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Statistics > Chapter 12: ANCOVA > Flashcards

Chapter 12: ANCOVA Flashcards

1
Q

What is ANCOVA?

A

When we measure covariates and include them in an analysis of variance we call it analysis of covariance: ANCOVA

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2
Q

Covariates

A

Continuous variables that are not part of the main experimental manipulation but have an influence on the dependent variable

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3
Q

Reasons for including covariates in ANOVA

A

1- To reduce within-group error variance

2- Eliminate confounds

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4
Q

ANCOVA: Equation

A

Outcome=bo+b1Dummy1+b2Dummy2+b3covariate+error

  • Covariate: added as a predictor in ANCOVA
  • this model tests the difference in group means adjusted for covariate
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5
Q

ANCOVA:

ANOVA table: Model 1

A
  • how well the model fits when only the covariate is used in model
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6
Q

ANCOVA:

ANOVA table: Model 2

A
  • the goodness of fit of model when covariate & dummy variables are used is used in model
  • difference in R^2: the individual contribution of experimental groups
    —> R^2 (M.2)- R^2 (M.1)
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7
Q

Constant

A

bo in ANCOVA

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8
Q

ANCOVA as ‘controlling’ for the covariate

A
  • compares the predicted group means at the average value of the covariate, so the groups are being compared at a level of the covariate that is the same for each group
  • ‘controlling for covariate’ analogy is not a good one
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9
Q

Assumptions of ANCOVA

A
  • Linearity
  • Normality
  • Independence of error
  • Homoscedasticity
  • Independence of the covariate and experimental groups
  • homogeneity of regression slopes
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10
Q

Independence of the covariates and predictor groups

A
  • covariate must be independent of categorical predictor
  • this situation arises mostly when participants aren’t randomly assigned to experimental conditions
  • covariance must share no variance with experimental groups: the expected value of covariance will be the same for every group
    —> group means for covariance will be equal
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11
Q

Solution for violation of:

The independence of covariate and experimental effect

A
  • assign participants randomly to experimental groups

- or: match experimental groups on the covariate

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12
Q

Statistical Requirement:

Independence of Covariate and experimental effect

A
  • no statistical requirement for experimental effect to be independent of covariate
  • this assumption makes interpretation more straightforward
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13
Q

Temporal Additivity

A
  • assumption that all experimental groups would experience the same change in covariate over time if the experimental groups had no effect
  • according to Senn: the idea that ANCOVA is biased unless experimental groups are equal on the covariate applies only when there is temporal additivity
  • when we have temporal additivity: make sure that the covariate is same in all experimental groups
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14
Q

Homogeneity of Regression Slopes

A
  • relationship between outcome (dependent variable) & covariate is the same in each of our treatment groups
  • visual representation: scatter plot of covariate vs outcome for each experimental group
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15
Q

Homogeneity of Regression Slopes

- how to check for it

A
  • When an ANCOVA is conducted we look at overall relationship between outcome (dependent variable) & covariate
  • fit a regression line to entire data set, ignoring to which group a person belongs
  • imagine plotting a scatterplot for each group of participants with covariate on one axis and outcome on the other
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16
Q

Heterogeneity of regression slopes

A
  • relationship between participant’s outcome and covariate is different in the different experimental groups
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17
Q

What are the consequences of violating the assumption of homogeneity of regression slopes?

A

I. Type I error rate is inflated and the power to detect effects is not maximized
—> This is especially true when group sizes are unequal and when the standardized regression slopes differ by more than .4

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18
Q

What to do when assumptions are violated?

A
  • bootstrap (robust)
  • post hoc (robust)
  • R (main bits of ANCOVA can not be done using bootstrap or post-hoc test)
19
Q

If assumption of Homogeneity of Regression Slopes is violated:

A
  • use a multilevel model
20
Q

ANCOVA: SPSS

  • Testing the independence of the treatment variable and covariate
A
  • Run ANOVA
  • Outcome or Dependent Variable: Covariate
  • Predictor or Independent Variable: Experimental groups
  • if F of predictor is non-significant then assumption has not been violated
21
Q

ANCOVA: Main Analysis: SPSS

A

I. Analyze
II. General Linear Model
III. Univariate

22
Q

ANCOVA: Contrasts

A
  • You can NOT enter your own codes

- Select one of the standard contrasts

23
Q

ANCOVA: Other Options

A
  • here you can get a limited range of Post-Hoc tests
24
Q

ANCOA:

How to specify Post-Hoc test?

