Question

Data from n = 113 hospitals in the United States are used to assess factors related to the likelihood that a hospital patients acquires an infection while hospitalized. The variables here are y = infection risk, x₁ = average length of patient stay, x₂ = average patient age, x₃ = measure of how many x-rays are given in the hospital. The Minitab output is as follows:

Regression Analysis: InfctRsk versus Stay, Age, Xray

Analysis of Variance

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	3	73.099	24.366	20.70	0.000
Stay	1	31.684	31.684	26.92	0.000
Age	1	1.126	1.126	0.96	0.330
Xray	1	13.719	13.719	11.66	0.001
Error	109	128.281	1.177
Total	112	201.380

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
1.08484	36.30%	34.55%	30.64%

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	1.00	1.31	0.76	0.448
Stay	0.3082	0.0594	5.19	0.000	1.23
Age	-0.0230	0.0235	-0.98	0.330	1.05
Xray	0.01966	0.00576	3.41	0.001	1.18

Regression Equation

InfctRsk

=

1.00 + 0.3082 Stay - 0.0230 Age + 0.01966 Xray

1. Set up the hypothesis test to decide whether there is a connection between the infection risk and the group of predictors.
2. Then, decide, using tests, which of the predictors and constant should be in the final relationship.
3. Give the value of the coefficient of determination and tell what it means.

Answer 1

The regression model that is being estimated is

where is the intercept, are the slope coefficients for stay, Age, Xray and is a random error

1. Set up the hypothesis test to decide whether there is a connection between the infection risk and the group of predictors.

The hypotheses are

The test statistics for this test has F distribution and it is obtained from the ANOVA table

Source	DF	Adj SS	Adj MS	F-Value	P-Value
Regression	3	73.099	24.366	20.70	0.000
Stay	1	31.684	31.684	26.92	0.000
Age	1	1.126	1.126	0.96	0.330
Xray	1	13.719	13.719	11.66	0.001
Error	109	128.281	1.177
Total	112	201.380

The test statistics is

F=20.70

The p-value =0.000

We will reject the null hypothesis if the p-value is less than the significance level.

Here, the p-value is 0.000 and it is less than the significance level 0.05. Hence we reject the null hypothesis.

We conclude that, at 5% level of significance, there is a connection between the infection risk and the group of predictors.

• Then, decide, using tests, which of the predictors and constant should be in the final relationship.

We need to test for each of the slopes the following hypotheses.

Test if the variable Stay is significant in the relationship

The test statistics for this 2 tailed test is in the coefficient table.

Coefficients

Term	Coef	SE Coef	T-Value	P-Value	VIF
Constant	1.00	1.31	0.76	0.448
Stay	0.3082	0.0594	5.19	0.000	1.23
Age	-0.0230	0.0235	-0.98	0.330	1.05
Xray	0.01966	0.00576	3.41	0.001	1.18

The test statistics is t=5.19 and the p-value is 0.000

We will reject the null hypothesis if the p-value is less than the significance level.

Here, the p-value is 0.000 and it is less than the significance level 0.05. Hence we reject the null hypothesis.

We conclude that, at 5% level of significance, there is a connection between the infection risk and the predictor Stay.

Test if the variable Age is significant in the relationship

The test statistics for this 2 tailed test is in the coefficient table.

The test statistics is t=-0.98 and the p-value is 0.330

We will reject the null hypothesis if the p-value is less than the significance level.

Here, the p-value is 0.330 and it is not less than the significance level 0.05. Hence we do not reject the null hypothesis.

We conclude that, at 5% level of significance, there is no connection between the infection risk and the predictor Age.

Test if the variable Xray is significant in the relationship

The test statistics for this 2 tailed test is in the coefficient table.

The test statistics is t=3.41 and the p-value is 0.001

We will reject the null hypothesis if the p-value is less than the significance level.

Here, the p-value is 0.001 and it is less than the significance level 0.05. Hence we reject the null hypothesis.

We conclude that, at 5% level of significance, there is a connection between the infection risk and the predictor Xray.

We can decide that we need to drop Age from the model and retain Stay and Xray in the final relationship.

The new model that we need to estimate would be

Give the value of the coefficient of determination and tell what it means

The value of the coefficient of determination is

Model Summary

S	R-sq	R-sq(adj)	R-sq(pred)
1.08484	36.30%	34.55%	30.64%

Ans: The value of the coefficient of determination is 0.3630

This indicates that 36.30% of the variation in infection risk is explained by the model (or is explained by the predictor variables).