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University of Toronto Mississauga

STA305 H5S: Experimental Design - Winter 2020

Final Project

Due Date: Friday April 3rd, 2020 at 12:00 midnight (end of the day). Late submission will not be

accepted. You need to submit an electronic copy as follows. If are currently enrolled in TUT0101,Otherwise if you are enrolled

in TUT0103 and TUT0104. Also, Submit your work as a single file. The name of the file should take

the form: LASTNAME FIRSTNAME STUDENT# SECTION#

Instructions:

❼ You can solve this project either individually or by a group of two. The solution that comes from a

group is expected to have a higher quality of work.

❼ Answer with quality, justify your work and with details.

❼ You should use R. Any other software will not be accepted.

❼ Start your R codes with the following information: Course Number, Final Project,

Question #, Your Last Name, First Name, Student Number. For Example,

# Course: STA305

# Final Project: # 1

# Last Name: ABCD, First Name: XYZW

# St. #: 0123456789

❼ You may not alter the output by typing. Output should be directly copied and pasted

to the project where needed. If you do not follow these rules, your project will not

be accepted.

❼ NO HANDWRITTEN! Your project should be word-processed and presently neatly.

Use the following format:

– Times New Roman or Arial size 12 font, single space.

– Attach the cover page (provided) at the front of your project. (Be sure to fill

in/circle all the information required). You are also allowed to design your own

cover page, which should contain same information.

– Include the question numbers/part letters (1a,b,c, etc.) in your answers.

– Do not simply hand in pages of R output without further explanation. Only

include the relevant tables or plots that are asked in each question. Make sure

to interpret the results in plain English in terms of the problem, quote relevant

numbers from the output, and give justifications as a part of your solutions.

– Do not include unnecessary code or output in the body of the project. At the

end, include an appendix with ALL your R code and output.

STA312- Instructor: Dr. Luai Al Labadi Page 1 of 5

Only general discussion is permitted between students. You

must hand in solutions in your own words. Do not let others

see your solutions or your selected article. It is plagiarism (a

serious academic offence) to submit solutions in other people’s

words (including but not limited to other students, the instructor’s,

solutions from previous years or courses, websites, etc).

You are responsible for knowing and adhering to the University

of Toronto’s Code of Behaviour on Academic Matters (see course

outline).

STA312- Instructor: Dr. Luai Al Labadi Page 2 of 5

Answer ALL of the following questions.

CASE STUDY I. Are the Fish Safe to Eat?

Assessing which Factors affect Mercury Levels in Fish in Maine Lakes.

Mercury is a heavy metal that occurs naturally in the environment through several forms (elemental,

organic, inorganic) and through human activity (burning of fossil fuels and incineration of waste). In

particular, large amounts of mercury are found in some fish.

Human consumption of mercury is known to lead to neurological and physical disorders. A study was

carried out in Maine to investigate which characteristics of lakes are associated with higher levels of

mercury in the fish in the lakes. Data were collected on 120 lakes. The data is attached. The variables in

the dataset are:

❼ Lake: name of the lake.

❼ Mercury: the mercury level in the fish in parts per million (ppm).

❼ Dam: whether a dam is present or not - an indicator variable which is 0 if there is no functional

dam present (so all water ow is natural) and is 1 if there is a man-made dam in the drainage area

of the lake.

❼ Type: each lake is classified as 1. ‘oligotrophic’ (sustains fish based on its vegetation and oxygen),

2. ‘eutrophic’ (has few fish), or 3. ‘mesotrophic’ (in between oligotrophic and eutrophic).

Some questions of interest are:

(A) Most states consider a mercury level of more than 0.5 ppm to be high enough to take action (issuing a

health advisory, clean-up methods, etc.). Are mercury levels in Maine high enough to be of concern?

(B) The industries that benefit from dams are concerned that environmentalists will claim that high

levels of mercury in fish are related to the presence of dams. Does the data support this claim?

(C) Do mercury levels vary by lake type? If so, specially how?

(D) Do mercury levels vary by lake type differently for lakes with dams than for lakes without dams

Answer any ALL of the following questions.

Let the significance level α = 0.1. For each of the following questions, include the necessary output/plots/numbers

to answer the questions and give a complete commentary on the results, written

in report style.

(a) It would be helpful to examine the data and do some preliminary analysis / descriptive statistics

before answering these questions. You can remove outliers and missing data before conducting the

analysis. Hint: one way to recognize outlier is through boxplot.

(b) Identify the following in terms of the above study. Be as specific as possible:

❼ Experimental unit(s)

❼ Population(s)

❼ Sample(s)

❼ Variables - identify which are response(s), and which are explanatory and classify each as

quantitative or categorical.

