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Researchers performed a clinical trial to compare two hypotensive drugs used to lower blood pressure during operations. The dependent variable, *y*, is the recovery time (in minutes) elapsing between the time at which the drug discontinued and the time at which the systolic blood pressure had returned to 100mmHg. The data used are from 53 patients relating to one of the two drugs used in the trial. The question of interest is the extent to which recovery time depends on the quantity of drug used (*x*1), controlling for the level to which blood pressure was lowered during hypotension (*x*2). Correlation Coefficients *yx*1*x*2*y*1.0000.335-0.108*x*10.3351.0000.469*x*2-0.1080.4691.000 *Y regressed on X1*VariableParameter EstimateStandard Errorp-valueIntercept-9.23312.7650.4728X116.0266.3170.0143 *Y regressed on X2*VariableParameter EstimateStandard Errorp-valueIntercept37.74719.5900.0596X2-0.2270.2930.4429 *Y regressed on X1 and X2*VariableParameter EstimateType I SSType II SSIntercept23.011X123.6391545.62623.4X2-0.7151237.71237.7

Does the correlation matrix suggest confounding among the independent variables? Explain.

- Verify your findings from part a. Is
*x*2 confounding the association between*x*1 and*y*? Explain.

Test for the association between *x*1 and *y*, controlling for *x*2. Write out the model you are using, the null hypothesis and conclusions. Interpret the parameter estimate for *x*1. (SSE = 11007.7)

The 53 patients were divided into three groups according to the type of operation. The groups are defined as: AMinor, non-thoracic, BMajor, non-thoracic, and CThoracic. The question of interest is differences between operation types in terms of the average recovery time, controlling for the quantity of drug used and the level to which blood pressure was lowered during hypotension. The mean quantity of drug used is 1.99 and the mean level to which blood pressure was lowered during hypotension is 66.34.*Unadjusted ANOVA Model*VariableParameter EstimateStandard Errorp-valueIntercept29.1544.451<.0001A-10.7545.7180.0658B-6.3545.7180.2718 *ANCOVA Model with Interaction*VariableParameter EstimateStandard ErrorType I SSIntercept6.46035.535 X132.00915.6281545.6X2-0.6540.3101237.7A14.44938.740214.2B21.87636.648207.1A*X1-10.68819.1600.4B*X1-13.20717.720126.3 *ANCOVA Model*VariableParameter EstimateStandard Errorp-valueIntercept27.99918.6540.1399X121.8827.0040.0030X2-0.6660.3040.0333A-7.3955.3940.1767B-5.1455.3090.3374

Perform a test to assess if the type of operation effect on recovery time depends on the quantity of drug used, controlling for the level to which blood pressure was lowered during hypotension. (SSE = 10459.7)

Assuming no interaction, perform a test to assess if there is a type of operation effect on recovery time, controlling for the quantity of drug used and the level to which blood pressure was lowered during hypotension.

Using the appropriate model, calculate the adjusted mean recovery times for each operation type.

Are *x*1 and *x*2 confounding the type of operation effect on recovery time?

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