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I have run a multiple regression in which the model as a whole is significant and explains about 13% of the variance. However, I need to find the amount of variance explained by each significant predictor. How can I do this using R?

Here's some sample data and code:

D = data.frame(
    dv = c( 0.75, 1.00, 1.00, 0.75, 0.50, 0.75, 1.00, 1.00, 0.75, 0.50 ),
    iv1 = c( 0.75, 1.00, 1.00, 0.75, 0.75, 1.00, 0.50, 0.50, 0.75, 0.25 ),
    iv2 = c( 0.882, 0.867, 0.900, 0.333, 0.875, 0.500, 0.882, 0.875, 0.778, 0.867 ),
    iv3 = c( 1.000, 0.067, 1.000, 0.933, 0.875, 0.500, 0.588, 0.875, 1.000, 0.467 ),
    iv4 = c( 0.889, 1.000, 0.905, 0.938, 0.833, 0.882, 0.444, 0.588, 0.895, 0.812 ),
    iv5 = c( 18, 16, 21, 16, 18, 17, 18, 17, 19, 16 ) )
fit = lm( dv ~ iv1 + iv2 + iv3 + iv4 + iv5, data=D )
summary( fit )

Here's the output with my actual data:

Call: lm(formula = posttestScore ~ pretestScore + probCategorySame + 
    probDataRelated + practiceAccuracy + practiceNumTrials, data = D)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6881 -0.1185  0.0516  0.1359  0.3690 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)
 (Intercept)        0.77364    0.10603    7.30  8.5e-13 ***
 iv1                0.29267    0.03091    9.47  < 2e-16 ***
 iv2                0.06354    0.02456    2.59   0.0099 **
 iv3                0.00553    0.02637    0.21   0.8340
 iv4               -0.02642    0.06505   -0.41   0.6847
 iv5               -0.00941    0.00501   -1.88   0.0607 .  
--- Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.18 on 665 degrees of freedom
 Multiple R-squared:  0.13,      Adjusted R-squared:  0.123
 F-statistic: 19.8 on 5 and 665 DF,  p-value: <2e-16

This question has been answered here, but the accepted answer only addresses uncorrelated predictors, and while there is an additional response that addresses correlated predictors, it only provides a general hint, not a specific solution. I would like to know what to do if my predictors are correlated.

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    $\begingroup$ Did you look at the answer by Jeromy Anglim in here? $\endgroup$
    – Stat
    Dec 11, 2013 at 22:47
  • $\begingroup$ Yes, that was the additional response that I was referring to. I was hoping for something more specific and step-by-step. I did download ppcor but was not sure what to do with the spcor output. Also, I am wondering whether there is a way to do this in core R? It seems like a common enough task that it would not require a special package. $\endgroup$
    – baixiwei
    Dec 11, 2013 at 22:54
  • $\begingroup$ The shortest answer to your question about correlated predictors is that their separate importance cannot be quantified, without at the very least further assumptions and approximations. Consider it this way: if this is straightforward, why is it not readily and easily available, because many researchers think they want it? $\endgroup$
    – Nick Cox
    Dec 12, 2013 at 11:43
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    $\begingroup$ I would suggest looking into the relaimpo package, and its accompanying paper: jstatsoft.org/index.php/jss/article/view/v017i01/v17i01.pdf I use the "LMG" method frequently. $\endgroup$
    – Phil
    Aug 21, 2016 at 18:37

4 Answers 4

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The percentage explained depends on the order entered.

If you specify a particular order, you can compute this trivially in R (e.g. via the update and anova functions, see below), but a different order of entry would yield potentially very different answers.

[One possibility might be to average across all orders or something, but it would get unwieldy and might not be answering a particularly useful question.]

--

As Stat points out, with a single model, if you're after one variable at a time, you can just use 'anova' to produce the incremental sums of squares table. This would follow on from your code:

 anova(fit)
Analysis of Variance Table

Response: dv
          Df   Sum Sq  Mean Sq F value Pr(>F)
iv1        1 0.033989 0.033989  0.7762 0.4281
iv2        1 0.022435 0.022435  0.5123 0.5137
iv3        1 0.003048 0.003048  0.0696 0.8050
iv4        1 0.115143 0.115143  2.6294 0.1802
iv5        1 0.000220 0.000220  0.0050 0.9469
Residuals  4 0.175166 0.043791        

--

So there we have the incremental variance explained; how do we get the proportion?

Pretty trivially, scale them by 1 divided by their sum. (Replace the 1 with 100 for percentage variance explained.)

