BS141 Lecture 5 - Linear Regression

What does variance additivity state

If distributions are independent, variance of their sum = sum of variances

What happens to variance as sample size increases

Increases

What does covariance measure

How two variables change together

What is the formula for joint variance

Variance of X + variance of Y + 2 x covariance of X and Y

What is the covariance of independent samples

Zero

Why is covariance limited as a measure

Depends on units and can’t show independence clearly

What is Pearson correlation coefficient?

R, standardized measure of association btwn 2 variables

What is the range of Pearson correlation coefficient

-1 to 1

What does R equal to zero indicate

No correlation between variables

What does R close to one indicate

Strong positive correlation

What does R close to minus one indicate

Strong negative correlation

What is the limitation of correlation

It shows association but does not allow prediction

What statistical method allows prediction of one variable from another

Linear regression

What method is used to fit a regression line

Least squares method

What does the least squares method minimize

Sum of squared differences btwn observed and predicted values

What is the equation of a linear regression line

y = a + bx

What does the intercept a represent

y when x = 0

What does the slope b represent

Change in y for a 1 unit increase in x

What is the null hypothesis in a regression test

The population slope beta equals zero

What is the alternative hypothesis in a regression test

The population slope beta is not equal to zero

What type of data is required for regression testing

Paired continuous measurements

What variances are computed in a regression test

Model variance and error variance

What test statistic is used in regression testing

The F statistic

Why is a regression test similar to ANOVA

It compares explained variance to unexplained variance

What does the model term represent in regression

How far the regression line is from the null hypothesis line

What does the error term represent in regression

How close observed values are to predicted values

What is the degrees of freedom for the model in regression

One

What is the degrees of freedom for error in regression

# of observations - 2

What is the coefficient of determination

R squared

What does R squared measure

Proportion of total variance explained by regression model

If R squared = 0.48, what does it mean?

48% of variance is explained by regression line

How is R squared related to Pearson correlation?

Square of the Pearson correlation coefficient

Why does correlation not imply causation

Association does not prove one variable causes the other

What is the difference between correlation and regression

Correlation measures association while regression predicts values

When is linear regression appropriate

When the relationship between variables is linear

What is an important limitation when making predictions

Predictions should only be made within the observed data range