Question

	c1	c2	c3	c4	c5	c6	c7	p1	p2	p3	p4	p5
Zone1	9	81	100	1000	0.1	21	53	97	0.7	1001	27	303
Zone2	11	9	700	7000	0.7	23	65	12	0.1	1807	31	411
Zone3	7	4	605	6035	1.9	34	88	1	0	991	39	120
Zone4	16	81	357	1972	4.9	22	99	99	0	301	43	140
Zone5	8	77	87	3315	8.9	26	49	88	0.1	5119	55	199
Zone6	11	69	420	497	8.7	25	87	87	0.5	8007	27	613
Zone7	9	7	199	4414	7.6	24	66	3	3.2	5000	31	810
Zone8	7	10	148	3937	3.3	25	77	14	6.1	98	33	98
Zone9	6	18	152	6163	3.7	22	81	17	0	9001	18	120
Zone10	1	1	433	999	0.1	21	78	2	0.1	6330	63	140

Drive characteristics root and canonical correlation weight from this table. how many root of them is significant?

Answer 1

1. The raw data

There are 7 variables in the criteria set, and 5 in the predictors, values of these variables being observed for 10 areas

YEAR ONE									YEAR TWO
CRITERIA									PREDICTORS
EMPLOYMENT                 VARIABLES									EMPLOYMENT       VARIABLES
		1	2	3	4	5	6	7	1	2	3	4	5
AREAS	Zone1	9	81	100	1000	0.1	21	53	97	0.7	1001	27	303
	Zone2	11	9	700	7000	0.7	23	65	12	0.1	1807	31	411
	Zone3	7	4	605	6035	1.9	34	88	1	0	991	39	120
	Zone4	16	81	357	1972	4.9	22	99	99	0	301	43	140
	Zone5	8	77	87	3315	8.9	26	49	88	0.1	5119	55	199
	Zone6	11	69	420	497	8.7	25	87	87	0.5	8007	27	613
	Zone7	9	7	199	4414	7.6	24	66	3	3.2	5000	31	810
	Zone8	7	10	148	3937	3.3	25	77	14	6.1	98	33	98
	Zone9	6	18	152	6163	3.7	22	81	17	0	9001	18	120
	Zone10	1	1	433	999	0.1	21	78	2	0.1	6330	63	140

1. Calculate means and standard deviations

Mean	8.5	35.7	320.1	3533.2	3.99	24.3	74.3	42	1.08	3765.5	36.7	295.4
Standard Deviation	3.89444	35.96	217.281	2372.88	3.43461	3.83116	15.8958	44.1286	2.01539	3332.71	13.6955	244.311

1. Standardise the raw data

Standardised scores are required for the calculation of the canonical scores in step 6. (Most computer programs use standardized data for computing the correlation coefficients.

Standard scores, Z, are given by

Standardised Z scores
YEAR ONE									YEAR TWO
CRITERIA									PREDICTORS
EMPLOYMENT                 VARIABLES									EMPLOYMENT       VARIABLES
		1	2	3	4	5	6	7	1	2	3	4	5
AREAS	Zone1	6.82	80.01	98.53	998.51	-1.06	14.66	48.33	96.05	0.16	999.87	24.32	301.79
	Zone2	8.82	8.01	698.53	6998.51	-0.46	16.66	60.33	11.05	-0.44	1805.87	28.32	409.79
	Zone3	4.82	3.01	603.53	6033.51	0.74	27.66	83.33	0.05	-0.54	989.87	36.32	118.79
	Zone4	13.82	80.01	355.53	1970.51	3.74	15.66	94.33	98.05	-0.54	299.87	40.32	138.79
	Zone5	5.82	76.01	85.53	3313.51	7.74	19.66	44.33	87.05	-0.44	5117.87	52.32	197.79
	Zone6	8.82	68.01	418.53	495.51	7.54	18.66	82.33	86.05	-0.04	8005.87	24.32	611.79
	Zone7	6.82	6.01	197.53	4412.51	6.44	17.66	61.33	2.05	2.66	4998.87	28.32	808.79
	Zone8	4.82	9.01	146.53	3935.51	2.14	18.66	72.33	13.05	5.56	96.87	30.32	96.79
	Zone9	3.82	17.01	150.53	6161.51	2.54	15.66	76.33	16.05	-0.54	8999.87	15.32	118.79
	Zone10	-1.18	0.01	431.53	997.51	-1.06	14.66	73.33	1.05	-0.44	6328.87	60.32	138.79

1. Calculate and partition the correlation matrix

These relationships can be expressed by combining the two sets of data and by calculating product-moment correlation coefficients for each pair of variables. In our employment example, there were 7 criteria variables and 5 predictor variables so that the dimensions of the correlation matrix R are 12 x 12.

