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I have done the following measurements:

  • Independent Variables IV1 and IV2 with a ordinal scale
  • Dependent Variable 'DV' with an ordinal scale
  • Tested 8 different values for IV1 and 6 different values for IV2 in a full factorial design with 5 repetition each and recorded the dependent variable. So in total 240 measurements were done.

Now an ANOVA test on these measurements show that with a very high probability (> 1 - 2e-16) both independent variables contribute to the variance of 'DV'. Thats what I expected, so everything OK until here.

> aov(DV~IV1+IV2, data=d)
Terms:
                IV1                  IV2 Residuals
Sum of Squares  1721.6289       918.6018  198.5668
Deg. of Freedom         7              5       227

Residual standard error: 0.9352772 
Estimated effects may be unbalanced

> summary(aov(DV~IV1+IV2, data=d))
                Df Sum Sq Mean Sq F value Pr(>F)    
IV1              7 1721.6  245.95   281.2 <2e-16 ***
IV2              5  918.6  183.72   210.0 <2e-16 ***
Residuals      227  198.6    0.87                   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

I now want to quantify how much influence a change in IV1 and a change in IV2 has on the dependent variable DV. How can I do this?

I'm not sure whether I'm asking the right question at all.

A solution which came to my mind was fitting a linear model to DV~IV1+IV2 (R syntax) an taking the coefficients as the influence. I'm not sure if this is the right way of looking at the data because I can't be sure that there is a linear dependency between DV and IV1, IV2.

> lm(DV~IV1+IV2, data=d)
Coefficients:
   (Intercept)       IV1            IV2  
     6.2580969       0.0002853      -0.0703225
theomega
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2 Answers2

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In order to see how the IVs influence the DV through the variance explained why don't you just look at the ANOVA table which for each variable provides the amount of remaining variance explained by that IV. In the newly provided ANOVA table in the edited question we see that the sum of squares explained by IV 1 is 1721.6 and 918.6 for IV 2. So in this sense IV 1 has a lot more influence on the DV then IV 2 when both are included in the model. Nevertheless you can see that both have huge values for their respective F statistics indicating that their coefficients are very significant for this model.

Michael R. Chernick
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    It would be all too easy (and natural) for many readers to draw incorrect conclusions from this answer. In the absence of further clarification, it appears to say that "amount of remaining variance explained" is the same thing as the coefficient in a linear model as mentioned in the question. To avoid such an unhappy ambiguity, this question needs an explanation of how we are meant to interpret "influence." – whuber Jul 12 '12 at 16:41
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    @whuber I edited my answer in a way that I hope clarifies my suggestion and eliminates any ambiguity of what we mean here by "influence". – Michael R. Chernick Jul 12 '12 at 16:49
  • Thanks for your answers. I editied my question and added a example for a ANOVA output. Could you tell me which values provide me with the information you are proposing? – theomega Jul 12 '12 at 17:20
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    The column Sum Sq right after the degrees of freedom is what you are looking for. – Michael R. Chernick Jul 12 '12 at 17:53
  • So can I tell from these numbers that IV2 has roughly half the influence on the DV as IV1? I know that my understanind of influence might be a little bit vague. Could you clarify what this means in this context? Perhaps by editing your answer so other users also befefit from your work! Thanks again! – theomega Jul 12 '12 at 17:59
  • Thanks for your edit! Would it be appropriate to calculate percentages from the sum of squares to quantify the influence in a easier way? Telling that 61% of the influence on the mode are of IV1, 32% are of IV2 and 7% are caused by the residuals. – theomega Jul 12 '12 at 19:42
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    I think that is okay. Keep in mind that if IV1 was not in the model IV2 could possible have a larger sum of squares and explain a larger percentage of the variance in the DV. – Michael R. Chernick Jul 12 '12 at 19:52
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The researcher-analyst may be better off subjecting the data to stepwise regression – a method whereby the predictive variables are added to the regression models in steps. In each step, a variable is considered for addition to, or subtraction from, the set of explanatory variables, based on some pre-determined and specified criterion. Usually, this takes the form of a sequence of F-tests or t-tests, but other techniques, such as adjusted R², Akaike information criterion, Bayesian information criterion, and Mallows's Cₚ are possible.

Hazel Traüffer
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