```{r setup, include=FALSE}
knitr::opts_chunk$set(echo=TRUE, fig.align="center")
options(width=110)
```

#### Statistics for Laboratory Scientists ( 140.615 )

## Linear Regression

#### Example from class: David Sullivan's heme data

```{r}
h2o2 <- rep(c(0,10,25,50),each=3)
pf3d7 <- c(0.3399,0.3563,0.3538,
           0.3168,0.3054,0.3174,
           0.2460,0.2618,0.2848,
           0.1535,0.1613,0.1525)
```

Plot the data, and add the least squares fit line.

```{r}
par(las=1)
plot(h2o2, pf3d7, xlab="H2O2 concentration", ylab="OD")
abline(lsfit(h2o2, pf3d7), col="red", lty=2)
```

##### The regression of optical density on hydrogen peroxide concentration, using the built-in 'lm' function

```{r}
lm.out <- lm(pf3d7 ~ h2o2)
lm.out
lm.sum <- summary(lm.out)
lm.sum
attributes(lm.sum)
```

The table of coefficients.

```{r}
lm.sum$coef
```

The parameter estimates.

```{r}
lm.sum$coef[,1]
```

The residual standard deviation.

```{r}
lm.sum$sigma
```

The coefficient of determination.

```{r}
lm.sum$r.squared
```

To see how to derive these statistics from scratch, please see the advanced code. It is simple arithmetic following the lecture notes and therefore not really advanced code, I just tucked it away there. 

Also, to avoid data glitches, it is a good habit to make a data frame.

```{r}
dat <- data.frame(conc=h2o2, od=pf3d7)
dat
str(dat)
```

The linear regression model.

```{r}
lm.out <- lm(od ~ conc, data=dat)
summary(lm.out)
```

Plot of the data with least squares regression line..

```{r}
par(las=1)
plot(dat$conc, dat$od, xlab="H2O2 concentration", ylab="OD")
abline(lm.out, col="red", lty=2)
```

##### Confidence intervals for the regression parameters

```{r}
confint(lm.out)
confint(lm.out,level=0.99)
```

To see how to derive these from scratch, also see the advanced code.

##### Checking model assumptions

```{r}
plot(lm.out,which=1)
plot(lm.out,which=2)
```