[R] How to interpret an ANOVA result?
Bryan Hanson
hanson at depauw.edu
Tue May 15 13:59:28 CEST 2012
I see that no one has replied on this, so I'll take a stab.
This is probably a matter of personal taste, but I would suggest a somewhat different and simpler approach. What you have done is not strictly an ANOVA, it's a linear model (they are related). But the particular way you've asked R to report gives you the answer in terms of the linear model. That means your significance stars refer to whether or not the slopes in the model differ significantly from zero. Perhaps you are aware of this.
Anyway, I thought your data set was interesting, so I took the approach that comes to my mind. Here it is. It might be pretty much self-explanatory, if not, try ?aov and ?TukeyHSD for details. Maybe it answers your questions about why things are significant or not. Hopefully I didn't misunderstand your questions.
Good Luck. Bryan
***********
Bryan Hanson
Professor of Chemistry & Biochemistry
DePauw University
Using your full data set, in variable x:
res <- aov(d~site*pos, data = x)
summary(res)
Df Sum Sq Mean Sq F value Pr(>F)
site 4 636.5 159.13 8.397 0.00308 **
pos 1 0.0 0.05 0.003 0.96005
site:pos 4 59.7 14.93 0.788 0.55886
Residuals 10 189.5 18.95
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
# So clearly site is the only significant factor.
# Use TukeyHSD to see which sites are different from each other:
TukeyHSD(res)
Tukey multiple comparisons of means
95% family-wise confidence level
Fit: aov(formula = d ~ site * pos, data = x)
