designs
Advanced Experimental Statistical Methods Chapter 16
if your subjects are exposed to only one of conditions, between-subjects design if your subjects are exposed to all conditions, within-subjects design
The Within-Subjects Single-Factor Design
Experimental designs
Between-subjects
Experiment Control
Within-subjects designs
Advantages: more efficient and convenient need less number of subjects
Within-subjects
Repeated measures Condition 1 Condition 2
Within-subjects designs
When they are used?
research question involves studying change over time (longitudinal study) nature of the study: sensation, perception studies (more sensitive to small differences between conditions)
Within-subjects designs
Disadvantages order effects
progressive effects (practice effects)
improved performance from trial to trial
carryover effects
impaired performance from trial to trial (e.g., fatigue)
How to control order effects?
counterbalancing!
1
Between-Subjects Single-Factor Design
Analysis of variance (ANOVA)
one-way ANOVA Simultaneously compare the mean differences among more than two groups basic idea:
ANOVA
Total variance Total variance A A S/A
F=
MS A MS S / A
=
MSbetween MS within
between - groups variablity = within - group variablity
S/A F ratio is small
=
Treatement effects + Experimental error Experimental error
F ratio is large
Within-Subjects Single-Factor Design
Analysis of variance (ANOVA) for repeated measures basic idea: Total variance = between-condition variance + between-subjects variance + within-subject variance
AxS
SS S / A = SS S + SS AS
between - conditions variance MS A F= = within - subject variance MS AS
S AxS
A
AxS
Statistical Assumptions
Between-subjects design
1. 2. 3.
Statistical Assumptions
Within-subjects design
One solution use more conservative level!
e.g., use .025 rather than .05
treatment populations are normality distributed treatment populations have equal variances individual observations are independent of (or uninfluenced by) any other observations within the same population or between treatment populations
Within-subjects design
1 through 3 The variances of differences among all conditions are equal in the population Sphericity assumption serious impact on F-test even with minor violations of this assumption
2
Lecture and Vocabulary Test: Between-Subject Design
Lecture Physical 53 49 47 42 51 34 44 48 35 18 32 27 480 Social 47 42 39 37 42 33 13 16 16 10 11 6 312 History 45 41 38 36 35 33 46 40 29 21 30 20 414
Between-Subjects Design: Basic Ratios
SS A = [A] [T ] SST = [Y ] [T ]
SS S / A = [Y ] [ A]
SST = SS A + SS S / A
ANOVA table (Between-Subject Design)
Source A Basic Ratios SS df MS F
ANOVA table (Between-Subject Design)
[A] =
S/A
A
j
a
Source
2 j
SS
df
MS
F
Sig.
[A] - [T] a-1
S
SS A df A
MS A MS S / A
A(LECTURE)
1194
2
597.00
4.28
.02
[Y = ] Y 2
[T ] =
T2 as
[Y] - [A] a(s-1)
SS S / A df S / A
S/A
4599
33
139.36
Total
[Y] - [T] as-1
Total
5793
35
Lecture and Vocabulary Test: Within-Subject Design
Lecture Subjects S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 A Physical 53 49 47 42 51 34 44 48 35 18 32 27 480 Social 47 42 39 37 42 33 13 16 16 10 11 6 312 History 45 41 38 36 35 33 46 40 29 21 30 20 414 S 145 132 124 115 128 100 103 104 80 49 73 53 1206
Within-Subjects Design: Basic Ratios
SS A = [A] [T ] SS S = [S ] [T ]
SS AS = SS S / A SS S = ([Y ] [ A]) ([ S ] [T ]) = [Y ] [A] [ S ] + [T ]
3
ANOVA table (Within-Subject design)
Source Basic Ratios SS A S AXS Total
[A] =
ANOVA table (Within-Subject Design)
df a-1 s-1
(a-1)(s-1)
MS
SS A df A SS S df S SS AS df AS
F
MS A MS A S
A2
s S
2
[A] - [T] [S] - [T] [Y]-[A][S]+[T] [Y] - [T]
Source A (LECTURE) S (Subjects) AXS Total
SS 1194.00 3531.67 1067.33 5793.00
df 2 11 22 35
MS 597.00 321.06 48.52
F 12.30
Sig. .000
[S ] =
a
[Y ] = Y 2
[T ] =
T2 as
(a)(s) -1
Single-df comparisons (p. 281)
Between-subjects design the error term from the omnibus analysis (MS S/A) Under the assumption that population treatment variances are equal. MS S/A provides a perfectly suitable estimate of error variance present in any comparison conducted on the treatment means. Within-subjects design the error term from the omnibus analysis (MS AXS) can not serve a similar function. need to find a specific error term that is based only on the data contributing to the particular comparisons under study.
Further analysis: Single-df comparisons
SS Acomp. for S j =
(c )
i
( Diff ) 2
2
AS matrix Subjects S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 a1 ...