Assignment 5 F23

3 Problem 1: Regression models The purpose of this problem is to look at fitting regression models using the shiny app https://shiny.math.uwaterloo.ca/sas/stat231/regressionmodels/ The app explores regression models by generating two variates ‐ x and y ‐ according to an underlying relationship. In the first setting ('Random') there is no relationship between x and y. In the second setting ('Linear') there is a linear relationship between x and y. In the third setting ('Quadratic') there is a quadratic relationship between x and y. If you select Linear or Quadratic you can specify the true parameters that relate x and y in the models. The data are generated as follows. First, a sample of the specified size is generated for x from a Uniform distribution. The y values are then generated depending on the true relationship and the specified parameters. For example, if you choose 'Linear' for the true relationship, and set the y intercept to 1 and the coefficient of x to 2, then the y values are observations from a Gaussian distribution with mean equal to 1 + 2x and standard deviation equal to 𝝈 . The variance of the error term can also be controlled with a slider, and you can specify whether the error is homoscedastic (constant variance) or heteroscedastic (non‐constant variance). Once the true relationship is specified, you can then choose whether to fit a linear or quadratic model to the data, and view the resulting model statistics as well as look at various residual plots. Try experimenting with the different inputs and see what happens in the results! Several questions below ask you to include the plots generated by the Shiny app. You can do this by either right‐clicking the graph and selecting 'save as', or by taking a screenshot. Include only the plot in your screenshot. All written answers must be in full sentences. Please do not include any instructions in your assignment submission to Crowdmark or the LEARN Dropbox. (a) On the Shiny app select quadratic as the true relationship, 125 as the sample size, ‐2 as the y intercept, 0 as the coefficient of x, ‐1 as the coefficient of x 2 , 0.6 as the standard deviation, homoscedastic as the variability behaviour, and linear as the model to fit. Increase the sample size up and down from 125 and see what happens. (Note: you do not need to write anything for this, just observe what changes in the plots.) Now, set the sample size back to 125 and answer the following questions: (i) Hit the resample! button serval times until you view a scatterplot which you think clearly displays a quadratic relationship between the explanatory variate x and the response variate y. Insert the following 3 plots in your assignment: (1) the scatterplot, (2) the plot of the standardized residuals versus x, and (3) the qqplot of the standard residuals.

6 In Problems 2 and 3 you will continue to analyse variates in your data set. Make sure that you use the same data set that you generated, saved, and uploaded to the LEARN Dropbox as part of the Prerequisite Assignment. Your commented R code should only be included in the R file that you upload to the LEARN Dropbox. Do not include your R code in your answers submitted to Crowdmark. All written answers must be in full sentences. Please do not include any instructions in your assignment submission to Crowdmark or the LEARN Dropbox. Problem 2: Simple Linear Regression – Wrist circumference and Right foot length The purpose of this problem is to use R to fit a simple linear regression model. You are also asked to check the model, test the hypothesis of no relationship, construct confidence intervals for the slope and the mean response, and construct a prediction interval for a future response. See Sections 6.1 – 6.3 of the Course Notes. You may find it helpful to refer to the material posted on LEARN in the 'Assignment 5 R Tutorial' folder. All written answers must be in full sentences. In this problem you will examine the data for the variate Wrist.circumference (What is the circumference of your left wrist? Answer in centimetres to one decimal place.) and the variate Right.foot.length (What is the length of your right foot, without a shoe? Answer to the nearest centimetre.) for students aged greater than 14. The following R code stores the subset of observations used in this analysis in the vectors y (right foot length) and x (wrist circumference). # determine the complete cases for Age, Wrist circumference and Right foot length cc<‐complete.cases(dataset$Age,dataset$Wrist.circumference,dataset$Right.foot.length) # # Right foot length for students aged greater than 14 y<‐dataset$Right.foot.length[cc&dataset$Age>14] # Their corresponding Wrist circumference x<‐dataset$Wrist.circumference[cc&dataset$Age>14]

