Field_2005_chapter_9

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Field_2005_chapter_9

• Analysis of covariance
• ANCOVA (GLM 2)?

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Recap t-test, one-way ANOVA
wrt with respect to

• What we can do so far is
• Compare 2 groups (of same or different subjects)
  wrt a single dep variable, e.g., whether being
  confronted with real spiders or pictures of
  spider changes the anxiety (t- test, chapter 7)?
• Compare more than 2 groups (of different people)
  wrt a single dep variable, e.g., whether
  receiving no, a low dose, or a high dose of
  viagra changes the libido (one-way ANOVA, GLM 1,
  chapter 8)?

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Covariates

• Now we expand the ANOVA to ANCOVA
• In an 'Analysis of Covariance', ANCOVA, we
  include continuous variables into the analysis
  which have a presumed influence on the dependent
  variable (they co-vary with it), so we want to
  control them by assessing them (rather than
  manipulating them).
• The covariates are entered into the regression
  model (which underlies ANOVA) first, then the
  independent variable. Thus, we can assess the
  pure influence of the independent variable by
  having partialled out the influence of the
  co-variates before.

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Why including covariates?

• To reduce the within-group error variance By
  controlling the covariate(s), we can explain part
  of the unsystematic error variance (residual sum
  of squares SSR), which reduces the error variance
  within the experimental groups. Thus, it makes
  our statistical model better.
• Elimination of confounds a covariate and a
  dependent variable are 'confounded' (i.e., it is
  impossible to tell from which the influence comes
  since they co-vary) unless the covariate is
  identified and assessed.

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Example Viagra

• A potential confounding covariate for the
  dependent variable 'libido of the subject' is the
  'libido of the partner' (measured by how often
  the partner tried to initiate sexual contact).
• Formally, the covariate is entered into the
  regression equation as follows
• Libidoi b0 b3Covariatei b2Highi b1Lowi
  ?i
• Libidoi b0 b3Partner's Libidoi b2Highi
  b1Lowi ?i

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Running ANCOVA on SPSS(using ViagraCovariate.sav)
?

• Data input organize the data in columns dummy
  variable (placebo, low, high dose) / subject's
  libido/ partner's libido

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Running ANCOVA on SPSS

• There is no special menue option for ANCOVA.
  Entering covariates is an option in various
  General Linear Models. The simplest one is
  conducting a Univariate ANOVA (One-way ANOVA)
  which offers the possibility of adding
  covariates.
• Analyze ? General Linear Model ? Univariate

dependent Variable
Independent Variable
Covariate
No posthoc tests available but contrasts
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Specifying contrasts

• You cannot enter codes as we did in ANOVA, but
  only choose from some pre-specified contrasts.
• Here, we choose 'simple' (non-orthogonal contrast
  between the control group and the experimental
  groups) and define the reference category as
  'first' (placebo is the first category).
• Click on 'change' so that the factors-box looks
  as depicted above.

Group - contrast
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Options

• Here, you can request post hoc tests.

Independent variable

• 3 adjustment levels for CI
• Tukey LSD
• Bonferroni
• Sidak

By placing the independent variable 'dose' in the
'Display Means' window, a table of estimated
marginal means for this variable will be
displayed. These are adjusted group means after
the co-variate has been removed.
Levene's test for homogeneity of variances
Plots of oberved-by- predicted-by- standardized re
sidual values
Regression coefficients and significance tests
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Excursion idák and Bonferroni

• If you apply many tests to the same data set, the
  probability increases that you detect a rare
  event.
• idák and Bonferroni are both used as corrections
  for such multiple comparisons, in order to
  control for the familywise error type I, i.e., to
  assume that there is an effect while there is
  none. (Abdi 2007)?
• idák is the more basic one of these corrections.
  It assumes independence of the tests.
• Bonferroni is a simpler approximation of the
  idák correction, which is easier to compute and
  therefor has become more popular. It is also
  somewhat more pessimistic.

