---
title: "STA 106 - TFA Part II"
author: "Dr. Erin K. Melcon"
date: "February 13, 2018"
output: html_document
---

### 1. The data
Lets use the dataset `ToothGrowth` (again).  It desribes how much guniea pig teeth grow, based on type of supplement, and the dose of the supplement.  We have to make the dosage level a factor (you do not have to worry about this for your data).

```{r}
the.data = ToothGrowth
the.data$dose = as.factor(the.data$dose)
```

### 2. Fitting models.
We now have potentially 5 models of interest.  The interaction model, the TFA no-interactions, the factor A effects model, the factor B effects model, and the "empty or null model" (which would be appropriate if there were no factor A or B effects).

To simpify things a bit, I will assume column 1 is numeric, column 2 is factor A, and column 3 is factor B.  I will rename my columns for more efficient typing:
```{r}
names(the.data) = c("Y","A","B")
```

Fitting the models are simple in R, and can be done with the following commands.  Note, I am using `AB` to denote the interation model, $A.B$ to denote the non-interation model, and `N` to denote the empty model.

```{r}
AB = lm(Y ~ A*B,the.data)
A.B = lm(Y ~ A + B,the.data)
A = lm(Y ~ A,the.data)
B = lm(Y ~ B,the.data)
N = lm(Y ~ 1, the.data)
```

For your data, you will either have to rename your columns (in the appropriate order to your data), or you will have to replace the values `the.data`, `Y`, `A`, `B` with your dataset names.

These have fit the "regression version" of ANOVA models, which I may go over in week 7.  However, we can use them to find SSE values, and perform the general F test. 

### 3. Finding SSE values
To find a particular SSE value for a particular model, we may use the general command:

`sum(the.model$residuals^2)`

I have a few commands that will take all the models above and find the SSE for all of them, and label them nicely:
```{r}
all.models = list(AB,A.B,A,B,N)
SSE = t(as.matrix(sapply(all.models,function(M) sum(M$residuals^2))))
colnames(SSE) = c("AB","(A+B)","A","B","Empty/Null")
rownames(SSE) = "SSE"
```

You can remove specific models if you so choose.

### 4. Hypothesis tests
For finding test-statistics and p-values for various tests, the general command is:

`anova(smaller.model, larger.model)`

This will return all relevant values of SSE, and d.f.  For example, if I wanted to test for interactions, I could use the following:

```{r}
anova(A.B, AB)
```

In row 1 we have (in order): $d.f\{ SSE_R\}$, $SSE_R$

In row 2 we have (in order): $d.f\{ SSE_F\}$, $SSE_F$, $d.f\{ SSE_R\} - d.f\{ SSE_F\}$, $SSE_R - SSE_F$, $F_S$, and the p-value.

You can access a specific row and column by saving the table, and using [i,j] for the ith row, jth column.  For example, if I want the test-statistic and p-value, we can use:
```{r}
results = anova(A.B,AB)
```
Then the p-value is `r results[2,6]`, with test-statistic `r results[2,5]`.

If you wanted to do a different hypothesis test, you would simply change what you defined as the "small" model, and what you defined as the "larger" model.  I.e., change the names of the models.

### 5. Finding partial R^2.

I have built a function (surprise!) to find partial R^2 values for you, although it is not strictly necessary.  You can find the values of SSE and calculate it yourself, but the function definition is below:
```{r}
Partial.R2 = function(small.model,big.model){
  SSE1 = sum(small.model$residuals^2)
  SSE2 = sum(big.model$residuals^2)
  PR2 = (SSE1 - SSE2)/SSE1
  return(PR2)
}
```

Then, after you have copied the definition into R, to find a partial R^2 you can use:

`Partial.R2(smaller.model, larger.model)`

For example, if I wanted to find $R^2\{A | \cdot\}$, I could use:

```{R}
Partial.R2(N,A)
```