```{r setup, include=FALSE}
knitr::opts_chunk$set(echo=TRUE, fig.align="center")
```

#### Statistics for Laboratory Scientists ( 140.615 )

## Regression and Correlation - Advanced

Loading the SPH.140.615 package.

```{r}
library(SPH.140.615)
```


#### Fathers' and daughters' heights

The Pearson & Lee (1906) data.

```{r}
example(pear)
```

Calculate the regression of daughter's height on father's height (i.e., for predicting daughter from father).

```{r}
lm.outA <- lm(daughter ~ father, data=pear)
summary(lm.outA)
```

Calculate the regression of father's height on daughter's height (i.e., for predicting father from daughter).

```{r}
lm.outB <- lm(father ~ daughter, data=pear)
summary(lm.outB)
```

The intercept and slope for regression A.

```{r}
coA <- lm.outA$coef
coA
```

The intercept and slope for regression B.

```{r}
coB <- lm.outB$coef 
coB
```

Transform regression B coefficients: y = mx + b $\Rightarrow$ x = y/m - b/m

```{r}
coB[1] <- -coB[1]/coB[2]
coB[2] <- 1/coB[2]
coB
```

Plot the data with the two regression lines.

```{r}
plot(pear)
abline(coA, lwd=2, col="green")
abline(coB, lwd=2, col="orange")
```

### Span and height example

The data.

```{r}
plot(span, xlab="span [ inches ]", ylab="height [ inches ]")
abline(lsfit(span$span,span$stature), col="red", lty=2, lwd=2)
```

Predicting height from span.

```{r}
lm.fit <- lm(stature~span, data=span)
summary(lm.fit)
summary(lm.fit)$coef[,1]
```

Residual standard deviation.

```{r}
summary(lm.fit)$sigma
```

The correlation.

```{r}
r <- cor(span$span, span$stature)
r
cor(span)
```

Slope of the regression line.

```{r}
r * sd(span$stature)/sd(span$span)
```

Height standard deviation.

```{r}
sd(span$stature)
```

Typical prediction error using span to predict height.

```{r}
sd(span$stature) * sqrt(1-r^2)
```