1. In each situation, determine whether one-way analysis of variance (ANOVA) is
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Question
1. In each situation, determine whether one-way analysis of variance (ANOVA) is an appropriate method for analyzing the data described. Briefly explain why or why not for each part.[1]
a. A researcher compares mean serum cholesterol levels by ABO blood type. To do so, she samples 100 men of age over 50 in each blood type (A, B, or O) and measures the serum cholesterol of each man.
b. An oncologist examines four chemotherapy regimens for breast cancer. A randomized experiment is done in which 80 participants are randomly assigned to one of the 4 treatments to form equal size treatment groups. The response variable is five year survival – whether the participant is alive or not 5 years after treatment.
c. Three methods for memorizing information are compared. Participants (n = 45) are randomly divided into three groups, and each group memorizes information with a different method. All participants then take a test on the memorized information, and the scores (0-100) are used to compare the methods.
d. A Hospital wishes to assess patient satisfaction. Fifty individuals take a survey and rate their experience for each of three questions (quickly seen, courteous staff, questions sufficiently answered) on a scale of 0-10. The mean scores for each question are compared.
2. A researcher is studying treatments for back pain. 120 patients were randomly divided into four groups of 30 each. One group was assigned to placebo and the other three groups were assigned to one of the other treatments. After 5 weeks on treatment a measure of pain was evaluated (high score indicating a higher level of pain). Assume the data from the four groups are independent and the responses are approximately normal.
The researchers did an ANOVA test of the data and obtained the following results.
Source
DF
SS
MS
F
P-value
Groups
3727.8
0.0094
Error
310.87
Total
Fill in the missing pieces of the ANOVA table.
Assume the data from the four groups are independent and the responses are approximately normal. What is one additional assumption that we make for the ANOVA procedure to be valid?
What is the value of the estimate for the pooled variance within each group?
Write the null and alternative hypothesis that we are testing with the ANOVA procedure.
At a = 0.05, do we reject the null hypothesis?
What percent of the overall variation in the outcome is explained by group membership?
If we wish to make all pair-wise comparisons using a t-test, what is the new value we need to compare our p-value to for conclusions in order keep our overall type I error rate at 0.05 using the Boneferroni technique?
3. Recommendations regarding how long infants in developing countries should be breast fed are controversial. Introduction of other foods risks infection from contaminated water, possible loss of maternal-infant transmission of antibodies, or other factors. However, if maternal nutrition is grossly inefficient then breast fed infants may fail to thrive. One controversy concerns whether infants who have other foods introduced into the diet at different ages have different levels of energy intake. Part of one study compared energy intakes, measured in kilocalories per day (kcal/d) for infants who were breast-fed exclusively for 4, 5, or 6 months.
The data are available on the course Moodle page for this lab in the file infant.csv.
a. Identify the response variable and the populations to be compared.
b. Give the values for k, ni (for each i, 1ik), and for N where k is the number of populations studies, N is the total sample size and niis the sample size of each group 1ik.
c. Prepare summary statistics for each group. By informally examining the means for each group, does there appear to be a difference in energy intake across groups? Explain your reasoning.
d. From your summary statistics above, do you think that we need to be concerned that a possible lack of Normality in the data will invalidate the conclusions that we might draw using ANOVA to analyze the data? Explain your reasoning.
e. Is it reasonable to use a pooled standard deviation sp for these data? Explain your reasoning. If it is reasonable then compute sp from your summary statistics above.
f. State the null and alternative hypotheses for this situation in terms of the population parameters. Define your notation (what do the symbols represent?)
g. Carry out the one-way ANOVA for these hypotheses. Report your ANOVA table, including your F-statistic, the numerator and denominator degrees of freedom and the p-value (Final conclusions will be reported in the last question below).
h. What is the estimate of the between-groups variance? Show the work on how you obtained it from the output.
i. What is the estimate of the within-groups variance? Show the work on how you obtained it from the output.
j. Calculate spfrom the entries in the ANOVA table (it should obviously match your answer above).
k. Using the reported F-statistic, write code in your statistical software (i.e. using CDF or pf) to calculate the p-value. Show that it works by providing the output.
l. Is the use of post-hoc multiple comparison procedures advisable in this experiment? Explain briefly.
m. Based on the hypothesis test results from question 13, what do you conclude? Be sure to write your conclusion in statistical terms and in the context of the problem.
[1] Adapted from Utts, J.M., & Heckard, R.F. (2006). Statistical Ideas and Methods. 1st ed. Belmont: Thomson Brooks/Cole.
Source
DF
SS
MS
F
P-value
Groups
3727.8
0.0094
Error
310.87
Total
Explanation / Answer
a) ABO Blood groups are there. 3 groups in total. Each group has 50 members
The response variable, serum cholesterol level is a continuous variable
ANOVA can be performed on three groups to find out if there is a difference in the serum cholesterol levels between the 3 groups.
b) 80 participants are assigned to 4 groups. This is fine.
But the response variable is binary (0 / 1) and hence this will hinder the ANOVA process.
ANOVA cannot be performed here
c) 3 groups. 45 members in each group. Random sample.
Score the response variable is (0-100) is a continuous variable.
ANOVA can be performed to compare the scores between the 3 groups.
d) Response variable is patient satisfaction.
Same group is used, it is not random. And the group is answering 3 questions. What we can deduce is – if there is any difference in the scores of “quickly seen”, “courteous staff”, “questions answering”.
We wouldn’t be able to find any patient satisfaction using the difference in scores between the 3 parameters.
ANOVA will not give any good results.
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