3. Say we have a continuous response variable of systolic blood pressure Y and c

Question

3. Say we have a continuous response variable of systolic blood pressure Y and categorical yes/no explanatory variables of alcohol drinking status X1/Drink (yes-1 or no-0) and tobacco smoking status X2/Smoke (yes-1 or no-0). In a regression setting of the population mean model, say you fit the model: E(Y) = 0 + 1 Drink, + 2Smoke, + 3Drink,Srnokei. would mean in context of this problem. Is this a meaningful a. Interpret what the estimate interpretation? b What does the 3 term signify in quantifying how Drink is associated with the response of blood pressure? c. Given estimates of A, A, 2, and A, interpret the effect of drinking on the response of blood pressure (will have to consider smokers and non smokers) d. Say you want to test if Smoke should be in the model in any way (main effect and interaction) Write down the null and alternative hypothesis for this test. What two models would you need to fit to be able to conduct this test?

Explanation / Answer

(a) Beta_0: If a person neither drinks nor smokes, what is the systolic blood pressure (SBP). This is meaningful because in real life it can be incentivizing to know the average SBP of a person who doesn't drink and smoke

(b) Beta_3: what is the change in or effect on (SBP) of drinking when done together with smoking

(c) Effect of drinking on SBP:-

Case 1: Smokers

Avg. SBP rises from Beta_0 + Beta2 to Beta_0 + Beta_1 + Beta2 + Beta3

=> Effect: Increase of Beta_1 + Beta3

Case 2: Non-Smokers

Avg. SBP rises from Beta_0 to Beta_0 + Beta_1

=> Effect: Increase of Beta_1

WE CAN SAY THAT FOR A SMOKER, THE SBP INCREASES MORE THAN THAT IN CASE OF A NON-SMOKER

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(d)

Null hypothesis = There's NO significant effect of Smoke on the SBP response

Alternative hypothesis = There's a significant effect of Smoke on the SBP response

NOTE: Reject null hypothesis if p-value << 0.05

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Two models we would need to fit to be able to conduct this test are:-

1. E(Yi) = Beta_0 + Beta_1*(Drink)i + Beta2*(Smoke)i <-- Signifies main effect of Smoke variable
2. E(Yi) = Beta_0 + Beta_1*(Drink)i + Beta2*(Smoke)i + Beta3*(Drink)i*(Smoke)i <-- Signifies interaction effect of Smoke variable and will show effect of smoking WITH drinking as opposed to that WITHOUT (Main effect can be overshadowed by the interaction effect)

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