17. Multivariate Analysis with Path Statistics (35 Patients)

Multivariate analysis is a method that, simultaneously, assesses more than a single outcome variable. It is different from repeated measures analysis of variance and mixed models, that assess both the difference between the outcomes and the overall effects of the predictors on the outcomes. Multivariate analysis, simultaneously, assesses the separate effects of the predictors on one outcome adjusted for the other. E.g., it can answer clinically important questions like: does drug-compliance not only predict drug efficacy but also, independently of the first effect, predict quality of life.

Path statistics can be used as an alternative approach to multivariate analysis of variance (MANOVA) (Chap. 18), with a result similar to that of the more complex mathematical approach used in MANOVA.

Does the inclusion of additional outcome variables enable to make better use of predicting variables.

The effects of non compliance and counseling on treatment efficacy of a new laxative was assessed in the Chap. 16. But quality of life scores are now added as additional outcome variable. The first 10 patients of the data file is given underneath.

The entire data file is entitled “chapter17multivariatewithpath”, and is in extras.springer.com. Start by opening the data file in SPSS. For analysis the statistical model Linear in the module Regression is required.

The above table shows (1) the effect of counseling on therapeutic efficacy.

Similar commands produce

(2) the effect of counseling on quality of life (qol)

(3) the effect of compliance on qol

(4) the effect of compliance on therapeutic efficacy

(5) the effect of compliance on counseling.

Next similar commands are given to produce two multiple linear regressions:

(6) the effects of counseling and compliance on qol

(7) the effects of counseling and compliance on treatment efficacy.

The above tables show the correlation coefficients of the two multiple regressions (r = 0,813 and 0, 560), and their levels of significance. Both of them are significant, meaning that the correlation coefficients are much larger than zero than could happen by chance.

First, we have to check whether the relationship of either of the two predictors with the two outcome variables, treatment efficacy and quality of life, is significant in the usual simple linear regression: they were so with p-values of 0,0001, 0,005, 0,003 and 0,007. Then, a path diagram with standardized regression coefficients is constructed. The underneath figure gives the decomposition of correlation between treatment efficacy and qol.

The standardized regression coefficients of the residual effects are obtained by taking the square root of (1- R Square). The standardized regression coefficient of one residual effect versus another can be assumed to equal 1.00.

A path statistic of 1.29 is considerably larger than that of the single outcome model: 1.29 versus 0.46 (Chap. 16), 2.80 times larger. Obviously, two outcome variables make better use of the predictors in our data than does a single one. An advantage of this nonmathematical approach to multivariate regression is that it nicely summarizes all relationships in the model, and it does so in a quantitative way as explained in the above figure.

Multivariate analysis is a linear model that works with more than a single outcome variable. It, thus, simultaneously, assesses the separate effects of the predictors on one outcome adjusted for the other. E.g., it can answer clinically important questions like: does drug-compliance not only predict drug efficacy but also, independently of the first effect, predict quality of life. The current chapter shows that path statistics can be used as an alternative approach to multivariate analysis of variance (MANOVA) (Chap. 18), with a result similar to that of the more complex mathematical approach used in MANOVA.

More background, theoretical, and mathematical information of multivariate analysis with path statistics is given in Statistics applied to clinical trials 5th edition, Chap. 25, Springer Heidelberg Germany, 2012, from the same authors.