A 2016 article in JAMA reports the results of a study of treatment outcomes for children with mild gastroenteritis who were given oral rehydration therapy. Enrolled children were randomized to received either rehydration with diluted apple juice (DAJ), or an electrolyte maintenance solution (EMS). As per the study authors:
“The primary outcome was a composite of treatment failure defined by any of the following occurring within 7 days of enrollment: intravenous rehydration, hospitalization, subsequent unscheduled physician encounter, protracted symptoms, crossover, and 3%or more weight loss or significant dehydration at in-person follow-up. Secondary outcomes included intravenous rehydration, hospitalization, and frequency of diarrhea and vomiting.”
Of the 323 children randomized to DAJ, 54 experienced treatment failure. (17 %). Of he 324 children randomized to EMS, 81 experienced treatment failure. (25 %)
1. For this study, what is the outcome of interest?
2. For this study what is the primary exposure of interest?
3. Estimate the risk difference (difference in proportions) of treatment failure for children in the DAJ group compared to children in the EMS group. (DAJ-EMS)
4. Interpret the estimate from item 3 in a sentence.
5. Estimate the relative risk (risk ratio) of treatment failure for children in the DAJ group compared to children in the EMS group.
6. Interpret the estimate from item 5 in a sentence.
7. Estimate the relative odds (odds ratio) of treatment failure for children in the DAJ group compared to children in the EMS group.
8. Interpret the estimate from part f in a sentence.
9. Do the estimated risk difference, relative risk and odds ratio agree in terms of the direction of association?
Question 2
A pilot study was designed to evaluate the potential efficacy of a program designed to reduce prison recidivism amongst inmates who have a documented long-term history of drug and/or alcohol problems. A sample of 11 prisoners was followed for up to 24 months after their most recent release from prison. Six of the inmates returned to prison at 3, 7 9, 11, 14 and 21 months respectively. Five of the inmates had not returned to prison as of the last time they were last contacted which was at 4, 8, 16, 24, and 24 months respectively.
Use the Kaplan Meier approach to estimate the survival curve for this set of inmates
(which tracks the proportion who have not yet returned to prison over time). It will be
helpful to construct a table like the ones appearing in lecture 5: however, all you will
need to report in the quiz generator are certain quantities from this table for specific
times.
10. What is the estimated proportion of the total sample who had not returned to prison by 7 months after enrolling in the study?
11. What is the estimated proportion of the total sample who had not returned to prison by 11 months after enrolling in the study?
12. What is the estimated proportion of persons who did not return to prison at 11 months among those who were still at risk of returning to prison at 11 months?
13. What is the estimated percentage of the original sample had not return to prison by 16 months?
14. Why does the Kaplan-Meier curve not reach 0% by the end of the follow-up period?
Question 3
In a July, 2010 article published in the New England Journal of Medicine[footnoteRef:1], researchers report the results of a randomized clinical trial to evaluate mortality differences in HIV infected subjects in Haiti. Subjects were randomized to receive early versus the current standards for implementation of Antiretriviral therapy. [1: Sever P, et al. Early versus Standard Antiretroviral Therapy for HIV-Infected Adults in Haiti. New England Journal of Medicine. (2010). Vol 363, No 3. ]
As per the abstract:
In summarizing the findings, the researchers present the following Kaplan-Meier curve
Statistical Reasoning in Public Health 1, 2016: Homework 2 10
15. Why do the curves for both groups start at 1 (100%) at time = 0 month?
16. What is the estimated proportion of persons surviving (remaining alive) beyond 36 months in the Early Retroviral Treatment sample?
17. What is the estimated proportion of persons surviving (remaining alive) beyond 36 months in the Standard Retroviral Treatment sample?
18. Based only on this graphic, what can you say about the estimated incidence rate ratio of mortality for standard treatment group compared to the early treatment group? (greater than, less than, or equal to 1). Why?
Question 4
In an August 2013 article published in American Journal of Public Health[footnoteRef:2], researchers report the results of a two-site (San Francisco and NYC) randomized trial: here is a description of the trial and the sample from the article abstract: [2: Masson C, et al. A Randomized Trial of a Hepatitis Care Coordination Model in Methadone Maintenance Treatment. American Journal of Public Health. 2013. Published online ahead of print August 15, 2013]
Objectives. We evaluated the efficacy of a hepatitis care coordination intervention
to improve linkage to hepatitis A virus (HAV) and hepatitis B virus
(HBV) vaccination and clinical evaluation of hepatitis C virus (HCV) infection
among methadone maintenance patients.
