1 Correlations

  • Everything correlates with everything (Ambient Correlational Noise).
  • Our goal today is to determine how much
  • Deal with 2 variables: Next we explore the problems with 3 or more

Correlations between two variables

  • Pearson’s (interval by interval)
  • Spearman’s (interval by ordinal)
  • Point-by-serial (interval by dichotomous)
  • Polychoric (ordinal vs ordinal)
  • Tetrachroic (dichotomous vs dichotomous)

1.1 Pearson’s Correlation

\(r_{xy}=\frac{\sum{xy}}{\sqrt{\sum{x^2}\sum{y^2}}}\)

  • Numerator = How much they vary together (co-variance)
  • Denominator = How much they vary alone (variance)
  • values is bounded between -1 and 1
  • Let’s create two random normal variables that correlate with each other
  • to do this we need to make a covariance matrix with a correlation at .6
#packages we will need to conduct to create and graph our data
library(MASS) #create data
library(car) #graph data
r=.6 #Correlation value
Means.XY<- c(10,10) #set the means of X and Y variables
CovMatrix.XY <- matrix(c(1,r,r,1),2,2) # creates the covariate matrix 

CovMatrix.XY #nice and simple 2x2 matrix
##      [,1] [,2]
## [1,]  1.0  0.6
## [2,]  0.6  1.0
#build the correlated variables. 
# Note: empirical=TRUE means make the correlation EXACTLY r. 
# empirical=FALSE, the correlation value would be normally distributed around r
CorrData<-mvrnorm(n=100, mu=Means.XY,Sigma=CovMatrix.XY, empirical=TRUE)

#Convert them to a "Data.Frame", which is like SPSS data window
CorrData<-as.data.frame(CorrData)
#lets add our labels to the vectors we created
colnames(CorrData) <- c("Happiness","IceCream")
#Lets view the first few subjects
head(CorrData)
##   Happiness  IceCream
## 1 10.774760 10.907732
## 2  8.700742  8.816727
## 3  9.722170  9.617680
## 4 10.927843 10.499182
## 5 11.584289 10.692111
## 6  9.641194  9.557827
#make the scatter plot
scatterplot(Happiness~IceCream,CorrData, smoother=FALSE)

#calculate the Pearson's correlation 
cor.test(CorrData$Happiness, CorrData$IceCream, 
         method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  CorrData$Happiness and CorrData$IceCream
## t = 7.4246, df = 98, p-value = 4.193e-11
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4574985 0.7124547
## sample estimates:
## cor 
## 0.6

1.1.1 Pearson’s correlation is scale independent!

  • No matter the mean differences or range of scores, the Pearson’s r will give the same results.
  • We can also z-score the data and get the same result
  • However if they are scaled non-linearly (sqrt, ^2, log,…) the correlation will change.
#lets add (change the mean)
Happiness.big<-CorrData$Happiness+1000
IceCream<-CorrData$IceCream
cor.test(Happiness.big, IceCream, method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  Happiness.big and IceCream
## t = 7.4246, df = 98, p-value = 4.193e-11
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4574985 0.7124547
## sample estimates:
## cor 
## 0.6
#zscored
Happiness.z<-scale(CorrData$Happiness)
IceCream.z<-scale(CorrData$IceCream)
cor.test(Happiness.z, IceCream.z, method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  Happiness.z and IceCream.z
## t = 7.4246, df = 98, p-value = 4.193e-11
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4574985 0.7124547
## sample estimates:
## cor 
## 0.6
#what happens if I LINEARLY scale them differently?
cor.test(Happiness.big, IceCream.z, method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  Happiness.big and IceCream.z
## t = 7.4246, df = 98, p-value = 4.193e-11
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4574985 0.7124547
## sample estimates:
## cor 
## 0.6
#what happens if I NON-LINEARLY scale them differently?
Happiness<-CorrData$Happiness # orginal
IceCream.sq4<-(CorrData$IceCream)^4 #Non-linear

scatterplot(Happiness~IceCream.sq4, smoother=FALSE)

cor.test(Happiness, IceCream.sq4, method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  Happiness and IceCream.sq4
## t = 6.6871, df = 98, p-value = 1.409e-09
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4082241 0.6812735
## sample estimates:
##       cor 
## 0.5597593

