Question

Researchers studied 70 children periodically until age 18 on weights at ages 2, 9, and 18
(WT2, WT9, WT18 in kg) and BMI in age 18 (BMI1). Data is attached below.

a) Decide if there is collinearity in the data from scatter plots.

Use R to generate the ANOVA table and lm summaries for necessary models and answer the following question.

b)  What is SS(WT9|WT2), SS(WT18|WT9, WT2)

c)   Discuss the influence of collinearity on the parameters and the standard errors of WT2 in three models: regress with WT2 only, regress with WT2 and WT9, and regress with all three predictors.  

Please include R code

WT2	WT9	WT18	BMI18
13.6	32.5	56.9	22.53536
2	11.3	27.8	49.9
3	17	44.4	55.3
4	13.2	40.5	65.9
5	13.3	29.9	62.3
6	11.3	22.8	47.4
7	11.6	30	57.3
8	11.6	24.3	50
9	12.4	29.9	58.8
10	17	44.5	80.2
11	12.2	31.8	59.9
12	15	32.1	56.3
13	14.5	39.2	67.9
14	10.2	23.7	52.9
15	12.2	26	58.5
16	12.8	36.3	73.2
17	13.6	29.9	54.7
18	10.9	22.2	44.1
19	13.1	34.4	70.5
20	13.4	35.5	60.6
21	11.8	33	73.2
22	12.7	25.7	57.2
23	11.8	29.2	56.4
24	14.1	31.7	56.6
25	10.9	23.7	46.3
26	11.8	35.3	63.3
27	13.6	39	65.4
28	12.7	30.8	60.1
29	12.3	29.3	55
30	11.5	28	55.7
31	12.6	33	71.2
32	14.1	47.4	65.5
33	11.5	27.6	57.2
34	12	34.2	58.2
35	10.9	28.1	56
36	12.7	27.5	64.5
37	11.3	23.9	53
38	11.8	32.2	52.4
39	15.4	29.4	56.8
40	10.9	22	49.2
41	13.2	28.8	55.6
42	14.3	38.8	77.8
43	11.1	36	69.6
44	13.6	31.3	56.2
45	12.9	26.9	52.5
46	13.5	33.3	64.9
47	16.3	36.2	59.3
48	13.6	29.5	54.2
49	10.2	23.4	49.8
50	12.6	33.8	62.6
51	12.9	34.5	66.6
52	13.3	34.4	65.3
53	13.4	38.2	65.9
54	12.7	31.7	59
55	12.2	26.6	47.4
56	15.4	34.2	60.4
57	12.7	27.7	56.3
58	13.2	28.5	61.7
59	12.4	30.5	52.4
60	10.9	26.6	52.1
61	13.4	39	58.4
62	10.6	25	52.8
63	11.8	25.6	60.4
64	14.2	34.2	61
65	12.7	29.8	67.4
66	13.2	27.9	54.3
67	11.8	27	56.3
68	13.3	41.4	97.7
69	13.2	41.6	68.1
70	15.9	42.4	63.1

Answer 1

- To answer this question fir we need to know what is collinearity - It is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy.

Example - you went see a rock and roll band with two great guitar players. You want to know which plays better. But, they are both playing amazing leads at the same time! When they are both playing loud and fast, how can you tell which guitarist has the biggest effect on the sound? Even though they aren't playing the same notes, what they're doing is so similar it's difficult to tell one from the other. that's the problem with collinearity.

To check that we check VIF (variance inflation factor)

If the VIF is equal to 1 there is no collinearity among factors, but if the VIF is greater than 1, the predictors may be moderately correlated. A VIF between 5 and 10 indicates high correlation that may be problematic

Based on our data

a) Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	51.87	4.69	11.05	0.000
WT2	0.0497	0.0414	1.20	0.235	1.01
WT9	-3.125	0.429	-7.28	0.000	2.09
WT18	1.456	0.181	8.03	0.000	2.07

Since the VIF lies in the range of 1.01 to 2.09, we can say that collinearity is there but not to that extent that it may cause problem in data analysis

b) Regression Analysis: WT9 versus WT2

The regression equation is
WT9 = 13.56 - 0.01332 WT2

Analysis of Variance

Source	DF	SS	MS	F	P
Regression	1	4.942	4.94177	0.63	0.429
Error	68	529.944	7.79329
Total	69	534.886

As per the above data SS (WT9/WT2) is 4.942

Regression Analysis: WT18 versus WT9

The regression equation is
WT18 = 9.832 + 1.692 WT9

Source	DF	SS	MS	F	P
Regression	1	1530.94	1530.94	72.33	0.000
Error	68	1439.32	21.17
Total	69	2970.27

As per the above data SS (WT18/WT9) is 1530.94

Regression Analysis: WT18 versus WT2

The regression equation is
WT18 = 32.33 - 0.01000 WT2

Analysis of Variance

Source	DF	SS	MS	F	P
Regression	1	2.79	2.7882	0.06	0.801
Error	68	2967.48	43.6394
Total	69	2970.27

As per the above data SS (WT18/WT2) is 2.79

c) Standard error tells us the average distance that the observed values fall from the regression line, Conveniently, it tells you how wrong the regression model is on average using the units of the response variable. Smaller values are better because it indicates that the observations are closer to the fitted line, we can say that prediction is better for WT18 for WT2 as compared to WT9