Methods. We conducted a randomized controlled trial of 489 participants
from methadone maintenance treatment programs in San Francisco, California,
and New York City from February 2008 through June 2011. We randomized
participants to a control arm (n = 245) and an intervention arm (n = 244), which
included on-site screening, motivational-enhanced education and counseling,
on-site vaccination, and case management services.
Of the 150 participants in the intervention group who needed the combined HAV—
HBV vaccine, 115 received the vaccine within 30 days of the vaccine being recommended. Of the 150 participants in the control group who needed the combined HAV-HBV vaccine,18 received the vaccine within 30 days of the vaccine being recommended.
19. In the above results presented, what is the outcome ?
20. In the above results presented, what is the exposure (predictor)?
21. Estimate the risk difference (difference in proportions) of getting the vaccine within 30 day of recommendation for the intervention group compared to the control group.
22. Interpret the estimate from item 21 in a sentence.
23. Estimate the relative risk (risk ratio) of getting the vaccine within 30 day of recommendation for the intervention group compared to the control group.
24. Interpret the estimate from item 23 in a sentence.
25. Estimate the relative odds (odds ratio) of getting the vaccine within 30 day of recommendation for the intervention group compared to the control group.
26. Interpret the estimate from part c in a sentence.
27. Do the estimated risk difference, relative risk and odds ratio agree in terms of the direction of association?
28. How do the estimated relative risk and estimated odds ratios compare in value?
29. Suppose you were to misinterpret the odds ratio as the relative risk. What would this do to the reported efficacy of the intervention program with regard to the vaccination outcome (under estimate or over estimate the efficacy)?
Homework 2, Part B: (the question in part a will be presented as fill in the blank/short answer questions in the Quiz Generator version):
Question 1
An October 25, 2012 article in the New England Journal of Medicine reports the results of a study examining aspirin and survival among patients with colorectal cancer. The following pieces of text are taken directly from the article abstract: (my edits are in italics)
“METHODS
We obtained data on 964 patients with rectal or colon cancer from the Nurses’
Health Study and the Health Professionals Follow-up Study, including data on aspirin
use after diagnosis and the presence or absence of PIK3CA mutation……
RESULTS
Among patients with mutated-PIK3CA colorectal cancers, regular use of aspirin after
diagnosis was associated with superior colorectal cancer–specific survival (adjusted relative risk for cancer-related death, 0.18; 95% confidence interval [CI], 0.06 to
0.61; P<0.001 by the log-rank test) and overall survival (adjusted relative risk for
death from any cause, 0.54; 95% CI, 0.31 to 0.94; P = 0.01 by the log-rank test). In
contrast, among patients with wild-type PIK3CA, regular use of aspirin after diagnosis
was not associated with colorectal cancer–specific survival (adjusted relative risk,
0.96; 95% CI, 0.69 to 1.32; P = 0.76 by the log-rank test) ) or overall survival (adjusted relative risk, 0.94; 95% CI, 0.75 to 1.17; P = 0.96 by the log-rank test)”
The authors present the following graphic as part of the article: (on the next page)
1. What is the outcome of interest for this study?
2. What is the primary predictor of interest?
3. What type of study design is this?
4. Describe the findings with regards to aspirin and survival in patients with colorectal cancer with respect to the presence or absence of the PIK3CA mutation.
5. Even though the authors estimated the association between aspirin and survival separately for the mutated-PIK3CA and wild-type PIK3CA, each of the two associations was adjusted for multiple factors including age, sex, year of diagnosis etc… Why was it potentially necessary to do this adjustment?
Question 2
A 2003 article in New England Journal of Medicine[footnoteRef:3] reports the results from a randomized trial comparing weight change and comorbidity development between severely obese subjects randomized to either receive a low carbohydrate diet or a low fat diet regimen. Both diet regimens lasted for six months. The results with regards to weight change are in the following table: [3: Samaha F, et al. A Low-Carbohydrate as Compared with a Low-Fat Diet in Severe Obesity. New England Journal of Medicine 2003;348:2074-81. ]
With regards to the weight change portion of these study:
6. What is the main outcome of interest for this study?
7. What is the main exposure of interest for this study?
8. Did the subjects in the low-carb diet group gain or lose weight?
9. Did the subjects in the low-fat diet group gain or lose weight?
10. In which diet group were the individual weight change values more variable?
11. Estimate and report the mean difference in weight change for the low-carb diet group compared to the low-fat diet group.