1.1.2 Pearson’s: Let visualize our result

  • Overlap between the two variables is defined by \(r^2\)
library(VennDiagram) # lets us plot our results (like the book)
## Loading required package: grid
## Loading required package: futile.logger
## 
## Attaching package: 'VennDiagram'
## The following object is masked from 'package:car':
## 
##     ellipse
# calculate r-squared
overlap=r^2 

Simple.Corr.Venn<-draw.pairwise.venn(1, 1, overlap, c("Happiness", "IceCream"))
grid.draw(Simple.Corr.Venn)

1.2 Spearmen’s Correlation

  • Spearman is a Pearson Correlation on rank ordered data.
  • Let’s rank order our random correlated data
  • You have to rank each variable independently first
CorrData$X.rank<-rank(CorrData$Happiness)
CorrData$Y.rank<-rank(CorrData$IceCream)

scatterplot(Y.rank~X.rank,CorrData, smoother=FALSE)

cor.test(CorrData$Y.rank, CorrData$X.rank, method = c("pearson"))
## 
##  Pearson's product-moment correlation
## 
## data:  CorrData$Y.rank and CorrData$X.rank
## t = 6.7079, df = 98, p-value = 1.278e-09
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  0.4096673 0.6822010
## sample estimates:
##       cor 
## 0.5609481
# Or you can just use the built in spearman correlation and get the same result
cor.test(CorrData$Y, CorrData$X, method = c("spearman"))
## 
##  Spearman's rank correlation rho
## 
## data:  CorrData$Y and CorrData$X
## S = 73168, p-value = 1.923e-09
## alternative hypothesis: true rho is not equal to 0
## sample estimates:
##       rho 
## 0.5609481

1.3 Point-by-Serial

  • This time lets make up some data on the fly
library(polycor) #Advanced Correlations
## 
## Attaching package: 'polycor'
## The following object is masked from 'package:psych':
## 
##     polyserial
set.seed(42)
ratings<-c(rnorm(25,mean=5,sd = .5),rnorm(25,mean=2,sd = .5))
Flavors<-c(rep(0,25),c(rep(1,25)))
FlavorNames<-c(rep("cookie Dough",25),c(rep("Rum-Raisin",25)))

#Build data frame
Ice.Cream.Data<-data.frame(
  ratings = ratings,
  Flavors = Flavors,
  Names = FlavorNames)
head(Ice.Cream.Data)
##    ratings Flavors        Names
## 1 5.685479       0 cookie Dough
## 2 4.717651       0 cookie Dough
## 3 5.181564       0 cookie Dough
## 4 5.316431       0 cookie Dough
## 5 5.202134       0 cookie Dough
## 6 4.946938       0 cookie Dough
scatterplot(ratings~Flavors,Ice.Cream.Data, smoother=FALSE)

polyserial(Ice.Cream.Data$Flavors,Ice.Cream.Data$ratings)
## [1] -0.8868748

2 Regression

  • Correlation and regression are similar
  • Correlation determines the standardized relationship between X and Y
  • Linear regression = 1 DV and 1 IV, where the relationship is a straight line
  • Linear regression determines how X predicts Y
  • Multiple (linear) regression = 1 DV and 2+ IV (also straight lines)
  • Multiple regression determines how X,z, and etc, predict Y [next week]

2.1 Basic Regression Equation

  • Linear Regression equation you learned when younger was probably \(y = MX + b\)
  • \(y\) = predict value
  • \(M\) = slope
  • \(X\) = Variable used to predict Y
  • \(b\) = intercept

2.2 Modern Regression Equation

  • \(Y=B_{YX}X + B_0 + e\)
  • \(Y\) = predict value
  • \(B_{YX}\) = slope
  • \(B_{0}\) = intercept
  • \(e\) = error term (observed - predicted). Also called the residual.

