Table Of Contentkxkxk ANOVA/GLM Example
The data come from one of several studies of food preference among Nebraska turtle species of various ages and
husbandry histories within different ecological contexts and day-rhythms. This 3-way grew out of an interest in whether
food preference was moderated by the age of the turtle, and whether this was different for different species. The original
study used 6 foods, 7 species and 5 ages (N ≈ 1600 baby turtles). This example uses a subset of those data, 3 foods, 2
species and 2 ages. We learned early in the research that repeated measures designs produced order effects that could
not be counter-balanced (leading to a separate line of research on food preference malleability), so a between subjects
design was used here.
About the food stimuli used in this study.
The crickets were immature House Crickets, of 5-7 mm body length. Animals had been pithed before presentation, so
that they were able to move their limbs, but could not escape. Proximate crickets have a substantial odor! Four
crickets were presented.
In order to approximate “carrion” 5-7 mm bits of hamburger were “incubated” for 72 hours in a warm, damp,
unsterilized environment. It stank! Four bits were presented.
The lettuce was fresh mature outer leaves of Romaine lettuce, washed with sterilized water and shredded to a
standard size approximating the size of mature “duck weed.” 30-40 “shreds” were presented.
Process:
There are a lot of steps to a complete analysis of a 3-way design. Different patterns of significant and non-significant
effects will require different subsets of these. Here’s a preview…
Initial Analysis
Get descriptive means, plots & F-tests
Determine what effects are significant
Consider what lower-order effects we will need to check for descriptive/misleading patterns
Consider what lower-order effects are likely to be interesting – based on the aggregations involved
3-way Interaction
Select the set of simple effects that most directly addresses the research question or research hypothesis
Getting the related cell means & follow-up analyses
Describe the pattern of the 3-way
Compute LSDmmd based on full model error term
Get the related set of simple 2-way interaction F-tests
2-way Interactions
Get the 2-way interaction plot and the corresponding simple 2-way plots
Get 2-way estimated semi-marginal means & follow-up analyses to describe the 2-way interactions
Why are the “Descriptive” and “Estimated” semi-marginal means different & which are plotted/tested ?
Use the related depiction of the 3-way to check the 2-way interaction for descriptive/misleading patterns
Main Effects
Get the main effect and corresponding simple effect plots
Get estimated marginal means & follow-up analyses to describe the main effect
Why are the “Descriptive” and “Estimated” marginal means different ?
Use the related depiction of the 3-way to check the main effects for descriptive/misleading patterns
Initiaal Analysis
Get descriptivee means, ploots & F-testts
UNIAANOVA feedss2 BY speciess age foodoff dv BY ivvs -- order determines lefft-to-right ordeering of IVs inn
. the Desscriptive Statistics table
/MEETHOD=SSTYYPE(3) corrects each effect foor all other efffects
/PRRINT=DESCRRIPTIVE gets desscriptive cell, ssemi-marginaal and marginal means
/PLOOT=PROFILEE(foodoff*agee*species) gets plott of cell means ( x-axis *seeparate lines * graphs )
/DEESIGN= species age fooodoff specify ddesign
species*age speccies*foodoff aage*foodoff
species*age*foodooff.
Arranngement of DDescriptive Staatistics table aand the plots reflects, “Thiss 3-way greww out of an inteerest in whethher food
prefeerence was mmoderated by tthe age of thee turtle, and wwhether this wwas different ffor different species.”
Noticce that both loook at type of food, within eeach age group, for each sspecies.
Fromm the means aand plots, it loooks like younnger turtles make more feeeding attemptts and show liittle food prefeerence.
Older turtles showw a food prefeerence, but thee preference is somewhat different for PPaints & Snappers.
Deteermine what effects aree significannt
All 7 efffects are significant.
One thinng to notice aabout these
data is hhow small aree the
standarrd deviations!
Baby turtles are an aamazingly
homogeeneous groupp!
So, smaall mean differences are
reliably different & wee get lots of
significaant effects!!
Consider what lower-order effects wee will need tto check forr descriptivve/misleadinng patterns
With the significannt 3-way, the means patterrns of all 6 lowwer-order effeects -- all 3 2-way interactions and all 33 main effectss
-- will have to be ccarefully checcked against tthe corresponnding higher-oorder effects, to determine if they are deescriptive or
misleeading.
