# How to interpret p-values of 0 or 1?

I ran an ANOVA finding for example an interaction between gender and grade than I want to know in what grades boys and girls differ, but in many cases I find (adjusted) p-values of 0 and 1. How / why is this possible? Doesn't seem right...

as.factor(gender)                     1     16    16.2    2.6377  0.104396
as.factor(grade)                      7  50077  7153.9 1165.4184 < 2.2e-16 ***
as.factor(gender):as.factor(grade)    7    132    18.9    3.0795  0.003056 **
Residuals                          7747  47555     6.1
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Tukey multiple comparisons of means
95% family-wise confidence level

Fit: aov(formula = rating ~ as.factor(gender) * as.factor(grade), data = users_c[users_c$grade %in% 1:8, ])$as.factor(gender)
m-f -0.09135851 -0.2016276 0.01891058 0.1043964

$as.factor(grade) diff lwr upr p adj 2-1 0.3823566 -0.5454435 1.310157 0.9169296 3-1 1.9796023 1.1649854 2.794219 0.0000000 4-1 3.9558543 3.1534606 4.758248 0.0000000 5-1 5.7843111 4.9829529 6.585669 0.0000000 6-1 7.0752044 6.2708610 7.879548 0.0000000 7-1 8.4868609 7.6776332 9.296089 0.0000000 8-1 9.3867231 8.5626511 10.210795 0.0000000 3-2 1.5972457 1.0395026 2.154989 0.0000000 4-2 3.5734976 3.0337642 4.113231 0.0000000 5-2 5.4019544 4.8637616 5.940147 0.0000000 6-2 6.6928478 6.1502200 7.235476 0.0000000 7-2 8.1045042 7.5546625 8.654346 0.0000000 8-2 9.0043665 8.4329024 9.575831 0.0000000 4-3 1.9762520 1.6694948 2.283009 0.0000000 5-3 3.8047088 3.5006705 4.108747 0.0000000 6-3 5.0956021 4.7837806 5.407424 0.0000000 7-3 6.5072586 6.1830461 6.831471 0.0000000 8-3 7.4071208 7.0474558 7.766786 0.0000000 5-4 1.8284568 1.5588754 2.098038 0.0000000 6-4 3.1193501 2.8410202 3.397680 0.0000000 7-4 4.5310066 4.2388618 4.823151 0.0000000 8-4 5.4308688 5.0998193 5.761918 0.0000000 6-5 1.2908933 1.0155630 1.566224 0.0000000 7-5 2.7025498 2.4132612 2.991838 0.0000000 8-5 3.6024120 3.2738803 3.930944 0.0000000 7-6 1.4116565 1.1141985 1.709114 0.0000000 8-6 2.3115187 1.9757711 2.647266 0.0000000 8-7 0.8998622 0.5525763 1.247148 0.0000000$as.factor(gender):as.factor(grade)
m:1-f:1  0.005917865 -1.77842639  1.7902621 1.0000000
f:2-f:1  0.318074165 -1.28953805  1.9256864 0.9999988
m:2-f:1  0.442924925 -1.11597060  2.0018205 0.9998619
f:3-f:1  1.769000750  0.35262166  3.1853798 0.0020136
m:3-f:1  2.174229216  0.76569156  3.5827669 0.0000147
f:4-f:1  3.738998543  2.34268666  5.1353104 0.0000000
m:4-f:1  4.163719997  2.77146170  5.5559783 0.0000000
f:5-f:1  5.769586591  4.37599400  7.1631792 0.0000000
m:5-f:1  5.816721075  4.42497532  7.2084668 0.0000000
f:6-f:1  7.169439003  5.77317769  8.5657003 0.0000000
m:6-f:1  7.000924045  5.60308216  8.3987659 0.0000000
f:7-f:1  8.330142924  6.92683436  9.7334515 0.0000000
m:7-f:1  8.674488370  7.26930678 10.0796700 0.0000000
f:8-f:1  9.535307293  8.11198164 10.9586329 0.0000000
m:8-f:1  9.251081088  7.82191240 10.6802498 0.0000000
f:2-m:1  0.312156300 -1.12690148  1.7512141 0.9999959
m:2-m:1  0.437007060 -0.94741539  1.8214295 0.9995001
f:3-m:1  1.763082885  0.54136279  2.9848030 0.0000892
m:3-m:1  2.168311350  0.95569081  3.3809319 0.0000001
f:4-m:1  3.733080678  2.53468294  4.9314784 0.0000000
