[stats_class_2019.git] / problem_sets / week_06 / ps6-worked-solution.Rmd

---
title: 'Week 6 problem set: Worked solutions'
subtitle: "Statistics and statistical programming  \nNorthwestern University  \nMTS 525"
author: "Jeremy Foote"
date: "April 11, 2019"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, messages = F)
```

## Programming Questions

### PC0  

First we import the data.

```{r}
raw_df = read.csv("~/Desktop/DeleteMe/Teaching/owan03.csv") # Note that I saved the file as a CSV for importing to R
head(raw_df)
```

### PC1

Let's reshape the data

```{r}
library(tidyverse)

# Rename the columns
colnames(raw_df) <- c("None", "Low", "Medium","High")

# Gather gets all of the columns from the dataframe and 
# use them as keys, with the value as the value from that cell.
# This gives us a "long" dataframe
df <- gather(raw_df, dose, weeks_alive)

# We want to treat dose as a factor, so we can group by it easier
df$dose <- as.factor(df$dose)

# Finally, when we look at it, we see some missing data (NA); this is because not all group sizes were the same.
# We can safely remove these.
df <- df[complete.cases(df),]
```

### PC2  

Now we're going to calculate summary statistics and create some visualizations

```{r}

tapply(df$weeks_alive, df$dose, summary)

# Alternative way to do this using tidyverse

df %>% group_by(dose) %>% summarize_all(c('min','max','mean', 'IQR'))

```

When it comes to visualizations, we definitely want to use ggplot. We have lots of options for what to do.

```{r}

# Histograms

h_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize
                      fill = dose # How to group by color
                      )) + geom_histogram(position = 'dodge', bins = 5)
h_plot

# In this case, faceted histograms is probably better

h_facet = df %>% ggplot(aes(x=weeks_alive # What to summarize
                      )) + geom_histogram(bins = 5) + facet_grid(~dose)

h_facet

# Density plots
d_plot = df %>% ggplot(aes(x=weeks_alive, # What to summarize
                      fill = dose # How to group by color
                      )) + geom_density(alpha = .2)
d_plot
  

# Boxplots
box_plot = df %>% ggplot(aes(y=weeks_alive,
                      x = dose
                      )) + geom_boxplot()
box_plot

# My favorite - ridgeline plots

# install.packages('ggridges')

library(ggridges) 

ridge_plot = df %>% ggplot(aes(x=weeks_alive, y = dose)) + 
  geom_density_ridges(jittered_points = T, fill = 'orange') + theme_minimal()
ridge_plot

```

It's a bit tough to tell, but the overall assumptions of normality and equal variance seem reasonable.

The global mean is 

```{r}
mean(df$weeks_alive)
```

### PC3  

Anova! 
```{r}
summary(aov(weeks_alive ~ dose, data = df))

```

This provides evidence that the group means are different.

### PC4

T-test between None and Any, and between None and High.

```{r}


t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose
       df[df$dose != 'None','weeks_alive'] # Samples with any dose
       )

# Or, using formula notation
t.test(weeks_alive ~ dose == 'None', data = df)

# T-test between None and High

t.test(df[df$dose == 'None', 'weeks_alive'], # Samples with no dose
       df[df$dose == 'High','weeks_alive'] # Samples with high dose
       )

# Formula notation is a bit tricker. I would probably create a temprorary dataframe

tmp = df %>% filter(dose %in% c('None', 'High'))

t.test(weeks_alive ~ dose, data = tmp)

```

These t-tests both support the idea that receiving a dose of RD40 reduces lifespan. However, we should not completely trust these p-values, since we are doing multiple comparisons. One option is to do a Bonferroni correction, where we only cnsider things significant if $\alpha < .05/m$ where $m$ is the number of tests. In this case, we would fail to reject the null for the second test because we would set $\alpha = .025$.

The Bonferroni correction is more conservative than it needs to be, ane there are other approaches; for example,  the `TukeyHSD` function takes in an anova result and does post-hoc comparisons with corrections for all of the groups.

## Statistical Questions

Q1. 
a) It is a sample statistic, because it comes from a sample.
b) Confidence intervals for proportions are equal to

$$p \pm z * \sqrt{ \frac{p*(1-p)}{n}}$$

For a 95% confidence interval, $z = 1.96$, so we can calculate it like this:


```{r}

lower = .48 - 1.96 * sqrt(.48 * .52 / 1259)

upper = .48 + 1.96 * sqrt(.48 * .52 / 1259)

ci = c(lower, upper)

print(ci)

```

This means that we are 95% confident that the true proportion of Americans who support legalizing marijuana is between ~45% and ~51%.

c) We have a large enough sample, which is collected randomly, to assume that the distribution is normal.
d) The statement isn't justified, since our confidence interval include 50%

6.20
We can use the point estimate of the poll to estimate how large a sample we would need to have a confidence interval of a given width.

Basically, we want each half of the confidence interval to be 1%, i.e.,  $1.96 * \sqrt{\frac{.48 * .52}{n}} = .01$

We can solve for $n$:

$$\sqrt{\frac{.48 * .52}{n}} = .01/1.96$$
$$\frac{.48 * .52}{n} = (.01/1.96)^2$$
$$n = (.48 * .52)/(.01/1.96)^2$$
```{r}
(.48 * .52)/(.01/1.96)^2
```
So, we need a sample of approximately 9,589

6.38

The question is whether there has been a change in the proportion over time. While the tools we have learned could allow you to answer that question, they assume that responses are independent. In this case, they are obviously not independent as they come from the same students.

6.50

a) We can test this with a $\chi^2$ test.

