Multiple Regression Analysis The principles of Simple Regression Analysis can be extended to two or more explanatory variables. With two explanatory variables we get an equation Y = α + β 1 X 1 + β 2 X 2.. It is customary to write it as Y = β 0 +β 1 X 1 + β 2 X 2 As an example, if a hypotensive agent is administered prior to surgery, recovery time for blood pressure to normal value will depend on the dose of the hypotensive and the blood pressure during surgery. This can be modelled as Recovery time = log dose – Surgery B.P.

Categorical Explanatory Variables Binary variables are coded 0, 1. For example a binary variable x 1 (‘Gender’) is coded male = 0, female = 1. Binary variables are coded 0, 1. For example a binary variable x 1 (‘Gender’) is coded male = 0, female = 1.

Recovery time for Blood Pressure and dose of hypotensive The scatter plot shows a linear relationship. Blood Pressure takes longer to come back to normal value the larger the dose of the hypotensive. There are many outliers because of individual variability of subjects and because of different types of surgical operations.

Recovery time for Blood Pressure and lowest Blood Pressure reading during surgery The lower the blood pressure achieved during surgery the longer the time for it to reach normal value during recovery from anaesthesia

Multiple Regression Analysis The effects of the two explanatory variables acting jointly is described by the equation Recov. Time = Log dose – Surg. B.P. As noted on the scatter plots several observations had outliers or larger than expected X values.

Categorical Explanatory Variables Binary variables are coded 0, 1. For example a variable x 1 (Gender) is coded Binary variables are coded 0, 1. For example a variable x 1 (Gender) is coded male = 0 female = 1. Then in the regression equation male = 0 female = 1. Then in the regression equation Y = β 0 + β 1 x 1 + β 2 x 2 when x 1 = 1 the value of Y indicates what is obtained for female gender; and when x 1 = 0 the value of Y indicates what is obtained for males. If we have a nominal variable with more than two categories we have to create a number of new dummy (also called indicator ) binary variables

How many Explanatory Variables? As a rule of thumb multiple regression analysis should not be performed if the total number of variables is greater than the number of As a rule of thumb multiple regression analysis should not be performed if the total number of variables is greater than the number of subjects ÷ 10.

Analysis In the computer output look for: Adjusted R 2. It represents the proportion of variability of Y explained by the X’s. R2 is adjusted so that models with different number of variables can be compared. Adjusted R 2. It represents the proportion of variability of Y explained by the X’s. R2 is adjusted so that models with different number of variables can be compared. The F-test in the ANOVA table. Significant F indicates a linear relationship between Y and at least one of the X’s. The F-test in the ANOVA table. Significant F indicates a linear relationship between Y and at least one of the X’s. The t-test of each partial regression coefficient. Significant t indicates that the variable in question influences the Y response while controlling for other explanatory variables. The t-test of each partial regression coefficient. Significant t indicates that the variable in question influences the Y response while controlling for other explanatory variables.

Usefulness of Scatter Plots - I The scatter plot on the right illustrates the relationship between water hardness and mortality in 61 large towns in England and Wales. The scatter plot on the right illustrates the relationship between water hardness and mortality in 61 large towns in England and Wales. The regression line indicates inverse relationship between water hardness and mortality rates. The regression line indicates inverse relationship between water hardness and mortality rates.

Usefulness of Scatter Plots - II The inverse relationship between water hardness is till maintained. But For towns in the North the regression line is less steep than for towns in the South indicating that other causes of mortality are stronger in the North compared to the South.