9 (c) Now assume that the variate Right.foot.length is the explanatory variate y and the variate Wrist.cicumference is the response variate x. In other words, assume the simple linear regression model 𝑋 ௜ ~ 𝐺ሺ 𝑎 ൅ 𝑏 𝑦 ௜ , 𝜎 ௑ ሻ , 𝑖 ൌ 1 ,2 , … , 𝑛 independently where the 𝑦 ௜ ′𝑠 are assumed to be known constants (i) Complete the following table based on the output from the R command summary(lm(x~y)). Values in the table may be rounded to 3 decimal places for convenience. maximum likelihood estimate of the parameter a maximum likelihood estimate of the parameter b equation of the fitted line: 𝑥 ൌ 𝑎 ො ൅ 𝑏 ෠ 𝑦 unbiased estimate of σ X (ii) For your data set are the estimates of σ and σ X the same? Explain why or why not. ( Hint: Remember what these parameters represent in their respective models.) (iii) Use R to create a plot which includes all of the following:  a scatterplot of your data (x i , y i ), i=1,2,…,n  the fitted line 𝑦 ൌ 𝛼 ො ൅ 𝛽 መ 𝑥  the fitted line 𝑥 ൌ 𝑎 ො ൅ 𝑏 ෠ 𝑦 (Note: To plot this line on the scatterplot you need to rearrange this equation into the form 𝑦 ൌ 𝑏 ൅ 𝑚𝑥 .)  the point ሺ 𝑥̅ , 𝑦 തሻ clearly labeled To receive full marks:  plots must have a titles  axes must be labeled  lines must be in a different colours and labeled  point of intersection must be clearly labeled  plot must occupy 1/3 to 1/2 of a page both vertically and horizontally (iv) You will notice that the two fitted lines in (iii) intersect at a point. What are the coordinates of this point? (Note: If they do not, then you have done something wrong and you will need to go back and carefully check your work.) (v) Suppose you are given a set of data (x i , y i ), i=1,2,…,n and you fit the two lines 𝑦 ൌ 𝛼 ො ൅ 𝛽 መ 𝑥 and 𝑥 ൌ 𝑎 ො ൅ 𝑏 ෠ 𝑦 . Show why these two fitted lines intersect at the point ሺ 𝑥̅ , 𝑦 തሻ . Note: Your proof must be for a general set of points, not just for your data set.

12 Note: Even if you conclude the Gaussian models do not fit your data well, continue to do the following analyses based on the Gaussian models. Note that this would not be done in a real world analysis! (d) Determine a 95% confidence interval for σ 1 and a 95% confidence interval for σ 2 . Based on these intervals, is it reasonable to assume that σ 1 = σ 2 ? Give reasons for your answers. (e) Explain what the null hypothesis H₀: μ 1 = μ 2 means in real‐world terms, that is, explain what this null hypothesis means in words that a non‐statistician would understand. (f) Use the R command t.test to test the hypothesis H₀: μ 1 = μ 2 . Regardless of your answer to (d) assume that σ = σ 1 = σ 2 to do this question . Be sure to include the pooled estimate of σ, the observed value of the test statistic 𝐷 ൌ |𝑌 ത ଵ െ𝑌 ത ଶ | 𝑆 ௣ ට 1 𝑛 ଵ ൅ 1 𝑛 ଶ the degrees of freedom associated with the test statistic, the p‐value, and the conclusion. Does your conclusion agree with what you observed for the side‐by‐side boxplots in (c)? (g) If it is not reasonable to assume that σ 1 = σ 2 then the asymptotic pivotal quantity 𝐷 ൌ |𝑌 ത ଵ െ𝑌 ത ଶ | ඨ 𝑆 ଵ ଶ 𝑛 ଵ ൅ 𝑆 ଶ ଶ 𝑛 ଶ which has an approximate G(0,1) distribution if n 1 and n 2 are large, could be used to test H₀: μ 1 = μ 2 . Use this test statistic to test H₀: μ 1 = μ 2 . Be sure to include the observed value of the test statistic, the p‐value and the conclusion. Is your conclusion the same as in (f)?