QUESTION
A sports doctor wants to investigate factors affecting pulse rate . He has conducted a small study of ten
young athletes and has collected the following data:
Pulse (Y): Number of beats per minute
Weight (X1): Weight in Kilograms
BP(X2): diastolic blood pressure
Exercise (X3): Number of minutes of exercise per day
Age (X4): Age, in years
Gender (X5): Gender = 0 for female, 1 for male
The following simple correlation matrix has been obtained using Minitab
Based on the simple correlation matrix, discuss the problem of multicollinearity in a model with Pulse as
the dependent variable and containing all of the independent variables. Specify two different pairs of
variables that should not be included in the model if one wishes to avoid multicollinearity.
The model would exhibit significant multicollinearity because several of the independent variables are
highly correlated, e.g.,
R(weight,gender) = 0.913
R(weight, exercise) = 0.812
R(age, BP) = -0.737
Therefore, weight and gender should not appear in the model together.
Also, Age and BP should not appear in the model together.
Other combinations are possible.
Consider the following regression models:
Regression Analysis: Pulse versus BP
The regression equation is
Pulse = 28.7 + 0.521 BP
Predictor Coef SE Coef T P
Constant 28.67 14.76 1.94 0.088
BP 0.5208 0.1766 2.95 0.018
S = 4.28145 R-Sq = 52.1% R-Sq(adj) = 46.1%
Analysis of Variance
Source DF SS MS F P
Regression 1 159.35 159.35 8.69 0.018
Residual Error 8 146.65 18.33
Total 9 306.00
Regression Analysis: Pulse versus BP, Weight
The regression equation is
Pulse = 22.9 + 0.723 BP – 0.150 Weight
Predictor Coef SE Coef T P
Constant 22.87 15.19 1.50 0.176
BP 0.7232 0.2421 2.99 0.020
Weight -0.1503 0.1264 -1.19 0.273
S = 4.17453 R-Sq = 60.1% R-Sq(adj) = 48.7%
Analysis of Variance
Source DF SS MS F P
Regression 2 184.01 92.01 5.28 0.040
Residual Error 7 121.99 17.43
Total 9 306.00
Perform a test of hypothesis at the 10% level of significance to determine if Weight should be added to the
model containing ‘BP’.
Do not add Weight because the t-value is t-value is – 1.19 with a p-value of 0.273 which is not significant.
The sports doctor now decides to investigate an alternative model by regressing Pulse on Age and Gender.
The model is shown below:
Regression Analysis: Pulse versus Age, Gender
The regression equation is
Pulse = 113 – 2.42 Age + 0.57 Gender
Predictor Coef SE Coef T P
Constant 112.92 12.51 9.03 0.000
Age -2.42 0.736 -3.29 0.013
Gender 0.57 2.634 0.22 0.835
S = 4.13897 R-Sq = 60.8% R-Sq(adj) = 49.6%
Analysis of Variance
Source DF SS MS F P
Regression 2 186.08 93.04 5.43 0.038
Residual Error 7 119.92 17.13
Total 9 306.00
Based on the above output, and assuming a 5% level of significance, should both variables be retained in the
model? Explain your answer.
For Age, the t-value is -2.42/0.736 = -3.288
For Gender, the t-value is 0.57/2.634 = 0.216
The critical value of t is ± t.025;7 = ± 2.365 In this model, Age is significant, Gender is not significant.
Therefore, retain Age but drop Gender.
The complete model with all five independent variables is as follows:
Regression Analysis: Pulse versus Weight, BP, Exercise, Age, Gender
The regression equation is
Pulse = – 109 + 1.27 Weight + 0.703 BP – 0.0954 Exercise + 3.62 Age – 37.5 Gender
Predictor Coef SE Coef T P
Constant -109.09 67.67 -1.61 0.182
Weight 1.27 0.47 2.72 0.053
BP 0.70 0.32 2.17 0.096
Exercise -0.10 0.08 -1.20 0.295
Age 3.62 1.86 1.95 0.123
Gender -37.52 12.25 -3.06 0.038
S = 2.76381 R-Sq = 90.0% R-Sq(adj) = 77.5%
Interpret the coefficients of Weight and Exercise in the above model.
Weight: A 1-kilogram increase in weight will cause pulse rate, on average, to increase by 1.27 assuming all
other variable are fixed.
Exercise: one extra minute of exercise daily will cause pulse rate, on average, to decrease by 0.0954,
assuming all other variables are fixed
Based on this output, write up a brief recommendation advising young athletes regarding what actions they
can take to reduce their pulse rate. Be specific in interpreting the model.
Athletes cannot change their age or gender but they can control the factors weight and exercise.
Therefore they should lose weight and exercise more
Since BP is highly correlated with weight (r = 0.703) it may not be possible to directly lower BP, but losing
weight will likely result in a lower BP.
Construct a 95% confidence interval for the change in pulse rate (Pulse) as young athletes get older.
Interpret the interval.
4
4 4 .025;4
4
3.62 2.776(1.86) 3.62 5.16
1.54 8.78
b b t s
Interpretation: Since the interval contains zero we cannot reject H0: β4 = 0.
An extra year of age may decrease pulse rate by as much as 1.54 beats per minute or increase the rate by up
to 8.78 beats per minute.