Consider what lower-order effects aree likely to bbe interestinng – based on the aggrregations innvolved
Speccies
Paints andd Snappers reepresent just 2 of the 7 native Nebraskaan species annd of the 256 species worldwide. So
aggregating across theem doesn’t represent any important poppulation.
These twoo species do represent aboout 85% of “ppet” turtles solld in the US –– but most pets aren’t fed tthese foods at
these agees, so the lackk of external ppopulation & sstimulus validdity makes thee aggregate oof these conditions
uninterestting.
So, the main effect of aage (aggregating across sppecies & foodds), the main effect of foodd (aggregatingg across
species and age) and tthe age*food interaction (aaggregating across speciess) would onlyy be interesting as a
simplifyinng descriptioon if the meanns pattern waas descriptivee because it mmatched the ccorrespondingg simple effecct
means paatterns!
Age
There aree very few behhaviors that are age-related such that aggregating 3--day-old and 3-month-old jjuveniles tellss
us about aan interestingg population.
So, the main effect of sspecies (aggrregating acrosss age & foodds), the main effect of foodd group (aggreegating
across agge and species) and the sppecies*food innteraction (agggregating acrross age) wouuld only be intteresting as aa
simplifyinng descriptioon if the meaans pattern waas descriptivee because it mmatched the ccorrespondingg simple
effect meaans patterns!
Foodd Offered
These thrree foods were very carefully chosen to represent thee three majorr types of foodds available too juvenile
turtles: 1) available insects, 2) carrioon, & 3) vegetation (speciffically duckweeed). Therefoore, aggregatiing across
these threee foods into aan “available foods group”” would be inteeresting.
Still the thhe main effectt of species (aaggregating aacross age & foods), the mmain effect of aage (aggregaating across
species and foods) andd the species*age interactiion (aggregatting across fooods) would only be interessting as a
meaningfful aggregatiion if the meaans pattern wwas descriptivee because it mmatched the ccorresponding simple
effect meaans patterns!
Remmember – thouugh we have no such casees to address here – non-ssignificant loower-order eeffects that arre involved
in a ssignificant hhigher order eeffect must aalso be checcked whetherr they are deescriptive or misleading!!!!
3-waay Interactioon
Seleect the set oof simple efffects that mmost directlyy addressess the reseaarch questioon or researrch
hypoothesis
The sstatement thaat, “This 3-way grew out off an interest inn whether foood preference was moderated by the agge of the
turtlee, and whetheer this was diffferent for diffeerent speciess” makes the sselection of thhe simple effeects to use to describe the
interaaction straightforward.
Fromm this, we’ll waant to focus oon the simple effects of Foood Offered wiithin the simpple 2-way Foood Offered * AAge, and then
exammine how this 2-way is diffeerent for differrent Species.
Get the related cell meanss & follow-up analyses
/EMMMEANS=TABLES ( species * age * fooddoff ) comparee (foodoff) “comparee (foodoff)” assks for pairwisse
comparisonss among the 3 food off connditions – livee
crickets, groound meat & lettuce
“species**age*food” sppecifies that thhose pairwisee
comparisonss are made wwithin each agge group, for
each speciees
These ceell means “Esttimates” are the same as
the cell mmean “Descripptive Statisticss” given
above and the same mmeans that are plotted
above.
The F-tessts shown below test the simple effect
of Food OOffered for evvery combinattion of Age &
Species.
For both sspecies, theree are Food OOffered effectss
for youngger, but not for older, animaals.
Noticce that the MSSe for these FF-tests is
the ssame as for thhe full model –– based on
the fuull sample sizze. It is also ppossible to
get thhese tests by splitting the ffile into
thesee 4 groups annd getting the Foodoff
ANOOVA for each. However, that
approoach will use a separately composed
errorr term for eachh F, will have lower
poweer, greater chance of inconnsistent
effeccts & of Type II Errors.
Fromm the cell meaans given aboove and thesee pairwise results we can eexplicate the ppattern of the 3-way as…
Crickets v Ground Meatt Cricketts v Lettuce Groundd Meat v Lettuuce
>> > =
Paintted Turtles 3-days
== = =
3-months
== > >
Snappping Turtles 3-days
== = =
3-months
Desccribe the paattern of thee 3-way
In woords – followinng the SE of ffood for withinn each age grroup, for eachh species …
For ppainted turtless, 3-day-olds preferred cricckets to grounnd meat and lettuce, whereeas 3-month-olds showed no
prefeerence. Howeever, for snapppers, the 3-days preferredd crickets andd ground meaat to lettuce, wwhereas the 33-month-olds
showwed no prefereence.
