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Announcements this week: factorial ANOVAs (ch 14), correlation (ch 15) HW 5 due today, HW 5-R due Thurs 11/6 Quiz 6 fill-in question regraded today/ tomorrow (Quiz 5 almost done) Prelim 2 next week: Wed., Nov 12 focus on t-tests, one-way ANOVAs (both types), two-way ANOVAs (this week), some estimation and signal detection 1 1 Monday, November 3, 14
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  • Announcements this week: factorial ANOVAs (ch 14), correlation (ch

    15)

    HW 5 due today, HW 5-R due Thurs 11/6 Quiz 6 fill-in question regraded today/

    tomorrow (Quiz 5 almost done)

    Prelim 2 next week: Wed., Nov 12 focus on t-tests, one-way ANOVAs (both types),

    two-way ANOVAs (this week), some estimation and signal detection

    1

    1Monday, November 3, 14

  • Test multiple independent variables (factors) at multiple levels

    Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels: students and faculty)

    2

    What is a factorial design?

    What is the effect of caffeine on reaction time?

    Caffeine might influence people differently (age, lifetime exposure): we can test dosage in combination

    with age

    2Monday, November 3, 14

  • Test multiple independent variables (factors) at multiple levels

    Factor 1: Dosage (2 levels: 1 mg, 3 mg)Factor 2: Age (2 levels: students and faculty)

    3

    What is a factorial design?

    0

    15

    30

    45

    60

    75

    90

    1 mg 3 mg

    Effect of caffeine and age on reaction timestudentsfaculty

    Dosage

    Rea

    ctio

    n tim

    e

    3Monday, November 3, 14

  • can look for relationships between the variables

    The variables interact:As caffeine dosage increases, reaction time decreases more for faculty than for students

    4

    Advantages over repeated two-level designs?

    Dosage

    Rea

    ctio

    n tim

    e

    0

    15

    30

    45

    60

    75

    90

    1 mg 3 mg

    Effect of caffeine and age on reaction timestudentsfaculty

    4Monday, November 3, 14

  • Sampling from how many populations?

    Dosage

    Age

    Student Faculty

    Low

    High

    Factorial designs

    5

    5Monday, November 3, 14

  • Sampling from how many populations?

    Student Faculty

    Dosage

    Age

    Student Faculty

    Factorial designs

    6

    6Monday, November 3, 14

  • Sampling from how many populations?

    Low High

    Dosage

    Age

    Low

    High

    Factorial designs

    7

    7Monday, November 3, 14

  • 0.7 0.6

    0.8 0.2Dosage

    Age

    Student Faculty

    Low

    High

    Multiple populations?

    8

    To see if there is an effect of just one variable, collapse (average) across the other variable

    8Monday, November 3, 14

  • Main effect of Dosage? Collapse across Age

    0.70 0.60

    0.80 0.20

    .65

    .5Dosage

    Age

    Student Faculty

    Low

    High

    Main effects

    9

    9Monday, November 3, 14

  • Main effect of Age? Collapse across Dosage

    0.70 0.60

    0.80 0.20

    .75 .4

    Dosage

    Age

    Student Faculty

    Low

    High

    Main effects

    10

    10Monday, November 3, 14

  • Identifying relationships among variables Interaction: when the effect of one variable

    depends on the level of another variable

    Does the relationship between the reaction times observed in high and low caffeine dosages depend on age?

    Interactions

    11

    11Monday, November 3, 14

  • Interaction?

    12

    Relationship between variables:Moving from 1mg to 3mg, the student reaction time increases slightly, but faculty reaction time decreases.

    0

    15

    30

    45

    60

    75

    90

    1 mg 3 mg

    Effect of caffeine and age on reaction timestudentsfaculty

    Dosage

    Rea

    ctio

    n tim

    e

    12Monday, November 3, 14

  • No Can have an interaction without main effects Can also have main effects without an

    interaction

    Are main effects a prerequisite for an interaction?

    13

    13Monday, November 3, 14

  • 2882

    Dosage

    AgeStudent Faculty

    LowHigh

    Interaction without main effects

    5

    55 5

    14

    Relationship between variables:Moving from low to high, reaction time for students increases, while faculty reaction time decreases.

    0

    2

    4

    6

    8

    low high

    studentsfaculty

    Dosage

    Rea

    ctio

    n tim

    e

    14Monday, November 3, 14

  • 10662

    Dosage

    AgeStudent Faculty

    Low

    High

    Main effects without interaction

    4

    84 8

    15

    Relationship between variables:Moving from 1 mg to 3mg, reaction time for students and faculty increases by the same amount.

