question regarding summary output with three-way interactions
Hi again, you're right. usually when terms are missing in our output - it means the're not identifiable from our data (e.g. there's not enough replication or specific combinations of factors are missing) - that's why your'e getting some interactions tested after removing some other terms. As for which terms to include - my conservative way of doing things is to include lower terms always when I want to have higher terms. What happens when you remove the triple interactions? Also I wouldn;t say that interpretation of continous:categorical interactions ?should be avoided in cases similar to yours: such interactions quantify differences in betas at respective levels of the categorical variable and provide useful valuable insight to how your response changes with the covariate. What however should be avoided (or better said - applied with caution) is drawing conclusions about categorical variables (i.e. A or B in your model) if the're involved in significant interactions with continous variables (problem similar to heterogenous slopes in classical ANCOVA). Cheers, sz. On 6 January 2012 20:40, Stefanie Kuchinsky
<stefanie.kuchinsky at gmail.com> wrote:
I just realized a probable explanation. For the time3 term for which it only displayed half of the interaction term, the lower order time3:A and time3:B had been included in the model. However, there was no time4:A in the model (though there was time4:B). When I removed this effect, I now get both parts of the interaction term output [and the F test df are now correct (2, 9294) instead of (1, 9294)]. I assume that this means the higher order terms are redundant with the lower order terms in the model. Is it then appropriate to remove the lower order interactions from the model (even though they appear significant when I run anova() and given their betas)? I believe I should not be theoretically interpreting time3:A when there is a time3:A:B interaction, anyway. So, I would interpret the betas for these terms very generally: time3:A1:B0 -- The effect of the cubic polynomial modulated by A (1 > 0) when B is held constant at 0. time3:A1:B1 -- The effect of the cubic polynomial as modulated by A (1>0) when B is held constant at 1. Thanks again for your input-- I wouldn't have thought of this is you hadn't mentioned looking at the lower order terms. On Fri, Jan 6, 2012 at 1:55 PM, Stefanie Kuchinsky <stefanie.kuchinsky at gmail.com> wrote:
Thanks so much for your response. I had not included some of the output table to save space in the original email, but the model does include the lower order terms. I've pasted the full table at the bottom of this email. The polynomial terms were indeed created with the poly() function. You were exactly right about the continuous vs. factor error in my second dataset. I had a typo in my code and so B had never been converted to a factor. I really appreciate you catching that! However, I'm still confused about the interpretation of time4:A1:B1-- specifically why there is no beta for time4:A1:B0 when there is both a beta for time3:A1:B1 and time3:A1:B0? Again, snippet of dataset 1 output: time3:A1:B1?? -68.6999? 33.14614 9294? -2.07264? 0.0382 time4:A1:B0?? 132.5765? 23.43786 9294?? 5.65651? 0.0000 time4:A1:B1? -124.2845? 23.43786 9294? -5.30272? 0.0000 Corrected output for dataset 2: time3:A1:B1?? 135.0679?? 46.2636 9294?? 2.91953? 0.0035 time4:A1:B0??? 80.8813?? 32.7133 9294?? 2.47243? 0.0134 time4:A1:B1? -108.9340?? 32.7133 9294? -3.32996? 0.0009 Full table for dataset 1(minus the correlation tables):
