# Analysis of Variance Repeated measures

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```					                                                          Repeated-Measures ANOVA:

Each subject participates in all conditions in the
experiment (which is why it is called repeated
Analysis of Variance: Repeated measures                   measures).

A repeated-measures ANOVA is equivalent to a
repeated-measures t-test, except that you have more
than two treatment conditions.

From last week:
systematic variation
F=
random variation (“error”)               Analysis of variance implies analyzing or breaking down
variance. We start by breaking down ‘Sum of Squares’ or
SS. We saw these first when we calculated SD.
Large value of F: a lot of the overall variation in
scores is due to the experimental manipulation,
rather than to random variation between                                                              2
sum of squares = ∑ (X − X )
participants.

Small value of F: the variation in scores produced by
We divide SS by the appropriate "degrees of
€
the experimental manipulation is small, compared to
freedom" (usually the number of treatments or subjects
random variation between participants.
minus 1) to get variance.

1
Effects of sleep-deprivation on vigilance
One-way Repeated-Measures ANOVA:                                         in air-traffic controllers:

Use this where you have:
(a) one independent variable (with 2 or more levels);                    No deprivation vs. 12 hours' deprivation:
One Independent Variable, 2 levels – use
(b) one dependent variable;                                              repeated-measures t-test.

(c) each participant participates in every condition in
the experiment (repeated measures).
No deprivation vs. 12 hours vs.
24 hours:
A one-way repeated-measures ANOVA is equivalent to
a repeated-measures t-test, except that you have more                   One Independent Variable, 3
than two conditions in the study.                                       levels (differing quantitatively) –
use one-way repeated-measures
ANOVA.

Effects of sleep deprivation on vigilance:
Independent Variable: length of sleep deprivation (0, 12 hours and 24        "Partitioning the variance" in a one-way repeated-measures
hours). Dependent Variable: 1 hour vigilance test (number of planes          ANOVA:
missed).
Each participant does all 3 conditions, in a random order.

Participant   0 hours   12 hours   24 hours     0 hours:
1             3         12         13           Mean = 4.6
2             5         15         14           standard deviation = 1.43.
3             6         16         16
4             4         11         12           12 hours:
5             7         12         11           Mean = 13.0
6             3         13         14           standard deviation = 2.31.
7             4         17         16
8             5         11         12           24 hours:
9             6         10         11           Mean = 13.30
10            3         13         14           standard deviation = 1.83.

2
within treatments             between treatments
variability                   variability             step 1                    The null hypothesis:
Participant       0 hours       12 hours            24 hours   0 hours:                                 H 0 : µ1 = µ2 = µ3 = µ4
1                 3             12                  13
Mean = 4.6
2                 5             15                  14                                                         No treatment effect
3                 6             16                  16                                                               α = .05
4                 4             11                  12
12 hours:
5                 7             12                  11                           €                           steps 2, 3, 4, 5 & 6
Mean = 13.0
6                 3             13                  14
7                 4             17                  16                                          Calculate 5 SS values:
8                 5             11                  12                                     1)  Total 2) Between treatments
24 hours:
9                 6             10                  11                                3) Within treatments 4) Between subjects
Mean = 13.30
10                3             13                  14                                                  5) Error

Between treatments SS
step 2                                        Total SS                         step 3
P#          0 hours          12 hours        24 hours
Participant            0 hours 12 hours          24 hours
1           3                12              13
1                      3             12          13
2           5                15              14
2                      5             15          14
3           6                16              16
3                      6             16          16
4           4                11              12
4                      4             11          12                                 5           7                12              11
5                      7             12          11                                 6           3                13              14
6                      3             13          14                                 7           4                17              16
7                      4             17          16                                 8           5                11              12
8                      5             11          12                                 9           6                10              11
9                      6             10          11                                 10          3                13              14
10                     3             13          14                                              X1 = 4.6         X 2 = 13       X 2 = 13.3
SStotal =    ∑ (X         − G)      2
SSbetween treatments = n[(X1 − G ) 2 + (X 2 − G ) 2 + (X 3 − G ) 2 ]
i                                        G = 10.3
€            €             €
SSTotal = 584.3                                                 SSbetween treatments = 487.8
€
€
€                          €
3
Within treatments SS                                                  step 5                      Between subjects SS
step 4
P#   0 hours                 12 hours                  24 hours                    P#       0 hours              12 hours             24 hours
1    3                       12                        13                          1        3                    12                   13                 P1 = 9.33
2    5                       15                        14                          2        5                    15                   14                 P2 = 11.33
3    6                       16                        16                          3        6                    16                   16                 P3 = 12.67
€
4    4                       11                        12                          4        4                    11                   12                 P4 = 9
€
5    7                       12                        11                          5        7                    12                   11                 P5 = 10
€
6    3                       13                        14                          6        3                    13                   14                 P6 = 10
€
7    4                       17                        16                          7        4                    17                   16                 P7 = 12.33
€
8    5                       11                        12                          8        5                    11                   12                 P8 = 9.33
€
9    6                       10                        11                          9        6                    10                   11                 P9 = 9
€
10   3                       13                        14                          10       3                    13                   14                 P10 = 10
€
X1 = 4.6                X 2 = 13                X 2 = 13.3                                                                               €
SS1 = ∑ (X i − X1 ) 2   SS2 = ∑ (X i − X 2 ) 2   SS3 = ∑ (X i − X 3 ) 2        SSbetween subjects = n [(P1 − G ) + (P2 − G ) + (P3 − G )€...+ (P10 − G ) ]
2           2           2               2

