LL4

This chunk of code shows how to do a simple confidence interval in R.

data(women)
attach(women)
mod <-  lm(weight ~ height, data = women )
summary(women)

##      height         weight     
##  Min.   :58.0   Min.   :115.0  
##  1st Qu.:61.5   1st Qu.:124.5  
##  Median :65.0   Median :135.0  
##  Mean   :65.0   Mean   :136.7  
##  3rd Qu.:68.5   3rd Qu.:148.0  
##  Max.   :72.0   Max.   :164.0

confint(mod, level=.9)

##                    5 %       95 %
## (Intercept) -98.030599 -77.002734
## height        3.288603   3.611397

This shows that no matter which way the variables are listed the correlation is still the same.As you can see there is a strong positive linear relationship.

cor(height, weight)

## [1] 0.9954948

cor(weight, height)

## [1] 0.9954948

Next we looked at the difference between confidence intervals and predicting intervals. This was new to me and an important point I learned was that predicting intervals are always larger than confidence intervals. We see this to be true in the next section of code. I also checked that the two are centered at the same place.

newdata <- data.frame(height = 52)
weight.mod <- lm(weight ~ height)
(predy <- predict(weight.mod, newdata, interval="predict") )

##        fit     lwr      upr
## 1 91.88333 87.6255 96.14116

(confy <- predict(weight.mod, newdata, interval="confidence") )

##        fit      lwr      upr
## 1 91.88333 89.18613 94.58054

confy %*% c(0, -1, 1)

##       [,1]
## 1 5.394408

predy %*% c(0, -1, 1)

##       [,1]
## 1 8.515662

confy[1] == predy[1]

## [1] TRUE

The last thing we learned how to do was F-tests which I did an example of below. As you can see there is a strong linear relationship between height and weight.

weight <- rnorm(136.7, sd=15.49869)
height <- rnorm(65, sd=4.472136)
var.test(weight, height)

## 
##  F test to compare two variances
## 
## data:  weight and height
## F = 10.057, num df = 135, denom df = 64, p-value < 2.2e-16
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
##   6.478154 15.117040
## sample estimates:
## ratio of variances 
##           10.05742