Psychology 412

1
Psychology 412

• Instructor Adam Kramer
• Week 6

2
Reprisal

• Midterm due tomorrow 9am
• Ask for meeting extension if necessary
• Today Contrast coding
• Next week Interactions
• Multiple variables, continuous categorical
• Beyond Nested or repeated-measures or
  within-subjects designs

3
Review

• Multiple regression is great
• Estimates indicate the effect of a one-point
  increase when other variables equal zero
• Allows us to partial out confounds
• Allows for dummy coding to compare groups to a
  baseline

4
Beyond baselines

• Choosing a baseline does not answer every
  possible research question
• We can make as many comparisons as we have
  degrees of freedom
• Because we dont want to re-use variance
• The art
• What a unit increase means
• What zero means

5
Coding contrasts

• Solution A tug of war with zero as the line
  and the categories as tuggers
• No religion vs. General, specific, prac
• Weight them equally so that they sum to 0
• Three copies of no religion against one each of
  general, specific, prac. 1 1 1 3 0
• A one unit increase means ¼ of the way from
  none to some
• Zero means they balance out

6
Coding contrasts

• A one-unit increase is ¼ the distance from none
  to some
• Because its 4 steps from 1 to -3
• So the SLOPE changes, and the error with it, so
  the SIGNIFICANCE stays the same
• (could also code -3/4, 1/4)

7
Degrees of Freedom

• Religiosity has four categories, thus three
  degrees of freedom
• Three degrees of freedom means we can do three
  tests
• Logic extends from paired t-test
• Fewer than 3 means we have not analyzed all of
  the differences among our groups
• More than 3 means we have OVERanalyzed the
  existing differences

8
Orthogonality

• swl religNoneVSome
• What else would we test?
• We were free to choose our first contrast, but
  in the full model, contrasts control for each
  other
• Each contrast is effectively flattened out when
  testing other variables
• Equating none and some, now lets look at

9
Orthogonality

• Equating none and some, now lets look at
• any other contrast that doesnt care or isnt
  affected by whether none or some differ
• Unrelated or orthogonal contrasts are contrasts
  that are not statistically related
• Meaning, you cant predict a CATEGORYS level on
  one from its level on the other

10
Orthogonality illustrated

• Our dummy codes are correlated
• If youre basic we know youre NOT anything
  else, r-0.33
• When we control for Specific and Practicing,
  a one-unit increase in basic compares it to the
  baseline
• We need non-orthogonal contrasts!

11
Orthogonality illustrated

• We need non-orthogonal contrasts!
• but not all the time.
• If we change c2 to be our none versus some
  test, all of the interpretations change
• When there is no difference between none and
  some, is there a difference between none and
  basic?

12
Orthogonality illustrated

• We need orthogonal contrasts, too!
• If none and some differ (or if they dont),
  there might be differences in the some.
• Compare practicing to the other somes
• These contrasts are uncorrelated, r0.00
• One contrast left?

13
Orthogonality illustrated

• One contrast left
• Only one is orthogonal to both
• One way to be sure contrasts are orthogonal if
  they are nested, testing, in turn,
  undifferentiated groups
• C1 didnt differentiate B, S, P
• C2 did, but didnt differentiate B, S

14
Coding in R

• R uses the factor type for a categorical
  variable
• Converts strings, levels into factors
• kreligion is a factor variable, which has a
  contrasts attribute
• factor(kreligion) would factorize it if
  necessary
• gt contrasts(kreligion)
• general specific practicing
• none 0 0 0
• general 1 0 0
• specific 0 1 0
• practicing 0 0 1

15
Coding in R

• gt contrasts(kreligion)
• general specific practicing
• none 0 0 0
• general 1 0 0
• specific 0 1 0
• practicing 0 0 1
• The contrasts are themselves a data matrix
• column names (contrast names)
• a row for each category

16
Coding in R

• gt contrasts(kreligion)
• NoneVSome PracVNonprac
  GeneralVSpecific
• none 0 0
  0
• general 1 0
  0
• specific 0 1
  0
• practicing 0 0
  1
• We can rename and assign at will
• colnames( contrasts(kreligion) ) lt-
  c(NoneVSome,PracVNonprac,GeneralVSpecific)

17
Coding in R

• gt contrasts(kreligion)
• NoneVSome PracVNonprac
  GeneralVSpecific
• none 0 0
  0
• general 1 0
  0
• specific 0 1
  0
• practicing 0 0
  1
• We can rename and assign at will
• contrasts(kreligion),1 lt- c(-3,1,1,1)
• contrasts(kreligion),2 lt- c(0,1,1,-2)
• contrasts(kreligion),3 lt- c(0,1,-1,0)

