report-hw02.Rmd

---
title: "Homework 02"
author: "Spencer Pease"
date: "10/14/2019"
output: pdf_document
---

```{r setup, include=FALSE}

# Set output options
knitr::opts_chunk$set(echo = FALSE)
options(knitr.kable.NA = '-')


# Include libraries
library(magrittr)

# Source helper functions
source("functions/summary_table.R")
source("functions/population_estimates.R")


# Load data
mri <-
  read.table("data/mri.txt", header = TRUE) %>%
  dplyr::as_tibble() %>%
  dplyr::select(.data$crt, .data$obstime, .data$death)

mri_grouped <- mri %>%
  dplyr::mutate(
    `Vital Status at 5 Years` = dplyr::if_else(
      .data$obstime >= 5 * 365,
      "Survived 5+ years", "Died within 5 years"
    )
  ) %>%
  dplyr::group_by(.data$`Vital Status at 5 Years`)


```

# Overview of the Data

The data in this report comes from a study of the risk factors associated with
cardiovascular and cerebrovascular disease among generally healthier older
adults. Specifically, this study enrolled a cohort of older (65+) and generally
healthy adults by randomly sampling individuals enrolled in the United States
Medicare system. Overall **`r nrow(mri)`** adults agreed to participate in the
study.


This report focuses on three variables from the study data. They are briefly
described below (see the study documentation for more details):

```{r variable-description}

var_desc <- tibble::tribble(
  ~Variable, ~Units, ~Description,
  "crt", "mg/dl", "Measure of creatinine in the participant's blood",
  "obstime", "days", "Total time the participant was observed on the study",
  "death", "binary", "Indicator that the participant died during the study"
)

knitr::kable(var_desc, caption = "Variable Descriptions")

```

Of particular note _obstime_ is measured in days, but this report is interested
in grouping by vital status (_death_) at five years. This report treats a year
as exactly 365 days, and will still report results using units of days
($365 \frac{days}{year} \times 5 years = 1825 days$).


# Creatinine Levels Across Groups Defined by Vital Status at 5 Years

## Summary Statistics

We begin this report by looking at the summary statistics of the creatinine
level variable distribution for two subsets of the data: participants who
survived at least five years since the date of MRI, and participants who did not
survive at least five years.

```{r crt-distribution-summary}

summary_surv <- mri_grouped %>%
  dplyr::select(.data$crt, dplyr::group_cols()) %>%
  summary_table() %>%
  dplyr::select(-.data$n, -.data$variable)


knitr::kable(
  summary_surv,
  caption = "Summary of creatinine level distribution by vital status group",
  digits = 2
)

```

Between these two groups, we observe that the mean, median, IQR, and minimum
values are within $.1$ to $.2$ $\frac{mg}{dl}$ of each other. The bigger
differences can be seen in the number of observations in each group, with
the surviving group having approximately $5$ times as many observation as the
non-surviving group. The distribution of the non-surviving group also has a
heavier right-skew, leading to a higher maximum value and standard deviation.


## Population Inference

We can use the distribution of creatinine level in our sample to infer some
parameters of the larger population. Below, we look at the point estimate
(sample mean), standard error of the point estimate, and $95\%$ confidence
interval of the population for separate vital status groups at $5$ years.

```{r pop-inference-summary}

crt_pop_inference <- mri_grouped %>%
  dplyr::select(.data$crt, dplyr::group_cols()) %>%
  summary_table(summary_funcs = list(
    "point est." = ~mean(., na.rm = TRUE),
    "std. error" = ~standard_error(.),
    "95% CI (lower)" = ~confidence_interval(., .95)["lower"],
    "95% CI (upper)" = ~confidence_interval(., .95)["upper"]
  )) %>%
  dplyr::select(-.data$variable)

knitr::kable(
  crt_pop_inference,
  caption = "Population inference of creatinine level by vital status group",
  digits = 2
)

```

Looking at the $95\%$ confidence intervals for both vital status groups, we see
that there is no overlap in the intervals. Since the confidence interval is
interpreted as having _a confidence of 95% that the true population mean of
creatinine level for a given group lies in their respective intervals_, we can
say with some confidence that the difference in point estimates between the two
vital status groups is due to more than just variations in the sample mean.

