpar- ametric Multiple Regression Analysis of Dyadic Data. Soc. Networks, (1988) 10, 359 – 381.
[8] Krackhardt, D. and Kilduff, M. “Friendship Pat- terns and Culture: The Control of Organizational Di- versity”. American Anthropologist, (1990), 92, 142–
154.
[9] Legendre, P. Qualitative methods and biogeo- graphic analysis; Evolutionary Biogeography of the Marine Algae of the North Atlantic (eds Garbary DJ & South RG), Vol. G22, pp. 9 – 34; 1990. NATO ASI Series, Springer – Verlag, Berlin.
[10] Manly, B. J. F. Randomization, Bootstrap and

Monte Carlo Methods in Biology (Second Edition). London: Chapman and Hall, 1997.

[11] Legendre, P. Comparison of Permutation Methods for the Partial Correlation and Partial Mantel Tests. J. Statist. Comput. Simulation, (67), 37 – 73; 2000.

International Journal of Scientific & Engineering Research, Volume 4, Issue 7, July-2013 2273

ISSN 2229-5518

APPENDIX

Table 1: Presentation of students scores in three courses

	FACULTY PHYSICAL SCIENCES	FACULTY BIOSCI- ENCES	FACULTY ENGINEERING
COURSE/DEPARTMENTS	STAT	MATHS	PHY	MCB	BCH	ZOO	MECH	CIVIL	ELECT
GSS 101	78	53	66	60	66	56	47	21	40
	74	76	62	84	66	68	64	57	57
	68	69	69	84	69	89	59	63	64
	77	59	65	85	60	56	36	45	40
	78	78	78	77	83	86	31	22	32
	54	57	73	69	89	81	30	25	55
	75	76	70	80	86	57	56	43	67
	73	55	66	80	71	78	44	34	29
	56	57	66	75	62	89	24	35	46
	72	54	78	87	80	63	28	63	79
GSS 102	61	66	67	77	65	64	58	71	81
	39	62	39	78	65	66	69	56	86
	55	39	50	76	75	67	80	86	80
	53	61	53	62	67	79	80	76	73
	50	38	65	62	66	68	72	76	56
	58	43	37	59	61	62	62	57	75
	48	65	41	66	73	60	55	54	74
	39	43	38	67	66	68	57	63	86
	64	55	41	66	62	78	77	57	84
	41	39	57	59	68	67	78	61	92
MAT 102	79	72	71	38	32	51	64	75	63
	73	83	70	53	37	37	63	87	76
	67	77	65	52	47	31	85	63	64
	62	58	66	34	49	32	78	68	81
	71	57	71	56	49	43	83	89	81
	87	71	83	53	53	38	64	68	90
	70	80	76	33	34	55	83	61	86
	68	83	62	51	40	44	68	81	73
	69	81	57	34	36	42	68	72	65
	67	82	81	46	40	46	60	60	60

Source: Nnamdi Azikiwe University, Awka Departmental student records for 2012 session

Key: STAT= Statistics department students, MATHS= Mathemat- ics department students, PHY= Physics department student, MCB= Microbiology department students, BCH=Biochemistry department students, ZOO= Zoology department students, MECH= Mechanical engineering department students, CIVIL= Civil engineering department student, ELECT= Electrical engi- neering department students, GSS =General social studies and MAT= Mathematics.