```{r}
chisq.test(x = c(83,121,193,103), p = c(.18,.22,.37,.23))

# Or manually

# Calculate expected values
500 * c(.18,.22,.37,.23)

# Use the formula for chi-squared
chisq = (83-90)^2/90 + (121-110)^2/110 + (193-185)^2/185 + (103-115)^2/115

chisq

# We could then look up the chi-square distribution for 3 degrees of freedom

```

The p-value for this is large, meaning that we don't have evidence that the sample differs from the census distribution.

b) 
i) Opinion is the response and location is the explanatory variable, since it's unlikely that people move to a region based on their opinion.
ii) One hypothesis is that opinions differ by region. The null hypothesis is that opinion is independent of region, while the alternative hypothesis is that there is a relationship.
iii) We can again use a $\chi^2$ test.

```{r}

x <- matrix(c(29,54,44,77,62,131,36,67), nrow = 4, # this makes a matrix with 4 rows
            byrow=T) # And this says that we've entered it row by row
chisq.test(x)
```

## Empirical Questions


EQ0.

a) The unit of analysis is the customer. The dependent variable is the type of board purchased and the independent variable is gender. Males, females, and unkown gender customers are being compared. This is a two-way test.

b) The null hypothesis is that the board purchased is independent of the gender of the customer. The alternative hypothesis is that if we know the gender of the customer that will tell us something about the type of board they purchased.

c) A $\chi^2$ test found statistically significant evidence that board purchase behavior differs by gender. This difference is convincing, but it does directly not tell us what we really want to know, which is the difference between men and women. It could be possible that it is simply identifying a significant difference in the number of unknown gender customers across board types. Many of these concerns are addressed in the text and with additional tests, giving increased confidence in the reality of this difference.

d) Statistical tests help to give (or take away) confidence in a conclusion. People are not natively good at estimating how likely something is due to chance and tests help us to make these judgments. Choosing a statistical test is based on the question that you want to answer and the type of data that you have available to answer it. For example, if this were numeric data (e.g., the amount of money spent on electronics for men and women) then we could choose a t-test to compare those distributions.


EQ1

a) These are counts for two categorical variables, so the procedure used was a $\chi^2$ test. The null hypothesis is that whether or not a blog is governed by one person is independent of whether it is on the left or the right.

b) It would be surprising to see these results by chance and it make sense to believe that this difference is real. However, the main reason to be skeptical is the way that the data are grouped. The authors could have grouped them differently (e.g., 1-2 people, 3-4 people, and 5+ people); if the decision on how to group was made after seeing the data then we have good reason to be skeptical.

c) 

```{r}

# First we create the dataframe
df = data.frame(Governance=c('Individual','Multiple', 'Individual','Multiple'),
                Ideology = c('Left','Left','Right','Right'),
                Count = c(13,51,27,38))

# We can make sure it's the same by testing the Chi-squared                
chisq.test(matrix(df$Count, nrow=2))

percentage_data = df %>% 
  group_by(Ideology) %>% 
  summarize(individual_ratio = sum(Count[Governance=='Individual']) / sum(Count),
            group_count = sum(Count))

shaw_benkler_plot = percentage_data %>%
  ggplot(aes(x=Ideology, y=individual_ratio * 100)) + 
    geom_bar(stat='identity', aes(fill=c('red','blue')), show.legend=F) + 
    ylab('Percentage of Blogs') + theme_minimal()

shaw_benkler_plot

# If we want to add error bars, we need to calculate them (Note that ggplot could do this for us if we had raw data - always share your data!)

# I decided to use confidence intervals. The standard error is another reasonable choice

# Remember that for a binomial distribution (we can consider individual/non-individual as binomial), confidence intervals are mu +- x * sqrt((p*(1-p))/n)

ci_95 = 1.96 * sqrt(percentage_data$individual_ratio * (1 - percentage_data$individual_ratio)/percentage_data$group_count)

shaw_benkler_plot + geom_errorbar(aes(ymin=(individual_ratio-ci_95)*100, ymax=(individual_ratio + ci_95)*100),
                                  alpha = .3, 
                                  size=1.1, 
                                  width=.4)

```

The error bars do overlap in this case, but that actually doesn't tell us whether they are significantly different.

d) We don't need to be very worried about the base rate fallacy because the sizes of both groups are about the same.