# Interpretation of two-way ANOVA result

does anybody knows how to interpret two way ANOVA result. My data looks like:

Protein_name    holo_1      holo_2
A1              82.3965243  70.91176151
B1              27.26637961 47.63355456
C1              97.75786493 64.92764661
D1              115.9354513 127.4018061
E1              130.4860545 163.4261778
F1              57.13565305 142.0628876
G1              88.66907173 87.42791862
H1              184.2934171 150.3209662
I1              95.70968618 68.99684474
J1              53.80736258 79.40920466
K1              166.5425346 97.48123164

I have run two ANOVA test in R studio with this commands:

x = c(82.3965243, 27.26637961, 97.75786493, 115.9354513, 130.4860545, 57.13565305, 88.66907173, 184.2934171, 95.70968618, 53.80736258, 166.5425346, 70.91176151, 47.63355456, 64.92764661, 127.4018061, 163.4261778, 142.0628876, 87.42791862, 150.3209662, 68.99684474, 79.40920466, 97.48123164)

h = as.factor(c(1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2))
p = as.factor(c(1,2,3,4,5,6,7,8,9,10,11,1,2,3,4,5,6,7,8,9,10,11))

cbind(x,h,p)

AOV.OUT = aov(x~h+p)
summary(AOV.OUT)

shapiro.test(AOV.OUT$resi) The output is : > cbind(x,h,p) x h p [1,] 82.39652 1 1 [2,] 27.26638 1 2 [3,] 97.75786 1 3 [4,] 115.93545 1 4 [5,] 130.48605 1 5 [6,] 57.13565 1 6 [7,] 88.66907 1 7 [8,] 184.29342 1 8 [9,] 95.70969 1 9 [10,] 53.80736 1 10 [11,] 166.54253 1 11 [12,] 70.91176 2 1 [13,] 47.63355 2 2 [14,] 64.92765 2 3 [15,] 127.40181 2 4 [16,] 163.42618 2 5 [17,] 142.06289 2 6 [18,] 87.42792 2 7 [19,] 150.32097 2 8 [20,] 68.99684 2 9 [21,] 79.40920 2 10 [22,] 97.48123 2 11 > AOV.OUT = aov(x~h+p) > summary(AOV.OUT) Df Sum Sq Mean Sq F value Pr(>F) h 1 0 0.0 0.000 1.0000 p 10 29209 2920.9 3.367 0.0343 * Residuals 10 8674 867.4 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > shapiro.test(AOV.OUT$resi)

I am trying to compare my sample two way : (1) holo1 vs holo2 and (2) across the proteins as well.

Does anybody know how to interpret this result?

Shapiro-Wilk normality test

data:  AOV.OUT\$resi
W = 0.97786, p-value = 0.8795

I am having a bit hard time finding how to interpret this type of ANOVA output online. Has anybody have an explanation what W is and do I compare it with the p value? Also which part of the result gives me comparison of holo_1 vs holo_2 and which type of the output gives me comparison among protein_names?

• I see several problems here. 1/ In your original data, you seem to have paired data, i.e. 82.3965243 and 70.91176151 seem to be taken on the same "individual", but your model does not take that into account. Can you clarify what are your data and your question? 2/ Your factor p has 11 levels and only two individuals per level, which is very few. 3/ W is the test statistic to check normality of residuals. But the (global) Shapiro test (if performed alone) is not very useful here imho. Commented Apr 8, 2020 at 6:18
• @Philopolis Related to the first one: Yes the two numbers are taken from the same "individual" and I think that is the issue I am having to understand how to implement it as this test doesnt seem to be suitable. I have 11 proteins (Protein_name) and I have used the same method to obtain number for (holo_1 and holo_2). Holo_1 and holo_2 relate to the same individual protein but they differ in experimental method they are obtained. Since I am dealing with different 11 proteins, I have normalized my data and the numbers presented are normalized data. Commented Apr 8, 2020 at 6:30
• @Philopolis I want to perform comparison of holo_1 to holo_2 over 11 samples (I think that can be done because now my data is normalized) and see if holo_1 has statistically better performance then holo_2. Also, I want to see over 11 proteins, which Protein tend to have "better" performance or higher number that is statistically significantly higher across 11 proteins . Commented Apr 8, 2020 at 6:32
• Okay! So I think that, basically, a simple paired test (e.g., Wilcoxon) associated to any useful graphical representation (maybe a parallel coordinates plot?) would be sufficient here. Maybe other people out there will have a better suggestion. ;) Commented Apr 8, 2020 at 6:52
• This seems to be a continuation of your previous question. // Your output shows no significant difference between holo_1 and holo_2. But there are significant differences among proteins: This is hardly surprising with B1 averaging about 37 and H1 averaging about 160. If there is a way to classify proteins into established similar types it might be worthwhile to look for diffs among types. But maybe too many proteins to do post hoc comaprisons for all 11, Commented Apr 8, 2020 at 6:58