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Lab 6: Image analysis to predict malignant breast cancer using logistic regression

Biostatistics 1

Lab6.Rmd

Learning Objectives

  • Identify and correct a problematic data file
  • perform exploratory data analysis with a binary variable
  • perform logistic regression in R using the glm() function
  • identify collinearity among many numeric variables
  • interpret logistic regression coefficients in a predictive setting
  • use a logistic regression model to predict probabilities of a binary outcome
  • use tidyr to pivot from a wide format to a long format dataframe
  • use ggplot2’s facet_wrap() to create multiple plots
  • overlay a line plot of predicted probabilities over an xy plot of data points

Materials

  • RStudio or any R environment
  • Dataset: breast cancer.csv (download). This dataset contains an outcome variable “diagnosis” with values B (benign) and M (malignant), and a number of cellular pathologic features representing the properties of cells as calculated by an image analysis algorithm. Each row corresponds to a patient who underwent a tissue biopsy. Here is more information about the dataset if you are interested.
  • R packages: readr, dplyr, ggplot2, tidyr, possibly ComplexHeatmap

1. Importing the Dataset

  1. Import the dataset (breast cancer.csv) into R, using readr. What is the problem with the 33rd column causing a warning? Fix this problem in the data file, then try again. Import id as a character, and diagnosis as a factor with values B and M.
  2. Recode the “diagnosis” column to the more informative values “benign” and “malignant”, with reference level “benign”

2. Exploratory Data Analysis (EDA)

For each part of this question, write out something you notice from the data exploration, such as variable type, if and where there are missing values. There are no right/wrong answers for this, it is just to get practice interpreting EDA.

  1. Check the dimensions of the dataset using the dim() function

  2. Preview the first few rows of the dataset using the head() function

  3. Use summary() to summarize the dataset.

  4. Identify collinear variables.

The following command, run on a wide dataframe, will calculate 1 minus the pairwise Pearson correlation between each pair of numeric variables in the dataset. This represents a distance matrix between variables, where “0” means two variables that are identical or perfectly anti-correlated, and “1” means two variables have zero correlation. We use 1 minus the absolute value of the correlations so that it becomes a distance measure instead of a correlation measure.

Plot this distance matrix to identify the most highly correlated or anticorrelated variables. A few ways you could do this is (if your distance matrix is called d) are as follows - try them and choose your favorite. You may want to increase the size of the figure output again.

or:

ComplexHeatmap::Heatmap(d)

or:

ComplexHeatmap::pheatmap(d)

Note, ComplexHeatmap is a Bioconductor package. You can install it as follows:

install.packages("BiocManager")
BiocManager::install("ComplexHeatmap")

pheatmap (“pretty heatmap”) is a normal CRAN package that works well and you can use instead, but ComplexHeatmap is the most powerful heatmap package available at the time of writing this lab.

  1. Create a box plot for each column except for id. Each boxplot should have two boxes, one for benign and one for malignant, allowing visual comparison of the distribution of that variable for benign and malignant specimens. Do any variables clearly have an association with breast cancer diagnosis?

Hints for e:

  1. Create a new dataset without the id variable, then use the tidyr::pivot_longer function to create a “long” dataframe with 3 columns: 1. diagnosis, 2. the name of the column from the “wide” dataframe, and 3. a column containing all the numeric values. The first example from the pivot_longer help page (“Simplest case where column names are character data”) is exactly analogous to what you need to do.
  2. With this “long” dataset you can add a facet_wrap() command to your ggplot to create a box plot for each variable in a grid. Use the scales argument to use “free” scales so that each box plot can have a different scale, so that you can read each box plot.
  3. Use the knitr chunk options fig.width=10 and fig.height=10 to increase the size of the figure to make theaxis labels readable.

3. Building a Logistic Regression Model

  1. Fit a univariate logistic regression model using area_mean as the predictor and diagnosis as the outcome, using the glm() function

  2. Print the summary of the model using the summary() function

  3. Interpret the coefficients of the logistic regression model

4. Making predictions

  1. Create a xy plot with area_mean of the x axis and diagnosis on the y axis. Use geom_jitter(width = 0) to create some spread on the y-axis so you can see where the points are without changing the x values.

  2. Now add to this plot the predicted probabilities for each observed temperature, using the predict function with argument type="response".

Hint: First add a column of predicted probabilities to the breast_cancer dataframe. You will have to add 1 to these probabilities to make them on the same scale as the data points. Then you can add a geom_line() to the previous plot, with its own aes(), to add the line.

Suggestion: Change the line color and increase its width to make it more visible.

  1. What do you notice about the data points where the predicted probabilities are 0, 0.5, and 1?

  2. What are the predicted probabilities of a malignant diagnosis when area_mean is 300, 500, 700, 900, and 1100?