    Dosage

    Rea

    ctio

    n tim

    e

    0

    2

    4

    6

    8

    10

    low high

    studentsfaculty

    15Monday, November 3, 14

  • ANOVA answers two questions: Do the different levels of a factor represent

    real differences in the dependent variable?

    Is there an interaction between the factors?

    ANOVA: a statistical test of main effects and interactions

    16

    16Monday, November 3, 14

  • Sum of squares Sum of the variances from the grand mean for each level of a factor Degrees of freedom Number of independent observations Mean square Mean deviation from the grand mean of each observation in a factor Error Tendency for scores to vary from the overall mean

    Essential parts of an ANOVA

    17

    17Monday, November 3, 14

  • Variance (within a factor)

    errorF-ratio =

    Main effects

    18

    18Monday, November 3, 14

  • F-ratio: If between-group differences equal within-group

    differences: H0 true

    If between-group differences are larger than within-group differences: H0 false

    Main effects: Hypotheses

    19

    19Monday, November 3, 14

  • All the variance must be accounted for Any variance not due to the main effects or

    error is due to an interaction of the factors

    Interactions: the variance left over

    20

    20Monday, November 3, 14

  • Example:

    Two-Factor ANOVA

    70 degrees 80 degrees 90 degrees

    30% humidity M = 85 M = 80 M = 75 M = 80

    70% humidity M = 75 M = 70 M = 65 M = 70

    M = 80 M = 75 M = 70

    21

    21Monday, November 3, 14

  • Two-factor ANOVA will do three things:

    - Examine differences in sample means for humidity (factor A)

    - Examine differences in sample means for temperature (factor B)

    - Examine differences in sample means for combinations of humidity and temperature (factor A and B).

    Three F-ratios.

    Two-Factor ANOVA

    22

    22Monday, November 3, 14

  • Main effect for humidity (Factor A)Main effect for temperature (Factor B)

    The differences among the levels of one factor are referred to as the main effect of that factor.

    Main Effects and Interactions

    An example: 70 degrees 80 degrees 90 degrees

    30% humidity M = 85 M = 80 M = 75 M = 80

    70% humidity M = 75 M = 70 M = 65 M = 70

    M = 80 M = 75 M = 70

    23

    23Monday, November 3, 14

  • Evaluation of main effects two out of three hypothesis tests in two-factor ANOVA.

    Factor A (humidity - 2 levels):Hypotheses:

    H0: A1 = A2

    H1: A1 A2

    Main Effects and Interactions

    F = variance between means (Factor A)variance expected by chance/error

    24

    24Monday, November 3, 14

  • Factor B (temperature - 3 levels):

    Hypotheses:

    H0: B1 = B2 = B3

    H1: At least one is different.

    F-ratio:

    Main Effects and Interactions

    F = variance between means (Factor B)variance expected by chance/error

    25

    25Monday, November 3, 14

  • All the variance must be accounted for

    Any variance not due to the main effects or error is due to an interaction of the factors

    Interactions: the variance left over

    26

    26Monday, November 3, 14

  • variance not explained by main effectsvariance expected by chance/error

    Interaction Hypotheses

    H0: There is no interaction between factors A and B. (all mean differences are explained by main effects)

    H1: There is an interaction between factors A and B

    Main Effects and Interactions

    F = 27

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  • 28

    In a graph, lines that are non-parallel indicate the presence of an interaction between two factors.

    Main Effects and Interactions

    28Monday, November 3, 14

  • 29

    Two-factor ANOVA consists of three hypothesis tests. The outcomes of these tests are totally independent.

    All combinations of outcomes are possible:

    2 main effects and interaction

    1 main effect and interaction

    2 main effects and no interaction

    1 main effect and no interaction

    interaction but no main effects

    no main effects, no interaction

    Main Effects and Interactions

    29Monday, November 3, 14

  • 30

    Two-factor ANOVA hypothesis test

    Step 1: State hypotheses

    Step 2: Determine critical region (critical F values)

    Step 3: Calculate F-ratios

    Step 4: Make decision

    30Monday, November 3, 14

  • 31

    Notation and Formulas

    Three hypothesis tests three F-ratios four variances.

    Schematic view:

    31Monday, November 3, 14

  • 32

    Notation and Formulas

    32Monday, November 3, 14

  • 33

    Stage 1:

    Total variability:

    SStotal = X2 -G2

    N

    dftotal = N-1

    Notation and Formulas

    33Monday, November 3, 14

  • 34

    Stage 1:

    Between-treatments variability:

    dfbetween treatments = number of cells -1

    SSbetween treatments = T2

    nG2

    N-

    Notation and Formulas

    34Monday, November 3, 14