A
  • select the independent variable and drag it to the box labeled: Display Means
  • select compare main effects
  • Select either Bonferroni or Sidak
  • Sidak more power than Bonferroni
  • Descriptive
  • Parameter Estimates
  • Homogeneity test
25
Q

Planned Contrasts in ANCOVA

A
  • use regression
  • create all dummy variables
  • Compute a hierarchical regression where the covariate is entered first and then enter all dummy variables
              Dummy 1 Control     -2 G1               1 G2              1

          Dummy 2 Control        0 G1              - 1 G2                1
26
Q

ANCOVA: Bootstrap

A
  • useful for parameter estimates and post-hoc tests but not main F test
27
Q

ANCOVA: Options

  • Estimates of Effect Size
A
  • produces partial eta square
28
Q

ANCOVA: Options

  • Contrast Coefficient Matrix
A
  • useful to see which groups are compared in which contrast
29
Q

Spread vs Level plot

A
  • useful to check if there is a relationship between mean and standard deviation
  • if a relationship exists: transform
30
Q

Residual Plot

A
  • useful to assess homoscedasticity
31
Q

ANCOVA: Levene’s test

A
  • ANCOVA is more concerned about the homogeneity of residuals rather than that of variance
  • so if Levene’s test is significant: ignore
32
Q

Homogeneity of Residuals

A
  • use Residual plots
  • Post-Hoc tests
  • bootstrap
33
Q

ANCOVA: b

  • df of t-test
A
  • df(t-test)= N-p-1

- p: number of predictors including the covariate

34
Q

ANCOVA: Dummy Variables

A
  • experiment 1: all who have value 1
  • experiment 2 all who have value 2
  • experiment 3: control: coded with 0

Experiment 1: b: represents the difference of means for those who have 0 and those who have 1

Experiment 1: b: represents the difference of means for those who have 0 and those who have 2

35
Q

ANCOVA: Contrasts

A
  • if results in bootstrap are different from other contrast analysis: reflects that our data may be biased
36
Q

ANCOVA: Interpreting the Covariate

A
  • draw Scatter plot:
    x-axis: covariate
    Y-axis: outcome
    —> check their relationship
  • also use the parameter estimates table for Beta value of covariate
37
Q

ANCOVA:

  • Testing the assumption of homogeneity of regression slopes
A
  • relationship between covariate & outcome should be similar at different levels of the predictor variable
  • Rerun ANCOVA: Specify Model: Custom
  • Enter main effects as well as interaction!!
38
Q

ANCOVA: Calculating Effect Size

A
  • eta squared (mu squared): total variance that a variable explains
  • partial eta squared (partial mu squared: the proportion of variance that a variable explains that is not explained by other variables in the analysis
39
Q

Eta Square Formula

A
  • μ^2= SSeffect/SST
40
Q

Partial Eta Square Formula

A
  • partial μ^2= SSeffect/ (SSeffect+SSR)
41
Q

What is ANCOVA used for?

A
  • to compare several means adjusted for 1 or more other variables (covariates)
42
Q

ANCOVA: Effect Size:

  • Omega Squared (ω^2)
A
  • can be calculated only when we have equal number of participants in each group
43
Q

ANCOVA: Effect Size

  • Contrasts
A
  • r contrast = √[t^2/ (t^2 +df)]

- r covariate: same formula

44
Q

ANCOVA: Reporting Results

A
  • The covariate, partner’s libido, was significantly related to the participant’s libido, F(1, 26) = 4.96,
    p =.035, r =.40. There was also a significant effect of Viagra on levels of libido after controlling for the effect of partner’s libido, F(2, 26) = 4.14, p = .027, partial η2 =.24.
  • Planned contrasts revealed that having a high dose of Viagra significantly increased libido compared to having a placebo, t(26) = −2.77, p =.01, r =.48, but not compared to having a low dose, t(26) = −0.54, p =.59, r =.11.

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