A dataset with two variables contains what is called bivariate data. For example, the heights and weights of people (i.e. for the purposes of determining the extent to which taller people weigh more)
Common bivariate statistical analyses include
A scatter plot of two variables shows the values of one variable on the Y axis and the values of the other variable on the X axis. Scatter plots are well suited for revealing the relationship between two variables.
Scatterplots can be implemented in R using the command ***plot()
Exercise: Let us construct scatter-plots for the Immer and Iris data sets.
plot(immer$Y1,immer$Y2)
plot(iris[,1],iris[,3])More complex scatterplots, with better visual aesthetics, can be constructed. We will look at this more later on in the semester.
We can give a name to the model (e.g. \(FIT1\)), and view all of the results of the calculation, including the regression coefficients, hypothesis test results and information on the residuals (i.e. the differences between the estimated ‘y’ values and the observed ‘y’ values).
In common with all data structures we can use the names() function and ‘\(\$\)’ to access components.
FIT1 = lm(Y~X)
summary(FIT1)
names(FIT1)
names(summary(FIT1))
FIT1$coefficients
class(FIT1)
Strength of a linear relationship between \(X\) and \(Y\)
M=1000
CorrData=numeric(M)
for (i in 1:M)
{
CorrData[i] = cor(rnorm(10),rnorm(10))
}
The null hypothesis is that the correlation coefficient is zero.
The alternative hypothesis is that the correlation coefficients is greater than zero.
The slope and intercept estimates
These tests are given in the “Two Tailed” format. The one tailed format compares a null hypothesis where the parameter of interest has a true value of less than or equalt to one versus an alternative hypothesis stating that it has a value greater than zero.
%\end{framed} The null hypothesis is that the correlation coefficient is zero.
The alternative hypothesis is that the correlation coefficients is greater than zero.
The slope and intercept estimates
These tests are given in the “Two Tailed” format. The one tailed format compares a null hypothesis where the parameter of interest has a true value of less than or equalt to one versus an alternative hypothesis stating that it has a value greater than zero.