Or, aalternatively……
The 33-day-old painted turtles preferred crickkets to groundd meat or lettuuce, while he 3-day old snappers preferrred crickets
and gground meat to lettuce. Thhe 3-month-old turtles of neither speciess showed a ppreference.
For eeither descripttion, notice thhat the patternn of the pairwwise comparisoons guaranteees that the mmain effect of Food Offered
and tthe 2-way inteeraction of Foood Offered * Age will be mmisleading! Inn each case, ddifferent subssets of data show different
meanns patterns, aand, so, they ccannot all maatch the meanns pattern of tthe corresponnding main efffect or 2-way.
Commpute LSDmmmd based on full moddel error terrm
This is a BG modeel, so all the FF-tests and foollow-up analyyses are baseed on a singlee error term (MMSe=111.8522), though thee
Standdard Errors oof the follow-ups calculatedd by SPSS vary with samplle size. Why care? Becauuse, the followw-up analysess
are bbased on a t-test (that isn’t shown in thee output) that uses the stanndard error in the denominnator.
So, ddepending on whether the cells being coompared havee larger or smmaller samplee sizes, the standard error can be largerr
(smaaller ns) or smmaller (larger nns), and the ssame cell meaan difference can be signifficant for one comparison aand not
signifficant for another!!
An alternative is too use this “full model error term” as the basis for commputing an LSSDmmd valuee that is then uused to
comppare any two cell means. TThis is an exteension of the “homogeneityy of variance”” assumption that is made when we
comppute the ANOOVA error termm for BG models. That asssumption is thhat it makes ssense to combbine the within-group
variaability from thee different dessign cells, beccause they eaach representt a sample taken from diffeerent populatiions that all
havee the same vaariability, so thhe aggregate of them all is the best estimmate of the vvariability of eaach. The exteension in the
“full mmodel error teerm” approach is that sincee the best esttimate is derivved from usinng the full design sample, the
signifficance test sshould be bassed on the df ffrom all the paarticipants.
Why do people whho like this appproach like itt?
1. Itt is based on the same esttimate of variaability, but larrger sample ssize, and, so, uses a smalleer standard eerror than the
ppairwise errorr term approach. So, it provvides a more powerful signnificance test,, and more paairwise cell mmean
ccomparisons aare significanntly different using this approach (thoughh the reverse can happen on occasion)).
2. TThis approachh provides an easy methodd for the compparison of nonnadjacent cellls means. WWe might wantt to know
wwhether the mmean number of strikes maade by 3-montth-old paintedd turtles offereed lettuce (mean = 13.2500) is different
than by 3-dayy-old snapperss offered grouund meat (meean = 14.857)) there is no eeasy to get SSPSS to providde this
ssignificance teest, but the Computators wwill give us an LSDmmd thaat we can usee to compare these meanss. For this
aanalysis k = 12 conditions oof the 2x2x3 ddesign, n = NN/k = 85/12 = 7/083 & MSee = 3.589. Using the LSDmmmd value, wee
wwould concludde these two groups have equivalent peerformances.
Gettting the relaated set of ssimple 2-waay interactioon F-tests
Havinng decides too describe thee 3-way interaaction as “the different 2-waay interactionn pattern of Foood Offered aand Age for
each Species” wee might want to have those two simple 22-way F-tests..
SORRT CASES BYY species. sortss the data file into snapperrs and paintedd turtles
SPLIT FILE LAYEERED BY speecies. tells SPSS to commpute the requested analyses
. sepaarately for each split file.
UNIAANOVA feedss BY age fooddoff. requuests the age & food 2-wayy ANOVA
Of the various F-teests presenteed, only the 2--way interactions are intereesting/meaninngful. These 22-ways analyyses each usee
an errror term baseed on the dataa and samplee size from a ssingle species. As a resullt, each of theese 2-way F-tests has less
poweer than the 3-wway F-test theey are being used to exploore. So, it is ppossible to haave one of thoose “significant effects thatt
aren’’t anywhere”
We ccan also re-coompute these simple 2-wayy F-tests usinng the MS from the table above, and the MS from the 33-
age*fooodoff eerror
way ffull model (MSe = 3.589). This would leead to…
FFood Offered * Age for Painnted Turtles Food Offeered * Age forr Snappers
82.854 122.698
FF(2, 73) = ------------ = 223.086 p < .0001 F(2, 73) = ------------ = 34.187 p < ..0001
3.589 3.589
2-waay Interactioons
Usingg the Age*Sppecies 2-way aas an examplle, here is howw to get the pplot & follow-uups for that 2--way and the
correesponding simmple-2-way off Age*Speciess for each Foood Offered.
