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Statistics > Chapter 11 part b: GLM1 > Flashcards

Chapter 11 part b: GLM1 Flashcards

1
Q

contrasts

A

necessary after conducting an ANOVA to find out which groups differ

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2
Q

A way to contrast the different groups without inflating Type I error rate

A
  1. break down variance accounted by model into component parts (planned contrasts)
  2. compare every group (as if conducting several t-tests) but to use a stricter acceptance criterion such that the familywise error rate does not rise above .05 (Post-Hoc Test)
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3
Q

The difference between planned comparisons and post hoc tests:

A
  • planned comparisons are done when you have specific hypotheses that you want to test,
  • whereas post hoc tests are done when you have no specific hypotheses
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4
Q

Planned Contrasts

A
  • used when testing specific hypothesis
  • Example:

— H1: any dose of Viagra changes libido compared to the placebo group
— H2: high dose should increase libido more than a low dose

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5
Q

Standard Planned Comparisons

A
  • Contrast I: compare experimental conditions to control
  • Contrast II: check the difference between the experimental groups

—> when 2 experimental groups: C.2: E1 vs E2
—> when 3 experimental groups C.2: E1 vs E2, E3
C.3: E2 vs E3

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6
Q

Rules of Planned Contrasts

A

I. control group to compare it against other groups
II. Each contrast must compare only two ‘chunks’ of variation
III. Once a group has been singled out in a contrast it can’t be used in another contrast

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7
Q

Number of Planned Contrasts

A

k-1

[ # predictor categories - 1]

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8
Q

Why compare only 2 chunks of variation at a time in planned contrasts?

A
  • we can be sure that a significant result represents a difference between these 2 portions of variation
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9
Q

Planned Contrasts:

- If contrast I is significant, conclude that:

A

the average of experimental groups is significantly different from the average of control

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10
Q

Planned Contrasts:

- If standard errors (SE) are the same:

A
  • experimental group with the highest mean will be significantly different from the mean of placebo group
  • for experimental group with the lowest mean: do a post hoc to determine if it differs from placebo
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11
Q

Planned Contrasts: Weights

A
  • To carry out planned contrasts assign certain values to dummy variables in regression model
  • The values assigned to dummy variables: weights
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12
Q

Rules for assigning weights in Planned Contrasts:

A

Rule I:
Compare only 2 chunks of variation and that if a group is singled out in one comparison, that group should be excluded from any subsequent contrasts
Rule 2:
assign one chunk of variation positive (+) weights and the opposite chunk negative (-) weights
Rule 3:
The sum of weights for a comparison should be 0
Rule 4:
If a group is not involved in a comparison, automatically assign it a weight of zero
Rule 5:
Weights assigned to the group(s) in one chunk of variation should be equal to the number of groups in the opposite chunk
- assign control (+3) when three experimental groups (-1, -1, -1)

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13
Q

Planned Contrasts: Orthogonal

A

Independent Contrasts

  • if sum product of contrasts equals 0
  • contrast 1 x contrast 2 for each variable (including control) and add them all
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14
Q

Planned Contrasts: Equation

A

Outcome = bo + b1Contrast1 + b2Contrast2

Control Mean = Grand mean - 2b1
(bo: grand mean)
(Contrast 1: weight of the control group for contrast 1)
(Contrast 2: weight of control group for this contrast is 0)

Experiment Group 1 Mean = Grand mean + b1 + b2

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15
Q

Planned Contrast: Equation

A
  • depends on the weights we give to each chunk or dummy variables
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16
Q

Planned Contrast: Equation

  • What is b1?
A
  • the difference between the average of experimental groups and control group
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17
Q

Planned Contrast: Equation

  • What is b2?
A
  • difference between mean of each experimental group divided by the number of groups in this contrast
  • contrast 2: high dose vs low dose
    • b2: difference between mean of high dose and low dose divided by 2
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18
Q

Planned Contrast: Equation

  • Why divide difference of means of each experimental group by number of groups in that contrast to obtain b2?
A
  • to control for family wise error rate
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19
Q

Non-orthogonal Contrasts

A
  • non-independent contrasts
  • disobeying Rule I: Once a group has been singled out in a contrast it can’t be used in another contrast
  • Example:
    - Contrast 1: Compare experimental groups against control
    - Contrast 2: Compare high-dose group to control
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20
Q

Non-orthogonal Contrasts

A
  • sum of product of contrasts is not 0
  • not wrong
  • careful with interpretations because comparisons are related and so will the p values
  • hence, use a more conservative alpha
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21
Q