Abdi, Hervé (2007) The Bonferroni and idák
corrections for multiple comparisons. In Neil
Salkind (Ed.) Encyclopedia of measurement and
statistics. Ohousand Oaks, CA. Sage.
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(pre-analysis) 1-way-ANOVA

• Conduct a simple ANOVA first without the
  covariate
• Analyze ? General Linear Model ? Univariate
• (then remove the covariate)

The three groups do not seem to differ
SSM
SSR
SST
You may also run a one-way ANOVA.
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Now run ANCOVA with the covariateOutput
ANCOVADescriptives
Note This is a different data set than in
chapter_8! (more subjects)?
Means, SD's, N'S of sample
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Output of ANCOVA

• Levene's test of homogeneous variances is
  significant.
• An alternative measure for homogeneity of
  variances is the ratio of the highest and the
  lowest variance, here
• 4.49 (high dose) 2.11
• 2.13 (low dose)
• This value is slightly higher than the 'rule of
  thumb' (ratio of 2) but we should not worry...

s2 3.2 2.13 4.49
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Output of ANCOVA Main analysis
The covariate predicts the Dep Var significantly
SSM SSR
The indep Variable alone also predicts the
Dep Var significantly

• If we hadn't taken into account the covariate, we
  would have thought that the indep Var (dose) did
  not have an influence, yet it does (SSDose
  28.337)?
• The covariate (partner's libido) alone also
  explains a part of the overall variation
  (SSPartner 17.09)?
• Together, the Model explains SSM 34,75 units
  of variance. ? The SSR has been reduced.

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Contrasts Parameter estimates
Why is the contrast between low and high dose
although their means are almost identical?
Why is the contrast between low and high dose
although their means are almost identical?

• DOSE1 contrast placebo vs. High dose is
• DOSE2 contrast low vs. High dose is !!!
• (DOSE3 redundant contrast)?
• PARTNER b.483 (when the partner's libido raises
  1 unit, the subject's libido raises for .483
  units libido

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Contrasts and corrected estimates
Original values
3.22 4.88 4.85
Placebo vs. Low dose n.s.
Here are the adjusted group means, taking into
account the covariate. Note how the values
change! Only the high dose group has a
significantly higher mean as compared to the low
dose and placebo group.
Placebo vs. High dose significant

• The group means (4.88 for low dose 4.85 for high
  dose) had not been adjusted by taking into
  account the covariate.

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Post hoc tests

• Only the difference between high dose and placebo
  is
• The difference between the high and the low dose
  group is n.s., maybe due to loss in power from
  the post hoc test (post-hoc tests are 2-tailed
  since they are not directed whereas contrasts are
  1-tailed and directed).

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Interpreting ANCOVA

• Interpret the b-value of the covariate if it is
  positive, the covariate and the dependent var
  have a postive relationship if negative, then
  negative.
• Verify this with a scatterplot
• Graphs ? Interactive ? Scatterplot
• Libido as x-axis Libido of partner as y-axis

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ANCOVA as hierarchical multiple regression

• ANCOVA can also be understood as multiple
  regression with the covariate as the first
  predictor and the independent variable as the
  second, in a hierarchical regression model.
• Therefore, we have to supplement the data with
  some dummy coding.
• With two dummies, we can specify each group
  (placebo 00 low dose10 high dose01)
  unambiguously
• See ViagraCovariateDummy.sav

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ANCOVA as hierarchical multiple regression
Block 1 (Previous)? partner's libido
Method Enter
Block 2 (shown)? 2 dummy var's

• Analyze ? Regression ? linear. Libido is the
  predicted, dependent variable the covariate is
  the 1st predictor (1st block) the two dummy
  variables with which we have coded the 3 exp.
  Groups (placebo, low and high dose) is the 2nd
  predictor (2nd block)?

