# R Tutorial for ANOVA and Linear Regression

## ANOVA table

• Let's say we have collected data, and our X values have been entered in R as an array called data.X, and our Y values as data.Y. Now, we want to find the ANOVA values for the data. We can do this through the following steps:
1. First, we should fit our data to a model. > data.lm = lm(data.Y~data.X)
2. Next, we can get R to produce an ANOVA table by typing : > anova(data.lm)
3. Now, we should have an ANOVA table!

### Fitted Values

• To obtain the fitted values of the model from our previous example, we type: > data.fit = fitted(data.lm)
• This gives us an array called "data.fit" that contains the fitted values of data.lm

### Residuals

• Now we want to obtain the residuals of the model: > data.res = resid(data.lm)
• Now we have an array of the residuals.

### Hypothesis testing

• If you have already found the ANOVA table for your data, you should be able to calculate your test statistic from the numbers given.
• Let's say we want to find the F - quantile given by $$\large \mathbf{F} (.95; 3 , 24)$$. We can find this by typing > qf(.95, 3, 24)
• To find the t - quantile given by $$\large \mathbf{t} (.975; 1, 19)$$ , we would type: > qt(.975, 1, 19)

### P - values

• To get the p - value for the F - quantile of, say, 2.84 , with degrees of freedom 3 and 24, we would type in > pf(2.84, 3, 24)

### Normal Q-Q plot

• We want to obtain the Normal Probability plot for the standardized residuals of our data, "data.lm".
• We have already fit our data to a model, but we now need the studentized residuals:

> data.stdres = rstandard(data.lm)

• Now, we make the plot by typing: > qqnorm(data.stdres)
• Now, to see the line, type: > qqline(data.stdres)

## More on Linear Regression

### Fitting a Model

• Let's say we have two X variables in our data, and we want to find a multiple regression model. Once again, let's say our Y values have been saved as a vector titled "data.Y". Now, let's assume that the X values for the first variable are saved as "data.X1", and those for the second variable as "data.X2".
• If we want to fit our data to the model $$\large Y_i = \beta_1 X_{i1} + \beta_2 X_{i2} + \epsilon_i$$ , we can type:

> data.lm.mult = lm(data.Y ~ data.X1 + data.X2).

• This has given us a model to work with, titled "data.lm.mult"

### Summary of Model

• We can now see our model by typing > summary(data.lm.mult)
• The summary should list the estimates, the standard errors, and the t-values of each variable. The summary also lists the Residual Standard Error, the Multiple and Adjusted R-squared values, and other very useful information.

### Pairwise Comparison Scatterplot Matrix

• Let's say we have a model with three different variables (the variables are named "data.X", "data.Y", and "data.Z"). We can compare the variables against eachother in a scatterplot matrix easily by typing:

> pairs(cbind(data.X, data.Y, data.Z))

• If the variables are listed together in one data frame (let's say it's called "data.XYZ"), we can get the same matrix by typing: > pairs(data.XYZ)

## Further Questions

• If you would like more information on any R instructions to be added to this page, please comment, noting what you would like to see, and we will work on putting up the information as soon as possible.

## Contributors

• Valerie Regalia
• Debashis Paul