STA 6167 – Exam 3 – Spring 2012 PRINT Name __________________________________________________________
Conduct all tests at  = 0.05 significance level.
Q.1. A study was conducted to relate incidence of West Nile virus in horses among the counties of South Carolina. The dependent
variable was the number of cases of West Nile virus in horses, the exposure variable was the number of farms (t), and the predictor
(independent) variables were PBR (# Bird Cases of West Nile / Human Population) and Human Density (HD), and the interaction
between PBR and HD. Three Poisson Regression models are fit:



Model 0: ln(/t) = ln() = ln(t) + 
Model 1: ln(/t) = PBR + HD ln() = ln(t) + PBR + HD
Model 2: ln(/t) = PBR + HD + PBR*HD ln() = ln(t) + PBR + HD + PBR*HD
Note: In the following tables, Residual Deviance(ModelA)-Residual Deviance(ModelB) = -2(logLA – logLB) where model B has more
parameters (predictor variables) than model A.
Parameter
Beta0
Beta1
Beta2
Beta3
Res.Dev.
Res. Df
Model0
Estimate StdErr
-5.92
0.14
#N/A
#N/A
#N/A
#N/A
#N/A
#N/A
109.32
45
Model1
Estimate StdErr
-7.18
0.35
7492
1094
0.00260 0.00127
#N/A
#N/A
66.32
43
Model2
Estimate StdErr
-6.98
0.37
5330
1596
0.00059 0.00167
26.00
12.38
62.01
42
p.1.a. Give the (estimated) predicted values for Dorchester County, where t = 314 farms, PBR = 0.00022, and HD = 168.
p.1.a.i. Model 0: ln() = ______________________________________________
___________________
p.1.a.ii. Model 1: ln() = ______________________________________________
___________________
p.1.a.iii. Model 2: ln() = ______________________________________________
___________________
p.1.b. Use the likelihood ratio test to test whether the main effects of PBR and/or HD are significant, by comparing model
1 and model 0: H0:  versus HA: and/or≠ 0
Test Statistic _____________________________________ Reject H0 if the test statistic is larger/smaller than ________
p.1.c. Use the Wald test to test whether PBR and HD are significant effects, controlling for the other predictor:
p.1.c.i. H0:PBR Not associated with West Nile Incidence in horses, controlling for HD:
Test Statistic ______________________________________ Reject H0 if the test statistic is larger/smaller than ________
p.1.c.ii. H0:HD Not associated with West Nile Incidence in horses, controlling for PBR:
Test Statistic ______________________________________ Reject H0 if the test statistic is larger/smaller than ________
p.1.d. Test whether the PBR/HD interaction is significant (controlling for main effects):
LR Test Stat ____________ Wald Test Stat ____________ Reject H0 if the test statistic is larger/smaller than _________
Q.2. A study based on a random sample of 50 horror movies, identified 465 characters, and classified them in terms of if
they were Female (F, 1=Yes,0=No), whether they had Sexual Activity (SA, 1=Yes,0=No), and whether they Lived (L)
through the film. The dependent variable was whether the character Lived through the film (1=Yes, 0=No). The following
sequence of models were fit with respect to logit(P(L=1)) = ln(P(L=1)/(1-P(L=1))):
Model 0: Model 1: F
Model 2: F + SA
Model 3: F + SA + F*SA
Variables in the Equation
Va riables in the Equa tion
St ep 0
Constant
B
-1. 549
S. E.
.119
W ald
168.165
df
1
Sig.
.000
Step
a
1
Ex p(B)
.213
Female
Constant
Step
a
1
Female
SexAct
Constant
S.E.
.246
.289
.190
Wald
8.411
7.087
80.311
S.E.
.242
.182
Wald
6.937
106.558
df
1
1
Sig.
.008
.000
Exp(B)
1.894
.154
a. Variable(s) entered on s tep 1: Female.
Variables in the Equation
Variables in the Equation
B
.713
-.770
-1.699
B
.639
-1.874
df
1
1
1
Sig.
.004
.008
.000
a. Variable(s) entered on s tep 1: SexAct.
Exp(B)
2.040
.463
.183
Step
a
1
Female
SexAct
FmSA
Constant
B
.808
-.510
-.428
-1.749
S.E.
.279
.447
.583
.205
Wald
8.408
1.300
.537
72.980
df
1
1
1
1
Sig.
.004
.254
.463
.000
Exp(B)
2.243
.601
.652
.174
a. Variable(s) entered on s tep 1: FmSA.
P.2.a. Based on the Wald tests, which statement best describes the results (mark the best answer, making use of the order of models):
p.2.a.i. Neither F, SA, or their interaction significant
p.2.a.iii. Only F is significant
p.2.a.ii. Interaction is significant, neither of the main effects are significant
p.2.a.iv. Only SA is significant
p.2.a.v. F and SA significant, their interaction is not.
P.2.b. Based on Model 2, give the predicted probabilities of L=1 for each of the following groups:
p.2.b.i. Female/Sexually Active:
p.2.b.ii. Male/Sexually Active:
p.2.b.iii. Female/Not Sexually Active:
p.2.b.iv. Male/Not Sexually Active:
p.2.c. Based on your results from p.2.b., show the interpretation of exp(B2)=0.463 for model 2
p.2.c.i. Males: __________________________________ / ________________________________ = 0.463
p.2.c.ii. Females: __________________________________ / ________________________________ = 0.463
Q.3. A nonlinear regression model is fit, relating Y (responsiveness, as a percentage of maximum possible
responsiveness) to X (Dose) for 19 individuals. The model fit is:


