Slides to accompany Weathington, Cunningham & Pittenger (2010),

Chapter 12: Single Variable Between-Subjects Research

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Objectives• Independent Variable• Cause and Effect• Gaining Control Over the Variables• The General Linear Model• Components of Variance• The F-ratio• ANOVA Summary Table• Interpreting the F-ratio• Effect Size and Power• Multiple Comparisons of the Means

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Multi-level Independent Variable• More than 2 levels of the IV• Permits more detailed analysis

– Can’t identify certain types of relationships with only two data points (Figure 12.1)

• Can increase a study’s power by reducing variability within the multiple treatment condition groups

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Figure 12.1

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Searching for Cause and Effect• Identifying differences among multiple

groups is a starting point for causal study

• Control is the key:– Through research design– Through research procedure

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Control through Design• Most easily secured in a true

experiment• You manipulate and control the IV

– Control groups are possible isolating effects of IV

• You control random assignment of participants– Helps to reduce confounding effects

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Control through Procedure• Each participant needs to experience

the same process (except the manipulation)– Systematic

• Identifying and trying to limit as many confounding factors as possible

• Pilot studies are a great way to test your process and your control strategies

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General Linear Model• Xij = µ + αj + εij

• A person’s performance (score = Xij) will reflect:– Typical score in that group (µ)

– Effect of the treatment/manipulation (αj)

– Random error (εij)

• Ho: all µi equal8

Figure 12.2

ControlM = µ + α1

SD = σXi1 = µ + α1 + εi1

Sampling frameµ, σ

SampleM = µSD = σ

Xi = µ + ε

Random Assignment

Intelligence feedback Effort feedbackNo feedback

EffortM = µ + α3

SD = σXi3 = µ + α3 + εi3

IntelligenceM = µ + α2

SD = σXi2 = µ + α2 + εi2

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GLM and Between-Subj. Research• Goal is to determine proportion of

total variance due to IV and proportion due to random error

• Size of between-groups variance is due to error (εij) and IV (αj)

• If b-g variance > w-g variance IV has some effect

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ANOVA• Compares different types of variance

– Total variance = variability among all participants’ scores (groups do not matter)

– Within-groups variance = average variability among scores within a group or condition (random)

– Between-groups variance = variability among means of the different treatment groups•Reflects joint effects of IV and error

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F-ratio• Allows us to determine if b-g

variance > w-g variance• F = Treatment Variance + Error

Variance

Error Variance

• F = MSbetween/MSwithin

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F-ratio: No Effect• Treatment group M may not all be

exactly equal, but if they do not differ substantially relative to the variability within each group nonsignificant result

• When b-g variance = w-g variance, F = 1.00, n.s.

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Figure 12.4

|-----M1-----|

|-----M2-----|

|-----M3-----|

|---Moverall---|

Condition

Control

Intelligence

Effort

Between-groups

0 1 2 3 4 5 6 7 8 9 10Score range

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F-ratio: Significant Effect• If IV influences DV, then b-g

variance > w-g variance and F > 1.00

• Examining the M can highlight the difference(s)

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Figure 12.5

|-----M1-----|

|-----M2-----|

|-----M3-----|

|------------------------Moverall------------------------|

Condition

Control

Intelligence

Effort

Between-groups

0 1 2 3 4 5 6 7 8 9 10Score range

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F-ratio Distribution• Represents probability of various F-

ratios when Ho is true

• Shape is determined by two df– 1st = b-g = (# of groups) - 1 – 2nd = w-g = (# of participants in a

group) – 1

• Positive skew, α on right extreme region

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Figure 12.6

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Summarizing ANOVA Results• Figure 12.7• Using the critical value from

appropriate table in Appendix B, if Fobs > Fcrit significant difference among the M

• Rejecting Ho requires further interpretation– Follow-up contrasts

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Figure 12.7

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Interpreting F-ratio• Omega squared indicates degree of

association between IV and DV• f is similar to d for the t-test• Typically requires further M

comparisons– t-test time, but with reduced α to

limit chances of committing a Type I error

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Multiple Means Comparisons• You could consider lowering α to .01,

but this would increase your Type II probability

• Instead use a post-hoc correction for α:

– αe= 1 – (1 – αp)c

– Tukey’s HSD = difference required to consider M statistically different from each other

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What is Next?• **instructor to provide details

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