BSTA 512/612
This homework is ready! 2/6/25
You will need to download the datasets. Use this link to download the HW3 datasets needed in this assignment. If you do not want to make changes to the paths set in this document, then make sure the files are stored in a folder named “data” that is housed in the same location as your HW3 .qmd file.
Please upload your homework to Sakai. Upload both your .qmd code file and the rendered .html file
For each question, make sure to include all code and resulting output in the html file to support your answers
Show the work of your calculations using R code within a code chunk. Make sure that both your code and output are visible in the rendered html file. This is the default setting.
Write all answers in complete sentences as if communicating the results to a collaborator.
It is a good idea to try rendering your document from time to time as you go along! Note that rendering automatically saves your .qmd file and rendering frequently helps you catch your errors more quickly.
Please complete the midterm feedback along with this homework!!
This question and data are adapted from this textbook.
In an experiment designed to describe the dose–response curve for vitamin K, individual rats were depleted of their vitamin K reserves and then fed dried liver for 4 days at different dosage levels. The response of each rat was measured as the concentration of a clotting agent needed to clot a sample of its blood in 3 minutes. The results of the experiment on 12 rats are given in the following table; values are expressed in common logarithms for both dose and response.
clot = read_excel(here("./data/CH05Q09.xls"))
clot %>% gt() %>%
cols_label(RAT = md("**Rat**"),
LOGCONC = md("**Log10 Concentration (Y)**"),
LOGDOSE = md("**Log10 Dose (X)**"))| Rat | Log10 Concentration (Y) | Log10 Dose (X) |
|---|---|---|
| 1 | 2.65 | 0.18 |
| 2 | 2.25 | 0.33 |
| 3 | 2.26 | 0.42 |
| 4 | 1.95 | 0.54 |
| 5 | 1.72 | 0.65 |
| 6 | 1.60 | 0.75 |
| 7 | 1.55 | 0.83 |
| 8 | 1.32 | 0.92 |
| 9 | 1.13 | 1.01 |
| 10 | 1.07 | 1.04 |
| 11 | 0.95 | 1.09 |
| 12 | 0.88 | 1.15 |
Use the log-transformed values as given in the dataset.
Use the following scatterplot to build your answers off of:
Fit a linear regression model to the data and add the regression line to the plot.
Use R to create the ANOVA table for the regression described in the exercise.
Using the F-test, determine whether there is an association between the log10 concentration and log10 dose.
Make sure to include all needed steps for an F-test. Calculating the F test statistic (step 5) is not needed if you use the ANOVA table. Make sure your conclusion connects back the research context.
Rewrite your hypothesis test in Part c to show the null and alternative models that we are testing. Did we reject the smaller (reduced) model?
You do not need to go through the hypothesis test process again. A quick statement on rejecting or not is okay.
If you prefer to write out the models by hand, remember that you can take a picture of your work and insert it into this document. HW0 can be a good reference for how we’ve done this before.
We will continue to work with the study and dataset from Question 2 above.
Find the correlation coefficient between the two variables. Is the value consistent with your description of the relationship in Question 2? Why or why not?
Calculate the coefficient of determination using linear regression summary output. Can we also calculate the coefficient of determination from the ANOVA in Question 2?
Give an interpretation of the coefficient of determination in the context of the study.
A high respiratory rate is a potential diagnostic indicator of respiratory infection in children. To judge whether a respiratory rate is “high” however, a physician must have a clear picture of the distribution of normal rates. To this end, Italian researchers measured the respiratory rates (in breaths/minute) of 618 children between the ages of 15 days and 3 years (measured in months).
The data and problem framing came from the Sleuth3 package. Please make sure to run the following code to load the data. You can directly access the dataset ex0824 from the package. I have included a new assignment of the data to q1_data if you would like to use that.
if(!require(Sleuth3)) { install.packages("Sleuth3"); library(Sleuth3) }
q1_data = ex0824 Create a scatterplot of the dependent and independent variables with both the best-fit line and a smoothed curve through the points. Describe the relationship between the dependent and independent variables, and also comment on whether you think it is reasonable to use a linear regression to model the relationship.
