Orthogonality, Complex Contrasts and Two-way ANOVA

Question 1

INDEPENDANCE OF CONTRASTS

Answer 1

• multiple contrasts require unique info (independence): otherwise show bigger difs than true
• ie. ABC conditions
• A/C = STATSIG; ind of AB/C? NO!
• contrasts will influence each other

Question 2

INDEPENDENT/ORTHOGONAL CONTRASTS

Answer 2

• contrast pairs are independent/orthogonal if inner product calc = 0
• ie. (1, 1, 1, -3)/(0, 1, 1, -2) = (1x0)+(1x1)+(1x1)+(-3x-2) = 0+1+1+6 = 8 < 0 so NORTH

Question 3

TESTING ORTHOGONALITY

Answer 3

• 2+ contrasts = calc per individual pairs
• ie. C3 ORTH? = C3/C1, C3/C2
• if BOTH = 0, then ORTH

Question 4

PC: TRENDS

Answer 4

LINEAR 
- (-3, -1, 1, 3); straight slope going up/down
QUADRATIC
- (1, -1, -1, 1); double slopes/bends
CUBIC
- (1, -3, 3, -1); zigzag

Question 5

POLYNOMIAL CONTRASTS

Answer 5

• some studies are interested in general trends (ie. decline) rather than non-specific effects
• which weights to be combined with which trends are in booklet

Question 6

ONE-WAY ANOVA: EXAMPLE

Answer 6

• driving distance with weather; 4c; 3pp; 1-10 scale of distance estimate accuracy (10 = perfect; 1 = guessing)
  1. day/clear (9, 10, 9 = 28)
  2. night/clear (8, 9, 7 = 24)
  3. day/foggy (5, 6, 5 = 16)
  4. night/foggy (2, 1, 1 = 4)

Question 7

OWA: RESULTS

Answer 7

1. Mean day (22) > mean night (14)
2. Mean clear (26) > mean foggy (10)
3. Night-foggy deficit (-20) > day-foggy deficit (-12)

Question 8

OWA: STATS

Answer 8

• only shows performance was different statistically in a group
• does NOT mean performance is affected day VS night/clear VS foggy; no answers for detailed questions
• however, there is a STATSIG difference day VS night/clear VS foggy
• dif = -1 VS 1 for fogginess, so it DOES change depending on night/day
• contrasts subdivide into 3 orthogonal parts

Question 9

OWA: CONCLUSION

Answer 9

• if experiment has +2 factors (ie. day/foggy, night/foggy, etc) contrasts can be used to focus on one at a time to filter influence of others
• raises possibility to examine +2 factors separately BUT in same design and to see how the effects of each are moderated by others
• can be automated via 2-WAY ANOVA

Question 10

2-WAY ANOVA

Answer 10

• can examine independent effects on data of +2 IVs
• outputs give main effects which say whether each variable has an IV effect alone
• interactions say how affect of each variable is itself altered by influence of others

Question 11

2WA: EXAMPLE

Answer 11

• input data in c3
• input conditions in c1/2 (ie. day/night, foggy/clear)
• ANALYZE-GENERAL LINEAR MODEL-UNIVARIATE
• c3 = DV; c1/2 = FF = OK

Question 12

2WA: EXAMPLE RESULTS

Answer 12

1. Driving during day VS driving during night = reliable distance effect (F(1, 8) = 42.6, p<0.001)
2. Foggy STATSIG affected distance VS clear (F(1, 8) = 170.6 p<0.001)
3. Day VS night changes depending on fog/clear (F(1, 8) = 10.6, p<0.001)

Question 13

OWA VS 2WA

Answer 13

• both give inconsistent answers; OWA = day/night doesn’t influence distance; 2WA = it does
• OWA = bigger error variation; assumes differences are random variation; 2WA may show its significant (ie. fogginess); inflates error in OWA
• 2WA = more informative so preferred

Question 14

DEGREES OF FREEDOM

Answer 14

• ie. F (1,8)
• 1 = numerator DOF (df1); main effect requires -1 than level number in examined factor (ie. day/night = 2 levels; 2-1 = 1)
• 8 = denominator DOF (error on SOURCE/df2); main effect relates to data-points collected at each level; increases w/sample; big df2 = low target value for reliable f-ratio