Week 8 - Linear models: regression

Question 1

linear regression

Answer 1

Assumption that relationship between variables is linear
Normal distribution
Numerical response variable

Question 2

method of least squares

Answer 2

The line that fits the data dots best

Least squares regression line: line for which the sum of all squared deviations in Y is smallest.

Question 3

forumla of a line

Answer 3

Y = a + bX

Slope: of a linear regression is the rate of change in Y per unit of X

Question 4

populations and samples

Answer 4

Regression line from sample should be the population mean

Question 5

predicted values

Answer 5

Predicted values: of Y from a regression line estimate the mean of Y for all individuals having a given value of X.

Question 6

residuals

Answer 6

Residual: of a point is the different between its measured Y value and the Y value predicted by the regression line

Question 7

standard error of a slope

Answer 7

Uncertainty associated with the sample estimate

Question 8

testing hypotheses about a slope

Answer 8

Evaluate if the population slope equals the null hypothesised value which is typically zero
T statistic is used with degrees of freedom

Question 9

t-test for regression slope

Answer 9

Find the best slope - see if it is positive, negative or zero
Work our standard error of slope with mean square residual
Calculate t statistic and compare it to t distribution
See if p value is significant

Question 10

F statistic or anova approach

Answer 10

F test used instead of t
Null hypothesis is slope is 0
Can be used when the test is two-sided and the null hypothesis slope is zero
Does not mean we are using ANOVA, just the ANOVA table for the F statistic

Question 11

using R squared

Answer 11

Use R square to measure fraction of variation in Y that is explained by X in the linear regression
If R squared is close to one, then X predicts most of the variation in Y

Question 12

assumptions of regression

Answer 12

At each value of X there is a population of possible Y values whose mean lies on the true regression line
At each of the X values the distribution of Y values is normal
The variance of Y values is the same at all values of X
At each value of X the Y measurements represent a random sample from the population of possible Y values
Linear relationship
Residual plots and QQ plots check assumptions
Extreme residuals can violate variance assumption

Question 13

what do we do about violations?

Answer 13

Ignore if not too drastic
Transform variables if necessary (log etc)
Transformations can be linearising
If there are 0s in data set, add 1 to all data points so they are not lost when transformed to log
Square root is good for Y that are counts
Arcsine square root is good for Y that are proportions

Question 14

transformations

Answer 14

Log transformation is easiest
Power and exponential relationships are also common

Question 15

precision of predictions

Answer 15

Predicting the mean of Y or a data point of Y from X is a prediction
Mean predictions have higher precision than predicting a single data point

Question 16

confidence bands

Answer 16

measure the precision of the predicted mean Y for each value of X.

Question 17

prediction intervals

Answer 17

measure the precision of the predicted single Y values for each X value.

Question 18

extrapolation

Answer 18

a prediction of the value of a response variable outside the range of X values in the data.