Standard Contrasts

A
  • under most circumstances: you can design your own contrasts
  • standard contrasts: special contrasts to compare certain situations
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22
Q

Orthogonal Standard Contrasts

A
  • Helmert

- Difference (Reverse Helmert)

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23
Q

Non-orthogonal Standard Contrasts

A
  • Deviation (first, last)
  • Simple (first, last)
  • Repeated
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24
Q

Helmert Standard Contrasts

A
  • orthogonal
  • each category (except last) is compared to the mean effect of all subsequent categories

Example:

   - 3 Categories                 - 4 Categories
   - 1 vs (2,3)                        - 1 vs (2,3,4)
   - 2 vs 3                               - 2 vs (3,4)
                                                 - 3 vs 4
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25
Q

Difference Standard Contrasts

- Reverse Helmert

A
  • orthogonal
  • each category (except 1st) is compared to the mean effect of all previous categories
  • 3 Categories - 4 Categories
  • 3 vs (1,2) - 4 vs (1,2,3)
  • 2 vs 1 - 3 vs (1,2)
    - 2 vs 1
26
Q

Deviation Standard Contrast

First

A
  • non-orthogonal
  • compare the effect of each category (except 1st) to the overall experimental effect
  • 3 Categories - 4 Categories
  • 2 vs (1,2,3) - 2 vs (1,2,3,4)
  • 3 vs (1,2,3) - 3 vs (1,2,3,4)
    - 4 vs (1,2,3,4)
27
Q

Deviation Standard Contrast

Last

A
  • non-orthogonal
  • compare the effect of each category (except last) to the overall experimental effect
  • 3 Categories - 4 Categories
  • 1 vs (1,2,3) - 1 vs (1,2,3,4)
  • 2 vs (1,2,3) - 2 vs (1,2,3,4)
    - 3 vs (1,2,3,4)
28
Q

Simple Standard Contrast

First

A
  • non-orthogonal
  • each category is compared to the 1st category
  • 3 Categories - 4 Categories
  • 1 vs 2 - 1 vs 2
  • 1 vs 3 - 1 vs 3
    - 1 vs 4
29
Q

Simple Standard Contrast

Last

A
  • non-orthogonal
  • each category is compared to the last category
  • 3 Categories - 4 Categories
  • 3 vs 1 - 4 vs 1
  • 3 vs 2 - 4 vs 2
    - 4 vs 3
30
Q

Repeated Standard Contrasts

A
  • non-orthogonal
  • each category (except 1st) is compared to previous category
  • 3 Categories - 4 Categories
  • 1 vs 2 - 1 vs 2
  • 2 vs 3 - 2 vs 3
    - 3 vs 4
31
Q

Polynomial Contrasts

A
  • trend analysis in the categorical predictor
  • orthogonal
  • no need to construct your own codes

> Linear Trend: Proportionate
Quadratic Trend: at least 3 categories
Cubic Trend: at least 4 categories and 2 changes in the direction of a trend
Quartic Trend: at least 5 categories and 3 changes in the direction of the trend

  • Example: Viagra - control, high, low
    —> 3 categories: check if linear or quadratic
32
Q

Post-Hoc

A
  • consists of pairwise comparisons

- compares all different combinations of predictor variables

33
Q

Post-Hoc:

How do pairwise comparisons control for family-wise error rate?

A
  • by correcting level of significance for each test so that Type 1 remains .05 across all comparisons
  • Bonferroni Correction
34
Q

Which Post-Hoc test performs best?

A

Depends on 3 criteria:
I- Does the test control for Type 1
II- Does the test control for Type 2?
III- Is the test Robust?

35
Q

Type 1 and Type 2 error rate for Post-Hoc tests

A
  • conservative Post-Hoc test:
    • Type 1 error small
    • Type 2 error high: lack power
36
Q

Liberal Post-Hoc

A
  • least significant difference (LSD)
  • studentized Newman-Keuls

—> high power

37
Q

Conservative Post-Hocs

A
  • Bonferroni and Turkey’s: lack power
    ~ Bonferroni more powerful when number of comparisons is small
    ~ Tukey’s more powerful when testing large number of means
  • REGWQ: good power & type 1 control
38
Q

Are Post Hoc tests robust?