2.3 Ice cream example

  • Specify the model with the lm function.
  • We are going to predict happiness scores from ice cream!
scatterplot(Happiness~IceCream, smoother=FALSE)

Happy.Model.1<-lm(Happiness~IceCream,data = CorrData)
summary(Happy.Model.1)
## 
## Call:
## lm(formula = Happiness ~ IceCream, data = CorrData)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.54020 -0.62443 -0.01595  0.61440  1.93532 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.00000    0.81211   4.925 3.42e-06 ***
## IceCream     0.60000    0.08081   7.425 4.19e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.8041 on 98 degrees of freedom
## Multiple R-squared:   0.36,  Adjusted R-squared:  0.3535 
## F-statistic: 55.13 on 1 and 98 DF,  p-value: 4.193e-11

2.3.1 Intercept

  • 4 is where the line hit the y-intercept (when happiness = 0).

2.3.2 Slope

  • 0.6 is the rise over run
  • for each 0.6 change in ice cream value, there is a corresponding change in happiness!
  • so we can predict happiness from ice cream score:

(0.6 * 5 spoons of ice cream + baseline happiness intercept: 4) = 7

  • This is your predicted happiness score if you had 5 spoons of ice cream

2.3.3 Error for this prediction?

R will actually do all the prediction for us for each value of ice cream residuals = observed - predicted

  • Red dots = observed above predictor line
  • Blue dots = observed below predictor line
  • the stronger the color the more an impact that point has in pulling the line in its direction
  • Hollow dots = predicted
  • The gray lines are the distance between observed and predicted values!

What should the mean of the residuals equal?

CorrData$predicted <- predict(Happy.Model.1)   # Save the predicted values with our real data
CorrData$residuals <- residuals(Happy.Model.1) # Save the residual values


library(ggplot2) # Note: a more powerful graphing tool I am forced to use to show this to you
ggplot(data = CorrData, aes(x = IceCream, y = Happiness)) +
  geom_smooth(method = "lm", se = FALSE, color = "lightgrey") +  # Plot regression slope
  geom_point(aes(color = residuals)) +  # Color mapped here
  scale_color_gradient2(low = "blue", mid = "white", high = "red") +  # Colors to use here
  guides(color = FALSE) +
  geom_segment(aes(xend = IceCream, yend = predicted), alpha = .2) +  # alpha to fade lines
  geom_point(aes(y = predicted), shape = 1) +
  theme_bw()  # Add theme for cleaner look

2.3.4 Ordinary least squares (OLS)

  • Linear regression finds the best fit line by trying to minimize the sum of the squares of the differences between the observed responses those predicted by the line.
  • OLS computationally simple to get the slope value, but is actually inaccurate

\[B_{YX}=\frac{\sum{XY}-\frac{1}{n}\sum{X}\sum{Y}}{\sum{x^2}-\frac{1}{n}\sum{x}^2} = \frac{Cov_{XY}}{var_x}\]

  • Modern methods use an alternative (ML, REML) we will examine later when we get to GLM

2.4 SE on the terms in the models (how good is the fit?)

  • Residual Standard error = \(\sqrt\frac{\sum{e^2}}{n-2}\)
  • in R language:
n=length(CorrData$residuals)

RSE = sqrt(sum(CorrData$residuals^2) / (n-2))
RSE
## [1] 0.8040713
  • So our error on the prediction is 0.8040713 happiness points based on our model.

