Chapter 5, Problem 8TP

To determine

The correlation between the variables and formulate a matrix.

Answer to Problem 8TP

The correlation between XY, YZ and ZX are $r_{XY}=0.055719$, $r_{XZ}=-0.15432$ and $r_{YZ}=0.389126$ respectively. And correlation matrix is,

The matrix showing relationship between variables is,

	X	Y	Z
	1
Y	0.055719	1
Z	–0.15432	0.389126	1

Explanation of Solution

Given info:

The table below shows the value of three variables age of injury (X), level of treatment (Y), 12-month treatment score (Z),

Age at Injury	Level of Treatment	12-month Treatment score
X	Y	Z
25	1	78
16	2	66
8	2	78
23	3	89
31	4	87
19	4	90
15	4	98
31	5	76
21	1	56
26	1	72
24	5	84
25	5	87
36	4	69
45	4	87
16	4	88
23	1	92
31	2	97
53	2	69
11	3	79
33	2	69

Calculation:

The correlation between the variables X, Y and Z can be calculated with formulas given below,

${\text{r}}_{XY}=\frac{n\left({\displaystyle \sum XY}\right)-\left({\displaystyle \sum X}\right)\left({\displaystyle \sum Y}\right)}{\sqrt{\left[n{\displaystyle \sum X^2-{\left({\displaystyle \sum X}\right)}^2}\right]}\sqrt{\left[n{\displaystyle \sum Y^2-{\left({\displaystyle \sum Y}\right)}^2}\right]}}$ (1)

${\text{r}}_{XZ}=\frac{n\left({\displaystyle \sum XZ}\right)-\left({\displaystyle \sum X}\right)\left({\displaystyle \sum Z}\right)}{\sqrt{\left[n{\displaystyle \sum X^2-{\left({\displaystyle \sum X}\right)}^2}\right]}\sqrt{\left[n{\displaystyle \sum Z^2-{\left({\displaystyle \sum Z}\right)}^2}\right]}}$ (2)

${\text{r}}_{ZY}=\frac{n\left({\displaystyle \sum ZY}\right)-\left({\displaystyle \sum Z}\right)\left({\displaystyle \sum Y}\right)}{\sqrt{\left[n{\displaystyle \sum Z^2-{\left({\displaystyle \sum Z}\right)}^2}\right]}\sqrt{\left[n{\displaystyle \sum Y^2-{\left({\displaystyle \sum Y}\right)}^2}\right]}}$ (3)

Where,

• r is the coefficient correlation.
• $r_{XY}$ is the correlation between variable X and Y.
• $r_{XZ}$ is the correlation between variable X and Z.
• $r_{YZ}$ is the correlation between variable Y and Z.
• X is variable age of injury.
• Y is variable level of treatment.
• Z is variable 12-month treatment score.

Calculate the values of all the variables as shown in figure below,

	X	Y	Z	XY	YZ	ZX	X2	Y2	Z2
	25	1	78	25	78	1950	625	1	6084
	16	2	66	32	132	1056	256	4	4356
	8	2	78	16	156	624	64	4	6084
	23	3	89	69	267	2047	529	9	7921
	31	4	87	124	348	2697	961	16	7569
	19	4	90	76	360	1710	361	16	8100
	15	4	98	60	392	1470	225	16	9604
	31	5	76	155	380	2356	961	25	5776
	21	1	56	21	56	1176	441	1	3136
	26	1	72	26	72	1872	676	1	5184
	24	5	84	120	420	2016	576	25	7056
	25	5	87	125	435	2175	625	25	7569
	36	4	69	144	276	2484	1296	16	4761
	45	4	87	180	348	3915	2025	16	7569
	16	4	88	64	352	1408	256	16	7744
	23	1	92	23	92	2116	529	1	8464
	31	2	97	62	194	3007	961	4	9409
	53	2	69	106	138	3657	2809	4	4761
	11	3	79	33	237	869	121	9	6241
	33	2	69	66	138	2277	1089	4	4761
Total	512	59	1611	1527	4871	40882	15386	213	132149

Substitute 1527 for ${\displaystyle \sum XY}$, 512 for ${\displaystyle \sum X}$, 15386 for ${\displaystyle \sum X^2}$, 213 for ${\displaystyle \sum Y^2}$ and 59 for ${\left({\displaystyle \sum Y}\right)}^2$ in the equation 1 to get the correlation between X and Y.

$\begin{array}{c}{\text{r}}_{XY}=\frac{20\left(1527\right)-\left(512\right)\left(59\right)}{\sqrt{\left[20\left(15386\right)-{\left(512\right)}^2\right]}\sqrt{\left[20\left(213\right)-{\left(59\right)}^2\right]}}\\ =\frac{30540-30208}{\sqrt{307720-262144}\times \sqrt{4260-3481}}\\ =\frac{332}{213.4853\times 27.9105}\\ =0.055719\end{array}$

Thus, the correlation between X and Y is 0.055719.

$\begin{array}{c}{r}_{XZ}=\frac{20\left(40882\right)-\left(512\right)\left(1611\right)}{\sqrt{\left[20\left(15386\right)-{\left(512\right)}^2\right]}\sqrt{\left[20\left(132149\right)-{\left(1611\right)}^2\right]}}\\ =-0.15432\end{array}$

Thus the correlation between X and Z is -0.15432.

$\begin{array}{c}{\text{r}}_{ZY}=\frac{20\left(4871\right)-\left(1611\right)\left(59\right)}{\sqrt{\left[20\left(132149\right)-{\left(1611\right)}^2\right]}\sqrt{\left[20\left(213\right)-{\left(59\right)}^2\right]}}\\ =0.389126\end{array}$

Thus, the correlation between Z and Y is 0.389126.

Conclusion:

The correlation between variables age of injury and level of treatment and level of treatment and 12-month treatment score is positive, but the correlation between level of treatment and 12-month treatment score is negative. This indicates that with increase in the age of injury, the level of treatment done also increases. Whereas, with increase in level of treatment, the 12-months score of treatment do not increase.

The matrix of correlation can thus be formulated as given below:

	X	Y	Z
	1
Y	0.055719	1
Z	–0.15432	0.389126	1