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Output multiple regression, ANCOVA
Individual contribution of 'dose of Viagra'
.313 - .058 .255 Partner's libido accounts for
5.8 of libido Viagra for 25.5
Model 1 only covariate Model 2 dose of viagra
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Output multiple regression, as compared to ANCOVA
Output ANOVA from regression Entire model 2
accounts for 34.75 units of variance (SSM) SST
110.97 unexplained variance SSR 76.22
Output from previous ANCOVA, corrected model
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Regression coefficients of multiple regression,
as compared to parameters estimate in ANCOVA
Output from regression
Partner's libido is Placebo vs. Low is
n.s. Placebo vs. High is
Previous Output from ANCOVA
Partner's libido if partner's libido is changed
1 unit, subject's libido changes .483
units. Dummy 1 contrast low dose vs.
Placebo Dummy 2 contrast high dose vs. Placebo.
Dose 2 low vs. high
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Adjusted means

• The b-values of the dummy variables represent the
  difference between the means of the low-dose
  group vs. Placebo (dummy1) and the high-dose
  group and the placebo (dummy2).
• meanLow-dose meanPlacebo 4.88 3.22 1.66
• meanHigh-dose meanPlacebo 4.85 3.22 1.63
• However the b-values for dummies 1 and 2 are
  very different from those means! This is because
  they have been calculated from the adjusted means
  (see table)!
• meanLow-dose meanPlacebo 3.027 3.313
  -0.286
• meanHigh-dose meanPlacebo 5.920 3.313
  2.607

unadjusted
adjusted
Previous output of ANCOVA, adjusted means
B-values of dummies in regression
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Additional assumptions in ANCOVAhomogeneity of
regression slopes

• We assume throughout, that the relationship
  between the covariate (partner's libido) and the
  various groups (placebo, low dose, high dose) is
  the same. We fit the regression line of the
  covariate to the entire data set, assuming
  homogeneity of regression slopes for all groups.

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Checking for homogeneity of regression slopes

• We can visually check this assumption by drawing
  scatterplots for selected groups (data ? select
  cases if dose 1 ? group 1 if dose 2 ? group
  2 if dose 3 ? groups 3)?
• Relationship partner's libido x placebo
• Relationship partner's libido x low dose
• Relationship partner's libido x high dose
• Analyze ? graph ? scatter, simple

Placebo group
Low dose group
High dose group
While the regression slopes for the placebo and
the low dose group look quite similar, the one
for the high dose group, however, does not
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Testing for homogeneity of regression slopes

• We have to run ANCOVA again, but this time
  specifying a 'customized' (adapted) model.
• 1st step main window univariate ANOVA
• Analyze ? GLM ? univariate
• 2nd step open 'model' window for specifying the
  main effects for the independent variable (dose)
  and the covariate (partner), as well as the
  interaction of dosepartner (this is where the
  effect of the covariate on the 3 groups is
  assessed)?

1st step
2nd step
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Testing for homogeneity of regression slopes
Main effect dose Main effect partner
Interaction dosepartner !!
Main effect dose Main effect partner
Interaction dosepartner !!
Main effect dose Main effect partner
Interaction dosepartner !! The effect of the
partner's on the subject's libido is dependent
on the group

• The assumption of homogeneity of regression
  slopes has been violated ( interaction
  dosepartner).
• This raises doubts about the main analysis.
• Note the importance of testing this assumptions,
  although there are no proposals in the book as to
  how to how the problem should be addressed.

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Calculating the effect size for the various
single contrasts

• General equation
• rcontrast ? t2
• t2 df
• rCovariate ? 2.472 .42
• 2.472 28
• rHigh Dose vs. Placebo ??-3.0952
  .50
• -3.0952 28
• rHigh Dose vs. Low dose ??-2.0512
  .36
• -2.0512 28

Medium to big effect sizes
Medium to big effect sizes
Medium to big effect sizes
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Summary of effects

• Effect of the covariate, partner's libido, on
  subject's libido
• Effect of 'dose'
• High vs. Placebo
• High vs. Low
• Low vs. Placebo n.s.
• Effect of inhomogeneity of regression slopes

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Reporting ANCOVA (Field_2005_385f)?

• The covariate, partner's libido, was
  significantly related to the participant's
  libido, F(1,26) 6.11, p was also significant effect of Viagra on levels
  of libido after controlling for the effect of
  partner's libido, F (2,26) 4.83, p
• Planned contrasts revealed that having a high
  dose of Viagra significantly increased libido
  compared to having both a placebo, t (26)
  -3.10, p -2.05, p