2.4.1 SE on the Intercept

  • Intercept Standard error = \(RSE\sqrt {\frac{1}{n}+\frac{M_x^2}{(n-1)var_x}}\)
  • in R language:
ISE = RSE*(sqrt( 1 / n + mean(CorrData$IceCream)^2 / (n - 1)*var(CorrData$IceCream)))
ISE
## [1] 0.8121124

2.4.2 SE on the Slope

  • Intercept Standard error = \(\frac{sd_y}{sd_x}\sqrt{\frac{1 - r_YX^2}{n-2}}\)
  • in R language:
#lets extract the r2 from the model
r2.model<-summary(Happy.Model.1)$r.squared

SSE = sd(CorrData$Happiness)/sd(CorrData$IceCream) * sqrt((1- r2.model)/ (n - 2))
SSE
## [1] 0.0808122

2.4.3 t-tests on slope and intercept and \(r^2\) value

  • Values are tested against 0, so its all one sample t-tests
  • slope: \(t = \frac{B_{YX} - H_0}{SE_{B_{YX}}}\)
  • intercept: \(t = \frac{B_{0} - H_0}{SE_{B_{0}}}\)

\(r^2\) is a little different as its a correlation value

  • correlations are not normally distributed
  • Fisher created a conversion for r to make it a z (called Fishers’ \(r\) to \(Z\))
  • \(r^2\): \(t = \frac{r_{XY}\sqrt{n-2}-H_0}{\sqrt{1-r_{XY}^2}}\) , where df = n - 2
  • its often given for as an F value, remember \(t = F^2\)
#intercept
t.I= Happy.Model.1$coefficients[1]/ISE
t.I
## (Intercept) 
##    4.925427
#Slope
t.S= Happy.Model.1$coefficients[2]/SSE
t.S
## IceCream 
## 7.424621
# For r-squared
t.r2xy = r2.model^.5*sqrt(n-2)/sqrt(1-r2.model)
F.r2xy = t.r2xy^2
F.r2xy
## [1] 55.125

Note: We are testing null hypothesis value for slope, i.e., null = 0. But it’s a terrible guess. Everything correlates with everything, so it’s important to keep this in mind moving forward. So that would be the NILL hypothesis. NILL can be tested better with bootstrapping.

3 Power and Regression

  • We studied power last semester
  • I will skip ahead to calculating it in R.
  • for regression, we will need to convert our \(r^2\) into cohen’s \(f^2\)
  • \(f^2 = \frac{r^2}{1-r^2}\)
library(pwr) #power analysis
#power for GLM
# u  = degrees of freedom for numerator
# v = degrees of freedom for denominator
# f2 = effect size
# sig.level= (Type I error probability)
# power = (1 minus Type II error probability)
f2.icecream <- r^2 / (1-r^2)

pwr.f2.test(u = 1, v = n-2, f2 = f2.icecream, sig.level = 0.05, power = NULL)
## 
##      Multiple regression power calculation 
## 
##               u = 1
##               v = 98
##              f2 = 0.5625
##       sig.level = 0.05
##           power = 1

So we had a power of basically 1 given this sample size and our true effect size of 0.6

4 Final Notes

  • Testing between correlations using old fashion Fisher’s test is old fashion (Cohen et al., p. 49). The modern approach is the bootstrap, as the old method is under-powered.
  • Your book is little out of date: CIs are better but bootstrapped CIs are becoming more standard.
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output:
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```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
knitr::opts_chunk$set(fig.width=5)
knitr::opts_chunk$set(fig.height=3.75)
knitr::opts_chunk$set(fig.align='center') 
```

# Correlations

- Everything correlates with everything (Ambient Correlational Noise). 
- Our goal today is to determine how much 
- Deal with 2 variables: Next we explore the problems with 3 or more

Correlations between two variables 

- Pearson's (interval by interval)
- Spearman's (interval by ordinal)
- Point-by-serial (interval by dichotomous)
- Polychoric (ordinal vs ordinal)
- Tetrachroic (dichotomous vs dichotomous)


## Pearson's Correlation

$r_{xy}=\frac{\sum{xy}}{\sqrt{\sum{x^2}\sum{y^2}}}$

- Numerator = How much they vary together (co-variance)
- Denominator = How much they vary alone (variance)
- values is bounded between -1 and 1
- Let's create two random normal variables that correlate with each other
- to do this we need to make a covariance matrix with a correlation at .6

