one-way between subjects anova also called one-way randomized anova purpose: determine whether there...
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ONE-WAY BETWEEN SUBJECTS ANOVA
• Also called One-Way Randomized ANOVA• Purpose: Determine whether there is a
difference among two or more groups– used mostly with three or more groups– does not show which groups differ (unless
there are only two)
Design and Assumptions
• Design:– One way means one independent variable– Between subjects means different people in
each group.
• Assumptions: same as for independent samples t-test
Why not t-tests?
• Multiple t-tests inflate the experimentwise alpha level.
• experimentwise alpha level is the total probability of Type I error for all tests of significance in the study.
• ANOVA controls the experimentwise alpha level.
Pc = N!
r!(N - r)!
p rqN -r
If I am doing six t-tests, each with a .05 alpha level, what is the experimentwise alpha?
P(0 errors) = 6x5x4x3x2x1
(1)6x5x43x2x1
.050 .956
P(0 errors) = 1 1 .956 = .7351
So, the probability of making one or more errors is 1 - .7351 = .2649.
Concept of ANOVA
• Analysis of Variance• Variance is a measure of variability• Two step process:
– divides the variance into parts– compares the parts
About Variance
s 2 (x x)2
N 1
• Numerator is the Sum of Squares• Denominator is the Degrees of Freedom
Mean Square
• Variance is also called Mean Square• Formula for variance in ANOVA terms:
Mean Square Sum of Squares
degrees of freedom
Part I: Dividing the Variance
• Total Variance is divided into two parts:– Between Groups Variance - only differences
between groups.– Within Groups Variance - only differences within
groups.
• Between Groups + Within Groups = Total
Example of Between Groups variance only:
Group 1 Group 2 Group 34 6 84 6 84 6 8
Example of Within Groups variance only:
Group 1 Group 2 Group 34 6 48 4 86 8 6
What Influences Between Groups Variance?
• effect of the i.v. (systematic)• individual differences (non-systematic)• measurement error (non-systematic)
What Influences Within Groups Variance?
• individual differences (non-systematic)• measurement error (non-systematic)
Part II: Comparing the Variance
F = Between Groups Variance
Within Groups Variance
F = non - systematic + effect of i.v.
non - systematic
About the F-ratio
• Larger with a bigger effect of the IV• Expected to be 1.0 if Ho is true• Never significant below 1.0• Can’t be negative
Sampling Distribution of F
1.0
Computation of One-Way BS ANOVA
EXAMPLE: Twelve participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance.
(See data on next page)
No noise Low noise High noise
15 15 12
17 19 10
18 14 10
14 12 12
x=16 x=15 x=11
grand mean = 14
ANOVA Summary Table
Source SS df MS F p
Between
Within
Total
STEP 1: SS Total = S(x-xG)2 grand meanx x-x (x-x)2
15 1 117 3 918 4 1614 0 015 1 119 5 2514 0 012 -2 412 -2 410 -4 1610 -4 1612 -2 4 S = SS Total = 96
STEP 2: SS Between = S(xg-xG)2
group meanx x-x (x-x)2
16 2 416 2 416 2 416 2 415 1 115 1 115 1 115 1 111 -3 911 -3 911 -3 9 11 -3 9 S = SS Between = 56
STEP 3: SS Within = SS Total - SS Between
SS Within = 96 - 56 = 40
ANOVA Summary Table
Source SS df MS F p
Between 56
Within 40
Total 96
STEP 4: Calculate degrees of freedom.
df Total = N-1
df Total = 12-1 = 11
df Between = k-1 k=#groups
df Between = 3-1 = 2
df Within = N-k
df Within = 12-3 = 9
ANOVA Summary Table
Source SS df MS F p
Between 56 2
Within 40 9
Total 96 11
STEP 5: Calculate Mean Squares
28.00 2
56
Bet df
Bet SS =Between MS
4.44 9
40
Withindf
WithinSS = Within MS
ANOVA Summary Table
Source SS df MS F p
Between 56 2 28.00
Within 40 9 4.44
Total 96 11
STEP 6: Calculate F-ratio.
6.31 4.44
28.00
WithinMS
Between MS = F
STEP 7: Look up critical value of F.df numerator = df Betweendf denominator = df Within
F-crit (2,9) = 4.26
APA Format Sentence
A One-Way Between Subjects ANOVA showed a significant effect of noise, F (2,9) = 6.31, p < .05.
ANOVA Summary Table
Source SS df MS F p
Between 56 2 28.00 6.31 <.05
Within 40 9 4.44
Total 96 11
Computing Effect Size
2 =SS Between
SS Total
2 =56
96 .58
Eta-squared is the proportion of variance in the DV that can be explained by the IV.
KRUSKAL-WALLIS ANOVA
• Non-parametric replacement for One-Way BS ANOVA
• Assumptions:– independent observations– at least ordinal level data– minimum 5 scores per group
EXAMPLE: Fifteen participants were randomly assigned to one of three noise conditions and given a spelling test. Determine whether noise had a significant effect on spelling performance.
(See data on next page)
Calculating the Kruskal-Wallis ANOVA
No noise Low noise High noise
17 19 9
18 16 8
14 12 12
16 11 8
13 10 7
STEP 1: Rank scores.
No noise Low noise High noise17 13 19 15 9 418 14 16 11.5 8 2.514 10 12 7.5 12 7.516 11.5 11 6 8 2.513 9 10 5 7 1
STEP 2: Sum ranks for each group.SR1 = 57.5 SR2 = 45 SR3 = 17.5
STEP 3: Compute H.
H =12
15(16)
57.52
5
452
5
17.52
5
3(16)
H = (.0500) 661.25 + 405 + 61.25 48
H = (.0500) 1127.50 48
H = 56.38 - 48 = 8.38
H =12
N(N+ 1)
R1 2
n1
R2 2
n2
...
3(N 1)
STEP 4: Compare to critical value from 2 table.
df = 2, critical value = 5.99
A Kruskal-Wallis ANOVA showed a significant difference among the three noise conditions, H(2) =8.38, p < .05 .
ANOVA for Within Subjects Designs
• When the IV is within subjects (or matched groups), a Repeated Measures ANOVA should be used
• The logic of the ANOVA is the same• Calculation differs to take advantage of the
design
ANOVA for Within Subjects Designs
• The Friedman ANOVA is the non-parametric replacement for One-Way Repeated Measures ANOVA