Problem Set Stats Bootcamp - class 12

Hypothesis Testing

Author

Neelanjan Mukherjee

Published

October 21, 2024

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here() starts at /Users/jayhesselberth/devel/rnabioco/molb-7950

biochem <- read_tsv("http://mtweb.cs.ucl.ac.uk/HSMICE/PHENOTYPES/Biochemistry.txt", show_col_types = FALSE) |>
  janitor::clean_names()

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# simplify names a bit more
colnames(biochem) <- gsub(pattern = "biochem_", replacement = "", colnames(biochem))

# we are going to simplify this a bit and only keep some columns
keep <- colnames(biochem)[c(1, 6, 9, 14, 15, 24:28)]
biochem <- biochem[, keep]

# get weights for each individual mouse
# careful: did not come with column names
weight <- read_tsv("http://mtweb.cs.ucl.ac.uk/HSMICE/PHENOTYPES/weight", col_names = F, show_col_types = FALSE)

# add column names
colnames(weight) <- c("subject_name", "weight")

# add weight to biochem table and get rid of NAs
# rename gender to sex
b <- inner_join(biochem, weight, by = "subject_name") |>
  na.omit() |>
  rename(sex = gender)

Problem # 1

Is there an association between mouse calcium and sodium levels?

1. Make a scatterplot to inspect variable

2. Are they normal (enough)?

Which test will you use and why?

3. Declare null hypothesis \(\mathcal{H}_0\)

\(\mathcal{H}_0\) is that there is no dependency/association between \(calcium\) and \(sodium\)

4. Calculate and plot the correlation on a scatterplot

Problem # 2

Do mouse calcium levels explain mouse sodium levels? If so, to what extent?

Use a linear model to do the following:

1. Specify the Response and Explanatory variables — (2 pts)

The response variable y is The explantory variable x is

2. Declare the null hypothesis — (1 pts)

The null hypothesis is …

3. Use the `lm` function to create a fit (linear model)

also save the slope and intercept for later

4. Add residuals to the data and create a plot visualizing the residuals

5. Calculate the \(R^2\) and compare to \(R^2\) from fit

\(R^2 = 1 - \displaystyle \frac {SS_{fit}}{SS_{null}}\)

\(SS_{fit} = \sum_{i=1}^{n} (data - line)^2 = \sum_{i=1}^{n} (y_{i} - (\beta_0 \cdot 1+ \beta_1 \cdot x)^2\)

\(SS_{null}\) <80><94> sum of squared errors around the mean of \(y\)

\(SS_{null} = \sum_{i=1}^{n} (data - mean)^2 = \sum_{i=1}^{n} (y_{i} - \overline{y})^2\)

6. Using \(R^2\), describe the extent to which calcium explains sodium levels

7. Report (do not calculate) the \(p-value\) and your decision on the null