```{r, echo=TRUE, warning=FALSE}
#packages we will need to conduct to create and graph our data
library(MASS) #create data
library(car) #graph data
```

```{r, echo=TRUE}

r=.6 #Correlation value
Means.XY<- c(10,10) #set the means of X and Y variables
CovMatrix.XY <- matrix(c(1,r,r,1),2,2) # creates the covariate matrix 

CovMatrix.XY #nice and simple 2x2 matrix

#build the correlated variables. 
# Note: empirical=TRUE means make the correlation EXACTLY r. 
# empirical=FALSE, the correlation value would be normally distributed around r
CorrData<-mvrnorm(n=100, mu=Means.XY,Sigma=CovMatrix.XY, empirical=TRUE)

#Convert them to a "Data.Frame", which is like SPSS data window
CorrData<-as.data.frame(CorrData)
#lets add our labels to the vectors we created
colnames(CorrData) <- c("Happiness","IceCream")
#Lets view the first few subjects
head(CorrData)
#make the scatter plot
scatterplot(Happiness~IceCream,CorrData, smoother=FALSE)

#calculate the Pearson's correlation 
cor.test(CorrData$Happiness, CorrData$IceCream, 
         method = c("pearson"))

```

### Pearson's correlation is scale independent! 
- No matter the mean differences or range of scores, the Pearson's r will give the same results. 
- We can also z-score the data and get the same result
- However if they are scaled non-linearly (sqrt, ^2, log,...) the correlation will change. 


```{r, echo=TRUE}

#lets add (change the mean)
Happiness.big<-CorrData$Happiness+1000
IceCream<-CorrData$IceCream
cor.test(Happiness.big, IceCream, method = c("pearson"))

#zscored
Happiness.z<-scale(CorrData$Happiness)
IceCream.z<-scale(CorrData$IceCream)
cor.test(Happiness.z, IceCream.z, method = c("pearson"))

#what happens if I LINEARLY scale them differently?
cor.test(Happiness.big, IceCream.z, method = c("pearson"))

#what happens if I NON-LINEARLY scale them differently?
Happiness<-CorrData$Happiness # orginal
IceCream.sq4<-(CorrData$IceCream)^4 #Non-linear

scatterplot(Happiness~IceCream.sq4, smoother=FALSE)
cor.test(Happiness, IceCream.sq4, method = c("pearson"))


```

### Pearson's: Let visualize our result
- Overlap between the two variables is defined by $r^2$

```{r, echo=TRUE, warning=FALSE}
library(VennDiagram) # lets us plot our results (like the book)
```

```{r, echo=TRUE}
# calculate r-squared
overlap=r^2 

Simple.Corr.Venn<-draw.pairwise.venn(1, 1, overlap, c("Happiness", "IceCream"))
grid.draw(Simple.Corr.Venn)

```

## Spearmen's Correlation
- Spearman is a Pearson Correlation on rank ordered data. 
- Let's rank order our random correlated data
- You have to rank each variable independently first

```{r, echo=TRUE, warning=FALSE}
CorrData$X.rank<-rank(CorrData$Happiness)
CorrData$Y.rank<-rank(CorrData$IceCream)

scatterplot(Y.rank~X.rank,CorrData, smoother=FALSE)

cor.test(CorrData$Y.rank, CorrData$X.rank, method = c("pearson"))

# Or you can just use the built in spearman correlation and get the same result
cor.test(CorrData$Y, CorrData$X, method = c("spearman"))


```

## Point-by-Serial

- This time lets make up some data on the fly

```{r, echo=TRUE, warning=FALSE}
library(polycor) #Advanced Correlations
set.seed(42)
ratings<-c(rnorm(25,mean=5,sd = .5),rnorm(25,mean=2,sd = .5))
Flavors<-c(rep(0,25),c(rep(1,25)))
FlavorNames<-c(rep("cookie Dough",25),c(rep("Rum-Raisin",25)))