It is important to note that we can compare these confidence intervals in this
manner because we can approximate both the distribution of the sample means for
both of these subsets to a normal distribution because of the Central Limit
Theorem, which states _for a sufficiently large n, the distribution of sample
means approximates a normal distribution_. Our sample sizes for each subset are
sufficiently large (approximately $100$ and $600$), so we can say the
distributions are normal. Because they are normal, we can then standardize them
and use the a standard z-score to build a confidence interval, allowing the
two distributions to be compared.


## Exploring different levels of confidence

Narrowing our scope to look at only participants in the vital status group not
surviving at least $5$ years, we can further explore the confidence interval
of creatinine level by looking at the intervals created from different
confidence levels. Let's look at three common levels: $90\%$, $95\%$, and
$99\%$.

```{r alternate-confidence-levels}

crt_death <- mri %>%
  dplyr::filter(.data$obstime < 5 * 365) %>%
  dplyr::pull("crt")

crt_ci <-
  list(
    "90%" = confidence_interval(crt_death, alpha = .90),
    "95%" = confidence_interval(crt_death, alpha = .95),
    "99%" = confidence_interval(crt_death, alpha = .99)
  ) %>%
  tibble::enframe("Confidence Level") %>%
  tidyr::unnest_wider("value")

knitr::kable(
  crt_ci,
  caption = paste("Confidence intervals for creatinine level among",
                  "participants not surviving at least 5 years"),
  digits = 2
)

```

Notice the pattern of widening intervals for higher levels of confidence. This
pattern exists because, for the same set of data, we can only be more confident
that the true mean of a population parameter falls within an interval if that
interval encompasses a larger range of possible values. Conversely, we can set
a smaller interval, but we can't be as confident that the true value falls
within it.


# Geometric Mean Analysis

Instead of using the the **arithmetic** mean, standard error, and confidence
interval to build our population inference, we can also look at the
**geometric** mean, standard error, and confidence interval. These geometric
statistics are useful when the data has an underlying exponential nature. Below
we provide a table of the geometric population inference as an alternative to
the arithmetic equivalent.

```{r geometric-inference-summary}

geo_crt_inference <- mri_grouped %>%
  dplyr::select(.data$crt, dplyr::group_cols()) %>%
  summary_table(summary_funcs = list(
    "point est." = ~exp(mean(log(.), na.rm = TRUE)),
    "std. error" = ~exp(standard_error(log(.))),
    "95% CI (lower)" = ~exp(confidence_interval(log(.), .95))["lower"],
    "95% CI (upper)" = ~exp(confidence_interval(log(.), .95))["upper"]
  )) %>%
  dplyr::select(-.data$variable)

knitr::kable(
  geo_crt_inference,
  caption = paste("Geometric population inference of creatinine level by",
                  "vital status group"),
  digits = 2
)

```


# Examining "High" Creatinine Levels

With the population inferences we made above, we can begin to make some
statements regarding what a reasonable population parameter would be, and if any
of our target population groups include exceptional values.

This report defines a "high" creatinine level in an individual as anything over
$1.2 \frac{mg}{dl}$. Grouping our data by vital status at $5$ years, we are
able to examine if there are any differences in the inferred population mean of
creatinine level for each group.

Looking at the subset of individuals who survived at least $5$ years since the
date of MRI, we see from the prior tables that the confidence interval of both
the arithmetic and geometric point estimate fall below our defined threshold
of $1.2 \frac{mg}{dl}$. Since our confidence interval represents the range that
we are 95% certain includes the true population mean of creatinine level, we
can say that a similar population of individuals who survive at least $5$ years
would not be prone to high creatinine levels.

For the subset of individuals who die within $5$ years of their MRI date, we
again look at the prior tables to see that the confidence interval of the for 
both the arithmetic and geometric interpretations of the point estimate of mean 
creatinine level include the values above the defined threshold of 
$1.2 \frac{mg}{dl}$ for high creatinine level. Because of this, we can't say
definitively if a population of similar individuals who died within $5$ years of
their MRI have high mean creatinine level, since our 95% confidence interval
includes values on either side of the threshold, so the true population mean
could be anywhere in that range.