# Difference between effect size (partial $R^2$) and coefficients [duplicate]

I am working with spoken language data and use linear models do determine the relationship between different phonological processes in my data.

Background

Measures of the regularity of syllable durations are influenced by how syllable is defined. I used two methods to determine syllable boundaries. Method 1 and 2 give different results for regularity of syllable durations, but the difference between the two methods is not the same for every speaker.

Linear model

I think the frequency of certain phonological processes (vowel elision, insertion of glottal stops and others) might be responisble for these differences. I want to find out if that's true and if yes how much each of these phonological processes contributes to explaining the differences.

Measuring effect size with partial $R^2$/sum of squares

To measure effect size, I could use partial $R^2$. The data below is the ANOVA of my model, and from the variance explained by each factor (Sum Sq) I can derive how much of the total variance each factor explains.

Analysis of Variance Table

Response: varcos
Df Sum Sq Mean Sq F value    Pr(>F)
rel_gs                     1 395.48  395.48 46.9135 7.022e-07 ***
rel_all2_gs                1   9.47    9.47  1.1236    0.3006
rel_gs:rel_all_gs          1   1.37    1.37  0.1622    0.6910
rel_gs:rel_all2_gs         1  13.63   13.63  1.6168    0.2168
rel_all_gs:relDur.mean.y   1  15.94   15.94  1.8910    0.1829
rel_all2_gs:relDur.mean.y  1  69.19   69.19  8.2079    0.0090 **
Residuals                 22 185.46    8.43


Coeffcients

Beta coefficients can also be interpreted as a measure of effect size, as an answer to this question points out. I was expecting that factors that explain a large part of the variance would also have large coeffcients, and the other way around. But that's not the case. rel_gs:rel_all2_gs has a huge negative coefficient, but explains only a small part of total variance.

Call:
lm(formula = varcos ~ rel_gs + rel_all2_gs + rel_all_gs:rel_gs +
rel_all2_gs:rel_gs + relDur.mean.y:rel_all_gs + relDur.mean.y:rel_all2_gs,

Residuals:
Min      1Q  Median      3Q     Max
-5.7929 -1.5933 -0.5208  1.5569  5.4834

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)                  15.539      2.725   5.702  9.8e-06 ***
rel_gs                     -336.020    118.306  -2.840  0.00952 **
rel_all2_gs                 492.997    183.440   2.688  0.01345 *
rel_gs:rel_all_gs          2533.220    810.012   3.127  0.00490 **
rel_gs:rel_all2_gs        -2292.357    716.625  -3.199  0.00414 **
rel_all_gs:relDur.mean.y   -955.239    315.827  -3.025  0.00623 **
rel_all2_gs:relDur.mean.y   691.578    241.394   2.865  0.00900 **


Question

Should I only trust partial $R^2$ for effect size and ignore differences between the coefficients?

P.S.: I asked a related question some time ago, and it hasn't received much attention and left some issues open. I suppose that might be because the original question wasn't clear enough or was based on toy data. In this question I'm using actual data. If my question could be imporoved in some way I'd be happy to consider any suggestions.

• Thanks for your answer! I applied scaling in this question and there were still substantial differences between the size of the coefficients and partial $R^2$. Are there cases when you think scaling is not appropriate? Jul 29, 2013 at 16:55