Chapter 18 Four Multivariate Techniques Angela Gillis & Winston Jackson Nursing Research: Methods & Interpretation

Multiple Regression Multiple regression used when we wish to examine the impact of several variables on a dependent variable. It is may be used when you have a ratio level dependent variable and, preferably, ratio level independent variables. There are methods, however, for using independents measured at the nominal or ordinal levels.

Multiple Regression Cont. Multiple regression is a powerful tool because it allows the researcher to: –estimate the relative importance of independent variables in predicting variation in a dependent variable –identify an equation describing the relation between the independent and dependent variables

Multiple Regression Cont. Elements in the equation tell us the relative importance of each factor is in predicting the dependent variable. Recall from the correlation analysis (Chapter 11) the formula Y = a + bX Multiple Regression extends the equation where: Y = a + b 1 X 1 + b 2 X 2 + …b k X k

a This value represents the constant--the point where the regression line crosses the Y axis. b These coefficients represent the weightings for each of the independent variables.

Y = a + ß 1 X 1 + ß 2 X 2 + …ß k X k ß These values are knows as beta weights. A beta weight simply represents a standardized version of a b coefficient. Think of ßs as Z-score versions of the b coefficients. Recall that Z scores standardize variables

Y = a + ß 1 X 1 + ß 2 X 2 + …ß k X k To compute the relative importance of variables once we have the betas we can use the following formula: % Variance explained ß 1 x R 2 by each variable = x 100  ßs

Multiple Regression Cont. When you do your SPSS run the program will produce both b and ßvalues. The a value (called the Constant) will also be printed. R 2 This value will also be reported which tells you how much of the variance in the dependent variable is explained by the equation

Using Non-Ratio Variables Ordinal variables may be included in their raw form (un-recoded) but remember that the equation will underestimate the relative importance of non-ratio variables Nominal variables may be included by transforming them into “dummy variables” Dummy variables are recoded to “presence/absence” variables.

Dummy Variables Create new variables to replace the nominal variable so that you have one fewer variables than categories in the original variable. I.e., if you have a 3 category religion variable (Protestant, Catholic, Jew) then recode this into two new variables coded into presence/absence. (See p. 566 of text.) Presence = 1; Absence = 0.

Tips for Regression Analysis Ensure variables are independent of the dependent variable, not an alternative measure of it. Watch for highly correlated independent variables (multicollinearity). Either convert these into an index (if that makes sense) or simply select one of them.

Tips Cont. Try to achieve ratio level measurement Use Raw data: do not use recoded forms of ordinal or ratio variables Use the Backward option when using regression Interpret weightings with care.

Tips Cont. Monitor number of cases; watch out for cases where N is getting close to number of variables. (Cases = total df + 1 on table) –Repeat analysis eliminating those variables that were dropped early in the analysis: keep in last two or three before final equation –Try “Pairwise” solution –Try “Means” solution where missing values set to mean for the variable

Discriminant Analysis Very similar to Regression analysis but used in cases where the researcher has a nominal dependent variable. Results in the calculation of discriminant coefficients similar to a regression equation D = B 0 + B 1 X 1 + B 2 X B k X k

B 0 This is the constant B 1 The coefficient for the 1st variable To compute the “discriminant score” multiply the coefficient by the observed value (see Table 18.3, p. 572).

Discriminant Analysis Cont. Discriminant analysis assumes ratio level independent variables (similar to regression) but like regression dummy variables may be included. Both standardized and unstandardized coefficients are provided on the output. If you want to calculate relative contributions use the standardized version

Discriminant Analysis Cont. When discriminant is run you will get a report on the % of cases which can be correctly classified by using the information on the independent variables. The analysis relies on Lambda. This statistic measures the proportionate reduction in error that results with knowledge of the independent variables.