Question

The data represent measures on a random sample of 25 individuals with high cholesterol levels.

The variables are as follows:

Dependent variable Y: Systolic Blood Pressure (SBP)

Independent variable X_1: Body Size, measured by Quetelet (QUET) Index = 100(weight/height2)

Independent variable X_2: Age

1. For an individual of age 51 and QUET 3.30, obtain by hand calculation the predicted level of SBP (remember they should be mean centered). If that individual’s actual SBP was found to be 130, obtain the residual.

2. Using R, obtain the squared semi-partial correlation of Age and SBP and interpret the value.

3. Using R, obtain the squared partial correlation of Age and SBP and interpret the value.

ID   SBP   Size   Age
1   135   2.876   45
2   122   3.251   41
3   130   3.100   49
4   148   3.768   52
5   146   2.979   54
6   129   2.790   47
7   162   3.668   60
8   160   3.612   48
9   144   2.368   44
10   180   4.637   64
11   166   3.877   59
12   138   4.032   51
13   152   4.116   64
14   138   3.673   56
15   140   3.562   54
16   134   2.998   50
17   145   3.360   49
18   142   3.024   46
19   135   3.171   57
20   142   3.401   56
21   150   3.628   56
22   144   3.751   58
23   137   3.296   53
24   132   3.210   50
25   149   3.301   54

Answer 1

Using R, carry out MLR analyses to obtain raw and standardized regression coefficients. Mean center age and QUET. Write the regression equation

use lm function in R to fit a linear regression of SBP on size and age

Rcode:

tab1 =read.table(header = TRUE, text ="
ID SBP Size Age
1 135 2.876 45
2 122 3.251 41
3 130 3.100 49
4 148 3.768 52
5 146 2.979 54
6 129 2.790 47
7 162 3.668 60
8 160 3.612 48
9 144 2.368 44
10 180 4.637 64
11 166 3.877 59
12 138 4.032 51
13 152 4.116 64
14 138 3.673 56
15 140 3.562 54
16 134 2.998 50
17 145 3.360 49
18 142 3.024 46
19 135 3.171 57
20 142 3.401 56
21 150 3.628 56
22 144 3.751 58
23 137 3.296 53
24 132 3.210 50
25 149 3.301 54

"
)
tab1

lmod <- lm(SBP ~ Size+Age   ,data=tab1)

summary(lmod)

coefficients(lmod)

Output:

Call:
lm(formula = SBP ~ Size + Age, data = tab1)

Residuals:
Min 1Q Median 3Q Max
-11.204 -7.779 -3.752 5.099 18.042

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 67.4398 17.6773 3.815 0.000946 ***
Size 9.5636 5.9759 1.600 0.123783
Age 0.8328 0.4854 1.716 0.100278
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.672 on 22 degrees of freedom
Multiple R-squared: 0.4805,   Adjusted R-squared: 0.4333
F-statistic: 10.18 on 2 and 22 DF, p-value: 0.0007432

coefficients(lmod)
(Intercept) Size Age
67.4398006 9.5635874 0.8328064

SBP= 67.4398006 +9.5635874*size+ 0.8328064 *age

Interpret the intercept Interpret the slope for QUET Interpret the slope for age Interpret the hypothesis tests for each regression coefficient

For intercept,coefficient is 67.4398006

For size and age equals zero, predicted SBP is 67.4398006 on an average

For size,coeffcient is 9.5635874

As size increases by 1 unit,SBP increases by 9.5635874 on an average holding age constant

For Age coeffcient is 0.8328064

As Age increases by one year,SBP increases by 0.8328064 on an average,holding size constant.

Report the coefficient of multiple determination obtained in your MLR analysis and interpret the associated hypothesis test

coefficient of multiple determination=R sq=0.4805

Ho:beta=beta2=0

Ha:atleast one of the beta is not =0

F-statistic: 10.18

p-value: 0.0007432
p<0.05

Reject Ho

Accept Ha

Conclude that size and age are jointly significant in predicting SBP

That is there is a relationship between SBP and size and age

or an individual of age 51 and QUET 3.30, obtain by hand calculation the predicted level of SBP (remember they should be mean centered

we have

SBP= 67.4398006 +9.5635874*size+ 0.8328064 *age

= 67.4398006 +9.5635874*3.30+ 0.8328064 *51

=141.4728

Residual=observed SBP-predcited SBP

=130-141.4728

=-11.4728