❼ Factor(s) and their levels. What are the treatments and how many are there?

STA312- Instructor: Dr. Luai Al Labadi Page 3 of 5

❼ Is this study experimental or observational? Justify.

❼ Is the design balanced or unbalanced? Justify.

❼ Are the factor crossed and nested? Is this design complete? In this design balanced? Justify

your answer in each case.

❼ Name at least one aspect of the design that you think may be problematic. Suggest an improvement.

(c) Answer Question of Interest (A). Name the statistical method(s) you are using and justify. Specify

the parameter or test of interest and define your notation. Make a conclusion in practical terms.

Include all the steps of the hypothesis test.

(d) Use a One-Way ANOVA to investigate Question of Interest (B). Include all the steps of the hypothesis

test. Conduct an appropriate post-hoc analysis to if appropriate. Name the method you used.

Give a practical conclusion.

(e) Use an appropriate method to answer Question of Interest (C) using a completely randomized design

(CR). Include all the steps of the hypothesis test.

(f) Use a Two-Way ANOVA to investigate Question of Interest (D). Include all the steps of the hypothesis

test.

(g) If you were interested in answering Question of Interest (C), which variable would be suitable to

use as a blocking factor? Justify by giving a practical reason and also by using numerical evidence

from the output. Conduct the analysis using a randomized complete block design (RCB). Estimate

the relative efficiency of the RCB compared to the CR design. Interpret the numerical value of the

relative efficiency in plain English and conclude if blocking was useful in this case.

CASE STUDY II. Vitamin May Affect the Weight Gain?

A scientist wants to see if the type of four vitamin supplements (A, B, C, D) affects the weight gain (y

in grams) of laboratory animals. The experiment was conducted in a completely randomized design with

five separately caged animals for each treatment. Since the caloric intake (x in calories/10) will differ

among animals and influence y, she wants to take this variable into account in the analysis. The data are

given in the following table:

Supplement y x Supplement y x Supplement y x Supplement y x

A 48 35 B 65 40 C 79 51 D 59 53

A 67 44 B 49 45 C 52 41 D 50 52

A 78 44 B 37 37 C 63 47 D 59 52

A 69 51 B 73 53 C 65 47 D 42 51

A 53 47 B 63 42 C 67 48 D 34 43

Let the significance level α = 0.05. For each of the following questions, include the necessary output/plots/numbers

to answer the questions and give a complete commentary on the results, written in

report style:

(a) Identify the following in terms of the above study. Be as specific as possible:

❼ Experimental unit(s)

❼ Population(s)

❼ Sample(s)

❼ Variables - identify which are response(s), and which are explanatory and classify each as

quantitative or categorical.

❼ Factor(s) and their levels. What are the treatments and how many are there?

❼ Is this study experimental or observational? Justify.

❼ Is the design balanced or unbalanced? Justify.

STA312- Instructor: Dr. Luai Al Labadi Page 4 of 5

❼ Are the factor crossed and nested? Is this design complete? In this design balanced? Justify

your answer in each case.

❼ Name at least one aspect of the design that you think may be problematic. Suggest an improvement.

(b) Write out a parallel planes model that you will use to test hypotheses in this experiment, using

Dummy Coding. In your answer, consider the vitamin supplement A as the reference group.

(c) Using the model above from (b), in terms of the betas (regression parameters), what null hypothesis

would you test to answer the following.

(i) Controlling caloric intake, all types of vitamin supplements results in the same mean on weight

gain.

(ii) There is no caloric intake effect on weight gain.

(iii) Is the average score of types A and B the same as the average score for types C and D,

controlling for caloric intake?

(d) Now write out the same model, but this time allowing for the effect of caloric intake on weight gain

to vary by types of vitamin supplements. Use Dummy Coding again.

(e) Using the model above from (d), in terms of the betas (regression parameters), what null hypothesis

would you test to answer the following.

(i) Equal slopes for all types of vitamin supplements.

(ii) Interaction between types of vitamin supplements and covariate.

(iii) Controlling for caloric intake, types of vitamin supplements do not have a significant effect on

weight gain.

(f) Are the assumptions of ANCOVA seems to be satisfied? Don’t forget to plot the regression line for

each diet.

(g) Use one-way ANOVA to test whether diet influence calorie intake. Include all steps of the hypothesis

testing.

(h) Use ANCOVA (parallel planes model) to test whether diet influence calorie intake. Include all steps

of the hypothesis testing. Compare the results with part (f).

(i) Use ANCOVA to test the significance of the covariate? Comment in the result.

(j) Use ANCOVA (non-parallel planes model) to test whether diet influence calorie intake. Comment

on your result.

Good Luck

STA312- Instructor: Dr. Luai Al Labadi Page 5 of 5

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