ST503 Statistics Assignment

1. A psychologist conducted a study to examine the nature of the relationship, if any, between a person’s emotional stability (X) and their ability to perform a certain task (Y). Emotional stability was measured by a written test for which the higher the score, the greater the emotional stability. Ability to perform the task (Y=1 if able, Y=0 if unable) was evaluated by the psychologist. The results for 27 subjects can be found at: File name: ability.txt
   1. Fit a logistic regression model to the data.
   2. What are the estimated parameters of the regression model (i.e. compute the values of b0 and b1)?
   3. Perform a significance tests to determine whether emotional stability has a significant effect on the probability that a person can perform the task.
   4. Calculate exp(b1) and interpret this number.
   5. Provide a 95% confidence interval for exp(b1).
   6. Write an equation for estimating the probability that a person will be able to perform the task as a function of their emotional stability score.
   7. What is the estimated probability that a subject with an emotional stability test score of 510 will be able to perform the task?
2. An equal-rights group claims that females are less likely to be hired by a particular firm than males with the same background and experience. Data was collected on years of education, years of experience and gender for 28 former applicants. In addition, it was recorded whether or not the applicant was hired. The data can be found at File name: discrimination.txt
   Column 1: Hiring Status (1 if hired/0 if not hired)
   Column 2: Education (years)
   Column 3: Experience (years)
   Column 4: Gender (1 if male/0 if female)
   1. Fit a multiple logistic regression model to the data to determine whether the variables education, experience and gender have a significant impact on the probability of being hired.
   2. What are the estimated parameters of the regression model? Give an interpretation of their values.
   3. Is there sufficient evidence to conclude that gender has a significant impact on the probability somebody is hired?
   4. Assuming gender and years of education are held constant, how do the odds of being hired change for each additional year of experience?
   5. Write an equation for estimating the probability that a person will be hired.
   6. What is the estimated probability that a female with 3 years of education and 1 year of prior work experience will be hired? Provide a 95% confidence interval for the estimate.
   7. What is the estimated probability that a male with 3 years of education and 1 year of prior work experience will be hired? Provide a 95% confidence interval for the estimate.
3. A local health clinic sent fliers to its clients to encourage everyone, but especially older persons at high risk of complications, to get a flu shot. In a pilot follow-up study, 50 clients were randomly selected and asked whether they actually received a flu shot. In addition, data were collected on their age (X1) and health awareness. The latter data were combined into a health awareness index (X2), for which higher values indicate greater awareness. A client who received a flu shot was coded Y = 1, and a client who did not receive a flu shot was coded Y = 0. The data are available on the file name flu.dat.
   X1 = Age
   X2 = Health awareness index
   X3 = Gender
   X3 = 1: Male
   X3 = 0: Female
   Y = Client received a flu shot or not
   Y=1: Received a flu shot
   Y=0: Did not receive a flu shot
   1. Find the maximum likelihood estimates of ?0, ?1, and ?2. State the fitted response function.
   2. Obtain exp(b1), exp(b2) and exp(b3) and interpret these numbers.
   3. What is the estimated probability that male clients aged 55 with a health awareness index of 60 will receive a flu shot?
   4. Interpret the following R code:
      logodds = predict(model, list(Age = c(35, 45),
      Health.Aware = c(50,50),gender=c(1,1))
      exp(logodds)/(1 + exp(logodds))
      ORpi.hat = predict(model, list(Age = c(35, 45),
      Health.Aware = c(50,50),gender=c(1,1)), type = "response") pi.hat
   5. Obtain CI for exp(b1), exp(b2) and exp(b3) and interpret these numbers
   6. Use the Wald test to determine whether X3, client gender, can be dropped from the regression model; use ? = .05. State the alternatives, decision rules, and conclusion. What is the approximate P-value of the test?
   7. Use the likelihood ratio test to determine whether X3, client gender, can be dropped from the regression model; use ? =0.05. State the full and reduced models, decision rule, and conclusion. What is the approximate P-value? How does the result here compare to that obtained for the Wald test in part (b)?
   8. Fit the following three models
      Model 1: B0 + B1*age + B2*sex
      Model 2: B0 + B1*age + B2*sex + B3*age2
      Model 3: B0 + B1*age + B2*sex + B3*age*sex
      • Test logit response is linear in age
      • Test there is no interaction between age and sex
   9. Recode age in dummy variables. Age ranges from 48 to 84 years, and 1 choose number of intervals equal to 5, so the intervals are [48, 53], (53, 61], (61, 61.5], (61.5, 68], (68, 84]. Then testing linearity in age becomes comparing model 4 and model 1 (with age as continuous variable). Notice here the first age interval is the reference level. Use the following R codes
      summary(age)
      Min. 1st Qu. Median Mean 3rd Qu. Max.
      48.0 53.0 61.0 61.5 68.0 84.0
      ageint=ifelse(age>=48 & (age<=53),
      0,ifelse(age>53
      & (age<=61),1,ifelse(age>61 &
      (age<=61.5),2,ifelse(age>61.5 &
      (age<=68),3,ifelse(age>68 &
      (age<=84),4,"none")))))
      x1=ifelse(ageint==1,1,0)
      x2=ifelse(ageint==2,1,0)
      x3=ifelse(ageint==3,1,0)
      x4=ifelse(ageint==4,1,0)
      model4=glm(shot~sex+x1+x2+x3+x4,family=binomial)
      mod=glm(shot~sex,family=binomial)
      anova(mod1,model4,test="Chisq")
   10. ROC curve of the logistic regression function. (Use the best model with all significant predictor variables)
4. For many people who smoke, the most natural thing to do in the midst of a stressful situation is to reach for a cigarette. Many smokers will explain that smoking helps them to relax and relieves their feeling of stress. Their adamant belief that this truly works has introduced the question of whether smoking does indeed relieve the amount of stress perceived by a smoker. The goal of this study was to investigate the relationship between smoking and the amount of recent life stress perceived, using some other variables such as age, and gender as independent (or explanatory) variables or covariates. Stress is a dependent response variable at three levels. Smoking can be viewed as either independent or independent categorical variable at three levels, and age (3 levels) and gender (2 levels) are independent variables (factors) as follows:Penn State researchers explored the relationship (if any) between four variables in the dataset, stress.txt:
   smoking status (1 = nonsmoker, 2 = quitter, 3 = smoker) in column 1,
   gender (1 = male, 2 = female) in column 2,
   age (1 = young, 2 = middle, 3 = old) in column 3, and
   perceived stress level (1 = mild / none, 2 = moderate / some, 3 = Severe / a
   lot) in Column 4

Stress is a "polytomous response" — having three values. In this case study, Stress treated as response.

Polytomous Logistic Regression Models with One Response

Model1: Stress on Gender (simplest polytomous model)

Model2: Stress on Age (simplest polytomous model)

Model3: Stress on smoking (simplest polytomous model)

Model4: Stress on Smoking, Age, and Gender

Interpret the parameters of each model.

Does the explanatory variable (factor) Age affect the responses or does not affect?

Does the explanatory variable (factor) Gender affect the responses or does not affect?

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