A
  • most Post Hoc test perform relatively well under small deviations of normality
  • perform badly when group sizes and variances are NOT equal
39
Q

Post Hoc tests when group sizes are slightly unequal:

A
  • Hochberg’s GT2

- Gabriel’s

40
Q

Post Hoc tests when group sizes and variances are very different:

A
  • Tamhane’s T2
  • Dunnett’s T3
  • Games-Howell
  • Dunnett’s C
41
Q

Best Post Hocs:

  • Equal Sample Sizes and Variances
A
  • REGWQ or Turkey
42
Q

Best Post Hoc:

  • guaranteed control for Type 1 error
A
  • Bonferroni
43
Q

Best Post Hoc:

  • sample sizes slightly different
A

Gabriel’s

44
Q

Best Post Hoc:

  • sample sizes very different
A
  • Hochberg’s GT2
45
Q

Best Post Hoc:

  • variances: unequal
A
  • Games-Howell

- run this test in addition to any other post Hoc tests to control for homoscedasticity

46
Q

Running One-Way ANOVA: SPSS

A

I. Analyze
II. Compare Means
III. One-way ANOVA

47
Q

One-Way ANOVA Procedure

A

I. Check for assumptions, outliers, influential cases
II. Correct for outliers or other violations
III. Run One-Way ANOVA
IV. Follow-up: Contrasts
V. Calculate effect sizes

48
Q

Planned Contrasts: SPSS

A 
I. One-Way ANOVA
II. Contrasts
III. Tick Polynomial to find trends
—> important to have coded our groups in ascending order such as:
Control: 1
Low viagra: 2
High viagra: 3

IV: Degree: Quadratic or Cubic depends on how many categories in total

For planned contrasts:
—> Coefficients: Add (ascending order)
—> Click Next to add weights of Contrast 2
—> Do NOT forget to assign 0 to the group you’re not using in that contrast

49
Q

Post-Hoc: SPSS

A
  • either do planned contrasts or Post-Hoc tests: NOT both

I. One-Way ANOVA
II. Post-Hoc

  • Always select Dunnett because it is the only post Hoc test that enables us to compare means to control group mean
  • which group is your control?

III. Choose 2-sided
IV: Choose cases analysis by analysis

50
Q

One-Way ANOVA: Options

A
  • Descriptive
  • Homogeneity of variances test: Levene’s
  • Brown-Forsythe (concerned about unequal variances)
  • Welch (concerned about unequal variances)
  • means plot
  • Exclude cases analysis by analysis
51
Q

One-Way ANOVA: Bootstrapping

A
  • good way to overcome bias

- if sample size is very small: do NOT select Bootstrap

52
Q

Output: One-way ANOVA

A
  • Descriptives
  • Test of Homogeneity of Variances: check if Levene’s test is significant - violation of homoscedasticity
  • ANOVA
    > within groups: gives details on unsystematic variation within data (SSR)
53
Q

Output: One-way ANOVA

  • Trend Analysis -
A
  • look at linear component:

* This comparison tests whether the means increase across groups in a linear way: if p

54
Q

Output: One-way ANOVA

  • Welch and Brown
A

Report this F Statistics when there is a violation of homogeneity of variances

55
Q

Are we allowed to halve the significance of a two tailed test ANOVA?

A

No!

  • when comparing more than 2 means: no directional hypothesis
56
Q

Output: One-way ANOVA

  • Tukey’s and REGWQ
A
  • These tests display subsets of groups that have the same means
  • if p non significant: means of both groups are statistically similar
  • Harmonic mean: weighted version of mean by taking into account relationship between sample size and variances
    - to reduce bias brought by unequal sizes: still biased though
57
Q

Effect Size of ANOVA: Eta Squared (mu^2)

A
  • R^2 = r^2 = SSM/SST

- this measure slightly biased

58
Q

Effect Size of ANOVA: Omega Squared

A
  • Best way
  • w^2=[SSM-(dfM) x MSR]/(SST+ MSR)
  • small effect .01
  • medium effect .06
  • large effect .14
59
Q

Effect Size of Planned Contrasts

A
  • r contrast =square root [t^2/(t^2+df)]

- eta squared criteria

60
Q

Reporting Results: One-Way ANOVA

A
  • There was a significant effect of Viagra on levels of libido, F(2, 12) = 5.12, p =.025, ω = .60.
  • There was a significant linear trend, F(1, 12) = 9.97, p =.008, ω =.62, indicating that as the dose of Viagra increased, libido increased proportionately.
  • Planned contrasts revealed that having any dose of Viagra significantly increased libido compared to having a placebo, t(12) = 2.47, p =.029, r =.58, but having a high dose did not significantly increase libido compared to having a low dose, t(12) = 2.03, p =.065, r =.51.

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