C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Linear Regression and Linear Prediction Predicting the score on one variable when the score on another variable is called regression In general, statistical prediction is achieved through the production of a simplified statement of the relationship between two variables The most commonly assumed relationship is a linear (straight line) relationship

C82MCP Diploma Statistics School of Psychology University of Nottingham 2 The Linear Equation A linear equation is defined in the following way where X is the independent variable Y is the dependent variable b is the slope of the line a is the intercept

C82MCP Diploma Statistics School of Psychology University of Nottingham 3 An Example of a Positive Relationship The graph below shows the plot of an equation

C82MCP Diploma Statistics School of Psychology University of Nottingham 4 An Example of a Negative Relationship The graph below shows the plot of an equation

C82MCP Diploma Statistics School of Psychology University of Nottingham 5 Simple Linear Regression Coefficients Since we are trying to achieve an equation of the form We need to find coefficients, a and b, that lead the equation to pass through the mean of the dependent variable scores minimise the “error of prediction”

C82MCP Diploma Statistics School of Psychology University of Nottingham 6 Simple Linear Regression Coefficients The following values for the coefficients : and Minimise the “error of prediction”

C82MCP Diploma Statistics School of Psychology University of Nottingham 7 Example Data The data on the right is the mean number of words recalled by primary school children after listening to a spoken list of words Is there a linear relationship between these two variables

C82MCP Diploma Statistics School of Psychology University of Nottingham 8 Example Data When the data is plotted on a scattergraph the points do not all fit on a straight line We need to find a way to describe the best fitting straight line relationship.

C82MCP Diploma Statistics School of Psychology University of Nottingham 9 Example Linear Regression

C82MCP Diploma Statistics School of Psychology University of Nottingham 10 Calculating the Slope The slope is given by: From the example calculations we get Therefore there is a positive relationship between age and the mean number of recalled words

C82MCP Diploma Statistics School of Psychology University of Nottingham 11 Calculating the Intercept The intercept for the example data is given by: The intercept is For this data the regression line crosses the y axis at y=3.62

C82MCP Diploma Statistics School of Psychology University of Nottingham 12 Example Linear Equation For this example data the complete regression equation is given by If we look at one of the five year olds who scored a mean number of recalled words of 6 we find that the equation predicts that they should score 5.81 The residual (i.e. the difference between the predicted score and the actual score) for this five year old is 0.19 which is small.

C82MCP Diploma Statistics School of Psychology University of Nottingham 13 The Statistical Test of the Regression Equation "Does the regression equation significantly predict the data that have been obtained?" The way to approach this problem is on the basis of the variability in the Y scores that the regression equation accounts for.

C82MCP Diploma Statistics School of Psychology University of Nottingham 14 Estimates of Variability The differences between the predicted and the observed scores are known as the residuals We can use the residuals as a measure of variability of the scores around the regression line

C82MCP Diploma Statistics School of Psychology University of Nottingham 15 Testing the Regression Equation We can test the amount of variability that the regression equation accounts for using an F-ratio The estimate of variance used in the F-ratio is known as a Mean Square Mean Squares are defined as:

C82MCP Diploma Statistics School of Psychology University of Nottingham 16 Sum of Square of the Regression The sum of squares of the regression can be calculated using the following formula where

C82MCP Diploma Statistics School of Psychology University of Nottingham 17 Sum of Squares of the Residual The sum of squares of the residual can be calculated using the following formula where

C82MCP Diploma Statistics School of Psychology University of Nottingham 18 The Mean Squares The mean square for the regression is given by: The degrees of freedom for the residual are N-2, so the mean square for the residuals is:

C82MCP Diploma Statistics School of Psychology University of Nottingham 19 Testing the Regression Equation

C82MCP Diploma Statistics School of Psychology University of Nottingham 20 Sum of Squares of X The sum of squares for X are given by: For the example data the sum of squares of X are given by:

C82MCP Diploma Statistics School of Psychology University of Nottingham 21 The Sum of Squares of the Regression The sum of squares of the regression is given by: For the example data the sum of squares of the regression is:

C82MCP Diploma Statistics School of Psychology University of Nottingham 22 The Sum of Squares of the Residual The sum of squares of the residual is given by: where For the example data the sum of squares of the residual is:

C82MCP Diploma Statistics School of Psychology University of Nottingham 23 The Mean Squares The mean square for the regression is given by: The mean square for the residual is given by: The F-ratio is given by:

C82MCP Diploma Statistics School of Psychology University of Nottingham 24 Results of the Analysis The results of this analysis are presented in a summary table The F ratio is looked up in tables with the regression and residual degrees of freedom For this experiment, given 1 & 8 df, the critical value of F, 5.32, is exceeded. Thus the regression equation is a significant predictor of the data

C82MCP Diploma Statistics School of Psychology University of Nottingham 25 Proportion of Variability accounted for One index of the success of the regression equation is the proportion of variability accounted for: This means the 83% of the variability in the dependent variable scores can be accounted for by the regression equation:

C82MCP Diploma Statistics School of Psychology University of Nottingham 26 Summary The regression equation is a significant predictor of this data. There is a linear relationship between the mean number of words recalled and the age of the child