# [R] Interpreting summary.lm for a 2 factor anova

David Winsemius dwinsemius at comcast.net
Fri Dec 2 18:09:28 CET 2016

```> On Dec 2, 2016, at 6:16 AM, Ashim Kapoor <ashimkapoor at gmail.com> wrote:
>
> Dear Pikal,
>
> All levels except the interactions are compared to the Intercept. I'm a
> little confused as to what's going on in interaction terms eg. the cell
> wool B : tension M. It's mean is :
> 28.78 and 28.78 - 44.56 = -15.78 != 21.111.
>
> It's something like 44.56 (intercept) -16.333 (wool B) -.20.556 (tension
> M)  + 21.111 (woolB:tensionM) = 28.782.
>
> I don't know how to sum up the above line in terms of differences
> succinctly.

The aov estimate will not exactly equal the observed mean (this is _statistics_ after all). You should be comparing the mean of that cell to the estimate:

44.556 + (-16.33) +(-20.556) + (21.11)

The difference between the observed mean and the estimated mean is known as a 'residual' and the squared sum of the all residuals is what this being minimized ... over all the cells including the one implicitly associated with the Intercept.

This isn't really on-topic for Rhelp since you are not having difficulty in getting the R program to perform its duties, but are rather in need of statistical education. That not what this mailing list is set up for.

--
David.

>
>>
>>> -----Original Message-----
>>> From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Ashim
>>> Kapoor
>>> Sent: Thursday, December 1, 2016 2:48 PM
>>> To: r-help at r-project.org
>>> Subject: [R] Interpreting summary.lm for a 2 factor anova
>>>
>>> Dear all,
>>>
>>> Here is a small example : -
>>>
>>>> model <- aov(breaks ~ wool * tension, data = warpbreaks)
>>>> summary.lm(model)
>>>
>>> Call:
>>> aov(formula = breaks ~ wool * tension, data = warpbreaks)
>>>
>>> Residuals:
>>>     Min       1Q   Median       3Q      Max
>>> -19.5556  -6.8889  -0.6667   7.1944  25.4444
>>>
>>> Coefficients:
>>>               Estimate Std. Error t value Pr(>|t|)
>>> (Intercept)      44.556      3.647  12.218 2.43e-16 ***
>>> woolB           -16.333      5.157  -3.167 0.002677 **
>>> tensionM        -20.556      5.157  -3.986 0.000228 ***
>>> tensionH        -20.000      5.157  -3.878 0.000320 ***
>>> woolB:tensionM   21.111      7.294   2.895 0.005698 **
>>> woolB:tensionH   10.556      7.294   1.447 0.154327
>>> ---
>>> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>>>
>>> Residual standard error: 10.94 on 48 degrees of freedom
>>> Multiple R-squared:  0.3778,    Adjusted R-squared:  0.3129
>>> F-statistic: 5.828 on 5 and 48 DF,  p-value: 0.0002772
>>>
>>>> model.tables(model,"e")
>>> Tables of effects
>>>
>>> wool
>>> wool
>>>      A       B
>>> 2.8889 -2.8889
>>>
>>> tension
>>> tension
>>>     L      M      H
>>> 8.241 -1.759 -6.481
>>>
>>> wool:tension
>>>    tension
>>> wool L      M      H
>>>   A  5.278 -5.278  0.000
>>>   B -5.278  5.278  0.000
>>>
>>>
>>>> model.tables(model,"m")
>>> Tables of means
>>> Grand mean
>>>
>>> 28.14815
>>>
>>> wool
>>> wool
>>>     A      B
>>> 31.037 25.259
>>>
>>> tension
>>> tension
>>>    L     M     H
>>> 36.39 26.39 21.67
>>>
>>> wool:tension
>>>    tension
>>> wool L     M     H
>>>   A 44.56 24.00 24.56
>>>   B 28.22 28.78 18.78
>>>>
>>>
>>> I don't follow the output of summary.lm. I understand the output of
>>> model.tables for effects and means. For instance what does 44.556
>>> represent ? Is it the grand average ? The grand mean is 28.14815. Can
>>> someone help me understand the output of summary.lm ?
>>>
>>> Best Regards,
>>> Ashim
>>>
>>>      [[alternative HTML version deleted]]
>>>
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