m:4-m:1  4.157802132  2.96412989  5.3514744 0.0000000
f:5-m:1  5.763668726  4.56844048  6.9588970 0.0000000
m:5-m:1  5.810803210  4.61772882  7.0038776 0.0000000
f:6-m:1  7.163521138  5.96518233  8.3618599 0.0000000
m:6-m:1  6.995006180  5.79482611  8.1951862 0.0000000
f:7-m:1  8.324225059  7.11768240  9.5307677 0.0000000
m:7-m:1  8.668570505  7.45984987  9.8772911 0.0000000
f:8-m:1  9.529389428  8.29962271 10.7591561 0.0000000
m:8-m:1  9.245163223  8.00863850 10.4816879 0.0000000
m:2-f:2  0.124850760 -1.02282435  1.2725259 1.0000000
f:3-f:2  1.450926585  0.50586965  2.3959835 0.0000172
m:3-f:2  1.856155050  0.92289131  2.7894188 0.0000000
f:4-f:2  3.420924378  2.50621691  4.3356318 0.0000000
m:4-f:2  3.845645832  2.93713824  4.7541534 0.0000000
f:5-f:2  5.451512425  4.54096139  6.3620635 0.0000000
m:5-f:2  5.498646910  4.59092496  6.4063689 0.0000000
f:6-f:2  6.851364838  5.93673457  7.7659951 0.0000000
m:6-f:2  6.682849880  5.76580854  7.5998912 0.0000000
f:7-f:2  8.012068759  7.08671595  8.9374216 0.0000000
m:7-f:2  8.356414205  7.42822339  9.2846050 0.0000000
f:8-f:2  9.217233128  8.26179669 10.1726696 0.0000000
m:8-f:2  8.933006923  7.96888762  9.8971262 0.0000000
f:3-m:2  1.326075825  0.46649985  2.1856518 0.0000150
m:3-m:2  1.731304290  0.88471145  2.5778971 0.0000000
f:4-m:2  3.296073618  2.46998162  4.1221656 0.0000000
m:4-m:2  3.720795071  2.90157332  4.5400168 0.0000000
f:5-m:2  5.326661665  4.50517434  6.1481490 0.0000000
m:5-m:2  5.373796150  4.55544575  6.1921465 0.0000000
f:6-m:2  6.726514078  5.90050756  7.5525206 0.0000000
m:6-m:2  6.557999120  5.72932364  7.3866746 0.0000000
f:7-m:2  7.887217999  7.04935402  8.7250820 0.0000000
m:7-m:2  8.231563445  7.39056617  9.0725607 0.0000000
f:8-m:2  9.092382368  8.22140761  9.9633571 0.0000000
m:8-m:2  8.808156163  7.92766524  9.6886471 0.0000000
m:3-f:3  0.405228465 -0.13578346  0.9462404 0.4221367
f:4-f:3  1.969997793  1.46166478  2.4783308 0.0000000
m:4-f:3  2.394719246  1.89762897  2.8918095 0.0000000
f:5-f:3  4.000585840  3.49977062  4.5014011 0.0000000
m:5-f:3  4.047720325  3.55206739  4.5433733 0.0000000
f:6-f:3  5.400438253  4.89224417  5.9086323 0.0000000
m:6-f:3  5.231923295  4.71940255  5.7444440 0.0000000
f:7-f:3  6.561142174  6.03389412  7.0883902 0.0000000
m:7-f:3  6.905487620  6.37327442  7.4377008 0.0000000
f:8-f:3  7.766306543  7.18788499  8.3447281 0.0000000
m:8-f:3  7.482080337  6.88942637  8.0747343 0.0000000
f:4-m:3  1.564769328  1.07871270  2.0508260 0.0000000
m:4-m:3  1.989490781  1.51520464  2.4637769 0.0000000
f:5-m:3  3.595357375  3.11716862  4.0735461 0.0000000
m:5-m:3  3.642491860  3.16971239  4.1152713 0.0000000
f:6-m:3  4.995209787  4.50929846  5.4811211 0.0000000
m:6-m:3  4.826694830  4.33626022  5.3171294 0.0000000
f:7-m:3  6.155913709  5.65010831  6.6617191 0.0000000
m:7-m:3  6.500259155  5.98928021  7.0112381 0.0000000
f:8-m:3  7.361078078  6.80213257  7.9200236 0.0000000
m:8-m:3  7.076851872  6.50319055  7.6505132 0.0000000
m:4-f:4  0.424721453 -0.01192015  0.8613631 0.0668946
f:5-f:4  2.030588047  1.58971048  2.4714656 0.0000000
m:5-f:4  2.077722532  1.64271796  2.5127271 0.0000000
f:6-f:4  3.430440460  2.98119847  3.8796825 0.0000000