#Build data frame
Ice.Cream.Data<-data.frame(
  ratings = ratings,
  Flavors = Flavors,
  Names = FlavorNames)
head(Ice.Cream.Data)

scatterplot(ratings~Flavors,Ice.Cream.Data, smoother=FALSE)
polyserial(Ice.Cream.Data$Flavors,Ice.Cream.Data$ratings)
           
```

# Regression
- Correlation and regression are similar
- Correlation determines the standardized relationship between X and Y
- Linear regression = 1 DV and 1 IV, where the relationship is a straight line
- Linear regression determines how X predicts Y
- Multiple (linear) regression = 1 DV and 2+ IV (also straight lines)
- Multiple regression determines how X,z, and etc, predict Y [next week]

## Basic Regression Equation
- Linear Regression equation you learned when younger was probably $y = MX + b$
- $y$ = predict value
- $M$ = slope
- $X$ = Variable used to predict Y
- $b$ = intercept

## Modern Regression Equation
- $Y=B_{YX}X + B_0 + e$
- $Y$ = predict value
- $B_{YX}$ = slope
- $B_{0}$ = intercept
- $e$ = error term (observed - predicted). Also called the residual.

## Ice cream example
- Specify the model with the lm function. 
- We are going to predict happiness scores from ice cream! 



```{r, echo=TRUE, warning=FALSE}

scatterplot(Happiness~IceCream, smoother=FALSE)
Happy.Model.1<-lm(Happiness~IceCream,data = CorrData)
summary(Happy.Model.1)

```


### Intercept
- `r Happy.Model.1$coefficients[1]` is where the line hit the y-intercept (when happiness = 0). 

### Slope
- `r Happy.Model.1$coefficients[2]` is **the rise over run**
- for each `r Happy.Model.1$coefficients[2]` change in ice cream value, there is a corresponding change in happiness!
- so we can predict happiness from ice cream score: 

> (`r Happy.Model.1$coefficients[2]` * 5 spoons of ice cream + baseline happiness intercept: `r Happy.Model.1$coefficients[1]`) 
=  `r Happy.Model.1$coefficients[2] * 5 +Happy.Model.1$coefficients[1]`

- This is your *predicted* happiness score if you had 5 spoons of ice cream

### Error for this prediction?

R will actually do all the prediction for us for each value of ice cream
residuals =  **observed** - **predicted**

- Red dots = **observed** *above* predictor line
- Blue dots = **observed** *below* predictor line
- the stronger the color the more an impact that point has in pulling the line in its direction
- Hollow dots = **predicted**
- The gray lines are the distance  between **observed** and **predicted** values!

What should the mean of the residuals equal?


```{r, echo=TRUE, warning=FALSE}

CorrData$predicted <- predict(Happy.Model.1)   # Save the predicted values with our real data
CorrData$residuals <- residuals(Happy.Model.1) # Save the residual values


library(ggplot2) # Note: a more powerful graphing tool I am forced to use to show this to you
ggplot(data = CorrData, aes(x = IceCream, y = Happiness)) +
  geom_smooth(method = "lm", se = FALSE, color = "lightgrey") +  # Plot regression slope
  geom_point(aes(color = residuals)) +  # Color mapped here
  scale_color_gradient2(low = "blue", mid = "white", high = "red") +  # Colors to use here
  guides(color = FALSE) +
  geom_segment(aes(xend = IceCream, yend = predicted), alpha = .2) +  # alpha to fade lines
  geom_point(aes(y = predicted), shape = 1) +
  theme_bw()  # Add theme for cleaner look

```

### Ordinary least squares (OLS)
- Linear regression finds the best fit line by trying to minimize the sum of the squares of the differences between the observed responses those predicted by the line. 
- OLS computationally simple to get the slope value, but is actually inaccurate 

$$B_{YX}=\frac{\sum{XY}-\frac{1}{n}\sum{X}\sum{Y}}{\sum{x^2}-\frac{1}{n}\sum{x}^2} = \frac{Cov_{XY}}{var_x}$$

- Modern methods use an alternative (ML, REML) we will examine later when we get to GLM


## SE on the terms in the models (how good is the fit?)
- Residual Standard error = $\sqrt\frac{\sum{e^2}}{n-2}$
- in R language: 