summary(m.int4e)
Linear mixed-effects model fit by maximum likelihood ?Data: Data ?????? AIC????? BIC??? logLik ? 109842.6 110837.5 -54782.28 Random effects: ?Formula: ~time1 | subjAB ?Structure: General positive-definite, Log-Cholesky parametrization ??????????? StdDev??? Corr (Intercept)? 38.33670 (Intr) time1???????? 434.94747 0.488 Residual???? 75.40533 Fixed effects: pupil1000 ~ time1 + time2 + time3 + time4 + time5 + subj + subj:time1 + subj:time2 +????? subj:time3 + subj:time4 + subj:time5 + A:time1 + A:time2 + A:time3 +????? A:time5 + B:time3 + B:time4 + A:B:time3 + A:B:time4 ?????????????????????? Value Std.Error?? DF?? t-value p-value (Intercept)???????? 976.6327? 19.63384 9294? 49.74231? 0.0000 time1???????????????? 891.8779 226.37671 9294?? 3.93980? 0.0001 time2??????????????? -917.0701? 38.86729 9294 -23.59491? 0.0000 time3??????????????? -131.3754? 40.59556 9294? -3.23620? 0.0012 time4???????????????? 459.9003? 40.59556 9294? 11.32883? 0.0000 time5??????????????? -444.2458? 38.86729 9294 -11.42981? 0.0000 subj2??????????????? 47.7965? 27.76645?? 63?? 1.72137? 0.0901 subj3??????????????? 65.4712? 27.76645?? 63?? 2.35793? 0.0215 subj4?????????????? 130.2721? 27.76645?? 63?? 4.69171? 0.0000 subj5??????????????? 57.2987? 27.76645?? 63?? 2.06359? 0.0432 subj6??????????????? 25.6571? 27.76645?? 63?? 0.92403? 0.3590 subj7??????????????? -7.6969? 27.76645?? 63? -0.27720? 0.7825 subj8??????????????? 81.3031? 27.76645?? 63?? 2.92811? 0.0047 subj9??????????????? 15.1726? 27.76645?? 63?? 0.54644? 0.5867 subj10?????????????? 62.5996? 27.76645?? 63?? 2.25450? 0.0276 subj11????????????? -64.4270? 27.76645?? 63? -2.32032? 0.0236 subj12?????????????? 10.0465? 27.76645?? 63?? 0.36182? 0.7187 subj13?????????????? 43.7500? 27.76645?? 63?? 1.57564? 0.1201 subj14?????????????? 26.4845? 27.76645?? 63?? 0.95383? 0.3438 subj15????????????? -46.7323? 27.76645?? 63? -1.68305? 0.0973 subj16????????????? 137.6261? 27.76645?? 63?? 4.95656? 0.0000 subj17?????????????? 14.9735? 27.76645?? 63?? 0.53926? 0.5916 subj18?????????????? 26.3230? 27.76645?? 63?? 0.94802? 0.3467 subj19?????????????? 55.4912? 27.76645?? 63?? 1.99850? 0.0500 subj20?????????????? 98.4027? 27.76645?? 63?? 3.54394? 0.0007 subj21????????????? 162.8186? 27.76645?? 63?? 5.86386? 0.0000 time1:subj2????????? -848.2627 314.38569 9294? -2.69816? 0.0070 time1:subj3????????? -462.8857 314.38569 9294? -1.47235? 0.1410 time1:subj4????????? -286.9459 314.38569 9294? -0.91272? 0.3614 time1:subj5????????? -350.9655 314.38569 9294? -1.11635? 0.2643 time1:subj6???????? -1079.1256 314.38569 9294? -3.43249? 0.0006 time1:subj7????????? -958.1396 314.38569 9294? -3.04766? 0.0023 time1:subj8?????????? 396.6793 314.38569 9294?? 1.26176? 0.2071 time1:subj9?????????? 870.7418 314.38569 9294?? 2.76966? 0.0056 time1:subj10???????? 2433.2401 314.38569 9294?? 7.73967? 0.0000 time1:subj11????????? 518.0574 314.38569 9294?? 1.64784? 0.0994 time1:subj12???????? -544.9605 314.38569 9294? -1.73341? 0.0831 time1:subj13???????? -142.6669 314.38569 9294? -0.45380? 0.6500 time1:subj14???????? -447.2835 314.38569 9294? -1.42272? 0.1549 time1:subj15?????????? 73.0371 314.38569 9294?? 0.23232? 0.8163 time1:subj16???????? -147.8436 314.38569 9294? -0.47026? 0.6382 time1:subj17???????? -321.5527 314.38569 9294? -1.02280? 0.3064 time1:subj18??????? -1104.0885 314.38569 9294? -3.51189? 0.0004 time1:subj19???????? -500.0802 314.38569 9294? -1.59066? 0.1117 time1:subj20?????????? 