R11 = The matrix of intercorrelations among 7 criteria variables

R22 = The matrix of intercorrelations among 5 predictor variables

R12 = The matrix of intercorrelations among 7 criteria variables with the 5 predictor variables

R21 = Transpose of R12

I have used EXCEL> DATA > DATA Analysis > Correlation to calculate correlation

R11							R12
1	0.5828	0.10853	-0.039389	0.348	-0.048	0.2199	0.5922	-0.092	-0.379907	-0.326	0.279104
0.5828	1	-0.3814	-0.601244	0.39	-0.232	-0.142	0.9972	-0.31	-0.062693	0.0079	0.01485
0.1085	-0.381	1	0.29588	-0.348	0.3357	0.4172	-0.349	-0.385	-0.161804	0.1143	0.050929
-0.039	-0.601	0.29588	1	-0.158	0.3856	-0.04	-0.628	0.048	-0.107017	-0.329	-0.0866
0.3476	0.3895	-0.3482	-0.157827	1	0.1548	-0.011	0.3676	0.108	0.407298	-0.031	0.461749
-0.048	-0.232	0.33565	0.38558	0.155	1	0.1845	-0.25	0.015	-0.221449	0.057	-0.09938
0.2199	-0.142	0.41717	-0.040026	-0.011	0.1845	1	-0.112	-0.075	-0.005127	-0.066	-0.2163
0.5922	0.9972	-0.3486	-0.627746	0.368	-0.25	-0.112	1	-0.296	-0.077673	0.009	0.012367
-0.092	-0.31	-0.3845	0.047905	0.108	0.0153	-0.075	-0.296	1	-0.298424	-0.195	0.133835
-0.38	-0.063	-0.1618	-0.107017	0.407	-0.221	-0.005	-0.078	-0.298	1	-0.074	0.277928
-0.326	0.0079	0.11434	-0.328752	-0.031	0.057	-0.066	0.009	-0.195	-0.074312	1	-0.33107
0.2791	0.0149	0.05093	-0.086605	0.462	-0.099	-0.216	0.0124	0.134	0.277928	-0.331	1
R21							R22

1. Calculate the latent roots

To find the canonical roots, we first have to find the latent roots of the canonical equation:

a) Invert R22 and R11.

MATLAB command to find inverse

>>inv(r11)

3.652	-3.48	-1.2793	-2.10662	-0.85	0.933	-1.03
-3.48	5.247	1.3368	3.03531	0.255	-0.84	1.234
-1.28	1.337	2.1619	0.54131	0.879	-0.78	-0.26
-2.11	3.035	0.5413	3.07055	0.389	-1.01	0.982
-0.85	0.255	0.8794	0.38941	1.669	-0.69	0.016
0.933	-0.84	-0.7756	-1.0062	-0.69	1.671	-0.36
-1.03	1.234	-0.2562	0.98198	0.016	-0.36	1.614

>>inv(r22)

1.154	0.448	0.26349	0.054	-0.1297
0.448	1.37	0.53342	0.215	-0.2658
0.263	0.533	1.29971	0.061	-0.4157
0.054	0.215	0.06085	1.158	0.33687
-0.13	-0.27	-0.41573	0.337	1.26425

b) Multiply the sub-matrices

M	0.989	-0.11	0.0418	0.01571	0.067
	-0.03	0.721	0.1425	-0.04383	0.214
	-0.01	-0.06	0.9398	0.19715	-0.06
	0.009	0.03	0.2603	0.38713	0.314
	0.018	0.194	-0.1235	0.08511	0.821

c) Extract the roots

I have used MATLAB to calculate characteristic polynomial and roots of the 5X5 matrix, M

charpoly(m)

1.0000   -3.8578 5.7173 -4.0055    1.2899 -0.1441

The characteristic equation will be

Characteristic roots

eig(m)

   0.6286

   0.2292

   1.0000

   1.0000

   1.0000

The canonical roots are the square roots of those values so

R canonical 1 = 0.7928

R canonical 2 = 0.4787

R canonical 3 = 1

R canonical 4 = 1

R canonical 5 = 1

1. Calculate the canonical weights for vector B1

The weights B for the predictor variables are given by

BB = The cofactor of the matrix C

Weights are given by

Calculate the canonical weights for vector A1

and the weights A for the criteria variables by

I have written MATLAB code for above calculation

%r is the vector roots

%m is the 5X5 matrix mentioned above

%ir11 is the inverse of r11

AA = ir11*r12;
for i = 1:5
C = m - r(i,1)*eye(5,5);
BB = cof(C);
P = BB(1,:)*r22*BB(1,:)';
B(i,:) = 1/sqrt(P)*BB(1,:);
A11 = AA*B(i,:)';
A(i,:) = (1/sqrt(r(i,1))).*A11;
end

CANONICAL WEIGHTS

A1	B1	A2	B2	A3	B3	A4	B4	A5	B5
-0.46	-0.3918	-0.1211	0.05468	-0.1945	0.88358	-0.6473	-0.4517	1.68813	-0.1142
0.505	-0.9507	-0.2271	0.25338	1.36392	-0.2391	1.11855	-0.5746	-1.2544	-0.1194
1.401	-0.3045	-0.1517	-0.2299	0.11651	-0.0951	-0.0082	-0.2712	-0.3911	-1.0351
0.086	0.36918	-1.0005	0.82935	0.26551	-0.1222	1.08095	-0.527	-0.7479	-0.2023
0.39	0.63747	-0.5903	-0.252	-0.1855	-0.1625	-0.6998	-0.7348	-0.7059	0.4775
-0.26		0.73908		-0.031		0.01334		0.62828
-0.41		-0.1765		0.16558		0.62741		-0.6027