Reference no: EM132372428

Linear Models (LMR)

Assignment 1

Part A

Background: The data for this assignment was modified from the following article.

Article- Determination of iris thickness development in children using swept-source anterior-segment optical coherence tomography By Shunsuke Nakakura and Yuki Nagata.

It is not necessary to read this for the assignment or course, and the link is provided purely for your own interest.

This article uses a linear regression to understand the relationships between patient characteristics and the physical properties of the eye in a group of children. This assignment question uses modified data from the article and restricts the variables to the three below:

• pid: Participant id number

• axial: The axial length of the eye (mm) – the outcome for this assigment

• age: The age of the participant

• sex: Sex of the participant (0 = males, 1 = female)

The purpose of this assignment question is to investigate the relationship between the axial length and age, and see if sex confounds or modifies this relationship. Investigate this by answering the questions below:

Note: You do not need to investigate the assumptions of regression analysis for questions 1 – 3, as this will be done in question 4.

Question 1:

Ignoring sex for now, examine the evidence for an association between axial length (outcome) and age (predictor). Interpret the key results of this analysis.

Question 2:

Describe the conditions under which we would expect sex to confound the relationship in Question

1. Have these conditions been met in this data?

Now using regression methodology, examine how taking account of sex changes the relationship between axial length and age. Do you consider sex to be acting as a confounder here?

Question 3:

Use regression to test for interaction (effect modification) between age and sex on the outcome of axial length. Interpret each of the regression coefficients in this analysis.

Question 4:

Investigate the assumptions of the regression model you believe to be most informative to report on. This should include a clear indication of whether you believe the assumptions have been met, and justifications for these conclusions with reference to appropriate diagnostic figures. This should also include some discussion of any outliers, leverage points or influential points that might affect the conclusions, and your recommendations for handling any such data points.

Question 5:

Write a conclusion suitable for non-statistician researcher that summarises your findings and analysis. This researcher would be familiar with introductory statistics concepts such as P-values and confidence intervals but would be unfamiliar with the technical details of a regression analysis.

Part B

Background: The size of genomes across organisms vary enormously. For example, the human genome is approximately 3 billion nucleotides (nt) long (a nucleotide is one of the 4 letters that make up the genetic language), the genome of the Japanese flower Paris Japonica is approximately 150 billion nucleotides long, and the genome size of a nematode is about 100 million nucleotides long. In bacteria and viruses, the genomes are much smaller, but still vary in size considerably. For example, Influenza (the flu) is only 2400nt, whereas Pandoravirus is approximately 2.4 million nucleotides long. What drives this variation in genome sizes is an active area of research. The motivation for this assignment question comes from the article which does not need to be read for this assignment, and is included for your interest only.

Article- An Allometric Relationship between the Genome Length and Virion Volume of Viruses By Jie Cui, Timothy E. Schlub and Edward C. Holmes.

A study investigating this topic sampled the genome size of 82 independent viruses. For each of these viruses, the volume of the virus (in terms of the physical space it occupies) was measured. The data has the following columns

• glength: The genome length of the virus in nucleotides (nt)

• vvolume: The volume of the virus (nm3)

(Note: These data has been modified from the original research study)

The study would like to investigate whether there is a relationship between genome length and volume of a virus.

Question 1:

Examine the evidence for a relationship between genome length and virus volume where genome length (untransformed) is the outcome and virus volume (untransformed) is the predictor variable. Investigate the validity of the assumptions of this regression analysis. Now examine the evidence for this relationship, and the assumptions of this analysis when both variables are log transformed. Which analysis (log transformed or untransformed) do you prefer and why?

Question 2:

Provide a suitable interpretation of your preferred analysis in terms that you could explain to a nonstatistician who is familiar with introductory statistics concepts such as P-values and confidence intervals but would be unfamiliar with the technical details of a regression analysis.

Question 3

You are interesting in predicting the genome length of a virus with a volume of 32000nm³. Use your final model interpreted in question 2 to give the predicted mean genome length and its 95% confidence interval on the log-scale; then provide a 95% confidence interval on the original scale.

Part C

Background: This part of the assignment requires some more theoretical work based on fitting a linear regression model to investigate the effect of three dosage levels on an outcome. Suppose a clinical investigator is interested in examining the relationship between the effect of increasing doses of a Vitamin D supplement given to individuals who are Vitamin-D deficient. They perform a randomised trial in which they allocate (at random) volunteers to three groups, 1000 IU (International Units), 2000 IU and 3000 IU of supplement, per day for a period of three months, after which the serum levels of a key metabolite of Vitamin D called 25(OH)D are measured in each participant.

Question 1

One possible analysis of the data described is to estimate the linear effect of dose, i.e. to assume a linear relationship of expected outcome (labelled Y, as usual) to dose level, which for simplicity we will represent as X = 1,2,3 representing doses of 1000IU, 2000IU and 3000UI respectively. To estimate the average rate of change in Y with dose we would fit the simple linear regression model with the standard assumptions for the error term:

Y_i = β₀ + β₁X_i +  ∈_i

To objective is to show (algebraically) that if the sample size allocation between group1 1, group 2 and group 3 is 1:1:4 (i.e. n₁ = n, n₂ = n, n₃ = 4n), then the least squares estimate of β₁ is

β₁ = (4Y^‾₃-3Y^‾₁-Y^‾₂)/7

Where Y^‾_i  is the mean of the outcome in dose group i. For simplicity we will assume that
X_i= 1 for i = 1, … , n, X_i = 2 for i = n + 1, … ,2n and X_i = 3 for i = 2n + 1, … ,6n

i) Show that the overall means for X is X^‾ = 5/2

ii) Show that ∑( X_i− X)² = 7n/2

iii) Use these results to prove the formula above for β₁

Question 2

The dataset provided contains some simulated data that might have arisen from the study just described, with 15 participants in groups 1 and 2, and 60 participants in dose group 3. Fit the regression model discussed above and demonstrate that the result obtained for β₁ in question 1 is true in this sample.

Attachment:- Data File.rar