$site
diff lwr upr p adj
2-1 9.50 -0.6304407 19.6304407 0.0686646
3-1 6.75 -3.3804407 16.8804407 0.2569519
4-1 14.25 4.1195593 24.3804407 0.0064949
5-1 -0.50 -10.6304407 9.6304407 0.9998140
3-2 -2.75 -12.8804407 7.3804407 0.8930095
4-2 4.75 -5.3804407 14.8804407 0.5603830
5-2 -10.00 -20.1304407 0.1304407 0.0533983
4-3 7.50 -2.6304407 17.6304407 0.1824748
5-3 -7.25 -17.3804407 2.8804407 0.2049583
5-4 -14.75 -24.8804407 -4.6195593 0.0051227
$pos
diff lwr upr p adj
R-N -0.1 -4.437723 4.237723 0.960045
$`site:pos`
diff lwr upr p adj
2:N-1:N 11.5 -5.7326665 28.732667 0.3077877
3:N-1:N 12.0 -5.2326665 29.232667 0.2662504
4:N-1:N 17.0 -0.2326665 34.232667 0.0539873
5:N-1:N 1.0 -16.2326665 18.232667 0.9999999
1:R-1:N 4.5 -12.7326665 21.732667 0.9818788
2:R-1:N 12.0 -5.2326665 29.232667 0.2662504
3:R-1:N 6.0 -11.2326665 23.232667 0.9093271
4:R-1:N 16.0 -1.2326665 33.232667 0.0750508
5:R-1:N 2.5 -14.7326665 19.732667 0.9997412
3:N-2:N 0.5 -16.7326665 17.732667 1.0000000
4:N-2:N 5.5 -11.7326665 22.732667 0.9418966
5:N-2:N -10.5 -27.7326665 6.732667 0.4048556
1:R-2:N -7.0 -24.2326665 10.232667 0.8191838
2:R-2:N 0.5 -16.7326665 17.732667 1.0000000
3:R-2:N -5.5 -22.7326665 11.732667 0.9418966
4:R-2:N 4.5 -12.7326665 21.732667 0.9818788
5:R-2:N -9.0 -26.2326665 8.232667 0.5798223
4:N-3:N 5.0 -12.2326665 22.232667 0.9658140
5:N-3:N -11.0 -28.2326665 6.232667 0.3540251
1:R-3:N -7.5 -24.7326665 9.732667 0.7639472
2:R-3:N 0.0 -17.2326665 17.232667 1.0000000
3:R-3:N -6.0 -23.2326665 11.232667 0.9093271
4:R-3:N 4.0 -13.2326665 21.232667 0.9915584
5:R-3:N -9.5 -26.7326665 7.732667 0.5185806
5:N-4:N -16.0 -33.2326665 1.232667 0.0750508
1:R-4:N -12.5 -29.7326665 4.732667 0.2293315
2:R-4:N -5.0 -22.2326665 12.232667 0.9658140
3:R-4:N -11.0 -28.2326665 6.232667 0.3540251
4:R-4:N -1.0 -18.2326665 16.232667 0.9999999
5:R-4:N -14.5 -31.7326665 2.732667 0.1224175
1:R-5:N 3.5 -13.7326665 20.732667 0.9966588
2:R-5:N 11.0 -6.2326665 28.232667 0.3540251
3:R-5:N 5.0 -12.2326665 22.232667 0.9658140
4:R-5:N 15.0 -2.2326665 32.232667 0.1041067
5:R-5:N 1.5 -15.7326665 18.732667 0.9999963
2:R-1:R 7.5 -9.7326665 24.732667 0.7639472
3:R-1:R 1.5 -15.7326665 18.732667 0.9999963
4:R-1:R 11.5 -5.7326665 28.732667 0.3077877
5:R-1:R -2.0 -19.2326665 15.232667 0.9999581
3:R-2:R -6.0 -23.2326665 11.232667 0.9093271
4:R-2:R 4.0 -13.2326665 21.232667 0.9915584
5:R-2:R -9.5 -26.7326665 7.732667 0.5185806
4:R-3:R 10.0 -7.2326665 27.232667 0.4599257
5:R-3:R -3.5 -20.7326665 13.732667 0.9966588
5:R-4:R -13.5 -30.7326665 3.732667 0.1683967
# Normally you can plot a TukeyHSD object but this one is too large.
# The end.
> Hello all,
>
> here's a real-world example: I'm measuring a quantity (d) at five
> sites (site1 thru site5) on a silicon wafer. There is a clear
> site-dependence of the measured value. To find out if this is a
> measurement artifact I measured the wafer four times: twice in the
> normal position (posN), and twice rotated by 180 degrees (posR). My
> data looks like this (full, self-contained code at bottom). Note that
> sites with the same number correspond to the same physical location on
> the wafer (the rotation has already been taken into account here).
>
>> head(x)
> d site pos
> 1 1383 1 N
> 2 1377 1 R
> 3 1388 1 R
> 4 1373 1 N
> 5 1386 2 N
> 6 1394 2 R
>
>> boxplot (d~pos+site)
>
> This boxplot (see code) already hints at a true site-dependence of the
> measured value (no artifact). OK, so let's do an ANOVA to make this
> more quantitative:
>
>> summary(lm(d ~ site*pos)
>
> Coefficients:
> Estimate Std. Error t value Pr(>|t|)
> (Intercept) 1378.000 3.078 447.672 < 2e-16 ***
> site2 11.500 4.353 2.642 0.02466 *
> site3 12.000 4.353 2.757 0.02025 *
> site4 17.000 4.353 3.905 0.00294 **
> site5 1.000 4.353 0.230 0.82294
> posR 4.500 4.353 1.034 0.32561
> site2:posR -4.000 6.156 -0.650 0.53050
> site3:posR -10.500 6.156 -1.706 0.11890
> site4:posR -5.500 6.156 -0.893 0.39264
> site5:posR -3.000 6.156 -0.487 0.63655
>
> Now I think that I see the following:
> - The average of d at site1 in pos. N (first in alphabet) is 1378.
> - Average values for site2, 3, 4 (especially 4) in pos. N deviate
> significantly from pos. 1. For instance, values at site4 are on
> average 17 greater than at site1.
> - The average value at site5 does not differ significantly from site1.
> OK, that was the top part of the result table. Now the bottom part:
> - In reverse position(posR) the average of d at site1 is 4.5 bigger,
> but that's not significant.
> - The average of d at site3:posR is 10.5 smaller than something, but
> smaller than what? And why does this -10.5 deviation have a p-value of
> .1 (not significant) vs the .02 (significant) deviation of 11.5
> (site2, top part)?
>
> Let's see if I can figure that out. Difference between posN and posR
> at site3 is not so big:
>> mean(d[site==3&pos=="R"])-mean(d[site==3&pos=="N"])
> [1] -6
> Is this what makes it insignificant?
>
> Shuffling around the numbers until I get to -10.5:
>
>> mean(d[site==3&pos=="R"])-mean(d[site==3&pos=="N"])-(mean(d[site==1&pos=="R"])-mean(d[site==1&pos=="N"]))
> [1] -10.5
>
> OK, one has to keep track of all the differences and stuff.
>
> So I think I have understood about 80% of this simple example. The
> reason I'm going after this so stubbornly is that I'm at the beginning
> of a DOE which will take several weeks of measuring and will end up
> being analyzed with a big ANOVA (two response and about six
> explanatory variables, some continuous, some factorial). Already in
> the DOE phase I want to understand what I will be doing with the data
> later (this is for a Six Sigma project in an industrial production
> environment, in case anybody wants to know).
>
> Thanks,
> robert
>
> Here's the full dataset:
>
> x <- structure(list(d = c(1383L, 1377L, 1388L, 1373L, 1386L, 1394L,
> 1386L, 1393L, 1390L, 1382L, 1386L, 1390L, 1395L, 1396L, 1392L,
> 1395L, 1378L, 1382L, 1379L, 1380L), site = structure(c(1L, 1L,
> 1L, 1L, 2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 5L, 5L,
> 5L, 5L), .Label = c("1", "2", "3", "4", "5"), class = "factor"),
> pos = structure(c(1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L,
> 2L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 1L), .Label = c("N",
> "R"), class = "factor")), .Names = c("d", "site", "pos"), row.names = c(NA,
> -20L), class = "data.frame")
> attach(x)
> head(x)
> boxplot (d~pos+site)
>
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