Here I've displayed it as an added column to the anova table:

 af <- anova(fit)
 afss <- af$"Sum Sq"
 print(cbind(af,PctExp=afss/sum(afss)*100))
          Df       Sum Sq      Mean Sq    F value    Pr(>F)      PctExp
iv1        1 0.0339887640 0.0339887640 0.77615140 0.4280748  9.71107544
iv2        1 0.0224346357 0.0224346357 0.51230677 0.5137026  6.40989591
iv3        1 0.0030477233 0.0030477233 0.06959637 0.8049589  0.87077807
iv4        1 0.1151432643 0.1151432643 2.62935731 0.1802223 32.89807550
iv5        1 0.0002199726 0.0002199726 0.00502319 0.9468997  0.06284931
Residuals  4 0.1751656402 0.0437914100         NA        NA 50.04732577

--

If you decide you want several particular orders of entry, you can do something even more general like this (which also allows you to enter or remove groups of variables at a time if you wish):

 m5 = fit
 m4 = update(m5, ~ . - iv5)
 m3 = update(m4, ~ . - iv4)
 m2 = update(m3, ~ . - iv3)
 m1 = update(m2, ~ . - iv2)
 m0 = update(m1, ~ . - iv1)

 anova(m0,m1,m2,m3,m4,m5)
Analysis of Variance Table

Model 1: dv ~ 1
Model 2: dv ~ iv1
Model 3: dv ~ iv1 + iv2
Model 4: dv ~ iv1 + iv2 + iv3
Model 5: dv ~ iv1 + iv2 + iv3 + iv4
Model 6: dv ~ iv1 + iv2 + iv3 + iv4 + iv5
  Res.Df     RSS Df Sum of Sq      F Pr(>F)
1      9 0.35000                           
2      8 0.31601  1  0.033989 0.7762 0.4281
3      7 0.29358  1  0.022435 0.5123 0.5137
4      6 0.29053  1  0.003048 0.0696 0.8050
5      5 0.17539  1  0.115143 2.6294 0.1802
6      4 0.17517  1  0.000220 0.0050 0.9469

(Such an approach might also be automated, e.g. via loops and the use of get. You can add and remove variables in multiple orders if needed)

... and then scale to percentages as before.

(NB. The fact that I explain how to do these things should not necessarily be taken as advocacy of everything I explain.)

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    $\begingroup$ @ Glen_b: no offense, but this is not an answer to the question. There is no $R^2$ in your output. And by the way, the anova(fit) will give similar out put as yours even without defining $m_0$-$m_5$! Cheers. $\endgroup$
    – Stat
    Dec 11, 2013 at 23:53
  • $\begingroup$ This revised answer is really useful. I think I'm getting there. One question: If I calculate the proportion of variance explained for iv5 (the last variable) in the manner you described, is this mathematically the same as the difference in R^2 values returned by summary applied to the model fits with and without iv5? I am in fact getting the same values and just wanted to check whether these are conceptually the same thing. $\endgroup$
    – baixiwei
    Dec 12, 2013 at 22:50
  • $\begingroup$ And one more question: is there any reason I couldn't do what I just described in the previous comment once for each of two different ivs? Would that be equivalent to your second proposed method involving different orders of entering variables? $\endgroup$
    – baixiwei
    Dec 12, 2013 at 22:52
  • $\begingroup$ baixiwei - yes, those two differences in $R^2$ will be identical. Yes, you could do a 'fitted last' contribution for each. (In fact the t-test in the regression table from summary.lm gives you a test of the significance of each variable fitted last already.) $\endgroup$
    – Glen_b
    Dec 13, 2013 at 0:34
  • $\begingroup$ @Glen_b I know this is a very old answer.. But I was wondering to what extent your (not necessarily advocated) answer, translates to non-linear models. I posted a new question about this (linking to this page) here: stats.stackexchange.com/questions/522588/… $\endgroup$
    – Tom
    May 3, 2021 at 15:44
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I proved that the percentage of variation explained by a given predictor in a multiple linear regression is the product of the slope coefficient and the correlation of the predictor with the fitted values of the dependent variable (assuming that all variables have been standardized to have mean zero and variance one; which is without loss of generality). Find it here:

https://www.researchgate.net/publication/306347340_A_Natural_Decomposition_of_R2_in_Multiple_Linear_Regression

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    $\begingroup$ user128460 welcome, but this is a Question and Answer site, not a Question and Link-To-Answer site. $\endgroup$ Aug 21, 2016 at 19:34
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    $\begingroup$ Isn't that the Pratt score? $\endgroup$
    – Brett
    Aug 14, 2018 at 20:30
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You can use hier.part library to have goodness of fit measures for regressions of a single dependent variable to all combinations of N independent variables

library(hier.part)
env <- D[,2:5]
all.regs(D$dv, env, fam = "gaussian", gof = "Rsqu",
     print.vars = TRUE)
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I am just re-posting the comment of @Phil here because this is clearly the best answer:

I would suggest looking into the relaimpo package, and its accompanying paper: jstatsoft.org/index.php/jss/article/view/v017i01/v17i01.pdf I use the "LMG" method frequently.

I have been searching the answer to this question for 5 hours now, skim-reading some papers, and indeed relaimpo::lmg seems a great solution. One can also use relaimpo::pmvd, relaimpo::pratt (the latter corresponds to @user128460 's answer, and has the problem of sometimesyielding negative shares), or methods relying on random forest. See these papers for more info: https://doi.org/10.1198/000313007X188252
https://doi.org/10.7275/5FEX-B874
https://www.sciencedirect.com/science/article/pii/S0951832015001672
https://www.jstor.org/stable/25652309

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