# MPH0822HW22022

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MPH 0822 Sooyun Lee MPH0822 - Applied Linear Models II Homework 2 The Bay State study data set consists of a sample of 200 subjects who were part of a much larger study on survival of patients following admission to an adult intensive care unit ( ICU ). The major goal of this study was to develop a logistic regression model to predict the probability of survival to hospital discharge of these patients . Explanatory variables were: AGE (continuous), GENDER (Female/Male * ), RACE (White * /Black/Other), SYS (systolic pressure at ICU admission [continuous]), LOC2 (level of consciousness at ICU admission [Full vs. Stupor/Coma * ]), TYPE (type of admission [elective/emergency * ]), and CAN (cancer presence [yes * /no]). The outcome of interest is a binary variable STA, representing vital status at time of discharge (0=Living, 1=Dead). Use this data to answer Problems 1 and 2. SAS Dataset name : hw1_problem1and2_data *For all analyses use reference cell coding for categorical variables . * indicates the level of variable to use as reference , if needed Problem 1. a. Write down the equation for the logistic regression model of STA with explanatory variables AGE, GENDER, RACE and SYS. log ( p ( STA = Lived ) 1 p ( STA = Lived ) ) = -0.8751 + 0.0276 * AGE + 0.0310 (Sex=Female) -0.8872 (Race=Black) + 0.5092 (Race=other) - 0.0171 * SYS b. Write down the expression for odds of death given the explanatory variables
MPH 0822 Sooyun Lee 1 unit increase in age is associated with 2.8% increase in odds of death in ICU. 1 unit increase in systolic blood pressure is associated with 1.7% decrease in odds of death in ICU. Female is 3.1% higher odds of death in ICU compared to Male. Black race is 58.8% lower in odds of death in ICU compared to white. Other race is 66.4% higher in odds of death in ICU compared to white. Odds of death= Pr(STA=1)/[1-Pr(STA=1)]= = p /(1-p) = = exp(α + β AGE AGE + β GENDER GENDER + β RACE RACE + β SYS SYS) c. Write down the expression for probability of death given the explanatory variables, i.e. Pr(STA="Dead" | AGE,GENDER,RACE,SYS). Pr(STA=1| AGE,GENDER,RACE,SYS) = = p = 1/[1 + exp{-(α + β AGE AGE + β GENDER GENDER + β RACE RACE + β SYS SYS)}] P (STA=Dead) = 1 1 + e {− ( 0.8751 + 0.0276 AGE + 0.0310 ( Sex = Female )− 0.8872 ( Race = ¿ )+ 0.5092 ( Race = other ) - 0.0171 SYS ) } Age: 1/(1+ 1.028)= 0.493 1 unit increase in age is associated with 49.3% percent chance of death Female: 1/(1+ 1.031) = 0.492 1 unit increase in age is associated with 49.2% percent chance of death Race Black: 1/(1+ 0.412) = 0.708 Being Black is associated with 70.8% percent chance of death Race Other: 1/(1+ 1.664) = 0.375 Being Other in race is associated with 37.5% percent chance of death. SYS: 1/(1+ 0.983) = 0.504 1 unit increase in SYS is associated with 50.4% percent chance of death.
MPH 0822 Sooyun Lee d. Fit a logistic regression to the model specified in (b). I don't understand what is "fit a logistic regression" exactly. But if this is fit the regression model with significant p value, then remove gender (p=0.9356), and race (black vs White p=0.4107, other vs white p=0.5485 ) . The fitted model will be log ( p ( STA = Lived ) 1 p ( STA = Lived ) ) = -0.9622 + 0.0284 * AGE -0.0168* SYS e. Using results from (d) and equation from (c), compute the probability of ICU death for an individual with AGE=45, SYS=200, RACE="Other" and GENDER="M". For this same individual, compute (by HAND) the odds of ICU death (using equation from (b)). Pr(STA=1| AGE=45,GENDER=M,RACE=Other, SYS=200) = 1/[1 + exp{-(-.8751 + .0276 × 45 + 0 + .5092 - .0171 × 200)}] = 0.073 Odds of death = exp(-.8751 + .0276 × 45 + 0 + .5092 - .0171 × 200)=0.079 (Alternate Calculation) Odds of death = Pr(STA=1)/[1-Pr(STA=1)] = .073/(1-.073) = 0.079 1/(1+exp ( -0.8751 + 0.0276 * 45 + 0.5092- 0.0171 * 200 )) = 0.927163 92.72% precent chance of ICU death f. Repeat the calculations in (e) for a 55 year old individual with identical values of the other explanatory variables. In other words, compute (by HAND) both probability of, and the odds of ICU death for an individual with AGE=55, SYS=200, RACE="Other" and GENDER="M". exp ( -0.8751 + 0.0276 * 55 + 0.5092 - 0.0171 * 200 )= 0.103529 89.64% less likely to be ICU death 1/(1+exp ( -0.8751 + 0.0276 * 55 + 0.5092 - 0.0171 * 200 )) = 0.906183 90.6% percent chance of ICU death By hand; 1/ (1 + 0.10) = 0.909 Pr(STA=1| AGE=55,GENDER=M,RACE=Other,SYS=200) = 1/[1 + exp{-(-.8751 + .0276 × 55 + 0 + .5092 - .0171 × 200)}] = 0.0938 Odds of death = exp(-.8751 + .0276 × 55 + 0 + .5092 - .0171 × 200)= 0.104 (Alternate Calculation) Odds of death = Pr(STA=1)/[1-Pr(STA=1)] = .0938/(1-.0938) = 0.104