12. Interpret the estimate from item 11 in a sentence.
13. What is the estimated mean difference in weight change for comparing the low-fat diet group to the low-carb diet group? How does this estimate compare to the estimate from item 11?
14. Write a sentence describing the findings conveyed by the following with regards to weight-loss and diet group over the study follow-up period. (While this resembles a Kaplan-Meier curve, it is not)
Suppose the researchers had been able to randomize 300 severely obese
subjects into the two weight-loss groups , such that 154 received the low-carb diet, and 146 the low-fat diet. How should the following quantities compare in value (larger, smaller etc..) to the estimates from the actual study of 132 subjects. Explain your reason for each answer.
15. The standard deviations of the individual weight change values in each diet group
16. The mean difference in weight change for the low-carb group compared to the low fat group.
Question 3
A 2015 article in JAMA Psychiatry5 investigates factors associated suicide, including a history of self-harm via poisoning.
The primary outcome of interest was suicide. Subjects were followed until this outcome or censoring (still alive, death by other causes, lost to follow-up).
The following Kaplan-Meier curves show the time-to-outcome curves for the exposure (self-poisoning) and control groups. Time-zero was the discharge date for the self-poisoning subjects. (and the corresponding matched control)
17. Suppose the incidence rate ratio (IRR) of suicide is computed for the DS cohort compared to the Control group. How will this IRR compare to 1 (<1, >1. =1)?
18. There are 65,784 subjects who had a self-poisoning episode. However, at
10 years of follow-up, only 21 are at-risk of suicide. In other words, less that 0.1 %
is still at-risk at 10 years. However, the corresponding Kaplan-Meier curve estimate
at 10 years is approximately 98% (98% had not committed suicide). How is this
possible?
statistics
Using moving averages for n = 2 and n = 3 will produce what types of forecasts?
If forecasting using 33 months of historical data, how many months should you forecast if using a 95% Confidence Interval?
If forecasting using 33 months of historical data, how many months should you forecast if using exponential smoothing, SC=.3?
If forecasting using 33 months of historical data, how many months should you forecast if using a regression forecast?
In a single channel single server system, if the service rate is 4 per hour and the arrival rate is 6 per hours, what is the average time a unit spends waiting in the system (Wq)?
In a single channel single server system, if the service rate is 6 per hour and the arrival rate is 4 per hour, what is the average time a unit spends waiting in the system (Wq)
Statistics
The board of directors at a large corporation wants to base their division managers’ pay raises on the profit performance of their respective divisions. They have asked you to evaluate the performance and raises at other companies and propose a formula for calculating the percentage increase in base pay based on the percentage change in the division’s profit. You collected information from 50 divisions at similar companies and performed a linear regression on the percentage change in the division profits vs. the percentage change in the manager’s salary.
Use what you have learned about linear regression to answer the following questions. Click here to download the output from the Excel ToolPak, Regression Tool.
Response Parameters
What is the regression equation from the Summary Output? Is this a useful model? How do you know?
Are the assumptions of regression satisfied? How did you verify them?
Does change in division profit appear to be a good predictor for the manager’s pay raise? Why do you think that?
One of your company’s divisions had a –0.51 percent change in profits last year, while another had a 20 percent increase. What is the predicted percentage change in salary for these two division managers?
Statistics
Using one of the Data Tools and Apps from the U.S. Census Bureau, choose some interesting census data and perform a statistical analysis on it in order to answer a question you want to answer with the data. Do not just copy and paste or summarize the data that you find. Use an appropriate statistical tool and vide your thought process behind the tool you chose as well as the results of your statistical analysis. Use this week’s lecture to aid your analysis. Cite the data source that you chose in your post.
250-300 words.
MUST cite a source used for the statistical analysis
Statistics
Statistics
Prompt: A major tire manufacturer claims their heavy-duty truck tires have an average usage life of 71,000 miles. The shipping department of the company you work for has been using these tires for several years and feels they are not getting the mileage promised. The manager pulled 25 maintenance records and found an average tire life of 68,050 miles, with a standard deviation of 11,602 miles. He asks you to conduct a test of hypothesis to determine if the actual life of the tires is less than the manufacturer’s claim.
Response Parameters
Use what you have learned about hypothesis testing to answer the following questions.
