psyc 6130 one-way independent anova. psyc 6130, prof. j. elder 2 generalizing t-tests t-tests allow...
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PSYC 6130
One-Way Independent ANOVA
PSYC 6130, PROF. J. ELDER 2
Generalizing t-Tests
• t-Tests allow us to test hypotheses about differences between two groups or conditions (e.g., treatment and control).
• What do we do if we wish to compare multiple groups or conditions simultaneously?
• Examples:
– Effects of 3 different therapies for autism
– Effects of 4 different SSRIs on seratonin re-uptake
– Effects of 5 different body orientations on judgement of induced self-motion.
PSYC 6130, PROF. J. ELDER 3
Reinterpreting the 2-Sample t-Statistic
2
1 2
222
p
X Xn
ts
2 2
2 2 21 2
The is an estimate of the variance of the population,
derived by averaging the variances the two samples:
denominator
within
1( )
2
p
p
s
s s s
PSYC 6130, PROF. J. ELDER 4
Reinterpreting the 2-Sample t-Statistic
2 2To see this, recall that X X
ss s ns
n
2 22
1 1 2 2 1 2
1 1Thus, ( ) ( )
2 2Xs X X X X X X
2 2 21 2 1 2
1For 2 groups, ( ) ( ) , where ( )
2G G GXs X X X X X X X
2
1 2
222
p
X Xn
ts
2The is also an estimate of the variance of the population,
derived from the
numerator
betwvariance the sample mee .en ans
2 2
1 2 2 1
1 1( ) ( )
2 2X X X X
21 2
1( )
2X X
PSYC 6130, PROF. J. ELDER 5
-10.2 4.8-1.8 6.715.2 -0.8-0.4 8.912.3 23.1-7.0 5.20.1 -0.1
-7.8 9.15.9 0.6
-2.5 -11.1
Mean 0.4 4.6 2.5Std Dev 8.4 8.8
Example
1X 2X
GX
PSYC 6130, PROF. J. ELDER 6
The F Distribution 2
1 2
222
p
X Xn
ts
2
Thus, under the null hypothesis, the numerator and denominator are
estimates of the same population varindependent iance .
The ratio of 2 independent, unbiased estimates of the same variance
foll distriows an .butionFF distribution for 2 groups of size n=13
0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
PSYC 6130, PROF. J. ELDER 7
Within and Between Variances
• Recall that the variance is, by definition, the mean squared deviation of scores from their mean.
• Since the numerator of the t2 statistic estimates the variance from the deviations of group means, it is called the mean-square-between MSbet.
• Since the denominator of the t2 statistic estimates the variance from the deviations within groups, it is called the mean-square-within MSW.
• These definitions allow us to generalize to an arbitrary number of groups.
Thus bet
W
MSF
MS
PSYC 6130, PROF. J. ELDER 8
Generalizing to > 2 Groups
bet
W
MSF
MS
2
i i iall scores
( ) 1 1, where X n Xi i G
bet Gbet T T
n X XMS X
df N N
2( 1)i iW
w
n sMS
df
PSYC 6130, PROF. J. ELDER 9
Degrees of Freedom
• Recall that the sample variance follows a scaled chi-square distribution, parameterized by the degrees of freedom.
• Thus the F distribution is a ratio of two chi-square distributions, each with different degrees of freedom.
1, where number of groups.betdf k k
, where = total number of subjects over all groups.
1W T T
i
df N k N
n
1tot bet W Tdf df df N
PSYC 6130, PROF. J. ELDER 12
Testing Hypotheses
3.32 for .05 (Appendix F)critF
bet
W
MSF
MS
20
Large values of suggest that differences between the groups
are inflating the estimate of reject .bet
F
MS H
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
p(F
)
F distribution for 3 groups of size n=13
PSYC 6130, PROF. J. ELDER 13
PSYC 6130, PROF. J. ELDER 14
When k=2
• ANOVA will give exactly the same result as two-tailed t-test.
• One-tailed tests must be done using t-tests.
PSYC 6130, PROF. J. ELDER 15
Example
From the Canadian Generalized Social Survey, Cycle 6 (1992)
PSYC 6130, PROF. J. ELDER 16
Example
DescriptivesDuring 12 months-Number of contacts: Psychologist
N Mean Std. Deviation
MARRIED 6601 0.185 2.034WIDOWED 1630 0.082 1.023SEPARATED OR DIVORCED 1012 0.900 4.688SINGLE 2568 0.620 4.012Total 11811 0.326 2.811
PSYC 6130, PROF. J. ELDER 17
Reporting Results
• A one-way ANOVA demonstrates that frequency of
contact with clinical psychologists depends on marital
status. Widowed individuals had the least contact
(M=0.082). Married individuals (M=0.185) had
somewhat more contact. Single (M=0.620) and
separated or divorced (M=0.900) had substantially more
contact. F(3,11807)=33.3, MSE = 7.8, p<.001.
PSYC 6130, PROF. J. ELDER 18
Summary Table (SPSS)
ANOVA
During 12 months-Number of contacts: Psychologist
783.673 3 261.224 33.332 .000
92531.091 11807 7.837
93314.764 11810
Between Groups
Within Groups
Total
Sum ofSquares df Mean Square F Sig.
