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I am setting up a linear analysis of a single continuous dependent variable, with multiple (up to 11!) continuous independent variables. I am interested in doing something like an ANCOVA that will show the relative contribution and co-variation among the variables. But my understanding is that ANCOVA requires having at last one categorical variable, which my data does not have. I have already done multiple regression analysis that yields a correlation grid, but does not show covariation among variables. Is there another analysis I should consider? Should I force my data to fit into an ANCOVA analysis?

Here's a sample of the data for clarity.

DV      iv1     iv2     iv3     iv4     iv5     iv6
-0.34   2.05    7.77    0.00    3.91    2.23    10.80
-0.40   2.23    0.53    3.14    1.67    3.34    13.93
-0.24   6.98    2.04    2.82    9.30    2.33    13.95
-0.39   0.00    1.48    0.00    2.62    3.06    10.04
-0.32   0.79    0.00    0.00    5.51    3.94    10.24
0.24    0.00    0.00    0.00    0.00    6.15    13.85
1.58    1.34    0.00    0.00    3.41    2.20    13.05
1.07    4.05    2.99    0.00    2.89    3.47    8.09
-0.44   2.31    1.86    0.00    1.54    4.62    13.08
-0.49   2.94    2.68    0.00    10.29   1.47    11.76
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  • $\begingroup$ Just for clarity - do you mean you'd like to see the covariation among the variables or among the coefficient estimates? $\endgroup$
    – jbowman
    Commented Dec 7, 2011 at 22:30
  • $\begingroup$ @jbowman I might not fully understand the distinction, but the latter (covariation among coefficient estimates of each IV) is what I had in mind. Could you explain the distinction between the two? $\endgroup$
    – ted.strauss
    Commented Dec 7, 2011 at 23:05
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    $\begingroup$ Covariation among the variables is the covariance of the columns of the X matrix in the regression, i.e., your $iv1 \dots iv6$ variables in your data sample. The covariance of the coefficient estimates is the covariance between $\hat{\beta}_0 \dots \hat{\beta}_6$ ($\hat{\beta}_0$ being the estimate of the constant term.) The vcov function applied to the result of your regression (vcov(lm(dv+iv1+iv2+iv3+iv4+iv5+iv6)), for example) will give you the covariance matrix of the parameter estimates. $\endgroup$
    – jbowman
    Commented Dec 8, 2011 at 1:11

1 Answer 1

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You said you want "the relative contribution and co-variation among the variables." Multiple regression can give you what you're looking for, if you take advantage of what it has to offer. A correlation matrix is a very small part of what regression can yield. Assuming the predictors are relatively independent--which can be tested using tolerance or variance inflation factor statistics from a regression--the strength of the standardized coefficients will indicate the strength of each predictor's association with the outcome. T-statistics and partial and semipartial correlations can add information about this as well.

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  • $\begingroup$ What if the opposite is the case and the predictors are correlated or there is one predictor with which several others are correlated to some degree? Is it possible to factor out those correlations and get a better estimate for the influence a predictor has on the target without the contribution of the influence another predictor has on it? $\endgroup$
    – jpp1
    Commented Sep 28, 2020 at 14:56
  • $\begingroup$ Yes. You will want to look into partial correlation, collinearity/multicollinearity, tolerance, and variance inflation. But your use of "influence" sounds like you are seeking causal relationships, which are prized but hard to legitimately obtain using most designs. $\endgroup$
    – rolando2
    Commented Sep 28, 2020 at 21:42
  • $\begingroup$ Thank you so much for your answer @rolando2! In my case I know that one indep variable likely causally influences other indeps and that this variable influences the target. But what I do not know and want to figure out is the additional "separate" correlation of the other independent vars. So in a way I want to "factor out" the influence of that one specific variable (at least it is not hidden/latent in my case!). Could you point me to methods or algorithms or papers for learning more about how to do this? $\endgroup$
    – jpp1
    Commented Sep 29, 2020 at 13:42

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