Write out the population regression model for the simple linear regression model. Please leave the variables untransformed for now.
Fit the regression model, display the regression table, and write out the fitted regression line.
Assess the normality of the model’s fitted residuals by creating a histogram, density plot, and boxplot of the residuals to visually inspect the distribution of the residuals, and describe any deviations from normality.
Assess the normality of the model’s fitted residuals by creating QQ plot of the residuals.
Bonus work, but not required: Compare the QQ plot to 4 such plots simulated from normal data, and discuss why or why not the residuals could have come from a normal distribution.
Test the normality of the model’s fitted residuals and comment on whether the test’s conclusion is consistent with your visual inspection or not. Make sure to include the hypotheses, needed R code, and a conclusion to the test based on the p-value (as shown in these slides).
Create a residual plot using ggplot and the residuals. Discuss what this means in the context of our model assumptions.
Determine whether there are any observations with high leverage. Please use the cutoff, \(h_i > 6/n\). If there are observations with high leverage, print the observations and state how many high leverage points there are.
Print the 10 observations with highest Cook’s distance. If there are observations with high Cook’s distance (\(d_i >1\)), state how many observations have high Cook’s distance.
Create a histogram for rate. Describe its distribution shapes.
Using the above histogram, and Tukey’s ladder of transformations, discuss the range of transformations that will be appropriate for Rate. Explain your reasoning.
Then use gladder() to decide on two possible transformations. Explain your reasoning.
Note: questions below will ask about model fit with the transformations. For now, just explain why you chose the two that you did.
Add the two rate transformations you chose above to the dataset. You do not need to print any output, just make sure the code is visible.
Create scatterplots using two transformed rates and age. Discuss if either transformation potentially improves the model fit. Explain why or why not. Note: including lines will help!
Using one of the transoformed outcomes, fit the regression model, display the regression table, and write out the fitted regression line.
Assess the normality of the model’s fitted residuals by creating QQ plot of the residuals. Does the transformation improve the QQ plot?
Create a residual plot using ggplot and the residuals. Discuss what this means in the context of our model assumptions. Does the transformation improve our model assumptions?
Between the model with the untransformed outcome and the transformed outcome, which would you recommend using for analysis? (Hint: there are pros and cons to both models)
This question uses the same dataset as HW 2, question 1.
This question is based on data collected as part of an observational study of patients who suffered from stroke.
Dataset: The main goal was to study various psychological factors: optimism, fatalism, depression, spirituality, and their relationship with stroke severity and other health outcomes among the study participants. Data were collected using questionnaires during a baseline interview and also medical chart review. More information about this study can be found in the article Fatalism, optimism, spirituality, depressive symptoms and stroke outcome: a population based analysis.
The dataset that you will work with is called completedata.sas7bdat. The two variables we are interested in are:
Covariate 1: Fatalism (larger values indicate that the individual feels less control of their life)
Covariate 2: Optimism (larger values indicate that the individual feels higher levels of optimism)
Covariate 3: Spirituality (larger values indicate that the individual has more belief in a higher power)
Outcome: Depression (larger values imply increased depression)
For our homework purposes we will assume each variable is continuous.
dep_df = read_sas(here("./data/completedata.sas7bdat"))Fit the regression model with all the covariates (Fatalism, Optimism, Spirituality), display the regression table, and write out the fitted regression line.
Interpret each coefficient (\(\beta_0\), \(\beta_1\), \(\beta_2\), \(\beta_3\)).
Does the intercept make sense for the range of values that each covariate can take? Explain.
Recall in Homework 2, we ran a simple linear regression model for Depression vs. Fatalism with the following interpretation for the coefficient: For every 1 point higher fatalism score, there is an expected difference of 0.25 points higher depression score (95%CI: 0.17, 0.32).
Does the addition of Optimism and Spirituality change our coefficient estimate for Fatalism? (No need for an official hypothesis test here. I just want us to note some differences.)
From the fitted regression model, calculate the regression line when Optimism score is 10 and Spirituality score is 6.