PSYC 6130, PROF. J. ELDER 19
Interpreting the F Ratio
+ between-group estimate of error variance
within-gro
est
up
imate of treat
estimate of er
ment
ror v
eff
ari
e
c
c
e
t
anF
PSYC 6130, PROF. J. ELDER 20
Effect Size and Proportion of Variance Accounted For
2Proportion of variance accounted for (sample): bet
tot
SS
SS
PSYC 6130, PROF. J. ELDER 21
(Approxiately) Unbiased Effect Size
2 ( 1)Proportion of variance accounted for (population): bet W
tot W
SS k MS
SS MS
PSYC 6130, PROF. J. ELDER 22
Reporting Results
• A one-way ANOVA demonstrates that frequency of
contact with clinical psychologists depends on marital
status. Widowed individuals had the least contact
(M=0.082). Married individuals (M=0.185) had
somewhat more contact. Single (M=0.620) and
separated or divorced (M=0.900) had substantially more
contact. F(3,11807)=33.3, p<.001. However, the size of
the effect was relatively small: 2 0.008.
PSYC 6130, PROF. J. ELDER 23
Planning a Study: ANOVA and Power
Estimating power for ANOVA: Xn
can be used to plan experiments, relating , and (Appendix ncF)n k
.05 :
PSYC 6130, PROF. J. ELDER 24
Example
• You are interested in whether there is a link between PSYC 6130 final grades and the professor teaching the section.
• Grades typically have a standard deviation of about 15%
• There are typically 3 sections, each with around 12 students.
• What is the probability you would pick up an effect if the standard deviation of the mean grade is around 5%?
PSYC 6130, PROF. J. ELDER 25
Advantages of ANOVA
• Avoid inflation in error rate due to multiple comparisons
• Can detect an effect of the treatment even when no 2 groups are significantly different.
PSYC 6130, PROF. J. ELDER 26
6-Step Process for ANOVA
1. State the hypotheses
2. Select the statistical test and significance level
3. Select the samples and collect the data
4. Find the region of rejection
5. Calculate the test statistic
6. Make the statistical decision
0 1 2: ...
: , [1,..., ] :n
A i j
H
H i j n
PSYC 6130, PROF. J. ELDER 27
Sums of Squares Approach
bet
W
MSF
MS
2, where ( )betbet bet i i G
bet
SSMS SS n X X
df
2, where ( 1)WW W i i
w
SSMS SS n s
df
:
total bet
total bet
W
WSS SS SS
MS MS M
NB
S
PSYC 6130, PROF. J. ELDER 28
ANOVA Assumptions
• Independent random sampling
• Normal distributions
• Homogeneity of variance
PSYC 6130, PROF. J. ELDER 29
More on Homogeneity of Variance
21
1 2 22
2: ( , )s
k F df dfs
1Where larger of the 2 std devss
2 :k 2max
max 2min
Hartley's s
Fs
Problem: sensitive to deviations from normality.
Levene's test:
More robust
Used by SPSS
Test of Homogeneity of Variances
During 12 months-Number of contacts: Psychologist
115.537 3 11807 .000
LeveneStatistic df1 df2 Sig.
PSYC 6130, PROF. J. ELDER 30
Levene’s Test: Basic Idea
1 21. Replace each score , ,... with its absolute deviation from the sample mean:i iX X
1 1 1
2 2 2
| |
| |
i i
i i
d X X
d X X
1 22. Now run an analysis of variance on , ,... :i id d
SPSS reports an F-statistic for Levene’s test
• Allows the homogeneity of variance for two or more variables to be tested.
bet
W
MSF
MS
2, where ( )betbet bet i i G
bet
SSMS SS n d d
df
2, where ( 1)WW W i di
w
SSMS SS n s
df
PSYC 6130, PROF. J. ELDER 31
What to do if Homogeneity of Variance Assumption is Rejected
• Some adjustment procedures are available in SPSS (e.g., Welch 1951).
• We will not cover the theory behind these adjustments.
PSYC 6130, PROF. J. ELDER 32
Fixed vs Random Effects
• Fixed Effects: interested only in the specified levels of the independent variable
(e.g., single/married/divorced/widowed)
• Random Effects: interested in a large number of possible levels of the independent variable – randomly sampling only a few of these.
e.g.,
– Does the order of questions on a questionnaire effect the results?
– Does the order of stimuli in a psychophysical experiment effect the results?
PSYC 6130, PROF. J. ELDER 33
Fixed vs Random Effects
• One-Way Independent ANOVA calculation is the same for fixed and random effect designs.
• Power and effect size calculations differ.
• More complex ANOVA designs differ.
• We restrict our attention in this course to fixed effect designs.
PSYC 6130, PROF. J. ELDER 34
Qualitative vs Quantitative Independent Variables
• In principle, ANOVA can be applied to either qualitative or quantitative variables.
• If IV is quantitative and effect is roughly linear, usually have more power using regression (only using up 2 degrees of freedom, instead of k).
• If effect is complex (e.g., non-monotonic):
– Use a higher-order regression model (e.g., quadratic)
– Use ANOVA (makes no smoothness assumptions)
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