1

Y   0 1 
1
 1  X 
 

 2 








The estimates and standard errors are given below.
Parameter Estimates
Parameter
b0
b1
b2
Es timate
100.340
6.480
4.816
Std. Error
1.174
.194
.028
95% Confidence Interval
Lower Bound Upper Bound
97.851
102.829
6.068
6.892
4.756
4.875
p.3.a. Give the fitted values for the following X-values:
p.3.a.i. X=0
p.3.a.ii. X=1
p.3.a.iii. X=8
p.3.a.iv. X=9
p.3.b. At what dose does the curve reach halfway to its maximum responsiveness?
120
100
e vs Y-hat
80
3
2
Y
40
residual
Y-hat
60
1
0
-1
20
0
20
40
60
80
100
120
-2
0
-3
0
2
4
6
8
Y-hat
10
p.3.c. Based on the fitted curve and the residual plot, do you feel the model fits well?
Fantastic /
Pretty Good / Not so good.
Q.4. An Analysis of Covariance was conducted to compare the effects of an experimental treatment (use of interactive
vocabulary learning) in secondary school German students learning a second language. In the Experimental condition,
children made a video fashion show, while in the Control condition, the children completed worksheet assignments. The
author fit the following two models, where Y represents post-treatment score, X represents pre-treatment score, and Z1 is
1 if student was in the experimental group, and 0 if in control group. Students were randomly assigned to treatment
condition, with nE = 142 and nC = 130. The models fit are:
Model 1: E(Y) = X
Model 2: E(Y) = X + 1Z1
Model 1: Pre-test score only predictor: Y  8.88  0.70 X
SSR1  9145 SSE1  8006
Model 2: Pre-test score and Experimental Condition Dummy Variable (Z1=1 if Experimental, 0 if Control):
Y  7.17  0.73 X  2.54Z1
SSR2  9564 SSE2  7587
p.4.a. Test H0: No treatment effect, after controlling for baseline score (1 = 0)
HA: 1 ≠ 0
p.4.a.i. Test Statistic:
p.4.a.ii. Reject H0 if the test statistic is larger/smaller than ______________________________________
p.4.b. The following table gives the sample means for X and Y by treatment and overall. Compute the adjusted means for
the experimental and control groups.
Experimental
Control
Overall
X-Mean Y-Mean
11.83
18.37
15.26
18.34
13.47
18.36
Adjusted Mean:
Experimental __________________________ Control ___________________________________