```{r, echo=TRUE, warning=FALSE}
n=length(CorrData$residuals)

RSE = sqrt(sum(CorrData$residuals^2) / (n-2))
RSE

```

- So our error on the prediction is `r RSE` happiness points based on our model.  

### SE on the Intercept
- Intercept Standard error = $RSE\sqrt {\frac{1}{n}+\frac{M_x^2}{(n-1)var_x}}$
- in R language: 
  
```{r, echo=TRUE, warning=FALSE}
ISE = RSE*(sqrt( 1 / n + mean(CorrData$IceCream)^2 / (n - 1)*var(CorrData$IceCream)))
ISE
```

### SE on the Slope
- Intercept Standard error = $\frac{sd_y}{sd_x}\sqrt{\frac{1 - r_YX^2}{n-2}}$
- in R language: 
  
```{r, echo=TRUE, warning=FALSE}

#lets extract the r2 from the model
r2.model<-summary(Happy.Model.1)$r.squared

SSE = sd(CorrData$Happiness)/sd(CorrData$IceCream) * sqrt((1- r2.model)/ (n - 2))
SSE
```

### t-tests on slope and intercept and $r^2$ value
- Values are tested against 0, so its all one sample t-tests
- slope: $t = \frac{B_{YX} - H_0}{SE_{B_{YX}}}$
- intercept: $t = \frac{B_{0} - H_0}{SE_{B_{0}}}$

$r^2$ is a little different as its a correlation value

- correlations are not normally distributed
- Fisher created a conversion for r to make it a z (called Fishers' $r$ to $Z$)
- $r^2$: $t = \frac{r_{XY}\sqrt{n-2}-H_0}{\sqrt{1-r_{XY}^2}}$ , where df = n - 2
- its often given for as an F value, remember $t = F^2$ 

```{r, echo=TRUE, warning=FALSE}
#intercept
t.I= Happy.Model.1$coefficients[1]/ISE
t.I
#Slope
t.S= Happy.Model.1$coefficients[2]/SSE
t.S

# For r-squared
t.r2xy = r2.model^.5*sqrt(n-2)/sqrt(1-r2.model)
F.r2xy = t.r2xy^2
F.r2xy

```

Note: We are testing null hypothesis value for slope, i.e., null = 0. But it's a terrible guess. Everything correlates with everything, so it's important to keep this in mind moving forward. So that would be the NILL hypothesis. NILL can be tested better with bootstrapping. 


# Power and Regression
- We studied power last semester
- I will skip ahead to calculating it in R. 
- for regression, we will need to convert our $r^2$ into cohen's $f^2$
- $f^2 = \frac{r^2}{1-r^2}$

```{r, echo=TRUE, warning=FALSE}
library(pwr) #power analysis
#power for GLM
# u	 = degrees of freedom for numerator
# v	= degrees of freedom for denominator
# f2 = effect size
# sig.level= (Type I error probability)
# power = (1 minus Type II error probability)
f2.icecream <- r^2 / (1-r^2)

pwr.f2.test(u = 1, v = n-2, f2 = f2.icecream, sig.level = 0.05, power = NULL)
```

So we had a power of basically 1 given this sample size and our true effect size of `r r`


# Final Notes 
- Testing between correlations using old fashion Fisher's test is old fashion (Cohen et al., p. 49). The modern approach is the bootstrap, as the old method is under-powered. 
- Your book is little out of date: CIs are better but bootstrapped CIs are becoming more standard. 



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