m:6-f:4  3.261925502  2.80779484  3.7160562 0.0000000
f:7-f:4  4.591144381  4.12045589  5.0618329 0.0000000
m:7-f:4  4.935489827  4.45924616  5.4117335 0.0000000
f:8-f:4  5.796308750  5.26892973  6.3236878 0.0000000
m:8-f:4  5.512082545  4.96913148  6.0550336 0.0000000
f:5-m:4  1.605866594  1.17800058  2.0337326 0.0000000
m:5-m:4  1.653001078  1.23118920  2.0748130 0.0000000
f:6-m:4  3.005719006  2.56923916  3.4421989 0.0000000
m:6-m:4  2.837204048  2.39569420  3.2787139 0.0000000
f:7-m:4  4.166422928  3.70789927  4.6249466 0.0000000
m:7-m:4  4.510768373  4.04654394  4.9749928 0.0000000
f:8-m:4  5.371587296  4.85503631  5.8881383 0.0000000
m:8-m:4  5.087361091  4.55492128  5.6198009 0.0000000
m:5-f:5  0.047134485 -0.37906079  0.4733298 1.0000000
f:6-f:5  1.399852412  0.95913504  1.8405698 0.0000000
m:6-f:5  1.231337454  0.78563790  1.6770370 0.0000000
f:7-f:5  2.560556334  2.09799705  3.0231156 0.0000000
m:7-f:5  2.904901779  2.43669086  3.3731127 0.0000000
f:8-f:5  3.765720703  3.24558412  4.2858573 0.0000000
m:8-f:5  3.481494497  2.94557538  4.0174136 0.0000000
f:6-m:5  1.352717928  0.91787572  1.7875601 0.0000000
m:6-m:5  1.184202970  0.74431204  1.6240939 0.0000000
f:7-m:5  2.513421849  2.05645683  2.9703869 0.0000000
m:7-m:5  2.857767295  2.39508230  3.3204523 0.0000000
f:8-m:5  3.718586218  3.20341827  4.2337542 0.0000000
m:8-m:5  3.434360013  2.90326187  3.9654582 0.0000000
m:6-f:6 -0.168514958 -0.62249009  0.2854602 0.9968060
f:7-f:6  1.160703921  0.69016548  1.6312424 0.0000000
m:7-f:6  1.505049367  1.02895400  1.9811447 0.0000000
f:8-f:6  2.365868290  1.83862318  2.8931134 0.0000000
m:8-f:6  2.081642085  1.53882109  2.6244631 0.0000000
f:7-m:6  1.329218879  0.85401081  1.8044269 0.0000000
m:7-m:6  1.673564325  1.19285330  2.1542753 0.0000000
f:8-m:6  2.534383248  2.00296656  3.0657999 0.0000000
m:8-m:6  2.250157043  1.70328327  2.7970308 0.0000000
m:7-f:7  0.344345446 -0.15203755  0.8407284 0.5648416
f:8-f:7  1.205164369  0.65953016  1.7507986 0.0000000
m:8-f:7  0.920938164  0.36023867  1.4816377 0.0000022
f:8-m:7  0.860818923  0.31038540  1.4112524 0.0000101
m:8-m:7  0.576592718  0.01122178  1.1419637 0.0401330
m:8-f:8 -0.284226205 -0.89329509  0.3248427 0.9688007

-
7747 residual degrees of freedom is a lot; is it possible that your data set has multiple responses per individual? If that's the case, you might want to either collapse each person's responses to a mean (automatically done by ezANOVA from the ez package), or use something like mixed effects models, which permit you to account for the repeated measurements (check out ezMixed from the ez package). –  Mike Lawrence Sep 5 '11 at 19:17
I meant to say "or use something more powerful like mixed effects models". Also, for the latest version of the ezMixed code (which permits powerful evaulation of possibly non-linear effects of continuous variables like grade, not to mention visualization via ezPlot2), source and run this ezDev function while connected to the internet: raw.github.com/mike-lawrence/ez/master/R/ezDev.R –  Mike Lawrence Sep 5 '11 at 19:41

## migrated from stackoverflow.comSep 5 '11 at 17:03

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