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k-G
test probpracapprprobprac
Group ANC
The purpusing practic
blems (Similactice groups ropriate difficblems as thectices comple
Here are
COVA
pose of the sce problemsar), and 3) hand given a culty) and a y liked, and eted (numpr
the results
study was to s that were: 1arder than thpacket that set of 5 test then comple
ract) and tes
of ANOVAs
compare th1) easier thahe test itemshad instructproblems.
eted the testt performanc
comparing t
e Test Perfoan the test ites (Harder). tions for the Students reat problems. ce (testperf
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ad the instruPractice groas a %) wer
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students whr) , 2) similarere randomly10 practice puctions, comoup (practgrre recorded
and numprac
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ere is a signiof practice a
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rm an ANCO
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OVA on these
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Data Preparation – Mean-centering the Covariate It is a good idea to work with “mean-centered” quantitative covariate scores. Mean-centering simplifies the math involved in constructing and plotting the results of the analysis, as well as limiting collinearities among the models terms that can lead to mis-estimation and statistical conclusion errors. Mean-centering is just what it sounds like… You compute a new variable for each person that is their covariate score minus the mean of covariate. compute numpract_cen = numpract - 6.6458. exe. “Kinds” of ANCOVA models Even for this, the simplest type of ANCOVA with a 2-group IV and a single covariate, there are different possible models. Main Effects ANCOVA models include the IV and the Covariate. A main effects model makes the “homogeneity of regression slope” assumption. That is, the model is constructed assuming that the slope of the linear relationship between the covariate and the DV is the same for all IV groups. Put differently, this is an assumption that there is no interaction between the covariate and the IV as they related to the DV. This regression slope homogeneity assumptions makes the comparison of the IV groups simpler, in that, it assumes that the corrected mean DV difference between the groups is the same for all values of the covariate. In terms of this example, the assumption is that the test performance difference between the Easy, Similar and Harder difficulty practice groups is the same for every amount of practice. Full Model ANCOVA models include the IV, the Covariate, and the IV-Covariate interaction. This model does not make the homogeneity of regression slope assumption, and allows there to be different corrected mean DV difference between the groups for different values of the covariate. Just like with factorial ANOVA, often the most important part of the model is the interaction! Also, sometimes, without careful attention to the pattern of the interaction, one or both main effects are misleading. Getting the Main Effects ANCOVA Model Some of the useful output isn’t available using the SPSS GUI, so we will use SPSS syntax code for these analyses. The simplest code for an ANCOVA is shown below. UNIANOVA testperf BY practgrp WITH numpract_cen /METHOD = SSTYPE(3) /EMMEANS = TABLES(practgrp) WITH(numpract_cen= mean) COMPARE (practgrp) /PRINT = DESCRIPTIVE PARAMETER /DESIGN = practgrp numpract_cen.
dv BY iv WITH covariate be sure to use the mean-centered cov uses formulas that work well with ≈n gets dv means for each group corrected for the mean covariate value gets simple effects test for that cov value gets descriptive/uncorrected means and the regression model parameters (we will use to plot the model) specifies that the IV and the Covariate are both in the model (notice the period)
Main
n Effects ANNCOVA output
These arperformaANOVA.
The F tapractgrp of practic
There is betweengroup me
Notice thin this ANANOVA didn’t adwon’t cha
The para“expresstable, buregressiothis table
Becauseslope asthe numpgroups. For eachANOVA
re the same ance means
ble shows theffect after
ces.
not a signifin numpract aembership i
hat the SSerNCOVA modmodel, tellin
dd much to thange the gro
ameter estimsion” of the inut presented on weights. e to plot the
e of the homsumption thpract_cen te
h t-test, df = model just a
(uncorrectewe got from
hat we havecontrolling f
cant relationand testperf, nto account
rror is not mudel than in thng us that thhe model, anoup compari
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ogeneity of e regression
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uch smaller he original e covariate nd probably ison much.
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and their C
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Comparison
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g program fo
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ting Practiceecause of the
esults
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include the Iers from that deviation of
describing th
t of the resu
from the F-tbest perform
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uses the “kx
IV group namtable – put the covariate
he Y- and X-
lts…
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mance is pren slope homo
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-axes of the
Please Use thethe “z1 wdifficulty Use thethe “z1 wdifficulty Use thethe “z1 wdifficulty
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note:
e IV group orwt = 1 & z2 y in this exam
e IV group orwt = 0 & z2 y in this exam
e IV group orwt = 0 & z2 y in this exam
ifficulty grou
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than the
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Getting the Full Model ANCOVA There only a couple of differences when asking SPSS for the full model ANCOVA including the interaction term. First, you will include the interaction term in the “DESIGN” subcommand. Represent this by listing the IV and Covariate, with “*” between them numpract_cen*practgrp (be sure to use the centered covariate) Second, since the model allows for an interaction, and the slopes of the regression lines might be different, the corrected group mean difference may be different for different values of the covariate (i.e., different practgrp simple effects for different values of numpract). So, it is usually a good idea to ask for group comparisons at several values of the covariate. For this analysis, it makes sense to ask for group comparisons for 1, 3, 5, 7, & 9 practices. However, remember that the number of practices variable we’ve included in the model has been mean-centered. So, we have to take that mean centering into account!