What type of test should you perform? Which of the three equations for hypothesis testing should you use? Why did you choose that one? You may assume tire life is normally distributed.
State your null and alternate hypotheses. Why did you choose those values and mathematical operators?
What is the value of your test statistic? (Clearly show how you arrived at this value.)
Interpret the test statistic: Choose an appropriate confidence level, then evaluate the test statistic using either the critical value or the p-value approach. Why did you choose the confidence level that you did?
Clearly state the outcome of your test of hypothesis.
What does your outcome mean in statistical terms?
What does your outcome mean in terms of the problem?
Strict Deadline no extensions
Use the example format uploaded.
Statistics
The board of directors at a large corporation wants to base their division managers’ pay raises on the profit performance of their respective divisions. They have asked you to evaluate the performance and raises at other companies and propose a formula for calculating the percentage increase in base pay based on the percentage change in the division’s profit. You collected information from 50 divisions at similar companies and performed a linear regression on the percentage change in the division profits vs. the percentage change in the manager’s salary.
Use what you have learned about linear regression to answer the following questions. Click here to download the output from the Excel ToolPak, Regression Tool.
Response Parameters
What is the regression equation from the Summary Output? Is this a useful model? How do you know?
Are the assumptions of regression satisfied? How did you verify them?
Does change in division profit appear to be a good predictor for the manager’s pay raise? Why do you think that?
One of your company’s divisions had a –0.51 percent change in profits last year, while another had a 20 percent increase. What is the predicted percentage change in salary for these two division managers?
statistics
For each of the following questions, identify: whether it involves one-variable analysis or two-variable analysis; whether the variable(s) are discrete or continuous. What are the 4 most popular songs played on the radio last week? How does the amount of fertilizer affect plant growth? How did the students in Mr. Datahead’s class perform in the recent Data Management test? How does temperature affect the number of ice-cream cones sold?
statistics
0.1151
0.0362
0.8750
0.2158
- As the size of the sample increases, what happens to the shape of the distribution of sample means?
It cannot be predicted in advance.
It is negatively skewed.
It approaches a normal distribution.
It is positively skewed.
- What is the following table called?
| Ages | Number of Ages |
| 20 up to 30 | 16 |
| 30 up to 40 | 25 |
| 40 up to 50 | 51 |
| 50 up to 60 | 80 |
| 60 up to 70 | 20 |
| 70 up to 80 | 8 |
Frequency polygon
Frequency distribution
Histogram
Cumulative frequency distribution
- How is the t distribution similar to the standard z distribution?
Both are continuous distributions.
Both are skewed distributions.
Both are families of distributions.
Both are discrete distributions.
- In a distribution, the second quartile corresponds with the __________.
Median
Variance
Mean
Mode
- A random sample of 85 supervisors revealed that they worked an average of 6.5 years before being promoted. The population standard deviation was 1.7 years. Using the 0.95 degree of confidence, what is the confidence interval for the population mean?
6.99 and 7.99
6.14 and 6.86
4.15 and 7.15
6.49 and 7.49
x±z(S/Ön)
- Incomes of 50 loan applicants are obtained. Which level of measurement is income?
Ordinal
Nominal
Ratio
Interval
- Which of the following is an example of a continuous variable?
Zip codes of shoppers.
Number of students in a statistics class.
Tons of concrete to complete a parking garage.
Rankings of baseball teams in a league.
- For the normal distribution, the mean plus and minus two standard deviations will include about what percent of the observations?
99.7%
68%
50%
95%
- The main purpose of descriptive statistics is to:
Summarize data in a useful and informative manner.
Determine if the data adequately represents the population.
Make inferences about a population.
Gather or collect data.
- Which of the following is true regarding the normal distribution?
The points of the curve meet the x-axis at z = -3 and z = 3.
It is asymmetrical.
It has two modes.
The mean, median, and mode are all equal.
- What is the relationship among the mean, median, and mode in a symmetric distribution?
The mean is always the smallest value.
They are all equal.
The mean is always the largest value.
The mode is the largest value.
- The probability distribution for the number of automobiles lined up at a Lakeside Olds dealer at opening time (7:30 a.m.) for service is:
| Number | Probability |
| 1 | 0.05 |
| 2 | 0.30 |
| 3 | 0.40 |
| 4 | .25 |
On a typical day how many automobiles should Lakeside Olds expect to be lined up at opening time? Use expected value or mean of the probability distribution.
2.85
1.00
10.00
1.96
- Which of the following is a point estimate for the population mean (µ)?
