Week 8 - Linear models: regression

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

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.

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

Regression line from sample should be the population mean

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

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

Uncertainty associated with the sample estimate

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

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

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

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

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

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

Log transformation is easiest Power and exponential relationships are also common

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

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

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

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