€                   €                        €
SS
€ within treatments=
€        SS1 + SS2 +SS3 = 96.5
€                                                                        SSbetween subjects = 48.97
€

step 6                              Error SS                                                   step 7                           Calculating df
SSerror = SSwithin treatments - SSbetween subjects= 96.5 - 48.97 = 47.53

dftotal = All scores – 1 = 29

dfbetween treatments = Number of treatments – 1 = 2

dfwithin treatments = df1 + df2 + df3 = 27

dfbetween subjects = Number of subjects – 1 = 9

dferror = dfwithin treatments – dfbetween subjects = 27 – 9 = 18

4
step 8
The ANOVA summary table:
Source:                    SS              df         MS      F
Total SS: reflects the total amount of variation amongst all the
Total                 584.3              29        20.15                  scores.

Between treatments 487.8                2        243.9   92.36          Between treatments SS: a measure of the amount of systematic
variation between the treatments.
Within treatments    96.5               27       3.57
€           €                                    Within treatments SS: a measure of the amount of unsystematic
Between subjects
€           48.97                 9    5.44                   variation inside each treatment
Error                 €
47.53        €        18      €
2.64
€                                                                    Between subjects SS: a measure of the amount of
€      €                                    unsystematic variation between the subjects. (This is not due to our
€                                                           experimental manipulation).
€   €
€                €    €                                              Error SS: a measure of the amount of unsystematic
€                                                          variation within each subject’s set of scores.
MSbetween treatments
F=                        = 243.9 ÷ 2.64 = 92.36                          Total SS = Between subjects SS + Within subjects SS
MSerror

Here, look up the critical F-
value for 2 and 18 degrees of
freedom
Assessing the significance of the F-ratio (by hand):
The bigger the F-ratio, the less likely it is to have arisen
merely by chance.                                                   Columns correspond to
TREATMENTS degrees of
Use the between-treatments and error degrees of                     freedom
freedom to find the critical value of F.                            Rows correspond to ERROR
degrees of freedom
Your F is significant if it is equal to or larger than the
critical value in the table.                                        Here, go along 2 and down 18:
critical F is at the intersection

Our obtained F, 92.36, is bigger
than 3.55; it is therefore
significant at p<.05. (Actually
it’s bigger than the critical
value for a p of 0.0001)

5
Interpreting the Results:                                To pinpoint the source of the difference:
A significant F-ratio merely tells us that there is a
statistically-significant difference between our         (a) planned comparisons - comparisons between groups
experimental conditions; it does not say where the       which you decide to make in advance of collecting the
difference comes from.                                   data.

In our example, it tells us that sleep deprivation       (b) post hoc tests - comparisons between groups which
affects vigilance performance.                           you decide to make after collecting the data:
Many different types - e.g. Newman-Keuls, Scheffé,
Bonferroni.

Data
entry

Using SPSS for a one-way repeated-measures
ANOVA on effects of fatigue on vigilance

6
Go to: Analyze > General Linear Model > Repeated Measures…                         Tell SPSS about your within-subjects Independent Variable (i.e. number
of levels; and which columns the levels of the independent variable are
in):

Move VAR 4, VAR 5 and VAR 6 into the ‘Within-Subjects Variables’ box by pressing the
top arrow; then press ‘options…’ button

7
The SPSS output (ignore everything except what's shown
here!):

Similar to Levene's test -
if significant, shows
inhomogeneity of
variance.

Then click continue and OK

SPSS ANOVA results:
This is not too interesting; this just tells us that the
subjects are significantly different from each other.

Use Sphericity Assumed F-ratio if Mauchly's test was NOT significant.
Significant effect of sleep deprivation (F 2, 18 = 92.36, p<.0001)
OR, (if Mauchly’s test was significant) use Greenhouse-Geisser (F 1.18,
10.63 = 92.36, p<.0001).

8
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9

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