Day 7

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# Day 7
## Linear model - ANOVA
### Michael W. Kearney📊 School of Journalism Informatics Institute University of Missouri
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Build course directory on your computer:

``` r
## this time it should actually work
source(
  "https://goo.gl/F7sD3U"
)
```

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## Linear model - ANOVA

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## Overview
+ Hypothesis testing (review)
+ Covariance (review)

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## Hypothesis testing

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## Alternative hypothesis
+ Also referred to as **research hypothesis**, this hypothesis formally states a **theorized relationship or difference**
   - Relationship between two or more **variables**
   - Difference between two or more **groups**
+ Research hypotheses describe **patterns**, not a lack of patterns
   - Every research hypothesis has a null hypothesis
+ Alternative/research hypotheses are typically represented as `$H_{1}$`:...

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## Null hypothesis
+ The **null hypothesis** is the version of the research hypothesis that assumes **no relationship or difference**
   - **No relationship** between two or more variables
   - **No difference** between two or more groups
+ Null hypotheses are typically represented as `$H_{0}$`:...

> In other words, a null hypothesis restates the research hypothesis as though everything in the world **only varies at random**

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## P-values
+ Calculate a statistic using sample data related to hypothesis
+ Convert test statistic to p-value
+ **p-value**: Likelihood of observing the data given a true null hypothesis
   - Absurdity of observing data if we assume null (no relationship)
+ **Significance**: p-value less than pre-set alpha level
   - Convention is .05

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## Test decisions

+ There are two types of errors
+ Alpha = significant
+ Beta = power

|Decision       |$H_{0}$_true                 |H0_false              |
|:--------------|:-----------------------|:---------------------|
|Fail to reject |Correct (1 - alpha)     |Incorrect Type I beta |
|Reject Null    |Incorrect Type II alpha |Correct (1 - beta)    |

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## Parametric vs non-parametric
**Parametric**

+ Assumes theoretical distribution

**Non-parametric**

+ Does not assume any particular distribution

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## Linear model - ANOVA

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## Anova

+ Study effect of one or more categorical variables on a numeric outcome
+ Predictors are categorical (sometimes called factors)
   - Factors are comprised of two or more levels

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## One-way ANOVA
+ A single factor/predictor (with mulitiple levels) 
+ Used because each value is classified in one way 
+ Null hypothesis is that mean at each level is equal

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## Assumptions

+ Independence
+ Normality
+ Homogeneity of variance

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## Anova
+ We can generalize one-way ANOVA to multiple factors/predictors
+ Just like regression, but we are modeling means
   - Intercept = **grand mean**
   
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