73.0061 314.38569 9294?? 0.23222? 0.8164 time1:subj21???????? -977.4932 314.38569 9294? -3.10922? 0.0019 time2:subj2?????????? 103.9564? 53.70288 9294?? 1.93577? 0.0529 time2:subj3?????????? 462.7823? 53.70288 9294?? 8.61746? 0.0000 time2:subj4?????????? 133.1718? 53.70288 9294?? 2.47979? 0.0132 time2:subj5?????????? 435.4522? 53.70288 9294?? 8.10854? 0.0000 time2:subj6?????????? 866.3608? 53.70288 9294? 16.13248? 0.0000 time2:subj7????????? 1040.2131? 53.70288 9294? 19.36978? 0.0000 time2:subj8?????????? -48.0412? 53.70288 9294? -0.89457? 0.3710 time2:subj9?????????? -33.9246? 53.70288 9294? -0.63171? 0.5276 time2:subj10????????? 575.5907? 53.70288 9294? 10.71806? 0.0000 time2:subj11???????? -534.9280? 53.70288 9294? -9.96088? 0.0000 time2:subj12????????? 180.5477? 53.70288 9294?? 3.36197? 0.0008 time2:subj13???????? -919.1235? 53.70288 9294 -17.11498? 0.0000 time2:subj14????????? 585.8640? 53.70288 9294? 10.90936? 0.0000 time2:subj15????????? 405.6596? 53.70288 9294?? 7.55378? 0.0000 time2:subj16??????? -1104.0865? 53.70288 9294 -20.55917? 0.0000 time2:subj17????????? 877.9724? 53.70288 9294? 16.34870? 0.0000 time2:subj18???????? 1828.7923? 53.70288 9294? 34.05389? 0.0000 time2:subj19????????? 820.0829? 53.70288 9294? 15.27074? 0.0000 time2:subj20?????????? 44.1932? 53.70288 9294?? 0.82292? 0.4106 time2:subj21????????? 845.2039? 53.70288 9294? 15.73852? 0.0000 time3:subj2?????????? 658.7997? 53.70288 9294? 12.26749? 0.0000 time3:subj3??????????? 30.3625? 53.70288 9294?? 0.56538? 0.5718 time3:subj4?????????? 595.8939? 53.70288 9294? 11.09612? 0.0000 time3:subj5?????????? 243.5716? 53.70288 9294?? 4.53554? 0.0000 time3:subj6?????????? 350.5870? 53.70288 9294?? 6.52827? 0.0000 time3:subj7??????????? 62.7208? 53.70288 9294?? 1.16792? 0.2429 time3:subj8?????????? 187.1384? 53.70288 9294?? 3.48470? 0.0005 time3:subj9????????? -745.5636? 53.70288 9294 -13.88312? 0.0000 time3:subj10???????? -937.4916? 53.70288 9294 -17.45701? 0.0000 time3:subj11????????? 349.5809? 53.70288 9294?? 6.50954? 0.0000 time3:subj12???????? -419.0377? 53.70288 9294? -7.80289? 0.0000 time3:subj13????????? 812.8245? 53.70288 9294? 15.13558? 0.0000 time3:subj14????????? 179.8396? 53.70288 9294?? 3.34879? 0.0008 time3:subj15???????? -150.8652? 53.70288 9294? -2.80926? 0.0050 time3:subj16???????? 1267.6097? 53.70288 9294? 23.60413? 0.0000 time3:subj17????????? 223.1138? 53.70288 9294?? 4.15460? 0.0000 time3:subj18?????????? 83.5245? 53.70288 9294?? 1.55531? 0.1199 time3:subj19???????? 1295.0211? 53.70288 9294? 24.11455? 0.0000 time3:subj20????????? 599.0144? 53.70288 9294? 11.15423? 0.0000 time3:subj21????????? 142.3725? 53.70288 9294?? 2.65111? 0.0080 time4:subj2????????? -357.4785? 53.70288 9294? -6.65660? 0.0000 time4:subj3????????? -343.2897? 53.70288 9294? -6.39239? 0.0000 time4:subj4????????? -705.7852? 53.70288 9294 -13.14241? 0.0000 time4:subj5????????? -445.1367? 53.70288 9294? -8.28888? 0.0000 time4:subj6????????? -602.6113? 53.70288 9294 -11.22121? 0.0000 time4:subj7????????? -421.4880? 53.70288 9294? -7.84852? 0.0000 time4:subj8??????????? 30.3527? 53.70288 9294?? 0.56520? 0.5720 time4:subj9?????????? 563.3041? 53.70288 9294? 10.48927? 0.0000 time4:subj10???????? -206.4712? 53.70288 9294? -3.84470? 0.0001 time4:subj11????????? 515.5052? 53.70288 9294?? 9.59921? 0.0000 time4:subj12????????? 184.6193? 53.70288 9294?? 3.43779? 0.0006 time4:subj13????????? 293.2901? 53.70288 9294?? 5.46135? 0.0000 time4:subj14???????? -478.9555? 53.70288 9294? -8.91862? 0.0000 time4:subj15???????? -240.5382? 