1 raw practices corresponds with a mean-centered value of 1 – 6.6458 = -5.6458 3 raw practices corresponds with a mean-centered value of 3 – 6.6458 = -3.6458 5 raw practices corresponds with a mean-centered value of 5 – 6.6458 = -1.6458 7 raw practices corresponds with a mean-centered value of 7 – 6.6458 = .3542 9 raw practices corresponds with a mean-centered value of 9 – 6.6458 = 2.3542
UNIANOVA testperf BY practgrp WITH numpract_cen /METHOD = SSTYPE(3) /EMMEANS = TABLES(practgrp) WITH (numpract_cen = -5.6458) COMPARE (practgrp) /EMMEANS = TABLES(practgrp) WITH (numpract_cen = -3.6458) COMPARE (practgrp) /EMMEANS = TABLES(practgrp) WITH (numpract_cen = -1.6458) COMPARE (practgrp) /EMMEANS = TABLES(practgrp) WITH (numpract_cen = .3542) COMPARE (practgrp) /EMMEANS = TABLES(practgrp) WITH (numpract_cen = 2.3542) COMPARE (practgrp) /PRINT = DESCRIPTIVE PARAMETER /DESIGN=practgrp numpract_cen numpract_cen*practgrp.
Full
Model ANCCOVA outpuut
The F tablesignificant after contropractices a There is noeffect of nufor practgrp There is a practgrp anmeans onemight be m Notice thatsmaller in tthan in the (5517.188) For each t-from the ANabove.
e shows thatpractgrp maolling for numand the intera
ot a significaumpract aftep and the int
significant innd numpracte or both ma
misleading!)
t the SSerrorthis ANCOVoriginal ANO
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ing regressio
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OVA table wV regressionrameter Estimded “3” (Har’t get the moded “1” (Eas
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ctgrpEHS npractgrpEHS
ating Numbe
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act”
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odel or signifsier) or the g
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r Similar gro
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2.645 * num
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iable for eac
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practgrpSHE
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e for the groucoded group
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erformance f
) + 85.940
) + 67.568
) + 45.713
is represente
to describe
which tells usthree groups
covariate-DVer the slope riate-DV reg
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up coded “1”p variable.
for the three
the slope is
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the slope is
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s whether ors
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”, we have to
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o recode the
:
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e Parameter esn’t change ameters fromIV group origIV group origIV group orig
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m the initial aginally codedginally codedginally coded
ults
rom either Amodel, just h
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Figure Plot of t
which group ihe regressionas “2”. Reme0” group (Ea
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A model
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Write-up for the ANCOVA
An ANCOVA was performed including Practice Difficulty Group (Easier, Similar & Harder Difficulty), Number of Practices and their interaction. The plot of the ANCOVA model is shown in Figure 1.
There is an interaction of Practice Item Difficulty and Number of Practices as they relate to Test Performance, F(2, 42) = 20.661, MSe = 65.574, p < .001. The pattern of the interaction is that, as can be seen in Figure 1, the Harder and Easier groups performed significantly better than the Similar group following 1 practice, the Harder group perform significantly better than the Same and Easier groups following 3 practices, and following 5, 7 & 9 practices, the Harder group performed best while the Easier group performed poorest.
An alternative description of the pattern of the interaction is that the slope of the Number of Practice regression line is positive for the Harder, b = 2.645, p = .02, and Similar, b = 3.292, p < .001, groups, while this slope is negative for the Easier group, b = -3.519, p < .001.
The main effect for Number of Practices was non-significant, F(1,42)= 2.363, MSe = 65.574, p = .132. However this main effect was not descriptive for any of the three Practice Difficulty groups, because of the pattern of the interaction. Although there is no relationship between number of practices and test performance on average, there was a positive relationship for the Harder and Similar groups and a negative relationship for Easier group.
The main effect of Practice Group was significant, F(1,42) = 69358, MSe = 65.574, p < .001. However, this main effect was not descriptive, as whether the pattern of Test Performance differences among the groups changed with the Number of Practices.