STA 6167 – Extra Problems
P1. A study considered the effects of different hypnotic conditions on pain intensity in subjects. There were 4 conditions: 1=Hypnotic analgesia suggestion, 2=imaginative analgesia suggestion, 3=placebo control, 4=no-treatment control. The following table gives means, standard deviations and sample sizes for each condition (x=pre-trt, y=post-trt).
An analysis of covariance (with no interaction) is fit, where: E(Y) = X + Z1 + Z2 + Z3 where Zs are dummy variables for treatments 1,2, and 3. Estimates of model parameters are given below. Give the adjusted means for each treatment (the overall mean of pre-treatment scores is 13.27).
The Analysis of Variance is given below. Test for treatment effects, controlling for Pre-Treatment scores.
P2. A Poisson Regression model (with a log link) was fit , relating the number of Road Traffic Crashes (RTC) to the following factors: Cigarette smoker, waterpipe smoker, and driving duration per day. The estimated regression coefficients and standard errors are given below.
Factor / Estimate / Standard errorCigarette Smoker (1=Yes, 0=No) / 0.33 / 0.152
Waterpipe smoker (1=Yes, 0=No) / 0.49 / 0.201
Driving Duration per day (hours) / 0.15 / 0.064
Constant / -2.14 / 0.168
Give the Wald test for each factor, controlling for the other 2 factors.
Give the predicted number of crashes for 1) a cigarette and waterpipe smoker who drives 8 hours/day and 2) a non-smoker (of either type) who drives 2 hours/day.
P3. A study was conducted to determine what factors correlated with children carrying a concealed gun. The authors reported the following variables, odds ratios and confidence intervals based on a multiple logistic regression model based on 986 individuals.
Explanatory Variable / Odds ratio / 95% CIMale (vs Female) / 5.1 / (3.1,8.1)
7th Grade (vs 10th) / 2.1 / (1.5,3.0)
Lives w/ both parents / 0.8 / (0.6,1.2)
Family owns home / 0.8 / (0.6,1.1)
Smokes Cigarettes / 5.5 / (3.4,9.0)
>4 drinks/month / 1.8 / (1.2,2.7)
Average or better student / 1.7 / (1.2,2.3)
Avoids Fighting / 2.7 / (1.9,3.8)
Family member shot / 2.3 / (1.6,3.4)
Shootings in ‘hood / 2.9 / (2.1,4.1)
Discussed guns w/ parents / 1.5 / (1.0,2.2)
a)Put asterisks by all factors that are associated with children carrying concealed weapons at the =0.05 significance level (controlling for all other factors).
b)Describe a kid who is most likely to carry a concealed weapon (based only on significant factors).
P4. A logistic regression model is fit, relating whether a soldier died during the civil war (Y=1 if he died, 0 if not) to two independent variables: rank (X1=1 if private, 0 if of higher rank) and duty (X2=1 if infantry, 0 if not).
Consider the following models for log(:
log( / Fitted equation / -2 log L / -2.15 / 2868.6
X1 / -2.58+0.46X1 / 2862.9
X1+2X2 / -3.28+0.50X1+0.87X2 / 2812.3
a)Test whether the probability of death is associated with either rank or duty at the =0.05 significance level.
i)Null hypothesis:
ii)Alternative hypothesis:
iii)Test Statistic:
iv)Rejection Region/Conclusion:
v)
b)Give the fitted values (in terms of probabilities) for each of the following two categories (based on the full model):
i)Private/Infantry
ii)Nonprivate/Noninfantry:
P5. A nonlinear regression model is fit, relating available chlorine (Y, proportion) to the length of time since it was produced (X, in weeks). Based on data considerations, the model fit is of the form with output below:
q.5.a. Give the predicted proportion of chlorine remaining at each of the following times since production:
q.5.a.i. 10 weeks:
q.5.a.ii. 20 weeks:
q.5.a.iii. 30 weeks:
q.5.b. Compute a 95% Confidence Interval for
q.5.c. Give the estimated rate of change in chlorine remaining as a function of age (using point estimates of parameters in the derivative