Get the 2-way interaction pplot and thee corresponnding simplle 2-way ploots
/PLOOT=PROFILE(( species * agge ) gets plot of estimateed semi-margginal means ffor Species * Age 2-way
. interacction ( x-axis **separate linees)
/PLOOT=PROFILE((species * age * foodoff ) gets plot of the cell mmeans arrangged to check if the pattern of the
. Speciees * Age interraaction is the ssame for each Food Offereed
( x-axiss * separate lines * separaate graphs)
2-waay plot of Speccies * Age Simple 2-way plot off Species * Agge for Live Crrickets
Simpple 2-way plott of Species * Age for Grouund Meat Simple 2-way plot off Species * Agge for Live Crrickets
Nonee of the simplee 2-ways lookks quite like thhe 2-way, butt it all hinges on the age efffects for Snaappers. Signifficance tests
can bbe comfortingg, concentratinng on the simmple effects off Age for eachh Species (forr each Food OOffered).
Get 22-way estimaated semi-maarginal meanns & follow-uup analyses tto describe tthe 2-way intteractions
/EMMMEANS=TABLES ( species * age ) commpare ( age ) gets pairwisse comparisoons of estimatted age groupp
. semi-marginal means foor each speciees – used to
. describe thhe pattern of tthe Species * Age 2-way
Whyy are the “Descriptive” and “Estimmated” semi-marginal mmeans diffeerent & whicch are plottted/tested ??
You sshould noticee the estimateed semi-marginal means shhown in the ggraphs above and the “Estiimates” table below are noot
the ssame as the raaw semi-margginal means ffrom the “Desscriptive Statisstics” up abovve. Those raw semi-marginal means
weree 3.41 for 3-daay Painted, 122.71 for 3-moonth Painted, 12.00 for 3-daay Snappers & 15.05 for 33-month Snapppers.
The ddifference bettween the raww and estimatted semi-marrginal means reflects that tthe effects in tthe design arre not
orthoogonal (becauuse of unequaal-n). The esttimated semi--marginal meaans are prediicted based oon the model ((not the raw
data)) and provide for the compparisons amonng these grouups, after corrrecting for thee other effectss in the model.
The F-ttests comparee the correcteed/estimated semi-
marginal means.
The paairwise compaarisons are reedundant with the F-tests
becausse these are aall 2-group coomparisons.
Be suree to check thee direction of each significcant
comparison !!!
The ppattern of thee 2-way interaaction is…
Painted 3 day <<<< 3 month
Snapper 3-day < 3 month
The AAge effect is tthe same direection for bothh Species, buut is much largger for Painteed turtles.
Use related deppiction of thhe 3-way to check the 22-way interaaction for ddescriptive/mmisleading patterns
/EMMMEANS=TABLES ( foodofff * species * aage ) comparee ( age ) gets pairwwise comparisons of age grroup cell
. means at each combination of species and food
. offered, too check whethher the patterrn of the
. species*age 2-way is ddescriptive or misleading
(from aabove) The ppattern of the 2-way interaction was…
Paiinted 3 dday <<< 3 mmonth
Snaapper 3-dday < 3 mmonth
For livve crickets, thhe corresponding simple 22-way is…
Paiinted 3 dday < 3 mmonth
Snaapper 3-dday = 3 mmonth
For grround meat, tthe corresponnding simple 22-way is…
Paiinted 3 dday <<< 3 month
Snaapper 3-dday = 3 month
For letttuce, the corrresponding ssimple 2-way is…
Paiinted 3 dday <<<< 3 month
Snaapper 3-dday <<< 3 month
So, thee 2-way interaaction patternn is descriptivve only for
cricketts, and is misleading for grround mean && lettuce.
Description:husbandry histories within different ecological contexts and day-rhythms. food
preference was moderated by the age of the turtle, and whether this was different
for study used 6 foods, 7 species and 5 ages (N ≈ 1600 baby turtles).