X”
x/n
s
σ
- When all the items in a population have an equal chance of being selected for a sample the process is called _________________.
Simple random sampling
Z-score
Non-probability sampling
Sampling error
- A student was interested in the cigarette smoking habits of college students and collected data from an unbiased random sample of students. The data is summarized in the following table:
| Males who smoke | 20 |
| Males who do not smoke | 30 |
| Females who smoke | 25 |
| Females who do not smoke | 50 |
What type of chart best represents relative class frequencies?
Pie chart
Scatter plot
Frequency polygon
Box plot
- A listing of all possible outcomes of an experiment and their corresponding probabilities of occurrence is called a ____________.
Probability distribution
Random variable
Frequency distribution
Subjective probability
- The distribution of a sample of the outside diameters of PVC pipes approximates a symmetrical bell-shaped distribution. The arithmetic mean is 14.0 inches, and the standard deviation is 0.1 inches. About 68% of the outside diameters lie between what two amounts?
13.8 and 14.2 inches
13.5 and 14.5 inches
13.9 and 14.1 inches
13.0 and 15.0 inches
- For the past week a company’s common stock closed with the following prices: $61.5, $62, $61.25, $60.875, and $61.5. What was the price range?
$1.750
$1.875
$1.250
$1.125
- Judging from recent experience 5% of the computer keyboards produced by an automatic high-speed machine are defective. If six keyboards are randomly selected, what is the probability that none of the keyboards are defective? Use binomial distribution.
0.500
0.001
0.735
0.167
- The National Center for Health Statistics reported that of every 883 deaths in recent years, 24 resulted from an automobile accident, 182 from cancer, and 333 from heart disease. What is the probability that a particular death is due to an automobile accident?
539/883 or 0.610
182/883 or 0.206
24/333 or 0.072
24/883 or 0.027
- The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95% of the observations lie between what two values?
350 and 650
475 and 525
400 and 600
480 and 520
- Which of the following is a characteristic of the normal probability distribution?
It’s bell-shaped.
It’s asymmetrical.
It’s rectangular.
It’s positively skewed.
- Refer to the following breakdown of responses to a survey of “Are you concerned about being tracked while connected to the Internet?”
| Response | Frequency |
| Very concerned | 140 |
| Somewhat concerned | 40 |
| No concern | 20 |
What is the class with the greatest frequency?
Very concerned
No concern
None apply
Somewhat concerned
- The names of the positions in a corporation such as chief operating officer or controller are examples of what type of variable?
Interval
Ratio
Quantitative
Qualitative
- The members of each basketball team wear numbers on their jerseys. What scale of measurement are these numbers considered?
Interval
Nominal
Ratio
Ordinal
- A portion or part of a population is called a:
Random survey
Frequency distribution
Sample
Tally
- Mileage tests were conducted on a randomly selected sample of 100 newly developed automobile tires. The results showed that the mean tread life was 50,000 miles, with a standard deviation of 3,500 miles. What is the best estimate of the mean tread life in miles for the entire population of these tires?
3,500
500
35
50,000
- A large manufacturing firm tests job applicants. Test scores are normally distributed with a mean of 500 and a standard deviation of 50. Management is considering placing a new hire in an upper-level management position if the person scores in the upper sixth percent of the distribution. What is the lowest score a new hire must earn to qualify for a responsible position?
625
50
460
578
- The following graph is a:
Box plot
Contingency table
Dot plot
Stem-and-leaf display
statistics
a. What is the Total Initial Investment in this project? (3 points) $240,000 You are evaluating a project for The Ultimate recreational tennis racket, guaranteed to correct that wimpy backhand. You estimate the sales price of The Ultimate to be $400 and sales volume to be 1,000 units in year 1, 1,250 units in year 2, and 1,325 units in year 3. The project has a 3 year life. Variable costs amount to $225 per unit and fixed costs are $100,000 per year. The project requires an initial investment in equipment of $165,000 which is depreciated straight-line to zero over the 3 year project life. The actual market value of the equipment at the end of year 3 is $35,000. Initial net working capital investment is $75,000. The tax rate is 35% and the required return on the project is 10%.
Use this information to answer questions (a)-(c) below
a. What is the Total Initial Investment in this project? (3 points) b. What is the operating cash flow for the project in year 1? (6 points) c. What total amount of terminal value in year 3? (4 points)
d. Should you accept/reject the project is you use the NPV method? (11 points)