Chapter 18Chapter 18Four Multivariate TechniquesFour Multivariate Techniques

Angela Gillis & Winston Jackson

Nursing Research: Methods & Interpretation

Multiple RegressionMultiple 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 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.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 + b1X1 + b2X2 + …bkXk

Y = Y = aa + + bb11XX11 + + bb22XX22 + … + …bbkkXXkk

• 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 + ß1X1 + ß2X2 + …ßkXk

• 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 + ß1X1 + ß2X2 + …ßkXk

• To compute the relative importance of variables once we have the betas we can use the following formula:

% Variance explained ß1 x R2

by each variable = x 100

ßs

Multiple Regression Cont.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.

• R2 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 VariablesUsing 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 VariablesDummy 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 AnalysisTips 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.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.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 AnalysisDiscriminant 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 = B0 + B1X1 + B2X2 + ... + BkXk

D = B0 + B1X1 + B2X2 + ... + BkXk

• B0 This is the constant

• B1 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 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.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.