statistics for laboratory science II
Biostatistics for Laboratory Scientists (2024) Homework Assignment 11 (Due Wednesday, May 8, 2024) The homework is due at 10.30am in the dropbox on the Course Plus page (you can find the dropbox under the ’Resources’ tab in the upper right). For exercises involving R code, please knit a document from your R markdown (Rmd) file. Generate a single pdf file for your entire submission and give it a name that makes it identifiable (calling it 140.615.HW.Number.Lastname.Firstname or similar). Show your work. 1. It is hypothesized that there are fluctuations in norepinephrine (NE) levels which accompany fluctuations in affect with bipolar affective disorder: during depressive states, NE levels drop; during manic states, NE levels increase. To test this relationship, researchers measured the level of NE by measuring the metabolite 3-methoxy-4-hydroxyphenylglycol (MHPG in microgram per 24 hour) in a patient’s urine experiencing varying levels of mania/depression. Increased levels of MHPG are correlated with increased metabolism (thus higher levels) of central nervous system NE. Levels of mania/depression were also recorded on a scale with a low score indicating mania and a high score depression. The data are: MHPG score 980 22 1209 26 1403 8 1950 10 1814 5 1280 19 1073 26 1066 12 880 23 776 28 (a) Plot the data. (b) Compute the Pearson and Spearman correlation coefficients. What do these statistics mean concerning the relationship between MHPG levels and affect score? (c) Model the affect score as a linear function of MHPG, and report your estimates. What percent of the variability in affect score is accounted for by MHPG? (d) What would be the predicted affect score if the individual had a MHPG level of 1,100? (e) Can you conclude that there is a relationship between affect score and MHPG in the population? 2. The copper data in the SPH.140.615 package concern an analysis of copper using flame atomic absorption spectroscopy. There are two columns: the concentration of copper (in ppm) in a set of standards, and corresponding measurements of percent transmittance (via the spectroscopy procedure). Let x be the concentration of copper and let y be the log10 of the percent transmittance. (a) Fit the model yi = β0 + β1 xi + ǫi , ǫi ∼ iid Normal(0, σ 2 ), and provide estimates of β0 , β1 , and σ. (b) Provide 95% confidence intervals for β0 and β1 . (c) Assess the appropriateness of the model. 3. The percent transmittance for a sample with unknown copper concentration was measured to be 35.6%. Use your fitted calibration line to estimate the copper concentration in this sample. Calculate a 95% confidence interval for the copper concentration in this sample. Make a figure (including the calibration line) that visualizes your results. 4. Forced exhalation volume (FEV) is a measure of how much air someone can exhale from their lungs. A study looked at the relationship between FEV and age in children and young adults. The data are available here. (a) Plot the data. How would you describe the relationship between age and FEV? (b) Subset the data to only include children between the ages of 5 and 12. (c) Derive estimates of the intercept and the slope for the linear regression model with log10 (FEV) as dependent variable, and age as independent variable. Also derive 95% confidence intervals for the intercept and the slope. Check your model assumptions and interpret your findings.
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