53.70288 9294? -4.47906? 0.0000 time4:subj16???????? -975.5266? 53.70288 9294 -18.16526? 0.0000 time4:subj17???????? -403.4663? 53.70288 9294? -7.51294? 0.0000 time4:subj18???????? -876.2704? 53.70288 9294 -16.31701? 0.0000 time4:subj19??????? -1979.2893? 53.70288 9294 -36.85629? 0.0000 time4:subj20???????? -534.4526? 53.70288 9294? -9.95203? 0.0000 time4:subj21???????? -448.4306? 53.70288 9294? -8.35022? 0.0000 time5:subj2?????????? -10.8598? 53.70288 9294? -0.20222? 0.8397 time5:subj3?????????? 356.3984? 53.70288 9294?? 6.63649? 0.0000 time5:subj4?????????? 321.0171? 53.70288 9294?? 5.97765? 0.0000 time5:subj5?????????? 159.0899? 53.70288 9294?? 2.96241? 0.0031 time5:subj6?????????? 277.8362? 53.70288 9294?? 5.17358? 0.0000 time5:subj7?????????? 392.0390? 53.70288 9294?? 7.30015? 0.0000 time5:subj8??????????? 76.0262? 53.70288 9294?? 1.41568? 0.1569 time5:subj9?????????? 675.6410? 53.70288 9294? 12.58109? 0.0000 time5:subj10????????? 739.3099? 53.70288 9294? 13.76667? 0.0000 time5:subj11???????? -218.8505? 53.70288 9294? -4.07521? 0.0000 time5:subj12????????? 315.8796? 53.70288 9294?? 5.88199? 0.0000 time5:subj13???????? -248.9957? 53.70288 9294? -4.63654? 0.0000 time5:subj14????????? 396.2154? 53.70288 9294?? 7.37792? 0.0000 time5:subj15????????? 231.0236? 53.70288 9294?? 4.30188? 0.0000 time5:subj16????????? 224.9218? 53.70288 9294?? 4.18826? 0.0000 time5:subj17????????? 209.4774? 53.70288 9294?? 3.90067? 0.0001 time5:subj18????????? 505.3070? 53.70288 9294?? 9.40931? 0.0000 time5:subj19????????? 369.0887? 53.70288 9294?? 6.87279? 0.0000 time5:subj20????????? 275.5678? 53.70288 9294?? 5.13134? 0.0000 time5:subj21????????? 480.7268? 53.70288 9294?? 8.95160? 0.0000 time1:A1??????????? 353.1338? 85.49229 9294?? 4.13059? 0.0000 time2:A1??????????? -68.5389? 16.57307 9294? -4.13556? 0.0000 time3:A1?????????? -154.4407? 23.43786 9294? -6.58937? 0.0000 time5:A1??????????? 123.8607? 16.57307 9294?? 7.47361? 0.0000 time3:B1??????? 100.1996? 23.43786 9294?? 4.27512? 0.0000 time4:B1??????? 201.6411? 23.43786 9294?? 8.60322? 0.0000 time3:A1:B1?? -68.6999? 33.14614 9294? -2.07264? 0.0382 time4:A1:B0?? 132.5765? 23.43786 9294?? 5.65651? 0.0000 time4:A1:B1? -124.2845? 23.43786 9294? -5.30272? 0.0000 On Fri, Jan 6, 2012 at 1:08 PM, Szymek Drobniak <geralttee at gmail.com> wrote:
Hi, for me it's unclear how you specify your data and your model. in general - it seems that you're using user-made polynomial terms? or have you used the poly() function to form them? I'm asking because you don't have most of the lower-order terms in your output. as for the interaction - it's interpretation is fairly straightforward. E.g. interaction of time4:A1:B1 contains the fourth order coefficient of regression for your relationship in the objects from both A1 and B1 groups. The lack of 0/1 next to B in the second example may be caused by accidental conversion of B into numerical (rather than factor) variable - in such a situation it would be treated as continous variable and just a single regression coefficient would be returned. cheers, sz. -- Szymon Drobniak || Population Ecology Group Institute of Environmental Sciences,?Jagiellonian University ul. Gronostajowa 7, 30-387 Krak?w, POLAND tel.: +48 12 664 51 79 fax: +48 12 664 69 12 www.eko.uj.edu.pl/drobniak
-- Szymon Drobniak || Population Ecology Group Institute of Environmental Sciences,?Jagiellonian University ul. Gronostajowa 7, 30-387 Krak?w, POLAND tel.: +48 12 664 51 79 fax: +48 12 664 69 12 www.eko.uj.edu.pl/drobniak