---
title: "Summary of statistical tests"
output: html_document
---

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

## Outline

At this point, we've learned quite a few statistical tests. Here's an outline with a few more details below:

  * Intro to hypothesis testing for proportions (Chapter 17)
  * More on inference for means (Chapter 18)
  * Comparing groups (Chapter 20)
    * Comparing two proportions (beginning of the chapter)
    * Comparing two means (end of the chapter)
  * Paired data (Chapter 21)
  * Binned data (Chapter 22)

### Still to come

After our exam and Thanksgiving, we have one or two more chapers to go

  * Chapter 23 on Inferences for regression and
  * Chapter 24 on Analysis of Variation

## Comments

### Commonalities

All of the tests have a few things in common. 

  * They all involve some well-formulated hypothesis - a null hypothesis $H_0$ vs an alternative hypothesis $H_A$.
  * Of course, they all involve data; the general question is - do the data support the alternative to the point where we should reject the null hypothesis?
  * The precise formulation of the general question involves a $p$-value which is
    * The probability of observing data at least as favorable to the alternative hypothesis as our current data set, if the null hypothesis is true.
    * The smaller the $p$-value, the less viable is the null hypothesis.

### Differences

Perhaps the most obvious difference centers on the type of data being considered: numerical vs categorical.

There are other differences too, though. 

 * How many data sets are under consideration? 
 * How large are the data sets?
 * What is the relationship between the variables?
 * What question are we asking about the data?

Understanding these helps you know which to apply in a certain situation.

## The tests

### Hypotheses for a single proportion

This is the simplest, first situation that we dealt with. We are dealing with proportions of categorical data. We often think of this in terms of a random variable $X$ that is *binomially* distributed; thus, we need to know the binomoial distribution *after* dividing through by $n$:

\begin{align}
\mu &= p
    &\sigma^2 &= p(1-p)/n
    &\sigma &= \sqrt{p(1-p)/n}
\end{align}


Our hypothesis can be written

\begin{align}
H_0 : \hat{p}=p_0 \\
H_A : \hat{p} \neq p_0
\end{align}

Ultimately, we compute the $p$ value using either a normal distribution (if the sample size is large) or a $t$-distribution (if the sample size is small). There are a couple of examples in our [intro notes on Hypothesis Testing](10.18.17.HypothesisTesting.html).


### Hypotheses for means

These are very much like our tests for proportions. The main difference is that we are now dealing with numerical data, rather than categorical dta. In the simplest case, we have one data sample - just a list of numbers. 

#### The hypothesis test

The question is - does that data support the hypothesis that the mean of the population from which is was drawn is some particular number? If our data has sample mean $\bar{x}$ and we suspec the population mean is $\mu_0$, then our two-sided hypothesis can be written

 * $H_0$: $\bar{x}=\mu_0$
 * $H_A$: $\bar{x}\neq\mu_0$

A one sided hypothesis can be written with a greater or less, rather than a not equal.

#### Conditions to check

  * Random sample of numeric data
    * Need less than 10% of population for independence
  * Large enough
    * Typically, at least 30

#### The $z$-score

The $z$-score for our mean is 
$$Z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}.$$
We use then compare this against the standard normal distribution or a $t$-distribution (depending on the sample size) to compute the $p$-value. There are a couple of examples in our notes on [the $t$-test](10.29.17.tDistribution.html).

### Tests for two sample means

We use these tests when we have two numerical data sets that are independent of one another and we want to compute the difference between their means.

If the sets have sizes $n_1$ and $n_2$, we analyze the difference of the two means using a $t$-test with

 * Mean $\bar{x}_1 - \bar{x}_2$,
 * Standard error 
 $$\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}},$$
 * and we use the minimum of $n_1-1$ and $n_2-1$ as the degrees of freedom.
 
 Our hypothesis test again looks like
 
$$
\begin{array}{ll}
H_0: & \mu_1 = \mu_2 \\
H_A: & \mu_1 \neq \mu_2 
\end{array}
$$


There are some examples of this in our notes on [Relating Data Sets](11.03.17.RelatingDataSets.html).

### Tests for two sample proportions

We use these tests when we have two categorical data sets that are independent of one another and we want to compute the difference between their proportions.

This is very similar to the difference of the two means but we now use

$$\hat{p} = \hat{p}_1 - \hat{p}_2$$
and
$$SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}.$$

We again use the minimum of $n_1-1$ and $n_2-1$ as the degrees of freedom.
 
 Our hypothesis test again looks like
 
$$
\begin{array}{ll}
H_0: & \hat{p}_1 = \hat{p}_2 \\
H_A: & \hat{p}_1 \neq \hat{p}_2 \\
\end{array}
$$


There are again some examples of this in our notes on [Relating Data Sets](11.03.17.RelatingDataSets.html).

### Tests for paired data

We use this when we have two data sets that are paired in a natural way; that is, each data point in one set corresponds to a particular data point in the other set.

Such a data set can be translated to a single data set by simply subtracting the data sets pair-wise.

Our hypotesis test looks like

$$
\begin{array}{ll}
H_0: & \mu_1 = \mu_2 \\
H_A: & \mu_1 \neq \mu_2 \\
\end{array}
$$

There are again some examples of this in our notes on [Relating Data Sets](11.03.17.RelatingDataSets.html).



## The $\chi^2$-test

The chi-square test is a method for assessing a model when the data are binned.

### The one-way test

In this situation, we have two data sets, call them 

  * $O_1$, $O_2$, ..., $O_k$, which represents observations in $k$ categories and
  * $E_1$, $E_2$, ..., $E_k$, which represents expected counts in $k$ categories.
  
Our hypothesis test looks like

  * $H_0$: The observations are representative of the expected counts
  * $H_A$: The observations are *not* representative of the expected counts

We then compute the $\chi^2$ statistic
$$\chi^2 = \frac{(O_1 - E_1)^2}{E_1} + \frac{(O_2 - E_2)^2}{E_2} + \cdots + \frac{(O_k - E_k)^2}{E_k}$$
and use the $\chi^2$ distribution with $k-1$ degrees of freedom.

There are some examples in our notes on [the $\chi^2$-test](11.13.17.ChiSquare.html).

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