18
Coding in R

• gt contrasts(kreligion)
• NoneVSome PracVNonprac
  GeneralVSpecific
• none -3 0
  0
• general 1 1
  1
• specific 1 1
  -1
• practicing 1 -2
  0
• lm respects this matrix

19
Contrast variables

• gt summary(lm(swls religion, datak))
• ...
• Coefficients
• Estimate Std. Error t
  value Pr(gtt)
• (Intercept) 64.3966 1.3203
  48.774 lt2e-16
• religionNoneVSome 0.8832 0.7049
  1.253 0.211
• religionPracVNonprac -0.2224 1.2016
  -0.185 0.853
• religionGeneralVSpecific -0.3885 1.7738
  -0.219 0.827
• ---
• Signif. codes 0 â 0.001 â 0.01 â 0.05 â 0.1 â 1
  
• Residual standard error 21.28 on 269 degrees of
  freedom
• (4 observations deleted due to missingness)
• Multiple R-squared 0.005973, Adjusted
  R-squared -0.005113
• F-statistic 0.5388 on 3 and 269 DF, p-value
  0.6561

20
Contrast variables

• (Intercept) 64.3966 1.3203
  48.774 lt2e-16
• religionNoneVSome 0.8832 0.7049
  1.253 0.211
• religionPracVNonprac -0.2224 1.2016
  -0.185 0.853
• religionGeneralVSpecific -0.3885 1.7738
  -0.219 0.827
• Residual standard error 21.28 on 269 degrees of
  freedom
• (4 observations deleted due to missingness)
• Multiple R-squared 0.005973, Adjusted
  R-squared -0.005113
• F-statistic 0.5388 on 3 and 269 DF, p-value
  0.6561
• R2, intercept, model p unchanged
• Because it is the same variable
• Slopes change, slope ps change
• Because we analyzed it differently

21
Polynomial Contrasts

• Instead of none versus some, maybe our
  hypothesis is linear More religion leads to more
  SWL
• colnames( contrasts(kreligion) )1 lt- linear
• contrasts(kreligion),1 lt- c(-3,-1,1,3)
• (why?)

22
Polynomial Contrasts

• Instead of none versus some, maybe our
  hypothesis is linear More religion leads to more
  SWL
• colnames( contrasts(kreligion) )1 lt- linear
• contrasts(kreligion),1 lt- c(-3,-1,1,3)
• Spaces the levels evenly and sums to zero
• Contrast tests where the linear effect is zero

23
Polynomial Contrasts

• contrasts(kreligion),1 lt- c(-3,-1,1,3)
• What else? Two more are possible, but not
  necessary
• Polynomial effects are orthogonal
• contrasts(kreligion),2 lt- c(1,-1,-1,1)
• colnames(contrasts(kreligion))2 lt- quad
• contrasts(kreligion),3 lt- c(-1,3,-3,1)
• why?

24
Polynomial Contrasts

• Fits a line to the contrasts

25
Polynomial Contrasts

• Why three?
• Allows for a test of the main effect
• summary(lm(swls religion, datak))

26
When to contrast code

• Whenever you have groups
• Define a set of contrasts
• Use all your contrast codes
• Otherwise, you are not fully analyzing the
  groups there is variance you could explain
• Use all the group codes together test the group
  fully
• So, the R2 for adding all groups tests the
  groupings worth

27
When to contrast code

• What if you only have one contrast?
• Then why did you have more groups?
• I only care whether differences exist!
• Choosing orthogonal contrasts haphazardly is ok
  just dont interpret them
• R2 is the same if all contrasts are used
• I have several controls!
• Test the control directly

28
Contrasts vs. Dummy Coding

• When is one good?
• Depends on your research question
• Contrast codes allow for random group comparison
• But orthogonality make each subsequent contrast
  less free to define
• Dummy coding for a single baseline

29
ANOVA

• is its own plural
• Is about analyzing where variance can be
  accounted for
• E.g., by groups, instead of by means
• Regression is the same thing
• But its about lines through means

30
ANOVA

• The F test for change in R2 is the F test for
  ANOVA
• Same DF, same result, same interpretation
• The t-test for each contrast is the explanation
  of group differences
• Each one is a contrast in ANOVA terms
• Tests a specific theory about groups

31
ANOVA

• Next week
• More than one categorical variable
• Just make as many contrast variables for each
  variable as you need
• Interactions
• Does the effect of one variable depend on the
  level of another?
• If the product of the variables is significant!
• Interpretations